UUniversit`a degli Studi di Modena e Reggio Emilia
DIPARTIMENTO DI SCIENZE FISICHE, INFORMATICHE E MATEMATICHECorso di Laurea Magistrale in Physics
Equivariant and supersymmetric localization in QFT
Relatore:
Prof. Diego Trancanelli
Laureando:
Paolo RossiAnno Accademico 2019-2020 a r X i v : . [ h e p - t h ] J a n bstract Equivariant localization theory is a powerful tool that has been extensively used in thepast thirty years to elegantly obtain exact integration formulas, in both mathematics andphysics. These integration formulas are proved within the mathematical formalism of equivari-ant cohomology, a variant of standard cohomology theory that incorporates the presence of asymmetry group acting on the space at hand. A suitable infinite-dimensional generalization ofthis formalism is applicable to a certain class of Quantum Field Theories (QFT) endowed with supersymmetry .In this thesis we review the formalism of equivariant localization and some of its applicationsin Quantum Mechanics (QM) and QFT. We start from the mathematical description of equiv-ariant cohomology and related localization theorems of finite-dimensional integrals in the caseof an Abelian group action, and then we discuss their formal application to infinite-dimensionalpath integrals in QFT. We summarize some examples from the literature of computations ofpartition functions and expectation values of supersymmetric operators in various dimensions.For 1-dimensional QFT, that is QM, we review the application of the localization principle tothe derivation of the Atiyah-Singer index theorem applied to the Dirac operator on a twistedspinor bundle. In 3 and 4 dimensions, we examine the computation of expectation values of cer-tain Wilson loops in supersymmetric gauge theories and their relation to 0-dimensional theoriesdescribed by “matrix models”. Finally, we review the formalism of non-Abelian localizationapplied to 2-dimensional Yang-Mills theory and its application in the mapping between thestandard “physical” theory and a related “cohomological” formulation.iii cknowledgments
First and foremost, I would like to express my sincere gratitude to professor Diego Tran-canelli, who supervised me during the draft of this thesis. I thank him for his constant avail-ability, his valuable advice and his sincere interest for my learning process, as well as his greatpatience in meticulously reviewing my work step by step. It is also a pleasure for me to thankprofessor Olindo Corradini, who initiated me to the wonderful world of QFT with two brilliantcourses, and was always available to discuss and answer my questions.I would like to thank all the friends and colleagues that grew up with me in this journey.Even though our paths and interests separated, they have always been a precious source ofinspiration to me. A heartfelt thanks goes to Giulia, whose presence alone makes everythingeasier. Finally I would like to thank my family, for the love and support, no matter what,during all these years. v ontents
Abstract iiiAcknowledgments v1 Introduction 12 Equivariant cohomology 7
A.1 Principal bundles, basic forms and connections . . . . . . . . . . . . . . . . . . . 151A.2 Spinors in curved spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
B Mathematical background on equivariant cohomology 161
B.1 Equivariant vector bundles and equivariant characteristic classes . . . . . . . . . 161B.2 Universal bundles and equivariant cohomology . . . . . . . . . . . . . . . . . . . 163B.3 Fixed point sets and Borel localization . . . . . . . . . . . . . . . . . . . . . . . 167B.4 Equivariant integration and Stokes’ theorem . . . . . . . . . . . . . . . . . . . . 168
Bibliography 171 hapter 1Introduction
Quantum Field Theory (QFT) is the framework in which modern theoretical physics de-scribes fundamental interactions between elementary particles and it is also central in thestudy of condensed matter physics and statistical mechanics. QFT has made the most pre-cise predictions ever in the history of science and it has been tested against a huge amountof experimental data. Nowadays, the most useful formulation of QFT is made in terms of path integrals , which are integrals over the space of all possible field configurations. A “field”mathematically speaking can be thought roughly as a function over spacetime, so these inte-grals are computed over functional spaces, that are infinite-dimensional. This makes the exactcomputation of such objects a complicated task, except for some very special cases, and in facttheir precise mathematical formulation is still an open problem. Despite these formal difficul-ties, many interesting results can be extracted from these objects, that can describe partitionfunctions and expectation values of physical observables in QFT. The favorite approach todeal with such computations is perturbation theory, applied in the case in which the QFT isweakly coupled. In this regime, one can compute approximately the expectation values in aperturbative expansion, order by order in the coupling constant. This method, applied to thecomputation of the partition function, is the infinite-dimensional analogous of a “saddle-point”or “stationary-phase” approximation. Intuitively it represents a semi-classical approach to thequantum dynamics.There are however many cases in which perturbation theory is not applicable, mainly whenthe QFT is strongly coupled, i.e. the coupling constant is of order 1. This is not a very raresituation. For example we know that one of the fundamental interactions of the Standard Modelof particle physics, the strong nuclear force, is well described by “quantum chromodynamics”(QCD), a QFT that is strongly coupled at low energies (so in the “phenomenological” regime).Understanding the behavior of QFT in the strong coupling regime is then a major problemfrom the physical point of view, and one is lead to develop techniques that permit to study thepath integral in a non-perturbative approach.In this thesis we describe features of one of these techniques, that has been exploited in thelast few decades for a class of special QFTs, those who exhibit some kind of supersymmetry .This technique is called supersymmetric localization , or equivariant localization , or simply lo-calization. Its name derives from the fact that, when one is able to use this method, the pathintegral of the QFT at hand simplifies (so “localizes”) to an integral over a smaller domain,sometimes even a finite-dimensional integral over constant field configurations. This result canbe viewed as an exact stationary-phase approximation, so that the full quantum spectrum ofthe localized theory is completely determined by its semi-classical limit. Without entering in1the technical details of this localization phenomenon, we just point out that this method fun-damentally relies on the presence of a large amount of symmetry of the theory, that can bedescribed by the presence of a group action on the space of fields. When the space of fields isgraded, i.e. there is a distinction between “bosonic” and “fermionic” degrees of freedom, thesymmetry group action can exchange these two types of fields and in this case it is called su-per symmetry. This situation arises mainly in BRST-fixed and topological field theories, wherethe grading is regarded as the ghost number , and in Poincar´e-supersymmetric QFT, where thegrading distinguishes between bosons and fermions in the standard sense of particle physics. Inboth cases, the action of a supersymmetry transformation “squares” to a canonical (bosonic)one, i.e. a gauge transformation or a Poincar´e transformation. It is reasonable that such a hugeamount of symmetry can simplify the dynamics of the theory, but it can be non-trivial a priorihow to translate this in a simplification of the path integral.At this point, one wishes to understand if there is a theoretical framework that allows tosystematically understand why and when such a drastic simplification of the path integral canoccur, and at which level this is related to (super-)symmetries in QFT. To be mathematicallymore rigorous, we can think in terms of integration over finite-dimensional spaces, and then tryto extrapolate and generalize the important results to the infinite-dimensional case. Integrals ofdifferential forms over manifolds are built up technically from the smooth (so local) structureof the space, but it is a well-known fact that their result can describe and is regulated bytopological (so global) properties. It is a consequence of de Rham’s theorem and Stokes’theorem that they really depend on the cohomology class of the integrand and not on theparticular differential form that represents the class. It is thus reasonable that a theory ofintegration that embeds the presence of a symmetry group action should arise from a topologicalconstruction. Indeed, the mathematical framework in which the localization formulas werefirstly derived is a suitable modification of the standard cohomology theory. This is called equivariant cohomology . As de Rham’s theorem relates the usual cohomology to differentialforms on a smooth manifold, equivariant cohomology can be associated to a modification ofthem, called equivariant differential forms .Equivariant cohomology theory was initiated in the mathematical literature by Cartan,Borel and others during the 50’s [1–4], but the first instance of a localization formula waspresented by Duistermaat and Heckman in 1982 [5]. In this paper, they proved the exactnessof a stationary-phase approximation in the context of symplectic geometry and HamiltonianAbelian group actions. Subsequently, Atiyah and Bott realized that the Duistermaat-Heckmanlocalization formula can be viewed as a special case of a more general theorem, that theyproved in the topological language of equivariant cohomology [6]. Almost at the same time,Berline and Vergne derived an analogous localization formula valid for Killing vectors on generalcompact Riemannian manifolds [7]. Roughly, the Atiyah-Bott-Berline-Vergne (ABBV) formulasays that the integral over a manifold that is acted upon by an Abelian group localizes as asum of contributions arising only from the fixed points of the group action.The first infinite-dimensional generalization of this localization formula was given soon afterby Atiyah and Witten in 1985 [8], applied to supersymmetric Quantum Mechanics (QM). Thisturns out to be an example of topological theory, since their localization formula relates thepartition function to the index of a Dirac operator. Many generalizations of this approachfollowed, first mainly in the context of topological field theories. Here localization allows toget closed formulas relating partition functions of physical QFTs to topological invariants ofthe spaces where they live. In all these cases, the BRST cohomology is interpreted as theequivariant structure of the theory, and is responsible for the localization of the path integral.hapter 1. Introduction 3In recent years, the formal application of the ABBV localization formula for Abelian sym-metry actions was employed in the context of Poincar´e-supersymmetric theories, whose explicitsupersymmetry is an extension of the spacetime symmetry that is common to all QFTs. Inthis case, the equivariant structure is generated by the cohomology of a supersymmetry charge ,that acts as a differential on the space of Poincar´e symmetric field configurations. This formalstructure of supersymmetric field theories was firstly realized by Niemi, Palo and Morozov inthe 90’s [9, 10]. Starting from the work of Pestun in 2007 [11], localization has been appliedto the computation of partition functions and expectation values of supersymmetric operatorson curved compact manifolds [12]. In many of these cases, the partition function localizes to afinite-dimensional integral over matrices, a so-called matrix model . To carry out this procedure,a number of technical difficulties have to be overcome, the most urgent being to understand howto define Poincar´e-supersymmetric theory on curved spaces. Nowadays there is a well-definedand well-understood procedure that allows to do that, essentially deforming an original theorydefined in flat space through the coupling to a non-trivial “rigid” supersymmetric background.This in general reduces the degree of supersymmetry of the original theory, but if some of it ispreserved on the new background then one is able in principle to perform localization. Besidethe general and abstract motivation of understanding strongly coupled QFT, the importance ofthese computations can be viewed in a string theory perspective, and in particular as possibletests of the so called
AdS/CFT correspondence [13].Some generalizations of the ABBV formula for non-Abelian group actions have been pro-posed both in the mathematical and physical literature. The first generalization of the Duister-maat-Heckman theorem to non-Abelian group actions was presented by Guillemin and Prato[14], restricting the localization principle to the action of the maximal Abelian subgroup. Aninfinite-dimensional generalization of the localization principle was proposed by Witten in 1992[15], applied to the study of 2-dimensional Yang-Mills theory, and a more rigorous proof ap-peared in the mathematical literature in 1995 by Jeffrey and Kirwan [16]. Of course theselocalization formulas find many interesting applications not only in physics but also in puremathematics, but this side of the story is far from the purpose of this work.
An exact saddle-point approximation
To give a feeling of what we mean by “exact saddle-point approximation”, we present avery simple but instructive example here, in the finite-dimensional setting. The saddle-point(or stationary-phase) method is applied to oscillatory integrals of the form I ( t ) = (cid:90) + ∞−∞ dx e itf ( x ) g ( x ) , (1.1)when one is interested in the asymptotic behavior of I ( t ) at positive large values of the realparameter t . In this limit, the integral is dominated by the critical points of f ( x ), where itsfirst derivative vanishes and it can be expanded in Taylor series as f ( x ) = f ( x ) + 12 f (cid:48)(cid:48) ( x )( x − x ) + · · · . (1.2)If F ⊂ R is the set of critical points, that for simplicity we assume to be discrete, the leadingcontribution to (1.1) is then given by a Gaussian integral, I ( t ) ≈ (cid:88) x ∈ F g ( x ) e itf ( x ) (cid:90) + ∞−∞ dx e it f (cid:48)(cid:48) ( x )( x − x ) = (cid:88) x ∈ F g ( x ) e itf ( x ) e iπ sign( f (cid:48)(cid:48) ( x )) (cid:115) πt | f (cid:48)(cid:48) ( x ) | . (1.3)If the integral is performed over R n , this last formula generalizes easily to I ( t ) ≈ (cid:18) πt (cid:19) n/ (cid:88) x ∈ F g ( x ) e itf ( x ) e iπ σ ( x ) | det(Hess f ( x )) | / , (1.4)where Hess f ( x ) is the matrix of second derivatives of f at x ∈ R n , and σ ( x ) denotes the sumof the signs of its eigenvalues.Of course, there is no reason for the RHS of (1.4) to be the exact answer for I ( t ), but theclaimed property of localization is that in some cases this turns out to be true! To see this,let us consider the integration over the 2-sphere S , defined by its embedding in R as the setof points whose distance from the origin is 1. For this example we chose f ( x, y, z ) = z , the“height function”, and g ( x, y, z ) = 1. The resulting oscillatory integral is then I ( t ) = (cid:90) S dA e itz , (1.5)where dA is the volume form on the sphere, normalized such that (cid:82) S dA = 4 π . The criticalpoints of the height function are the North and the South poles, where z ≈ ± (cid:18) −
12 ( x + y ) (cid:19) . (1.6)The volume form at the poles is just dA = dxdy , so if we apply the saddle-point approximationto (1.5) we get (cid:90) S dA e itz ≈ e it (cid:90) dxdy e − it ( x + y ) + e − it (cid:90) dxdy e it ( x + y ) = 2 πit e it − πit e − it = 4 π sin( t ) t . (1.7)Now, since this integral is rather easy, in this case we can actually compare the result ofthe saddle-point approximation with its exact value. Using spherical coordinates, I ( t ) = (cid:90) +1 − d cos( θ ) (cid:90) π dϕ e it cos( θ ) = 4 π sin( t ) t . (1.8)As promised, the result coincides with the stationary-phase result (1.7). This is the simplestexample of equivariant localization! The scope of the first chapters of the thesis is to describe ingeneral the structure underlying this result, how it can be related to symmetry properties of thespecific function and space under consideration, and the localization theorems that generalizethis specific computation to possibly more complicated examples. In the remaining part, wewill deal instead with examples of the infinite-dimensional analog of this exact stationary-phaseapproximation.hapter 1. Introduction 5 Structure of the thesis
The aim of this project is to summarize some results in the context of equivariant localizationapplied to physics. We will draw a line from the first mathematical results concerning the theoryof equivariant cohomology and associated powerful localization theorems of finite-dimensionalintegrals, to the generalization to path integration in quantum mechanical models and finally toapplications in quantum field theories. One of the purposes of this work is to provide a suitablereference for other students in theoretical and mathematical physics with a background at thelevel of a master degree, who are interested in approaching the subject. In this spirit, we will tryto expose the material in a pedagogical order, being as self-contained as possible, and otherwisegiving explicit references to background material.In Chapter 2, we will review the basics of equivariant cohomology theory, starting fromits construction in algebraic topology. After having recalled some notions of basic homologyand cohomology, we will introduce group actions, and define equivariant cohomology with theso-called
Borel construction . Then, we will describe the most common algebraic models thatgeneralize de Rham’s theorem in the equivariant setup, the
Weil , Cartan and
BRST models .These give a description of equivariant cohomology in terms of a suitable modification of thecomplex of differential forms.In Chapter 3, we will describe the common rationale behind the localization property ofequivariant integrals in finite-dimensional geometry, the so-called equivariant localization prin-ciple . Then we will state and explain the Abelian localization formula derived by Atiyah-Bottand Berline-Vergne. In the final part of the chapter, we will connect the discussion with thecontext of symplectic geometry that, as we will see, can be rephrased in terms of equivariantcohomology. We will review the basic notions of symplectic manifolds, symmetries and Hamil-tonian systems, and then state the Duistermaat-Heckman localization formula as a special caseof the ABBV theorem.Chapter 4 can be viewed as a long technical aside. Here we will review some notions aboutsupergeometry that are needed to understand a proof of the ABBV theorem, and then specializethe discussion to Poincar´e-supersymmetric theories. We will discuss their construction from theperspective of superspace, give some practical examples, and then generalize their descriptionover general curved backgrounds. This is achieved by coupling the given theory to the super-symmetric version of Einstein gravity, supergravity , and then requiring the gravitational sectorof the resulting theory to decouple from the rest in a “rigid limit”, analogous to G N →
0. Thismethod will bring up to the notion of
Killing spinors , a special type of spinorial fields whoseexistence ensures the preservation of some supersymmetry on the curved background. Finally,we will comment about a possible “super-interpretation” of the models of equivariant cohomol-ogy described in Chapter 2, that connects them to the usual BRST formalism for quantizationof constrained Hamiltonian systems.In Chapter 5 we will discuss examples of Abelian supersymmetric localization of path inte-grals in the infinite-dimensional setting of QFT. The first case we will report is 1-dimensional, i.e.
QM. In an Hamiltonian formulation on phase space, we will describe how it is possible to givea supersymmetric (so equivariant) interpretation to the path integral in a model-independentway, the supersymmetry arising as a “hidden” BRST symmetry which is linked to the Hamil-tonian dynamics. This results in a localization of the path integral over the space of classicalfield configurations or over constant field configurations, that applied to supersymmetric QMgives an alternative proof the Atiyah-Singer index theorem for the Dirac operator over a twistedspinor bundle. Next, we will review two modern applications of the localization principle to thecomputation of expectation values of
Wilson loop operators in supersymmetric gauge theories.The first application computes the expectation value in N = 4 , , ∗ Super Yang-Mills theoryon the 4-sphere S , the second one in N = 2 Super Chern-Simons theory on the 3-sphere S .In both cases, the partition function and the Wilson loop expectation value can be reduced tofinite-dimensional integrals over the Lie algebra of the gauge group of the theory. This makesthe supersymmetric theories in exam equivalent to a suitable “matrix model”, whose path in-tegral can be computed exactly with some special regularization. We will give an example ofcomputations of such a matrix model, since this class of objects arises in many important areasof modern theoretical physics.In Chapter 6 we will introduce Witten’s non-Abelian localization formula, and briefly de-scribe its possible application in the study of 2-dimensional Yang-Mills theory. This moregeneral formalism is able to show the mapping between the standard “physical” version ofYang-Mills theory and its “cohomological” ( i.e. topological, in some sense) formulation, and isat the base of the localization of the 2-dimensional Yang-Mills partition function.Some technical asides are relegated to the appendices. Appendix A is devoted to somebackground in differential geometry, concerning principal bundles and the definition of spinorsin curved spacetime. In Appendix B we report more details about equivariant cohomology,equivariant vector bundles and characteristic classes. This can be seen as a completion of thediscussion of Chapter 2, from a more mathematical point of view. hapter 2Equivariant cohomology In this chapter we review the theory of equivariant cohomology , as a modification of thestandard cohomology theory applied to spaces that are equipped with the action of a symmetrygroup G on them, the so-called G -manifolds. First, we will review the basic notions aboutcohomology and homology, from their algebraic definition to the application in topology anddifferential geometry. The main result that we need to care about, and extend to the equivariantcase, is de Rham’s theorem [17], that gives an algebraic model for the cohomology of a smoothmanifold in terms of the complex of its differential forms. This and Stokes’ theorem relate thetheory of cohomology classes to the integration on smooth manifolds. Next, we will extend thisto the equivariant setting, giving a topological definition of equivariant cohomology, and thendiscussing, in the smooth case, an equivariant version of de Rham’s theorem. This, analogouslyto the standard case, will give an equivalence between the topological definition of equivariantcohomology and the cohomology of some suitable differential complex built from the smoothstructure of the space at hand. There are different, but equivalent, possibilities of such algebraicmodels for the equivariant cohomology of a G -manifold: we will see the Weil model , the
Cartanmodel and the
BRST model , and discuss how they are related one to each other, since at theend they have to describe the same equivariant cohomology.The purpose of all this, from the physics point of view, is that with equivariant cohomologywe can describe a theory of cohomology and integration over manifolds that are acted uponby a symmetry group, the standard setup of classical mechanics and QFT. In the next chapterwe will review one of the climaxes of this theory applied to the problem of integration over G -manifolds: the famous localization formulas of Berline-Vergne [7] and Atiyah-Bott [6], thatpermit to highly simplify a large class of integrals thanks to the equivariant structure of theunderlying manifold. The aim and the core of this thesis will be then the description of somegeneralizations and application of those theorems to the context of QM and QFT, where theintegrals of interest are the infinite-dimensional path integrals describing partition functions orexpectation values of operators.For this introductory chapter, we follow mainly [18–21]. Another classical reference is [22].Some background tools from differential geometry that are needed can be found in AppendixA. In this section we will review some of the basic facts about standard homology and co-homology theory, and in particular its application to topological spaces with the definition of7 2.1. A brief review of standard cohomology theory singular homology and cohomology groups. Since this is after all standard material, we refer toany book of topology/geometry/algebra (for example [23–26]) for the various proofs, while wewill give some intuitive examples to help making concrete the various abstract definitions. Themain result that we aim to recall is de Rham’s theorem , that relates the cohomology theory todifferential forms over smooth manifolds, and that will be extended in the next sections to themodified equivariant setup.We start with the abstract definition of homology and cohomology as algebraic construc-tions. From this point of view, (co)homology groups are defined in relation to (co)chain com-plexes (or differential complexes ). Definition 2.1.1.
Given a ring R , a chain complex is an ordered sequence A = ( A p , d p ) p ∈ N of R -modules A p and homomorphisms d p : A p → A p − such that d p − ◦ d p = 0. A cochain complex has the same structure but with homomorphisms d p : A p → A p +1 , and d p +1 ◦ d p = 0.Chain complex : · · · A p − d p ←− A p d p +1 ←−− A p +1 · · · Cochain complex : · · · A p − d p − −−→ A p d p −→ A p +1 · · · An element α of a (co)chain complex is called (co)cycle or closed if α ∈ Ker( d p ) for some p . It is instead called (co)boundary or exact if α ∈ Im( d p ) for some p . By definitionIm( d p ) ⊆ Ker( d p ± ) , where the − is for chain and the + for cochain complexes, so the quotient sets Ker (cid:30) Im of p -(co)cycles modulo p -(co)boundries are well defined. Definition 2.1.2.
Given a (co)chain complex A , the p th (co)homology group of A is (cid:16) H p ( A ) := Ker( d p ) (cid:30) Im( d p +1 ) (cid:17) H p ( A ) := Ker( d p ) (cid:30) Im( d p − ) . (Co)homology groups are called like that because they inherit a natural Abelian groupstructure (or equivalently Z -module structure) from the sum in the original chain complex. As usual, once we have a definition of a class of mathematical objects, a prime interest lies in thestudy of structure preserving maps between them. A morphism between (co)chain complexes A and B is then a sequence of homomorphisms ( f p : A p → B p ) p ∈ N such that, schematically, f ◦ d ( A ) = d ( B ) ◦ f . It is easy to see that every such a morphism induces an homomorphismof (co)homology groups, since for example f ∗ : H p ( A ) → H p ( B ) such that f ∗ ([ α ]) := [ f ( α )] iswell defined.Notice that, in many applications one considers (co)chain complexes defined by graded modules or algebras with a suitable differential . An example of this type is the complex ofdifferential forms (Ω( M ) , d ) over a smooth manifold, that we will recover later on. For ageneral (co)chain complex ( A p , d p ) p ∈ N , we can always see A := (cid:76) p A p as a graded R -module,whose elements as α ∈ A p ⊂ A are said to have pure degree deg( α ) := p . A generic elementwill be a sum of elements of pure degree. Definition 2.1.3. A differential graded algebra ( dg-algebra for short) over R is then an R -algebra with the decomposition (grading) A = (cid:76) p A p , the product satisfying A p A q ⊆ A p + q ,and a differential d : A → A such that Notice that [ α ] + [ β ] := [ α + β ] is well defined. hapter 2. Equivariant cohomology 9(i) it has degree deg( d ) = ±
1, meaning that for every α of degree p , deg( dα ) = p ± d = 0;(iii) it satisfies the graded Leibniz rule, d ( αβ ) = ( dα ) β + ( − deg( α ) α ( dβ ).Every such an algebra clearly defines an underlying complex (cochain if deg( d ) = 1, chainif deg( d ) = −
1) and thus has associated (co)homology groups. Even if the algebra structure(the product and the Leibniz rule) are not needed to define the complex, we included it in thedefinition because this is the kind of structure that arises in physics or in differential geometry.Morphisms of dg-algebras are naturally defined as structure preserving maps between them,analogously to the above discussion.Beside the abstract algebraic definitions of above, one of the most important applicationsof homology and cohomology groups is in the study and classification of topological spaces. Inorder to define these groups in a topological setup, the complexes one takes into considerationare the simplicial complexes , that intuitively represents a formal way of constructing “polyhe-dra” over R n , and that can be used in turn to study properties of topological spaces. Given R ∞ with the standard basis { e i } i =0 , , ··· ( e = 0), a standard q-simplex is∆ q := (cid:40) x = q (cid:88) i =0 λ i e i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q (cid:88) i =0 λ i = 1 , λ i ∈ [0 , ∀ i = 0 , · · · , q (cid:41) . (2.1)Although this definition takes into account any possible dimensionality, we can embed thesesimplices in in finite-dimensional Euclidean spaces, giving them a more practical interpretation.Given q + 1 points v , · · · , v q ∈ R n , the associated affine singular q-simplex in R n is the map[ v · · · v q ] : ∆ q → R nq (cid:88) i =0 λ i e i (cid:55)→ q (cid:88) i =0 λ i v i . (2.2)This is the convex hull in R n generated by the vertices ( v i ). Geometrically, the 0-simplex isjust the point 0 ∈ R n , 1-simplices are line segments, 2-simplices are triangles and so on. Noticethat ∆ q − ⊂ ∆ q , and its image through [ v · · · v q ] is a “face” of the resulting polygon. Moreprecisely, [ v · · · ˆ v i · · · v q ] : ∆ q − → ∆ q (the hat means we take away that point from the list) isregarded as the i th face map , denoted concisely as F ( i ) q .The same idea can be used to embed the simplices in a generic topological space M , changingthe codomain of the simplex map. A singular q-simplex in M is then a continuous map σ q : ∆ q → M (2.3)where now { σ q ( e ) , · · · , σ q ( e q ) } are the vertices of σ q . Two simplices are said to have thesame/opposite orientation if the vertex sets are respectively even/odd permutations of eachother. The word “singular” is there because only continuity is required, thus from a “smooth”point of view these simplices can present singularities. The standard topology on R q is induced on ∆ q . (a) (b) Figure 2.1: (a) First standard simplices. (b) An oriented affine 2-simplex, its face maps and asingular 2-simplex σ on a 2-dimensional topological space.With this setup, we can construct chain complexes on topological spaces in terms of singularsimplices. In fact, defining the sum of two singular simplices σ q , ρ p as( σ q + ρ p ) : ∆ q (cid:116) ∆ p → Mλ (cid:55)→ (cid:40) σ q ( λ ) if λ ∈ ∆ q ρ q ( λ ) if λ ∈ ∆ p , (2.4)whose image is the (disjoint) union in M of the images of the two starting simplices. Since the“+” is clearly commutative, C q ( M ) := C (∆ q , M ) is an Abelian group, called the (singular)q-chain group of M . We can define a boundary operator as a group homomorphism ∂ : C q ( M ) → C q − ( M ) σ (cid:55)→ ∂σ := q (cid:88) i =0 ( − i (cid:0) σ ◦ F ( i ) q (cid:1) , (2.5)that restricts to the (oriented) sum of faces of a given simplex, and happens to satisfy thenilpotency condition ∂ ◦ ∂ = 0. This means that C ( M ) := (cid:76) ∞ q =0 C q ( M ) with the operator ∂ defines a dg-module over Z , and an associated chain complex, that we use to define the homolgygroups of M . Definition 2.1.4.
The singular q th homology group of M is H q ( M ; Z ) := Ker( ∂ q ) (cid:30) Im( ∂ q +1 ) . It is often useful to work with homology groups with coefficients in some Z -module A (like thereal numbers), that is considering H q ( M ; A ) as defined from the simplicial complex C ( M ) ⊗ A . Example 2.1.1 (Homology of spheres) . In practice, the strategy to get H ∗ ( M ) is the so-called triangulation of M , i.e. constructing a suitable simplicial complex K in R dim( M ) as aset of standard simplices, whose union gives a polyhedron that is homeomorphic to M . Then,one can count and classify all the cycles and the boundaries in K , and then get H ∗ ( K ) ∼ = H ∗ ( M ). Some examples of this rigorous approach can be found in [23]. We can still give somehapter 2. Equivariant cohomology 11examples, less rigorously, by looking directly at simple topological spaces, just to help buildingsome intuition. Remember that a q -cycle σ on M is a boundary-less singular q -simplex upto continuous deformations , and it is also a boundary if it can be seen as the border of a( q + 1)-simplex.( S ) On the circle there is no place for simplices of dimension higher than 1, so we look at the1-simplices. There are two inequivalent ways of deforming the standard 1-simplex ontothe circle: it can join at the end points covering all S or not. In the first case, that we call σ , we have ∂σ = 0 since S has no boundaries, in the second the boundaries are the endpoints of the singular 1-simplex. The first boundary-less case cannot be seen as a boundaryof something else, by dimensionality, so the Abelian group H ( S ; Z ) = Ker( ∂ ) / Im( ∂ )is generated by a single element, [ σ ]. In other words, H ( S ; Z ) ∼ = span Z { [ σ ] } ∼ = Z .The case q = 0 is trivial, since we have only one way of drawing a point on the circle,and every point is boundary-less. We have just said that the boundary of a 1-simplex iseither zero or two points, so a single point is never a boundary. Thus the Abelian group H ( S ; Z ) ∼ = Z since it is generated by only one element. Generalizing a little, we canalready see from this example that the homology group in 0-degree will always follow thistrend for connected topological spaces. If the space has n connected components, therewill be n inequivalent ways of drawing a point on it, so n generators for the homologygroup, giving H ( M ( n ) ; Z ) ∼ = Z ⊕ · · · ⊕ Z ( n factors).( S ) The 2-sphere does not necessitate of much more work, at least with this level of rigor.Again, by dimensionality the homology groups in degrees higher than dim( S ) = 2 areempty. For q = 2, the only way we can construct a boundary-less figure on the spherefrom ∆ is joining the vertices and the edges together and cover the whole sphere. Allother singular 2-simplices have boundaries, and the 2-sphere cannot be seen as a boundaryof something else by dimensionality, so analogously to the previous case H ( S ; Z ) ∼ = Z .For the 1-simplices, we notice that the only two ways of drawing a segment on the sphere(up to continuous deformations) is to close it or not at the end points. In the first case,the 1-simplex has no boundary, but can be seen as the boundary of its internal area, so itis in fact exact. In the second case, the 1-simplex has boundaries so it is outside Ker( ∂ ).This means that every 1-cycle is also a boundary, and thus H ( S ; Z ) ∼ = 0. In 0-degreewe can argue in the same way as for the circle that H ( S ; Z ) ∼ = Z .( S n ) It turns out that all spheres follow this trend, giving homology groups H q ( S n ; Z ) ∼ = (cid:40) Z q = 0 , n . If interested in the case with real coefficient, the homology of spheres are again verysimple, since Z ⊗ R ∼ = R .Now we can turn to the construction of singular cohomology groups on topological spaces.This is done considering the dual spaces Hom( C q ( M ) , A ) with values in a Z -module A . Thesimplest choice is of course A = Z . Notice that Hom( C q ( M ) , A ) itself is a Z -module. The coboundary operator in this case is defined as the Z -module homomorphism δ : Hom( C q ( M ) , A ) → Hom( C q +1 ( M ) , A ) f (cid:55)→ δf s.t. δf ( σ q +1 ) := f ( ∂σ q +1 ) (2.6)2 2.1. A brief review of standard cohomology theoryand from the nilpotency ∂ = 0 we get easily δ = 0. Definition 2.1.5.
The singular q th cohomology group of M , with coefficients in A , is H q ( M ; A ) := Ker( δ q ) (cid:30) Im( δ q − ) . Note that for a commutative ring A (as for example R ), the cohomology groups are naturally A -modules. Although the definition is less practical than the one for homology groups, there isan important theorem that allows to relate the two, so that homology computations can be usedto infer the structure of singular cohomology groups. This is the so-called universal coefficienttheorem [24]. Since for applications to smooth manifolds we will be primarily interested incohomology groups with real coefficients (as it will become clearer later) we take A = R . Forthis special case, the theorem says H q ( M ; R ) ∼ = H q ( M ; Z ) ⊗ R ∼ = H q ( M ; R ) ∗ , (2.7)so that the cohomolgy groups are exactly the dual spaces of the homology groups. Notice that,from the example above, H q ( S n ; R ) ∼ = R in degree q = 0 , n .The above construction relates the topological properties of the space M to the algebraicconcept of (co)homology groups. In general we said that morphisms of complexes inducemorphisms of associated (co)homologies, and this extends to the present topological case: ifwe consider a continuous map (morphism of topological spaces) F : M → N between thetopological spaces M, N , we can lift it to F : C q ( M ) → C q ( N ) such that F ( σ ) := F ◦ σ , that isa morphism of dg-modules. This in turn induces the morphism of homology groups as descrbedabove. For the cohomology groups we have, analogously, the lifted map in the opposite direction F : Hom( C q ( N ) , A ) → Hom( C q ( M ) , A ) such that F ( f ) := f ◦ F , giving the morphism ofcochain complexes. This induces F ∗ : H q ( N ) → H q ( M ) such that F ∗ ([ f ]) := [ F ( f )]. Also,we notice that if we have two continuous maps
F, G between the topological spaces, then ( F ◦ G ) = F ◦ G ⇒ ( F ◦ G ) ∗ = F ∗ ◦ G ∗ ( F ◦ G ) = G ◦ F ⇒ ( F ◦ G ) ∗ = G ∗ ◦ F ∗ . (2.8)An important fact that permits to use homology and cohomology groups to classify andcharacterize topological spaces, is that these objects are topological invariants , meaning thatisomorphic spaces have the same (co)homology groups. Moreover, a stricter result holds: twohomotopy-equivalent topological spaces have the same cohomology and homology groups. Werecall that two continuous maps F, G : M → N between topological spaces are homotopic if it exists a continuous map H : [0 , × M → N that deforms continuously F in G , i.e. H (0 , x ) = F ( x ) and H (1 , x ) = G ( x ) for every x ∈ M . Homotopy of maps is an equivalentrelation, and we denote it by F ∼ G . Two topological spaces M, N are said to be homotopy-equivalent , or of the same homotopy type, if there exist two maps F : M → N and G : N → M such that ( G ◦ F ) ∼ id M . Homotopy-equivalence is also an equivalence relation, that we denotealso as M ∼ N . The result stated above is then, for cohomologies M ∼ N ⇒ H ∗ ( M ) ∼ = H ∗ ( N ) . (2.9) In the context of smooth manifolds and de Rham cohomology, this is analogous to the pull-back of differentialforms. In category theory language, we can summarize these properties saying that singular homology H ∗ ( · ) is a covariant functor between the categories Top of topological spaces and Ab of Abelian groups, and singularcohomology H ∗ ( · ) is a contravariant functor between Top and Ab . Anyway, we will not need such a terminologyfor what follows. See for example [27], Appendix A, for a quick introduction to the subject. hapter 2. Equivariant cohomology 13 (a) (b) Figure 2.2: (a) The 2 inequivalent non-exact 1-cycles σ and σ (cid:48) on the 2-torus. (b) The genus- g surface Σ g has 2 g inequivalent non-exact 1-cycles. Figures adapted from [23]. Example 2.1.2 (More singular homologies) . ( pt ) We can consider the very trivial case of M being just a point. In this case, the in-formal discussion of Example 2.1.1 can be carried out just for the 0-dimensional sim-plices: H ∗ ( pt ; Z ) ∼ = Z in degree 0. It follows by definition and by the universal coeffi-cient theorem that for any commutative ring A (as R ), H ( pt ; A ) ∼ = H ( pt ; A ) ∼ = A and H q ( pt ; A ) ∼ = H q ( pt ; A ) ∼ = 0 for q >
0. By the homotopy-invariance property discussedabove, any contractible space will have the same trivial cohomology and homology as thepoint!( C ) Let us look at another simple case, the cylinder C = S × [0 , ,
1] is contractible, S × [0 , ∼ S . This means that H q ( S × [0 , R ) ∼ = H q ( S ; R ) ∼ = H q ( S ; R ).( T ) A less trivial example is the 2-torus T = S × S . In this case no one of the factors iscontractible, so we cannot use the homotopy invariance to get the result from a simplerspace. We can anyway get the answer using the same method of Example 2.1.1. Startingfrom the top-degree homology group, we notice that the only boundary-less surface onthe torus is the torus itself. Thus analogously to all the other cases, H ( T ; Z ) ∼ = Z or H ( T ; R ) ∼ = R . Since the torus is connected, in degree 0 we get trivially H ( T ; R ) ∼ = R .In degree 1 we see the difference with the other cases. On the torus there are twoinequivalent ways of drawing a closed line that is not a boundary of any 2-dimensionalsurface, following essentially the two factors of S (see figure 2.2). This means that the1 st homology group is generated by two elements, and thus H ( T ; Z ) ∼ = Z ⊕ Z . The samereasoning can be applied to higher genus surfaces Σ g , giving H (Σ g ; Z ) ∼ = ( Z ) ⊕ g .We leave now the purely topological setup, since in physics we are mostly interested instudying local properties, i.e. from the differential geometry point of view. We assume towork in the smooth setting and consider M to be a d -dimensional C ∞ -manifold. With T M we denote its tangent bundle , and with T ∗ M its cotangent bundle . At every point p ∈ M , Sections of any bundle E → M over M will be denoted in the following with Γ( M, E ), or Γ( E ) when thebase space is clear from the context. For example, vector fields are elements of Γ( T M ). T p M and T ∗ p M are the dual vector spaces of tangent vectors and 1-forms at p , respectively.We consider the exterior algebra (cid:86) ( T ∗ p M ), with the wedge product ∧ making it in a graded-commutative algebra, and the exterior derivative d : (cid:86) ( T ∗ p M ) → (cid:86) ( T ∗ p M ) acting as a gradedderivation of deg( d ) = +1. Extending these operations point-wise for every point p ∈ M , wehave the bundle of differential forms over M ,Ω( M ) := d (cid:77) k =0 Ω k ( M ) with Ω k ( M ) := (cid:71) p ∈ M k (cid:94) ( T ∗ p M ) . (2.10)(Ω( M ) , ∧ , d ) is thus a dg-algebra over the commutative ring C ∞ ( M ), and naturally defines acochain complex called the de Rham complex . The associated cohomology groups are the deRham cohomology groups , constituting the graded-commutative ring H dR ( M ) = d (cid:77) k =0 H kdR ( M ) with H kdR ( M ) := H k (Ω( M ) , d ) = Ker( d k ) (cid:30) Im( d k − ) . (2.11)The ring structure of H dR ( M ) is naturally inherited from the wedge product of differentialforms, that lifts at the level of cohomology classes. In fact, for two closed forms ω, η ∈ Ω( M ),[ ω ] ∧ [ η ] := [ ω ∧ η ] (2.12)is well-defined. The final important result that we state, and that will be crucial to extend tothe equivariant setting in the following section, is the so called de Rham’s theorem : Theorem 2.1.1 (de Rham) . The de Rham cohomology of the smooth manifold M is isomorphicto its singular cohomology with real coefficients: H dR ( M ) ∼ = H ∗ ( M ; R ) . The power of this theorem is that it allows to study topological properties of the manifold(recall that H ∗ ( M ; R ) are homotopy-invariants) using differential geometric (so local) objects,the differential forms. We say that the de Rham complex (Ω( M ) , d ) constitute an algebraicmodel for the singular cohomology of M . Notice that, by dimensionality reasons, we get triviallyalso in this case that the cohomology groups H qdR ( M ) for q > dim( M ) are automatically zero.Another important property that is intuitively very clear from the de Rham complex is the Poincar´e duality . For a closed connected manifold M this states that, as vector spaces H kdR ( M ) ∼ = H dim( M ) − kdR ( M ) . (2.13)A crucial tool for the proof of de Rham’s theorem is the so-called Stokes’ theorem , that relatesthe integral of an exact d -form over a d -dimensional manifold to the integral of its primitiveover the ( d − (cid:90) M dω = (cid:90) ∂M ω. (2.14)Notice that integration over M when ∂M = ∅ can be regarded as a function on the d th deRham cohomology (cid:82) : H ddR ( M ) → R . Analogously to the singular cohomology, in category theory language the de Rham cohomology H dR ( · ) is acontravariant functor between the categories Man of smooth manifolds and Ab of Abelian groups. This result can be seen also in the topological setup for singular cohomology groups, as it should be by deRham’s theorem. The operation that corresponds to the wedge product between singular cohomology classes iscalled cup-product [24]. It is important to remember that cohomology in general has a ring structure. hapter 2. Equivariant cohomology 15
Example 2.1.3 (Cohomology rings) . With the help of de Rham’s theorem, we can computesome of the previous example directly at the level of cohomology using differential forms andintegration. Let us consider the case of the tours T = ( S ) . We can parametrize it withcoordinates ( x, y ) taking values in [0 , ⊂ R . If we call α := dx and β := dy in Ω ( T ), anatural choice of volume form is ω := α ∧ β , that gives vol( T ) = 1. The volume form is of courseclosed by dimensionality, but it cannot be exact since otherwise by Stokes’ theorem the volumeof the torus would be 0, so it defines a non-trivial cohomology class [ α ∧ β ]. Any other 2-formis of the type ω (cid:48) = f ω for some f ∈ C ∞ ( T ), but closed forms must satisfy df = 0, so f ∈ R constant. We conclude that any other independent closed 2-form has to be “cohomologous” to[ α ∧ β ], so that in top-degree H dR ( T ) ∼ = span { [ α ∧ β ] } ∼ = R .In degree 1, any closed form must be a combination of α and β with real coefficients (sinceagain d ( f α ) = 0 ⇔ df = 0), so they are the only independent closed 1-forms (they correspondto the volume forms for the two S factors). To see whether or not they are exact, we can useStokes’ theorem: if they are, then their integral over any closed curve on T must be zero. Butwe can take the two curves σ ( t ) = ( t,
0) and σ (cid:48) ( t ) = (0 , t ) of Figure 2.2 and see that (cid:90) σ α = 1 = (cid:90) σ (cid:48) β, so they define two independent cohomology classes [ α ] and [ β ]. This means that H dR ( T ) ∼ =span { [ α ] , [ β ] } ∼ = R ⊕ R .Since the torus is connected, the only closed 0-form is a constant number, that we can choseto be 1. Thus, H dR ∼ = R . We can further easily get the ring structure of H dR ( T ) by lookingat the multiplication rules between the generators. If we call a := [ α ] and b := [ β ], the wedgeproduct of differential forms gives the following rules a ∼ , b ∼ , ab + ba ∼ . Thus we can rewrite the cohomology ring as a polynomial ring over the indeterminates ( a, b ),taken in degree 1, that satisfy the above rules: H ∗ ( T ; R ) ∼ = R [ a, b ] (cid:30) ( a , b , ab + ba ) , where ( a , b , ab + ba ) denotes the quotient by the ideal generated by the corresponding expres-sions.In the same fashion we can rewrite the cohomology rings of the other examples that we gaveabove for the n -sphere. Introducing an indeterminate u of degree n , and the multiplication rule u ∼
0, its cohomology ring can be expressed as H ∗ ( S n ; R ) ∼ = R [ u ] (cid:30) u . We quote another example, that will enter in the case of equivariant cohomology with respectto a circle action by U (1) ∼ = S . For the complex projective plane C P n , it turns out that H ∗ ( C P n ; R ) ∼ = R [ u ] (cid:30) u ( n +1) where deg( u ) := 2. In the limiting case n → ∞ , one has thus H ∗ ( C P ∞ ; R ) ∼ = R [ u ], thepolynomials in u . Another powerful tool to practically compute cohomology groups and rings, at the topological level, goesby the name of spectral sequences . See for example [18].
As already mentioned, equivariant cohomology is an extension of the standard cohomologytheory, partly reviewed in the last section, to the cases in which the space M is acted uponby some group G . This is the common setup in physics, from the finite-dimensional cases ofclassical Lagrangian or Hamiltonian mechanics to the infinite dimensional case of QuantumField Theory, where M can be the configuration space, the phase space, or the space of fields,and G is a Lie group representing a symmetry of the physical system. In gauge theory forexample, we want to identify those physical configurations that are equivalent modulo a gaugetransformations, so the moduli space of gauge orbits M/G . In Poincar´e-supersymmetric theo-ries, the group G is actually the Poincar´e group of spacetime symmetries. In all these cases weare interested in the cohomology of M modulo these symmetry transformations, since manyprimary objects of study (partition functions, expectation values...) are usually given in termsof integrals over M . Before moving to the technical definition of G -equivariant cohomology of M , we recall some terminology about group actions. Definition 2.2.1. (i) Given a group G and a topological space M , a G -action on M isgiven by a group homomorphism ( left action) or anti-homomorphism ( right action) ρ : G → Homeo( M ) (or Diff( M ) for smooth manifolds) . If m ∈ M, g ∈ G , the left action of g on m can be denoted ρ ( g ) m ≡ g · m , and the rightaction m · g , if this causes no confusion. M is said to be a (left or right) G -space .(ii) If M, N are two G -spaces, on the product M × N it is canonically defined the diagonal G -action ρ M × N ( g )( m, n ) := ( ρ M ( g ) m, ρ N ( g ) n ) for m ∈ M, n ∈ N, g ∈ G. (iii) Given a point m ∈ M , the orbit of m is the subset of M of all points that are reachedfrom m by the action of G . The orbit space with respect to the G -action is M/G . (iv) The stabilizer (or isotropy group , or little group ) of m is the subgroup of G of all elementsthat act trivially on m , i.e. g · m = m . The G -action is called free if the stabilizer ofevery point in M is given by the identity of G . The G -action is called locally free if thestabilizer of every point is discrete . The fixed point set F ⊆ M , is the set of all pointsthat are stabilized by the entire G .(v) Morphisms of G -spaces are called G -equivariant functions. f : M → N is G -equivariantif f ( g · m ) = g · f ( m ), for every m ∈ M, g ∈ G .In the following we will not care much about distinguishing between left and right actions,and assume all G -actions are from the left, unless otherwise stated. For the first part of thediscussion it is not needed, but we are going to assume M and G to be at least topologicalmanifolds, and then specialize to the case of smooth manifolds, since these are the most commonstructures arising in physics. Since, as we said above, we are interested in identifying those We are going to work practically always with smooth manifolds and (compact) Lie groups, but for themoment we do not need this level of structure on M and G . It is easy to check that m ∼ m (cid:48) ⇔ m (cid:48) = g · m for some g ∈ G is an equivalence relation. hapter 2. Equivariant cohomology 17elements in M that are equivalent up to a “symmetry” transformation by G , the first candidatefor the G -equivariant cohomology of M could be simply the cohomology of the orbit space M/G , H ∗ G ( M ) := H ∗ (cid:16) M (cid:30) G (cid:17) . (2.15)This definition has the problem that, if the G -action is not free and has fixed points on M , theorbit space is singular: in the neighborhood of those fixed points there is no well-defined notionof dimensionality. This kind of singular quotient spaces are called orbifolds . Example 2.2.1 (Some group actions and orbit spaces) . (i) Let us consider the Euclidean space R n . O ( n ) rotations (and reflections) act naturally onit, with the only fixed point being the origin. Any point but the origin identifies a directionin the Euclidean space, and thus is stabilized by the subgroup of n − O ( n − SO (2)-action is free on R \{ } . If we bring translations into the game, considering the Euclideanspace as an affine space acted upon by ISO ( n ) = R n (cid:111) O ( n ), then the stabilizer of anypoint is the entire O ( n ), since any point can be considered an origin after translation. Sothe action of ISO ( n ) is neither free nor locally free, but has no fixed points on the entire R n .Considering only rotations, the orbit space R n /O ( n ) is the space of points identified up totheir angular coordinates, that is an half-line starting from the origin, R n /O ( n ) ∼ = [0 , ∞ ).This is not a manifold, since the interval is closed on the left, giving a “singularity” onthe original fixed point of the action.Notice that, by embedding S n − in R n , an O ( n )-action descends on it, and the orbit ofany point of S n − is the sphere itself. The orbit of any point can be seen as the quotientof O ( n ) by the stabilizer of that point, so that one has O ( n ) (cid:30) O ( n − ∼ = S n − . This quotient describes the common situation of spontaneous symmetry braking inside O ( n )-models, in Statistical Mechanics.(ii) One can always consider circle actions on the spheres S n . Starting with n = 1, andconsidering the circle as embedded in the C -plane, U (1) acts on itself by multiplication: e iϕ (cid:55)→ e ia e iϕ for some a ∈ [0 , π ). This action has clearly no fixed points, and it is alsofree. Thus the quotient is well defined, giving simply S /U (1) ∼ = pt .The U (1)-action on the 2-sphere is already more interesting. Rotations around a givenaxis fix two points on S , that we identify with the North and the South poles. If weexclude the poles the resulting space is homeomorphic to a cylinder, the U (1)-actionbecomes free and indeed we have that S × (0 , → (0 ,
1) is a trivial principal U (1)-bundle. But considering the poles, the quotient space is singular since S /U (1) ∼ = [0 , n = 3. The 3-sphere S ∼ = SO (2) can be parametrized by apair of complex numbers ( z , z ) such that | z | + | z | = 1. The circle then acts naturallyby diagonal multiplication: ( z , z ) (cid:55)→ ( e ia z , e ia z ) for some e ia ∈ U (1). This action isclearly free, since the two coordinates z i cannot be simultaneously zero on the sphere,8 2.2. Group actions and equivariant cohomologyFigure 2.3: The circle acting on the 2-sphere, the orbits being the parallels. The orbit space S /U (1) is a meridian, homeomorphic to the interval [0 , U (1)-bundle known as the Hopf bundle . The equivalence classes [ z , z ] ∈ S /U (1) describe,by definition, points on the complex projective line C P ∼ = S , that is isomorphic to theRiemann 2-sphere. The Hopf bundle can be thus seen as S → S , with typical fiber S .Clearly this is not a trivial bundle, since S (cid:54) = S × S .The above case generalizes to any odd-dimensional sphere S n +1 , since they all can beembedded in complex spaces C n +1 . The circle acts always by diagonal multiplication, andthe resulting action is free. The bundles S n +1 → S n +1 /U (1) ∼ = C P n are all principal U (1)-bundles over the complex projective spaces C P n .(iii) The last case we mention is the possible U (1)-action on a torus T = ( S ) , by rotationsalong one of the two factors. This is the only possible free action on a closed surface,giving the well defined quotient T /U (1) ∼ = S . More examples can be found in [28].The example above showed that also in very simple cases singularities can appear in quotientspaces, so that one cannot define cohomology in a smooth way using the powerful de Rhamtheorem. It is thus more convenient to set up a definition of equivariant cohomology thatautomatically avoids this problem. This more clever definition is given by the Borel construction for the G -space M . Definition 2.2.2.
Considering a G -space M , its associated Borel construction , or homotopyquotient , is M G := ( M × EG ) (cid:30) G ≡ M × G EG where EG is some contractible space on which G acts freely, called the universal bundle of G (see Appendix B.2 for the precise definition). The G -equivariant cohomology of M is thendefined as H ∗ G ( M ) := H ∗ ( M G ) . We assume the action on the product M × EG to be the diagonal action. Notice that, since G acts freely on EG , the action on the product is automatically free. Indeed, if in the worstcase p ∈ M is a fixed point, for every point e ∈ EG , g · ( p, e ) = ( p, g · e ) (cid:54) = ( p, e ). This meansthat the homotopy quotient defines a smooth manifold, and we can hope for a generalizationof de Rham’s theorem, allowing to study this topological definition from its smooth structurein terms of something analogous to the differential forms on M . We will discuss this result inthe next section. From now on we always consider cohomologies with coefficients in R , unless otherwise stated. hapter 2. Equivariant cohomology 19Since the space EG does not need to either exist or be unique a priori, one could thinkthat the above definition contains some degree of arbitrariness, so a natural question is: isequivariant cohomology well defined? The answer is of course yes, and the crucial fact allowingthis stands in the contractibility of the space EG . The arguments that lead to this conclusionare summarized in Appendix B.2, together with some examples of universal bundles. Theimportant property that one has to keep in mind is that, intuitively, to get it acted freely by G and being contractible, one has to define it “so big” that for any other principal G -bundle P ,there is a copy of P sitting inside EG . This is why it is called “universal”. Here we just noticethat, if we assume a contractible free G -space EG to exist, we have an homotopy equivalence M × EG ∼ M , that descends also to the homotopy quotient( M × EG ) (cid:30) G ∼ M (cid:30) G, (2.16)since one can show that M × G EG → M/G is a fiber bundle with typical fiber EG [18]. Fromhomotopy invariance of cohomology, we see that at least in the case in which G acts freely on M and M/G is well defined, the equivariant cohomology reduces to the naive definition above, H ∗ G ( M ) = H ∗ ( M × G EG ) ∼ = H ∗ (cid:16) M (cid:30) G (cid:17) . (2.17)Notice that the contractible space EG alone has a very simple cohomology. Indeed, for what wepointed out in Example 2.1.2, it must be H ∗ ( EG ) ∼ = H ∗ ( pt ) ∼ = R in degree zero. When we takethe quotient, the base space BG := EG/G can have a less trivial cohomology. This space iscalled classifying space of the Lie group G . When, for example, the G -action on M is trivial (allpoints are fixed points), the homotopy quotient is just M × G EG ∼ = M × ( EG/G ) = M × BG ,and in this case we have H ∗ G ( M ) = H ∗ ( M × BG ) ∼ = H ∗ ( M ) ⊗ H ∗ ( BG ) , (2.18)so that the homotopy quotient by a trivial action does not bring any further information to thecohomology of M but for tensoring it with the cohomology of the classifying space. In Section2.4 we will see that the latter can be described in general by a very simple algebraic model,while here we carry on the example of the case G = U (1). Example 2.2.2 (A few U (1)-equivariant cohomologies) . To search for a suitable principal U (1)-bundle whose total space is contractible, we can first notice from Example 2.2.1 thatwe already described a class of principal U (1)-bundles, S n +1 → C P n , whose total spaces arethe odd-dimensional spheres. The bad news is that any of these total spaces are contractible,but this problem can be solved considering the limiting case n → ∞ , since it turns out that S ∞ = (cid:83) n S n +1 ≡ (cid:83) n S n is contractible [18]! Thus the universal bundle for U (1) can be chosento be EU (1) = S ∞ , and the classifying space BU (1) = C P ∞ . Being infinite-dimensional, theyare strictly speaking not manifolds, but S ∞ → C P ∞ is still a topological bundle, and this isenough for the definition of an homotopy quotient.(i) In Example 2.1.2 we quoted the resulting cohomology ring of the complex projectiveplanes, H ∗ ( C P n ) ∼ = R [ φ ] /φ n +1 and H ∗ ( C P ∞ ) = H ∗ ( BU (1)) ∼ = R [ φ ], with φ in degree 2. This is an application of the so-called
K¨unneth theorem [24]. Notice that any sphere S n can be embedded as the equator of S n +1 . Thus there is a sequence of inclusions S ⊂ S ⊂ · · · as well as C P ⊂ C P · · · , and the circle action is compatible with the inclusion. Thus, in thelimit, a free circle action induces on S ∞ . U (1)-equivariant cohomology of any space M on which U (1) acts trivially is,from (2.18), H ∗ U (1) ( M ) = H ∗ ( M ) ⊗ R [ φ ] . In particular if M is contractible, H ∗ U (1) ( M ) = H ∗ U (1) ( pt ) = R [ φ ].(ii) The opposite case is the one of a free action, for example the circle acting on itself. Aswe pointed out above, S /U (1) ∼ = pt , so simply H ∗ U (1) ( S ) ∼ = R .(iii) The last example that we mention is the case of U (1) acting on S . The equivariantcohomology H ∗ U (1) ( S ) is non-trivial a priori, as we remarked above, and it can be cal-culated easily for example using spectral sequences. We will not enter in the detail ofthe calculation but only describe the result. Consider first the standard cohomology ofthe 2-sphere that we already saw in various examples, being H ∗ ( S ) ∼ = R ⊕ R y , wherewe explicitly wrote a generator y for the term in degree 2, that can be identified in thede Rham model by a volume form y = [ ω ] , ω ∈ Ω ( S ). It turns out that its equivariantversion can be obtained simply by tensoring with the polynomial ring H ∗ ( BU (1)) = R [ φ ], H ∗ U (1) ( S ) ∼ = R [ φ ] ⊕ R [ φ ] y, although the generator y has now a different interpretation, that we will give in termsof an equivariant version of the de Rham model in the next sections. This equivariantcohomology actually can be given a ring structure, defining the multiplication y · y = aφy + bφ for some constants a, b . It turns out [18] that the correct constants are a = 1 , b = 0,making R [ y, φ ] (cid:30) ( y − φ ) → H ∗ U (1) ( S )into a ring isomorphism, where the denominator stands for the ideal generated by theexpression ( y − φ ) in R [ y, φ ]. Notice that the cohomology groups H nU (1) ( S ) are nownon-empty in every even-degree (while in odd-degree they are all trivial), even when n is bigger than the dimension of the sphere! This intuitively matches the fact that thequotient S /U (1) is singular, and thus simple dimensionality arguments do not make senseanymore at the fixed points. From now on, we specialize the equivariant cohomological theory to G being a Lie groupwith Lie algebra g , and M being a smooth G -manifold. We saw that de Rham’s theoremprovides an algebraic model for the singular cohomology (with real coefficients) of the smoothmanifold M , through the complex of differential forms. We now describe a way to obtain analgebraic model for the homotopy quotient ( M × EG ) /G , the so-called Weil model for the G -equivariant cohomology of M . From the discussion of the last sections, it is already imaginablethat this will contain in some way the de Rham complex of M , but modifying it through asomewhat “trivial” extension, in the sense of the triviality of the cohomology of EG . This isthus the most natural model that is connected to the topological definition of the last section, In terms of differential forms, ω will have to be equivariantly extended in the Cartan model, as discussed atthe end of Section 2.4. Some of what follows is only rigorous if G is compact, but the formal discussion can be applied generically. hapter 2. Equivariant cohomology 21but we will see that it is also overly complicated. In fact, in the next section we will describe asimpler but equivalent way to obtain the same equivariant cohomology, the Cartan model , thatis more intuitive from the differential geometry point of view, and that we will use to generalizethe theory of integration to the equivariant setting. This is what we often use in physics forpractical calculations.Before defining the Weil model, we notice that, in presence of a G -action, the de Rhamcomplex (Ω( M ) , d ) of differential forms on M has more structure than being a dg algebra. Infact, if ρ : G → Diff( M ) is the G -action, this induces an infinitesimal action of the Lie algebra g on any tensor space via the Lie algebra homomorphism g → Γ( T M ) X (cid:55)→ X := ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ρ (cid:0) e − tX (cid:1) ∗ (2.19)that defines for any X ∈ g the corresponding fundamental vector field X ∈ Γ( T M ). Then g acts infinitesimally on Ω( M ) via the Lie derivative and the interior multiplication with respectto the fundamental vector fields, L X ( α ) := L X ( α ) ι X α := ι X α for X ∈ g , α ∈ Ω( M ) , (2.20)with the additional property ( Cartan’s magic formula ) L X = d ◦ ι X + ι X ◦ d. (2.21)This makes Ω( M ) into a so-called g -gd algebra . In general, a g -gd algebra is defined as adifferential graded algebra (cf. definition 2.1.3) with two actions of g , denoted by analogy as ι and L , such that for any X ∈ g (i) ι X acts as an antiderivation of degree −
1, satisfying ( ι X ) = 0;(ii) L X acts as a derivation (of degree 0);(iii) the Cartan’s magic formula holds: L X = d ◦ ι X + ι X ◦ d .Morphisms of g -dg algebras are naturally defined as maps between g -dg algebras that commutewith all the above stated operations. It is not difficult to show that G -equivariant maps of G -manifolds induce pull-backs of differential forms that preserve the g -dg algebra structure.Now we can define the Weil model via an extension of (Ω( M ) , d ) that preserves this newstructure. We want this extension to be an algebraic analog of EG , so its cohomology mustbe trivial, but carrying information about g . To do this, we associate it to the characteristicdifferential structure of a generic principal G -bundle P (remember that any principal G -bundlesits inside EG ): its connection 1-form A ∈ Ω ( P ) ⊗ g and the associated curvature F = dA + [ A ∧ , A ] ∈ Ω ( P ) ⊗ g , satisfying the Bianchi identity dF = [ F, A ]. We first notice thatthe connection 1-form and the curvature 2-form can be seen as linear maps A : g ∗ → Ω ( P ) η (cid:55)→ A ( η ) := ( η ◦ A ) , F : g ∗ → Ω ( P ) η (cid:55)→ F ( η ) := ( η ◦ F ) . (2.22) Usually, in physics conventions, in the action of the exponential map one collects a factor of i at theexponent, in order to consider the Lie algebra element Hermitian for the most commonly considered groupactions. Left and right actions should be taken with different signs at the exponent. P ), if we start from algebras con-structed by g ∗ that respect the commutativity of 1- and 2-forms, respectively. This means that A has to “eat” an element of the (anticommutative) exterior algebra (cid:86) ( g ∗ ), while F has to“eat” an element of the (commutative) symmetric algebra S ( g ∗ ): A : (cid:94) ( g ∗ ) → Ω( P ) η ∧ · · · ∧ η k (cid:55)→ A ( η ) ∧ · · · ∧ A ( η k ) , F : S ( g ∗ ) → Ω( P ) η · · · η k (cid:55)→ F ( η ) ∧ · · · ∧ F ( η k ) . (2.23)We can combine the two maps in the homomorphism of graded-algebras f : S ( g ∗ ) ⊗ (cid:94) ( g ∗ ) → Ω( P ) η ⊗ ξ (cid:55)→ F ( η ) ∧ A ( ξ ) . (2.24)This captures the fact that a connection of P could be defined as a map S ( g ∗ ) ⊗ (cid:86) ( g ∗ ) → Ω( P ),and motivates the following definition. Definition 2.3.1.
The
Weil algebra of g is the graded algebra W ( g ) := S ( g ∗ ) ⊗ (cid:94) ( g ∗ )and the map f : W ( g ) → Ω( P ) is called the Weil map . We define the graded structure of W ( g )by assigning to the generators { φ a } of S ( g ∗ ) degree deg( φ a ) = 2, and to the generators { θ a } of (cid:86) ( g ∗ ) degree deg( θ a ) = 1.The generators { φ a , θ a } are two copies of a basis set for g ∗ , but taken in different degrees.With respect to this graded basis, the Weil algebra can also be written as W ( g ) = (cid:94) (cid:0) R [ φ , · · · , φ dim g ] ⊕ R [ θ , · · · , θ dim g ] (cid:1) , (2.25)and a generic element will be expanded as α = α + α a θ a + 12 α ab θ a θ b + · · · + α ( top ) θ θ · · · θ dim g with α I ∈ R [ φ , · · · , φ dim g ] , (2.26)since higher order terms vanish by the anticommutativity of the θ ’s. Here we suppressed tensorand wedge products to simplify the notation, as we will often do in the following. On thisbasis, the Weil map projects simply the connection and the curvature on the given Lie algebracomponents, f ( θ a ) = θ a ◦ A = A a , f ( φ a ) = φ a ◦ Ω = Ω a . (2.27)The Weil algebra is the central object to define an algebraic model for EG . We need todefine a g -dg algebra structure on it to properly take its cohomology, but this is naturally donerequiring the Weil map to be a morphism of g -dg algebras. This means introducing a differential d W : W ( g ) → W ( g ) and two g -actions L , ι such that the following diagram commutes for allthe three operations separately, W ( g ) Ω( P ) W ( g ) Ω( P ) . d W ι L f d ι L f (2.28)hapter 2. Equivariant cohomology 23One can check that, defining the Weil differential d W on the generators as d W θ a = φ a − f abc θ b θ c d W φ a = f abc φ b θ c or d W θ = φ −
12 [ θ ∧ , θ ] d W φ = [ φ ∧ , θ ] (2.29)where f cab are the structure constants of g , it commutes with f giving correctly the definitionof curvature and the Bianchi identity. Moreover, extending the differential on W ( g ) as anantiderivation of degree +1, it gives d W = 0 (since d W is a derivation, it is enough to check iton the generators). To be compatible with the properties of the connection and the curvature ι X A = A ( X ) = X, ι X F = 0 ∀ X ∈ g , (2.30)the interior multiplication must be defined as ι X θ a := θ a ( X ) = X a ι X φ a := 0 or ι X θ := θ ( X ) = Xι X φ := 0 (2.31)and extended as an antiderivation of degree −
1. Then the Lie derivative is simply defined viaCartan’s magic formula. We finally have defined the Weil algebra as a g -dg algebra. Theorem 2.3.1.
The cohomology of the Weil algebra is H ( W ( g ) , d W ) ∼ = R , H k ( W ( g ) , d W ) ∼ = 0 for k > . Proof.
The full proof can be found in [18]. Schematically it follows the proof of the Poincar´elemma: one has to find an cochain homotopy , i.e. a map K : W ( g ) → W ( g ) of degree -1, suchthat [ K, d W ] + = id . Then any cocycle ( d W α = 0) is also a coboundary, since α = [ K, d W ] + α = d W ( Kα ). This can be found for any degree k >
0. In degree zero W ( g ) ∼ = R by definition, soevery element is a cocycle, and no one is a coboundary for degree reasons. (cid:4) Example 2.3.1 (Weil model for torus and circle actions) . Consider the case of a compactAbelian group, i.e. a torus T = U (1) l for some l . Remember that a possible purpose of theWeil algebra is to describe the connection and the curvature of any principal T -bundle, so weare somewhat analyzing the structure of l electromagnetic fields, from the point of view of theLie algebra t . Since the structure constants are all zero, the Weil differential (2.29) and the t -actions (2.31) on the generators ( θ a , φ a ) simplify as d W θ a = φ a , d W φ a = 0 ,ι b θ a = δ ab , ι b φ a = 0 , L b θ a = 0 , L b φ a = 0 , where we denoted ι a ≡ ι T a and L a ≡ L T a , with { T a } the basis of t dual to the generators of W ( t ). We jump ahead a little and notice that the first line really resembles the structure of a“supersymmetry” transformation, with φ a being the “bosonic partner” of θ a . The remainingnon-Abelian piece of the generic case can be viewed as the action of a Chevalley-Eilemberg In the second column we introduced θ := θ i ⊗ T i and φ := φ i ⊗ T i in W ( g ) ⊗ g , where { T i } is a basis of g dual to the generators. Notice that this notation make the formulas independent on a choice of basis. Also,these are the objects that are really correspondent to the connection A and the curvature F on P , respectively. d W = d susy + d CE . We will return to this point in Section 4.5, after havingintroduced some technology about supergeometry and supersymmetry.Let us simplify again and prove theorem 2.3.1 for l = 1. In the case of U (1), the Lie algebrahas only one generator T ∼ = i , and the symmetric algebra is the algebra of polynomials in theindeterminate φ ∈ g ∗ , S ( g ∗ ) = R [ φ ], while the exterior algebra reduces to (cid:86) ( θ ) = R ⊕ R θ byanticommutativity. The Weil algebra is thus W ( u (1)) = R [ φ ] ⊗ ( R ⊕ R θ ) = R [ φ ] ⊕ R [ φ ] θ. The cohomology of W ( u (1)) = R in degree zero is as always trivial, since all constant numbersare closed, and none of them is exact, giving H ( W ( u (1)) , d W ) ∼ = R . In degree 1 we have W ( u (1)) = R θ , thus no one element (but zero) is closed. This extends to any odd-degree, since W ( u (1)) n +1 = R φ n θ , and d W ( φ n θ ) = φ n +1 (cid:54) = 0. This means that H n +1 ( W ( u (1)) , d W ) ∼ = 0. Indegree 2, we have W ( u (1)) = R φ , so that any element is closed but also exact, since φ = d W θ .This extends to any even-degree, since W ( u (1)) n = R φ n , and φ n = d W θφ n − = d W ( θφ n − ).Thus we have also H n ( W ( u (1)) , d W ) ∼ = 0, showing the triviality of the Weil algebra in thesimplest case of a circle action. Almost the same direct computation can be carried out in the l -dimensional case.Theorem 2.3.1 shows that we are in business: the Weil algebra is exactly an algebraicanalog of the universal bundle EG . Since the de Rham model for M is just Ω( M ), the product M × EG can be modeled by the complex W ( g ) ⊗ Ω( M ), since by the K¨unneth formula [24]and de Rham’s theorem H ∗ ( EG × M ) = H ∗ ( EG ) ⊗ H ∗ ( M ) = H ∗ ( W ( g ∗ ) , d W ) ⊗ H ∗ (Ω( M ) , d ) . (2.32)The differential and the g -actions are extended naturally on this complex as graded derivations,making it into a g -dg algebra too. Explicitly, d T := d W ⊗ ⊗ d,ι ≡ ι ⊗ ⊗ ι, L ≡ L ⊗ ⊗ L . (2.33)A model for the homotopy quotient M G can be guessed by the following argument. Since M G is the base of the principal bundle EG × G → M G , differential forms on M G identify the basicforms on EG × M (see Appendix A), i.e. those that are both G -invariant and horizontal . Itis thus reasonable that the homotopy quotient can be modeled by the basic subcomplex of theWeil model. Since the differential closes on the basic subcomplex, we are allowed to take itscohomology, giving the G -equivariant cohomology of M . This is exactly the content of the equivariant de Rham’s theorem . A recent original proof of it can be found in [18]. Theorem 2.3.2 (equivariant de Rham) . If G is a connected Lie group, and M is a G -manifold, H ∗ G ( M ) ∼ = H ∗ (( W ( g ) ⊗ Ω( M )) bas , d T ) . The equivariant de Rham’s theorem is telling us that the “correct” differential complex thatencodes the topology of the G -action on M is not anymore the complex Ω( M ) of differentialforms, but a modification of it through the presence of the Weil algebra. Remember always that, Remember that the C-E differential is the one that appears in BRST quantization of gauge theories. hapter 2. Equivariant cohomology 25via the Weil map, W ( g ) can be thought as in correspondence with the presence of a connectionand a curvature on some principal G -bundle. This means that the right extension of the deRham complex in presence of a G -action embeds somewhat the presence of a connection and acurvature with respect to G . We can analyze as an example the simplest case of a U (1)-action.The unrestricted Weil model is (from Example 2.3.1) W ( u (1)) ⊗ Ω( M ) = Ω( M )[ φ ] ⊕ Ω( M )[ φ ] θ, (2.34)thus any element can be written as α = α (0) + α (1) θ, (2.35)where α (0) , α (1) ∈ Ω( M )[ φ ] are polynomials in φ with differential forms as coefficients. Thesubcomplex of basic forms consists of those elements that satisfy both ι T α = 0 and L T α = 0,imposing the conditions α (1) = − ι T α (0) , L T α (0) = 0 . (2.36)Thus any basic element can be written as α = (1 − θι T ) (cid:80) i ( φ ) i α (0) i , where all the differentialforms α (0) i ∈ Ω( M ) G must be G -invariant, and the basic subcomplex can be identified withthe polynomials in φ with invariant differential forms as coefficients. In the next section wewill argue that this is not a special case, and that the Weil model can be simplified in general,producing another model for the same equivariant cohomology. As we said at the beginning of the last section, the cohomology of the Weil complex is notthe unique algebraic model for the G -equivariant cohomology of the G -manifold M . Moreover,although very transparent, the Weil model seems overly complicated for differential geometricapplications. In fact, the extreme simplicity of the basic subcomplex of the Weil algebra W ( g ) bas suggests that a simpler model for equivariant cohomology can be obtained simplifying this one.To see this, we analyze this basic subcomplex first. As we recalled at the end of the lastsection (for further details see Appendix A), a basic element α ∈ W ( g ) bas is both horizontal and invariant , i.e. ι X α = 0 = L X α. (2.37)The horizontal condition means that we pick only the symmetric algebra inside W ( g ), since bydefinition ι X φ a = 0. Imposing also the G -invariance we have W ( g ) bas ∼ = S ( g ∗ ) G , (2.38) i.e. the basic subcomplex is the algebra of Casimir invariants. It is easy to check that on thissubcomplex d W ∼ = 0, so that H ∗ ( W ( g ) bas , d W ) = H ∗ ( S ( g ∗ ) G , d W ) = S ( g ∗ ) G in degree 0, (2.39)since every element is closed, and no one element can be exact. Moreover, from the equivariantde Rham’s theorem H ∗ G ( pt ) ∼ = H ∗ ( W ( g ) bas , d W ), so the Casimir invariants are precisely thecohomology of the classifying space, H ∗ ( BG ) ∼ = S ( g ∗ ) G . (2.40)6 2.4. The Cartan modelMotivated by the above simplification, we can turn now to analyze the complete Weil model( W ( g ) ⊗ Ω( M )) bas . Let us see concretely what it means to restrict the attention to a basicelement α ∈ ( W ( g ) ⊗ Ω( M )) bas , starting from its expansion on a basis (2.26). Imposing thehorizontality condition means, using the multi-index notation I = ( a , · · · , a | I | ),0 = ι X α = ι X (cid:18) α + 1 | I | ! α I θ I (cid:19) ⇒ ι X α + 1 | I | ! ( ι X α I ) θ I + 1 | I | ! α I ( ι X θ I ) . (2.41)Equating the terms of the same degree and taking X to be a basis element, one arrives at thecondition on the various components, α a ··· a | I | = ( − | I | ι a · · · ι a | I | α (2.42)where we denoted ι k := ι T k , meaning that a horizontal element is fully determined by its firstcomponent α ∈ S ( g ∗ ) ⊗ Ω( M ), and it can be expressed as α = (cid:32) dim g (cid:89) k =1 (1 − θ k ι k ) (cid:33) α . (2.43)This comment, with some more checks (see again [18] for a complete proof), proves the followingtheorem, and extends the above discussion to the complete Weil model. Theorem 2.4.1 (Mathai-Quillen isomorphism) . There is an isomorphism of g -dg algebras,called the Mathai-Quillen isomorphism [29] (or
Cartan-Weil in [18]), ϕ : ( W ( g ) ⊗ Ω( M )) hor → S ( g ∗ ) ⊗ Ω( M ) α = α + 1 | I | ! α I θ I (cid:55)→ α (cid:32) dim g (cid:89) k =1 (1 − θ k ι k ) (cid:33) α (cid:55)→ α . The RHS of the isomorphism above inherits the g -actions and the differential from the Weilmodel on the LHS, making commutative the following diagram, similarly to (2.28),( W ( g ) ⊗ Ω( M )) hor S ( g ∗ ) ⊗ Ω( M )( W ( g ) ⊗ Ω( M )) hor S ( g ∗ ) ⊗ Ω( M ) . d T ι L ϕ d C ι L ϕ (2.44)In particular, the new differential is called Cartan differential , defined such that d C := ϕ ◦ d T ◦ ϕ − . (2.45)It is not difficult to get, directly from this definition, that it can be expressed more simply as d C = 1 ⊗ d − φ k ⊗ ι k (2.46)where d is the de Rham differential on Ω( M ). The two g -actions commute with ϕ withoutmodification, so they agree with their behavior in the Weil model, L X φ a = f abc φ b X c , ι X φ a = 0 . (2.47)hapter 2. Equivariant cohomology 27Using the nilpotency of d and ι , and Cartan’s magic formula, wee see that the Cartan dif-ferential on the horizontal subcomplex squares to a Lie derivative (an infinitesimal symmetrytransformation) d C = − φ k ⊗ L k , (2.48)so when we restrict to the G -invariant subspace, d C ∼ = 0 on ( S ( g ∗ ) ⊗ Ω( M )) G , as it should.Using the Mathai-Quillen isomorphism and the equivariant de Rham theorem we then have thefundamental result, H ∗ G ( M ) ∼ = H ∗ (cid:16) ( S ( g ∗ ) ⊗ Ω( M )) G , d C (cid:17) (2.49)that simplifies the algebraic model for the G -equivariant cohomology of M . Definition 2.4.1.
The g -dg algebraΩ G ( M ) := ( S ( g ∗ ) ⊗ Ω( M )) G with the Cartan differential d C is called the Cartan model for the equivariant cohomology of M . Elements of Ω G ( M ) are called equivariant differential forms on M . The degree of anequivariant form is the total degree with respect to the generators of Ω( M ) (in degree 1), andthe generators of S ( g ∗ ) (in degree 2).Equivariant forms will be in the next chapters the the principal object of study. In thephysical applications we are interested in, we will always search for an interpretation of thespace of interest as a Cartan model with respect to the action of a symmetry group G . TheCartan differential will be some object that squares to an infinitesimal symmetry, and onthe subspace of G -invariant forms (or “fields”, in the following) it will define a G -equivariantcohomology. Cartan differentials arise in Field Theory as supersymmetry transformations , thatwe will contextualize in Chapter 4 and relate to equivariant cohomology in Chapter 5. Thisinterpretation will be crucial in treating some of the most important objects in QM and QFTthat arise as (infinite-dimensional) path integrals over the space of fields. In fact, we will see inthe next chapter that integration of equivariant forms leads to powerful localization thorems ,that formally extended to the infinite-dimensional case greatly simplifying those integrals. Example 2.4.1 (Cartan model for U (1)-equivariant cohomology) . Until Chapter 6, we willactually deal with the equivariant cohomology with respect to a circle action of G = U (1), orat most a torus action of U (1) n for some n . As we saw also for the Weil model, this greatlysimplifies the problem, so we carry on that example also in the Cartan model for a U (1)-action.We recall from Example 2.3.1 that in the Weil model ι T φ = L T φ = d W φ = 0 , i.e. φ is automatically also U (1)-invariant. The equivariant forms are thusΩ U (1) ( M ) = ( R [ φ ] ⊗ Ω( M )) U (1) ∼ = Ω( M ) U (1) [ φ ] , so polynomials in φ with U (1)-invariant forms as coefficients. The Cartan differential is, sup-pressing tensor products, d C = d − φ ι T . In this 1-dimensional case, the indeterminate φ is just a spectator, and serves only to properlycount the equivariant form-degree. This is important of course, but for many purposes it creates8 2.5. The BRST modelno confusion to suppress its presence. More precisely, we often localize the algebra Ω G ( M ),substituting the indeterminate φ with a variable , and setting it for example to φ = − sothat d C = d + ι T . This differential squares to an infinitesimal symmetry generated by T , d C = L T . Equivariantdifferential forms after this localization are just U (1)-invariant forms.Often it is useful to generate equivariant forms from invariant differential forms in Ω( M ),for the purpose of integration for example. If α ∈ Ω n ( M ), an equivariant extension of α is˜ α ∈ Ω U (1) ( M ) such that ˜ α = α + f (2 n − φ + f (2 n − φ + · · · where any coefficient is an invariant form in Ω( M ). As an example, we can take the circleacting on the 2-sphere S , via rotations around a chosen axis. If θ is the polar coordinate and ϕ is the azimutal coordinate,( θ, ϕ ) (cid:0) e it · p (cid:1) := ( θ ( p ) , ϕ ( p ) + t ) for p ∈ S , e it ∈ U (1) , so that the fundamental vector field is T = ∂∂ϕ . Consider the canonical volume-form ω = d cos ( θ ) ∧ dϕ . It is obviously closed, and also U (1)-invariant, since L T ω = 0. Aiming to theextension of Ω( M ) to Ω U (1) ( M ), we can find an equivariantly closed extension of the volumeform ˜ ω = ω + f φ , with f ∈ C ∞ ( M ) such that d C ˜ ω = 0, so that it is closed in the “correct”complex. This imposes the equation df = ι T ω , and so f = − cos( θ ):˜ ω = ω − cos( θ ) φ. In this section we mention the last popular model for equivariant cohomology: the so-called
BRST model , or sometimes intermediate model . It is worth to mention it because we willsee in the next chapter that it is (as the first name suggests) intimately related to the BRSTmethod for gauge-fixing in the Hamiltonian formalism. Moreover, its complex is the one thatarises naturally in Topological Field Theories (TFT), as we will mention in Chapter 6. It isalso important because it provides (as the second name suggests) an “interpolation” betweenthe Weil and the Cartan models that we saw in the last sections, relating the latter more“physical”(or differential geometric) point of view with the former more “topological” one.As an algebra, the (unrestricted) complex of the BRST model is identical to that of theWeil model, B := W ( g ) ⊗ Ω( M ) , (2.50)but with the new differential (compare to (2.29) and (2.33)) d B = d W ⊗ ⊗ d + θ a ⊗ L a − φ a ⊗ ι a (2.51) This could seem harmless, but it is definitely a non-trivial move. We are really able to do this without spoil-ing the resulting equivariant cohomology because (algebraic) localization commutes with taking cohomology.More details on this are reported in Appendix B.3. Recall that this action has two fixed points, at the North and the South pole. ω is a symplectic form on S , and the df = ι T ω means that H := − f is the Hamiltonian function withrespect to the U (1)-action on the sphere. We will deepen this point of view in the next chapter. hapter 2. Equivariant cohomology 29that satisfies d B = 0 on B , and has the same trivial cohomology of the unrestricted Weil model.The idea that brought to the construction of this model in [30], was essentially to provealong the line we did in the last section the equivalence of the models, but from a slightlydifferent point of view. In fact one can construct d B using an algebra automorphism thatcarries the Weil model ( B, d W ) into the BRST model ( B, d B ), at the level of the unrestrictedalgebras. The restriction to the basic subcomplex gives then automatically the Cartan model.The automorphism is given by the map ϕ := e θ a ι a ≡ (cid:89) a (1 + θ a ⊗ ι a ) , (2.52)that looks very similar to the Mathai-Quillen isomorphism of theorem 2.4.1, but now is appliedto the whole algebra and not only on the horizontal part. Analogously to the definition of theCartan differential, d B is got as (2.51) from the commutativity of the diagram B BB B, d W ϕ d B ϕ (2.53)so that d B = ϕ − ◦ d W ◦ ϕ , as well as the two g -actions. In particular, it results ι ( B ) = ι ⊗ (cid:54) = ι ( W ) , L ( B ) = L ⊗ ⊗ L = L ( W ) , (2.54)where we called ι ( W ) , L ( W ) the one defined in (2.33). Thus the BRST differential carries thesame information of the Weil differential, giving the same trivial cohomology of the unrestrictedWeil model, H ∗ ( B, d B ) ∼ = H ∗ ( B, d W ) ∼ = H dR ( M ) (2.55)where the last equivalence follows from the triviality of the cohomology of the Weil algebra W ( g ∗ ). Of course, we have to restrict the the action of d B to the basic subcomplex, i.e. to theintersection with the kernels of ι ( B ) and L ( B ) , to get a meaningful G -equivariant cohomology.This reproduces again the Cartan model, as expected.This result shows that there is in fact a whole continuous family of g -dg algebras that giveequivalent models for the G -equivariant cohomology of M , because we can conjugate the Weildifferential through the modified automorphism ϕ t := e tθ a ι a with t ∈ R . (2.56)This produces, by conjugation, the family of differentials and g -actions on B , d ( t ) = d W ⊗ ⊗ d + tθ a ⊗ L a − tφ a ⊗ ι a + 12 t (1 − t ) f cab θ a θ b ⊗ ι c ,ι ( t ) = ι ⊗ − t )1 ⊗ ι, L ( t ) = L ( W ) ∀ t. (2.57)We see that for t = 0 we recover the Weil model, while for t = 1 we get the BRST model,as special cases. When restricted to the basic subcomplex, they all give the same equivariantcohomology. hapter 3Localization theorems infinite-dimensional geometry In this chapter we are going to introduce one of the most important results of the equivariantcohomology theory: the
Atiyah-Bott-Berline-Vergne (ABBV) localization formula for torusactions, discovered independently by Berline and Vergne [7], and by Atiyah and Bott [6]. Forthe most applications to QM and QFT, we will focus on the case of a circle action, and higher-dimensional generalizations will be postponed to Chapter 6. This formula can be viewed as ageneralization of an analogous result of Duistermaat and Heckman [5], that treats the specialcase in which the torus action is Hamiltonian on a symplectic manifold. We will expand onthis point of view in the second part of the chapter, since this is the situation we are morecommonly interested in when we treat dynamical systems in physics, at least at the classicallevel. The formal generalization of these formulas in the infinite-dimensional setting of QFTwill be discussed in Chapter 5.Since we are going to deal with integration of equivariant forms, we consider U (1)-equivariantcohomologies from the point of view of the Cartan model. The definition and notational conven-tions for integration of equivariant forms on a smooth G -manifold are reported in Appendix B.4,as well as an equivariant version of the Stokes’ theorem, needed for the proof of the localizationformulas that are presented in the following. Let U (1) act (smoothly) on a compact oriented n -dimensional manifold M without bound-aries, with fixed point set F ⊆ M , and consider the integral of a generic U (1)-invarianttop-form (cid:90) M α with α ∈ Ω n ( M ) U (1) . (3.1)As we saw in Example 2.4.1, in some cases we can find an equivariantly closed extension ˜ α ∈ Ω U (1) ( M ) such that d C α = 0, with d C = d + ι T (3.2) If not specified a manifold is always “without boundaries” since, strictly speaking, manifolds with boundarieshave to be defined in an appropriate separated way. In particular, near points at the boundary the manifold islocally homeomorphic not to an open set in R n , but to an half-open disk in R n .
312 3.1. Equivariant localization principleand T ∼ = i being the generator of U (1). Then we can deform the integral without changing itsvalue, I [ ˜ α ] := (cid:90) M ˜ α = (cid:90) M α (3.3)since only the top-degree component α is selected by integration. We are going to argue nowthat such integration of an equivariantly closed form is completely captured by its values at thefixed point locus F , using two different arguments. The first is cleaner, the second less explicitbut more common especially in the physics literature. We are going to need in both cases somepreliminary facts, that we collect in the following lemma. Lemma 3.1.1. (i) If G is a compact Lie group, any smooth G -manifold M admits a G -invariant Riemannian metric. In other words, G acts via isometry on M , and the funda-mental vector field T is a Killing vector field, L T g = 0 . (ii) If G is a connected Lie group, then the fixed point locus is the zero locus of all thefundamental vector fields: F ∼ = { p ∈ M | A p = 0 ∀ A ∈ g } . (iii) For any point p ∈ M , the stabilizer of p under the action of a Lie group G is a closedsubgroup of G . st argument: Poincar´e lemma For simplicity, suppose that F contains only isolated fixedpoints. From lemma (i), we can pick any U (1)-invariant metric on M , and define through it open balls of radius (cid:15) B ( p, (cid:15) ) around any fixed point p ∈ F . Then U (1) acts without fixedpoints on the complement ˜ M ( (cid:15) ) := M \ (cid:91) p ∈ F B ( p, (cid:15) ) , (3.4)that is a manifold with boundaries , them being the union of the surfaces of the balls at everyfixed point (oriented in the opposite direction to the usual one). From lemma (iii), the stabilizerof any point in ˜ M is a closed subgroup of U (1), but it cannot be U (1) since we excluded thefixed points, so it is discrete. This means that the U (1)-action on ˜ M is locally free.We would like to find an equivariant version of the Poincar´e lemma on ˜ M , where the actionis locally free. This means finding a map K : Ω( ˜ M ) U (1) → Ω( ˜ M ) U (1) of odd-degree suchthat [ d C , K ] + = id . If we are able to find such a map, then any equivariantly closed form η ∈ Ω( ˜ M ) U (1) is also equivariantly exact, η = ( Kd C + d C K ) η = K ( d C η ) + d C ( Kη ) = d C ( Kη ) . (3.5) Notice that we have localized the Cartan model and set φ = −
1, as discussed in Example 2.4.1. This willbe our standard convention up to Chapter 6. This follows from two facts: if a G -action on M is smooth and proper , then M admits a G -invariantRiemannian structure [31]; also, it is easy to prove that any smooth action of a compact Lie group is proper. This is just reasonable, see [18] for a proof. Connectedness is required because we passed from the actionof G to the action of g by the exponential map. By continuity of the action, every sequence inside the stabilizer of p converges inside the stabilizer. The closed subgroups of U (1) are U (1) and the finite cyclic groups { } , Z /n with n ∈ Z . hapter 3. Localization theorems in finite-dimensional geometry 33We can define the map K by multiplication with respect to an equivariant form ξ ∈ Ω( ˜ M ) U (1) of pure odd-degree such that d C ξ = 1, since[ ξ, d C ] + = ξd C + ( d C ξ ) + ( − deg( ξ ) ξd C = 1 . (3.6)This form can be defined using again a U (1)-invariant metric on M , that we call g . We definethe following 1-form away from the fixed point set, where T = 0, β := 1 g ( T , T ) g ( T , · ) (3.7)and notice that it is U (1)-invariant by invariance of g , and ι T β = 1, so that the action of theCartan differential on it gives d C β = dβ + 1. Then the odd-degree form ξ can be defined as ξ := β ( d C β ) − = β (1 + dβ ) − = β n − (cid:88) i =0 ( − i ( dβ ) i . (3.8)The inverse of the form (1 + dβ ) can be guessed pretending that dβ is a number, and using theTaylor expansion (1 + z ) − = ∞ (cid:88) i =0 ( − i z i . In the case of forms, the sum at the RHS stops at finite order, since by degree reasons ( dβ ) i = 0for i > ( n/ d C β ) − ( d C β ) = 1, d C ξ = 1, and deg( ξ ) is odd.Now we know that any equivariantly closed form in Ω( ˜ M ) U (1) is also equivariantly exact, sowe can simplify the integral I [ α ] of an equivariantly closed form α using an equivariant versionof Stokes’ theorem (see Appendix B.4): (cid:90) ˜ M α = (cid:90) ˜ M d C ( ξα ) = (cid:90) ∂ ˜ M ξα. (3.9)Taking the limit (cid:15) →
0, the domain of integration on the LHS covers all M , and the integralover the boundary on the RHS reduces to a sum of integrals over the boundaries of n -spherescentered at each fixed point p ∈ F (since ∂M = ∅ ). Thus the integral of an equivariantly closedform “localizes” as a sum over the fixed points of the U (1)-action, I [ α ] = (cid:90) M α = lim (cid:15) → (cid:90) ˜ M ( (cid:15) ) α = (cid:88) p ∈ F lim (cid:15) → (cid:18) − (cid:90) S n − (cid:15) ( p ) ( ξα ) (cid:19) = (cid:88) p ∈ F c p (3.10)for some contributions c p at each fixed point. The precise form of these contributions will bediscussed in the next section.2 nd argument: localization principle The second argument for the localization of theequivariant integral is less explicit, but more direct. Also, it is closer to the approach we willuse in the infinite-dimensional context of supersymmetric QFT.Again, we start from the integral I [ α ] of an equivariantly closed form α ∈ Ω( M ) U (1) . The ba-sic idea is to take advantage of the equivariant cohomological nature of the integral over M : thisdepends really on the cohomology class of the integrand, not on the particular representative.So we can deform the integral staying in the same class in a way that simplifies its evaluation,4 3.2. The ABBV localization formula for Abelian actionswithout changing the final result. To do this, we pick a positive definite U (1)-invariant 1-form β on M , and define the new integral I t [ α ] := (cid:90) M αe − td C β (3.11)with t ∈ R . It is again an integral of an equivariantly closed form, d C (cid:0) αe − td C β (cid:1) = ( d C α ) e − td C β − tα ( d C β ) e − td C β = 0 , (3.12)since d C = L T and β is U (1)-invariant. To show that this integral is equivalent to I [ α ], weshow that it is independent on the parameter t : ddt I t [ α ] = (cid:90) M α ( − d C β ) e − td C β = − (cid:90) M d C (cid:0) αβe − td C β (cid:1) (integration by parts)= 0 (equivariant Stokes’ theorem) . (3.13)Noticing that I [ α ] = I t =0 [ α ], from the t -independence it follows that I [ α ] = I t [ α ] for every valueof the parameter.We showed that the deformation via the exponential e − td C β does not change the equivariantcohomology class of the integrand, so we are free to compute the integral for any value of theparameter. In particular, in the limit t → ∞ , we see that the only contributions come fromthe zero locus of the exponential. This gives the “localization formula” (cid:90) M α = lim t →∞ (cid:90) M αe − td C β , (3.14)that will be the starting point for all the applications of the equivariant localization principleof the next chapters, also in the infinite-dimensional case in which M describes generically the“space of fields” of a given QFT.The 1-form β is usually called “localization 1-form”. Notice that choosing different localiza-tion 1-forms produces different practical localization schemes, but at the end of the computationthey must all agree on the final result! In particular, by lemma (i) we can pick a U (1)-invariantRiemannian metric g , and choose the 1-form as β := g ( T , · ) . (3.15)This makes it positive definite and produces the same localization scheme of the first argument,since its zeros coincide with the zeros of the fundamental vector field T and thus with the fixedpoint locus F of the circle action, by lemma (ii). Here we state the celebrated result by Atiyah-Bott and Berline-Vergne, about the localiza-tion formulas for circle and torus actions. The rationale of the last section showed that theequivariant cohomology of the manifold M is encoded in the fixed point set F of the symmetryaction, but left us with the evaluation of an integral over the fixed point set. We show thehapter 3. Localization theorems in finite-dimensional geometry 35result of this integration here, and we are going to give an argument for the proof in the nextchapter, with some tools from supergeometry. That proof is different from the original ones in[6, 7], but will introduce a method that can be easily generalized to functional integrals.To warm up, we consider first the simple case of isolated fixed point set F ⊆ M , and a U (1)-action. Notice that, at any fixed point p ∈ F , the circle action gives a representation of U (1) on the tangent space, since for any ψ ∈ U (1)( ψ · ) ∗ : T p M → T ψ · p M ≡ T p M, (3.16)so ( ψ · ) ∗ ∈ GL ( T p M ). Since T p M is finite dimensional, it can be decomposed in irreduciblerepresentations of U (1), T p M ∼ = V ⊕ · · · ⊕ V n . (3.17)The circle has to act faithfully on T p M , since if there was v ∈ T p M such that ( ψ · ) ∗ v = v , thenthe whole curve exp( tv ) = exp( t ( ψ · ) ∗ v ) = ψ · exp( tv ) would be fixed by U (1), thus p would notbe isolated. Recall that the irreducible representations of U (1) are complex 1-dimensional, andare labeled by integers, ψ = e ia ∈ U (1) , ρ m ( ψ ) := e ima with m ∈ Z . (3.18)This means that the irreducible representations in (3.17) are all non-trivial (of real dimension2), and that dim( M ) = 2 n . In other words, if a circle action on a manifold M has isolatedfixed points, M must be even-dimensional. Excluding the trivial representation with m = 0,the tangent spaces at the fixed points are thus labeled by a set of integers, T p M ∼ = V m ⊕ · · · ⊕ V m n (3.19)where ( m , · · · , m n ) ∈ Z n are called the exponents of the circle action at p ∈ F . They canbe regarded as maps m i : F → Z . We formulate now a simplified version of the localizationtheorem in term of this local data. The proof of this can be found in [18]. Theorem 3.2.1 (Localization for circle actions) . Let U (1) act on a compact oriented manifold M of dimension dim( M ) = 2 n , with isolated fixed point locus F . If m , · · · , m n : F → Z arethe exponents of the circle action, and α = α (2 n ) + α (2 n − φ + α (2 n − φ + · · · + α (0) is an equivariant top-form in Ω U (1) ( M ) such that d C α = 0, then (cid:90) M α (2 n ) = (cid:90) M α = (2 π ) n (cid:88) p ∈ F α (0) ( p ) m ( p ) · · · m n ( p )where the last component α (0) ∈ C ∞ ( M ). Example 3.2.1 (Localization on the 2-sphere) . Let us consider again the case of the heightfunction H : S → R such that, in spherical coordinates ( θ, ϕ ), H ( θ, ϕ ) := cos( θ ). In Example2.4.1 we related this function to the equivariantly closed extension of the volume form on the2-sphere, ˜ ω = ω + H. In terms of the Lie algebra representation, every exponent m coincide with the weight of the single generatorof U (1) in the fundamental representation. S . The 2-sphere has two isolated fixed points at the poles, and only one exponent m : F → Z . It is not difficult to see that the exponent of the action at the fixed points is m ( N ) = 1 at the North pole, and m ( S ) = − i.e. the integral of ω . Using the theorem we easily get the correct result, (cid:90) S ω = (cid:90) S ( ω + H ) = 2 π (cid:88) p ∈{ N,S } H ( p ) m ( p ) = 2 π (cid:18) cos(0)1 + cos( π ) − (cid:19) = 4 π. The second integral is the “partition function”on the sphere, Z ( t ) := (cid:90) S ωe itH = 1 it (cid:90) S e it ( H + ω ) where the second equality comes from degree arguments. This is the integral of an equivariantlyclosed form, since d C e it ( H + ω ) ∝ d C ˜ ω = 0, whose C ∞ ( S ) component is given by e itH . Using thelocalization theorem we get Z = 1 it π (cid:18) e it cos(0) e it cos( π ) − (cid:19) = 4 π sin( t ) t matching the result from the “semiclassical” saddle-point approximation (1.7).We now get to the main theorem, considering a more generic torus action with higherdimensional fixed point locus on M . Theorem 3.2.2 (Atiyah-Bott [6], Berline-Vergne [7]) . Let the torus T = U (1) l of dimension l act on a compact oriented d -dimensional manifold M , with fixed point locus F . If α ∈ Ω T ( M )is an equivariantly closed form, i.e. d C α = 0, and i : F (cid:44) → M is the inclusion map, then (cid:90) M α = (cid:90) F i ∗ αe T ( R ) | N where e T ( R ) | N is the T-equivariant Euler class of the normal bundle of F in M .This is the localization formula as originally presented for a torus action and fixed pointlocus F , that is generically an embedded (regular) submanifold of M . The normal bundle to F can be regarded as T N = T M (cid:30) i ∗ T F , (3.20)where the quotient is taken pointwise at any p ∈ F , so that the tangent bundle of M issplit as T M = i ∗ T F ⊕ T N . The finite sum is replaced by an integral over F , and the zero-degree component of α is replaced by the component with the correct dimensionality, thatmatches dim( F ), by pulling-back α on F . The product of the exponents at the denominator isrepresented in general by the equivariant Euler class of the normal bundle, e T ( R ) | N = Pf N (cid:18) R T π (cid:19) = Pf N (cid:18) R + µ π (cid:19) , (3.21)hapter 3. Localization theorems in finite-dimensional geometry 37where the pfaffian is taken over the coordinates that span the normal bundle T N , R is thecurvature of an invariant Riemannian metric on M , µ : t → Ω ( M ; gl ( d )) is the “moment map”that makes R T an equivariant extension of the Riemannian curvature in the Cartan model (seeAppendix B.1).As an example, let us apply the ABBV localization formula in the case of a discrete fixedpoint set F , so that we can recover at least the more readable version of theorem 3.2.1. Thenormal bundle in this case is the whole tangent bundle and, since F is 0-dimensional, therestriction of the equivariant curvature R T to F makes only its Ω component contribute, soPf( R T ) = φ a ⊗ Pf( µ a ). At an isolated fixed point p , as we said before, the tangent space T p is arepresentation space for the torus action. Since the torus is Abelian, analogously to the abovediscussion this representation can be decomposed as the sum of 2-dimensional weight spaces [22, 32], T p M ∼ = d/ (cid:77) i =1 V v i . (3.22)In Section 4.2 we will see that the moment map at an isolated fixed point encodes exactly theseweights, being the representation µ ( p ) : t → gl ( d ) ∼ = End( T p M ). The equivariant Euler classcomputes exactly the product of the weights, e T ( R ) p = 1(2 π ) d/ (cid:89) i v i = 1(2 π ) d/ (cid:89) i φ a ⊗ v i ( T a ) , (3.23)where T a are the generators of T . This recovers the formula for the circle action in theorem3.2.1, where the exponents play the role of the weights for the single generator of S .Notice that, as it is clear from the above example, in the generic l -dimensional case itis not so convenient to forget about the generators { φ a } of t ∗ , and the ABBV localizationformula should be thought as an equivalence of elements in H ∗ T ( pt ) = R [ φ , · · · , φ l ]. The LHSis clearly polynomial in φ a , so has to be the RHS. Since in the latter both the numerator and thedenominator are polynomials in φ a , some simplification has to occur in the rational expressionto give a polynomial as the final answer. Remark.
We anticipate that in QFT the pfaffian in the definition of the Euler class is usuallyrealized in terms of a Gaussian integral over Grassmann (anticommuting) variables, as wewill see in detail in Section 4.2. These “fermionic” Gaussian integrals arise naturally as “1-loop determinants” from some saddle-point (semi-classical) approximation technique to thepartition function of the theory, for example. In general, the differential form α will be an“observable” of the QFT, and the equivariantly closeness condition will be interpreted as itbeing “supersymmetric”. The localization locus F will be then the fixed point set of a symmetrygroup that is the “square” of this supersymmetry (as d C ∝ L T schematically), so a Poincar´esymmetry or a gauge symmetry. The integral then localizes onto the “moduli space” of gauge-invariant (or BPS) field configurations. In the context of Hamiltonian mechanics, the gaugesymmetry can be one generated by the dynamics of the theory itself, and in this case the pathintegral localizes onto the classical solutions of the equations of motion. The ABBV formulathus gives a systematic way to understand in which cases the semi-classical approximationresults to be exact. We will reexamine this point of view in the next section in the contextof finite-dimensional Hamiltonian mechanics, while in Chapter 5 we will describe the infinite-dimensional case of QM and QFT, giving some examples of the ABBV localization formula atwork.8 3.3. Equivariant cohomology on symplectic manifolds As we remarked at the beginning of the chapter, the localization formulas of the last sectioncan be seen as generalizing a similar result showed by Duistermaat and Heckman [5] in the con-text of Hamiltonian group actions on symplectic manifolds. This special case is of fundamentalimportance in physics, because this is the context in which classical Hamiltonian mechanicsis constructed. In some special cases also the quantum theory can be formally given such astructure, and thus some results from symplectic geometry can be extended to QM and QFTin general. We begin this section by quickly recalling some basic concepts about symplecticand Hamiltonian geometry, then we will describe how this can be seen as a special case ofequivariant cohomology theory from the point of view of the localization formulas.
The notion of phase space can be constructed in a basis-independent way in differentialgeometry through the definition of symplectic manifold . We suggest for example [28, 33, 34]for a complete introduction to the subject.
Definition 3.3.1. A symplectic manifold is a pair ( M, ω ), where M is a 2 n -dimensional smoothmanifold, and ω is a symplectic form on M :(i) ω ∈ Ω ( M );(ii) dω = 0;(iii) ω is non-degenerate.The fact that M is even-dimensional is not really a requirement but a consequence of itssymplectic structure. This is because any skew-symmetric bilinear map on a d -dimensionalvector space can be represented in a suitable basis by the matrix k − n n (3.24)with 2 n + k = d . To be non degenerate, it must be k = 0. The symplectic form is a skew-symmetric bilinear form on T p M at any point p ∈ M , so the even-dimensionality of M followsfrom its non-degeneracy. On manifolds, a stronger result than the above one holds: the so-called Darboux theorem . It states that, for every point p ∈ M , there exists an entire open neighborhood U p ⊆ M and a coordinate system x : U p → R n with respect to which ω µν = ω ( ∂ µ , ∂ ν ) has thecanonical form (3.24), with k = 0. The coordinates x are called Darboux coordinates . Noticethat from the non-degeneracy of ω we have a canonical choice for the volume form on M , theso-called Liouville volume form vol := ω n n ! = Pf || ω ( x ) µν || d n x = dp ∧ dp ∧ · · · dp n ∧ dq ∧ dq ∧ · · · ∧ dq n , (3.25)where ( q µ , p µ ) µ =1 , ··· ,n are Darboux coordinates. The closeness of ω implies that in some casesthere can be a 1-form θ ∈ Ω ( M ) such that dθ = ω. (3.26) This means that all symplectic manifolds look locally as the prototype R n with ω = (cid:80) ni =1 dx i ∧ dx i + n .This is a very strong property, compared for example with the Riemannian case. hapter 3. Localization theorems in finite-dimensional geometry 39Such a 1-form, if it exists, is called symplectic potential . In practice, sometimes it is usefulto locally define a symplectic potential even if ω is not globally integrable. Isomorphisms ofsymplectic manifolds are called symplectomorphisms or canonical transformations , defined asdiffeomorphisms that preserve the symplectic structure via pull-back.The standard example of a symplectic manifold is exactly the phase space associated tosome n -dimensional configuration space Q , i.e. its cotangent bundle M := T ∗ Q . A point p ∈ Q represents the “generalized position” of the system with coordinates q ( p ) = ( q µ ( p )) with µ = 1 , · · · , n , and a point p ∈ T ∗ Q represents the “generalized momentum”, with coordinates ξ ( p ) := ( q µ ◦ π ( p ) , ι µ ( p )) ≡ ( q µ , p µ ), where π : T Q → Q is the projection and ι µ ≡ ι ∂/∂q µ .The cotangent bundle has a canonical integrable symplectic form. In fact, the symplecticpotential is the so-called tautological 1-form given by the pull-back of the projection map, θ := π ∗ ∈ Ω ( T ∗ Q ). In Darboux coordinates, at a point p ∈ T ∗ Q , θ p = π ∗ ( p ) = p µ dq µ (3.27)where we denoted dq µ ≡ d ( q ◦ π ) µ = dξ µ with µ = 1 · · · , n , as 1-forms on the cotangent bundle.The canonical symplectic form is then just ω = dθ , and in Darboux coordinates ω = dp µ ∧ dq µ (3.28)where again we simplified the notation setting dp µ ≡ dξ µ for µ = n + 1 , · · · , n . Thus thecanonical coordinates on the cotangent bundle are Darboux coordinates. One can show thatcanonical symplectic structures over diffeomorphic manifolds are “canonically compatible”, i.e. if φ : Q → Q is a diffeomorphism, there is a lift of it as a symplectomorphism between( T ∗ Q , ω ) and ( T ∗ Q , ω ). If we take Q = Q , this means that there is a group homomorphismDiff( Q ) → Symp( T ∗ Q, ω ) . (3.29)This example showed that symplectic manifolds are the right generalization of the concept ofphase space in a fully covariant setting. It is thus common to call functions on a symplecticmanifold observables .Let us return to a generic symplectic manifold ( M, ω ). Giving to it some additional struc-ture, it is possible to define on it dynamics and symmetries in the sense of classical mechanics.Naturally, we call symmetry of (
M, ω ) a diffeomorphism φ : M → M that preserves thesymplectic structure, φ ∗ ω = ω , that is a symplectomorphism. At the infinitesimal level, adiffeomorphism can be generated by the flow of a vector field X ∈ Γ( M ), and the symmetrycondition is rephrased to L X ω = 0 . (3.30)Such a vector field is called symplectic vector field . It is easy to realize that a vector field issymplectic if and only if ι X ω = ω ( X, · ) is closed, by Cartan’s magic formula. More specialvector fields are those for which ι X ω is exact, so that it exists an observable f ∈ C ∞ ( M ) suchthat df = − ι X ω, (3.31)where the minus sign is conventional. The vector field X is called Hamiltonian vector field associated to the observable f . In components, ∂ µ f = ω µν X ν or X µ = ω µν ∂ ν f, (3.32)0 3.3. Equivariant cohomology on symplectic manifoldswhere ω µν is the “inverse” of the symplectic form. Of course Hamiltonian vector fields are sym-plectic, and the flow of the Hamiltonian vector field X preserves the value of the Hamiltonianfunction f , since L X ( f ) = X ( f ) = df ( X ) = ω ( X, X ) = 0. The flow of the Hamiltonian vectorfield is regarded as the “time-evolution” over the generalized phase space M , generated by theobservable f . Definition 3.3.2. An Hamiltonian (or dynamical) system is a tuple (
M, ω, H ), where (
M, ω ) isa symplectic manifold and H ∈ C ∞ ( M ) an observable called Hamiltonian . The time-evolution of points p ∈ M is defined by the flow of the Hamiltonian vector field X H of H , p ( t ) := γ Hp ( t )where γ Hp is the integral curve of X H with γ Hp (0) = p . In particular, the evolution of anobservable f ∈ C ∞ ( M ) is regulated by the equation of motion ˙ f ( p ) := ( f ◦ γ Hp ) (cid:48) (0) ≡ L X H ( f ) | p . The equation of motion can be rewritten in a more usual way introducing the
Poissonbrackets {· , ·} : C ∞ ( M ) × C ∞ ( M ) → C ∞ ( M ) such that { f, g } := ω ( X g , X f ), where X f , X g arethe Hamiltonian vector fields of f and g , respectively. In a chart and with respect to Darbouxcoordinates ( q µ , p µ ) on M , by the Darboux theorem the Poisson brackets take the usual form { f, g } = ∂f∂q µ ∂g∂p µ − ∂g∂q µ ∂f∂p µ . (3.33)With this definition we can write˙ f = −{ H, f } , ˙ q µ = ∂H∂p µ , ˙ p µ = − ∂H∂q µ , (3.34)recovering the Hamilton’s equations for the Darboux coordinates. The Poisson brackets areanti-symmetric and satisfy the Jacobi identity, so this turns ( C ∞ ( M ) , {· , ·} ) into a Lie algebra, and one can check that there is a Lie algebra homomorphism( C ∞ ( M ) , {· , ·} ) → (Hamiltonian v.f. , [ · , · ]) f (cid:55)→ X f , (3.35)where we also already used the fact that Hamiltonian vector fields form a Lie subalgebra withrespect to the standard commutator on Γ( T M ).We just reviewed that the concept of symmetry in symplectic geometry is correlated withthe concept of dynamics on the symplectic manifold. The next fact that we need is to connectthis formalism to the equivariant cohomology one, identifying these symmetries as generated bya group action on M . In particular, we would like to identify the Lie subalgebra of Hamiltonianvector fields as the Lie algebra of a Lie group that acts on the symplectic manifold. We canstart thus the discussion of symmetry by declaring that M is a G -manifold with respect to aLie group G of Lie algebra g . Denoting the G -action as ρ , this is called symplectic if it makes G act by symplectomorphisms on ( M, ω ), i.e. ρ : G → Symp(
M, ω ) . (3.36) In fact this is a
Poisson algebra , i.e. a Lie algebra whose brackets act as a derivation. hapter 3. Localization theorems in finite-dimensional geometry 41We can characterize again infinitesimally this action by saying that g acts on Ω( M ) via symplec-tic vector fields: if A ∈ g , the corresponding fundamental vector field preserves the symplecticstructure, L A ω = 0. We are interested in the special case analogous to the one above, in whichnot only a fundamental vector field is symplectic, but it is also Hamiltonian. This forces ageneralization of the concept of Hamiltonian function, because now there are more than oneindependent fundamental vector fields to take into account, if dim( g ) > Definition 3.3.3.
The G -action ρ : G → Symp(
M, ω ) on the symplectic manifold (
M, ω ) issaid to be an
Hamiltonian action if every fundamental vector field is Hamiltonian. In particular,there exists a g ∗ -valued function µ ∈ g ∗ ⊗ C ∞ ( M ) such that:(i) For every A ∈ g , µ ( A ) ≡ µ A ∈ C ∞ ( M ) is the Hamiltonian function with respect to A , dµ A = − ι A ω = ω ( · , A ) . (ii) It is G -equivariant with respect to the canonical (co)adjoint action of G on g ( g ∗ ), sofor any g ∈ G µ ◦ Ad ∗ g = ρ ∗ g ◦ µ or Ad ∗ g ◦ µ = µ ◦ ρ g , where in the first equation µ is considered as g µ −→ C ∞ ( M ), in the second one as M µ −→ g ∗ .If G is connected, this is equivalent to requiring µ : g → C ∞ ( M ) to be a Lie algebraanti-homomorphism with respect to the Poisson brackets, µ [ A,B ] = { µ B , µ A } ∀ A, B ∈ g . The map µ is called moment map , and ( M, ω, G, µ ) is called
Hamiltonian G -space .In general the job of the moment map is to collect all the “Hamiltonians” with respect towhich the system can flow. There are dim( G ) independent of them, one for every generator. Inthe 1-dimensional case, where G = U (1) (or its non-compact counterpart G = R ), the momentmap produces only one independent Hamiltonian, µ T ≡ H , and the above definition reducesto the Hamiltonian system ( M, ω, H ) of definition 3.3.2. Notice that for any Hamiltonianstructure we build on (
M, ω ), its flow preserves the symplectic form and thus the canonicalLiouville volume form ω n /n !. This is the content of the so-called Liouville theorem . We can first generalize what we noticed in examples 2.4.1 and 3.2.1, in the case of a circleaction on a symplectic manifold (
M, ω ). In the above examples the manifold was the 2-sphere S and the symplectic form was the canonical volume form. Rephrased in terms of symplecticgeometry, the existence of an equivariantly closed extension ˜ ω of the symplectic form is thecondition of U (1) acting in an Hamiltonian way, since d C ˜ ω = d C ( ω + H ) = ι T ω + dH = 0 (3.37)is satisfied if and only if dH = − ι T ω . This is readily generalizable to the multidimentional case,so that we can describe the Hamiltonian G -space ( M, ω, G, µ ) and its classical mechanics in If, for every g ∈ G , Ad g : G → G is the action by conjugation, the adjoint action Ad ∗ g on g is the push-forward of Ad g , while the coadjoint action Ad ∗ g on g ∗ is the pull-back of Ad g − . If g = exp( tA ) for some A ∈ g ,differentiating one gets the infinitesimal actions of g on g and g ∗ , ad A ( B ) = [ A, B ] and ad ∗ A ( η ) := η ([ · , A ]). G ( M ), ˜ ω := 1 ⊗ ω − φ a ⊗ µ a (3.38)where µ a ≡ µ T a ∈ C ∞ ( M ) and T a are the dual basis elements with respect to the generators φ a of S ( g ∗ ). It is straightforward to check that ˜ ω ∈ Ω G ( M ) is indeed G -invariant, and closedwith respect to d C thanks to the Hamiltonian property of the G -action, dµ a = − ι a ω .In the language of G -equivariant bundles (see Appendix B.1), the symplectic structure on M can be seen as the presence of a principal U (1)-bundle P → M whose connection 1-formis the symplectic potential θ (that has not always a global trivialization on M ), and whosecurvature is the symplectic 2-form ω = dθ (that instead transforms covariantly on M ). G actssymplectically if also θ is G -invariant, L X θ = 0 ⇒ L X ω = 0 ∀ X ∈ g , (3.39)so that P → M is a G -equivariant bundle. Thus, this equivariant extension to the curvature ω is the same as in [7, 35].We return for a moment to the symplectic geometric interpretation, to describe the resultsof Duistermaat and Heckman related to the localization formulas that we described in the lastsection. In [5] they proved an important property of the Liouville measure in the presenceof an Hamiltonian action by a torus T on ( M, ω ). Namely, defining a measure on g ∗ as the push-forward of the Liouville measure, µ ∗ (cid:18) ω n n ! (cid:19) ( U ) = (cid:90) µ − ( U ) ω n n ! ∀ U ⊆ M measurable , (3.40)they proved that µ ∗ ( ω n /n !) is a piecewise polynomial function . This, and an application ofthe stationary phase approximation showed a localization formula for the oscillatory integral (cid:90) M ω n n ! exp ( iµ X ) (3.41)for every X ∈ t := Lie ( T ) with non-null weight at every fixed point of the T -action. This canbe viewed as the Fourier transform of the Liouville measure, or as the partition function of a0-dimensional QFT with target space M . Let the fixed point locus F be the union of compactconnected symplectic manifolds M k (cid:44) → M of even codimension 2 n k , and denote ( m kl ) l =1 , ··· ,n k the weights of the T -action at a tangent space of a fixed point p ∈ M k . Then the
Duistermaat-Heckman (DH) localization formula is (cid:90) M ω n n ! exp ( iµ X ) = (cid:88) k vol( M k ) exp ( iµ X ( M k )) (cid:81) n k l ( m kl ( X ) / π ) (3.42)where µ X ( M k ) denotes the common value of µ X at every point in M k .In equivariant cohomological terms, we can see the above result as a localization formulafor the integral of an equivariantly closed form. In fact, if we fix the U (1) symmetry subgroup To be more precise, denoting µ ∗ ( ω n /n !) = f dξ with dξ the standard Lebesgue measure on g ∗ ∼ = R dim g , thefunction f is piecewise polynomial. The components of the fixed point set being symplectic is not an assumption, but a consequence of theHamiltonian action. See [28], proposition IV.1.3. hapter 3. Localization theorems in finite-dimensional geometry 43generated by X ∈ t , and consider the Cartan model defined by the differential d C = d + iι X ,the LHS can be rewritten as (cid:90) M ω n n ! exp( iµ X ) = (cid:90) M exp( ω + iµ X ) , (3.43)analogously to what we did in Example 3.2.1, and this is clearly the integral of an equivariantlyclosed form with respect to the differential d C . To see the correspondence with the ABBVformula, let us examine the case of a circle action and discrete fixed point locus F . In thiscase we have only one Hamiltonian function H := µ X , the weights are just the exponents m , · · · , m n of the circle action, and the sum over k runs over the isolated fixed points. TheDH formula thus recovers exactly the localization formula of theorem 3.2.1: (cid:90) M exp( ω + iH ) = (2 π ) n (cid:88) p ∈ F e iH ( p ) m ( p ) · · · m n ( p ) . (3.44)As we remarked in (3.23), the denominator can be expressed as the equivariant Euler class ofthe normal bundle to F (that is just the tangent bundle since F is 0-dimensional), recoveringthe DH formula as a special case of the ABBV localization formula for torus actions. Seealso [19] for an explicit correspondence between the two. We wish only to remark again that,especially in the context of Hamiltonian mechanics, this localization formula can be seen asthe result of an “exact” saddle-point approximation on the partition function (3.43). This isthe point of view we are going to take in the next chapters, when we are going to discuss thegeneralization of this formula to higher-dimensional QFT, where the integral of the partitionfunction is turned into an infinite-dimensional path integral .To see the correspondence with the saddle-point approximation, we recall that the isolatedfixed points of the U (1)-action are those in which X = 0, so dH = 0, and thus they are thecritical points of the Hamiltonian. We need to assume that the function H is Morse , so thatthese fixed points are non-degenerate, i.e. the Hessian Hess p ( H ) µν = ∂ µ ∂ ν H ( p ) at a given p ∈ F has non-null determinant. This Hessian can be expressed in terms of the exponents m k ( p ) via an equivariant version of the Darboux theorem [28, 37]: at any fixed point we canchoose Darboux coordinates in which the symplectic form takes its canonical form (3.28), andmoreover the action of the fundamental vector field at that tangent space decomposes as in(3.19). The latter can then be expressed as n canonical rotations of the type X = n (cid:88) µ =1 im µ (cid:18) q µ ∂∂p µ − p µ ∂∂q µ (cid:19) (3.45)with different weights m k . By the general form of the Hamilton’s equations (3.32), this meansthat the Hamiltonian near the isolated fixed point p ∈ F can be expanded as H ( x ) = H ( p ) + 12 n (cid:88) µ =1 im µ ( p ) (cid:0) p µ ( x ) + q µ ( x ) (cid:1) + · · · (3.46) This subject was in fact firstly connected with Morse theory by Witten in [36], where localization is appliedin the context of supersymmetric QM to prove Morse inequalities. We do not need to deepen this point of viewfor what follows, but a discussion about Morse theory and its connection with the DH formula can be found in[19], and references therein. (cid:90) M d n pd n q e iH ( p,q ) ≈ (cid:88) p ∈ F e iH ( p ) n (cid:89) µ =1 (cid:18)(cid:90) dp e − mµ ( p p (cid:90) dq e − mµ ( p q (cid:19) ≈ (cid:88) p ∈ F e iH ( p ) (2 π ) n (cid:81) µ m µ ( p ) (3.47)that, again, is exactly the result of the localization formula above. This motivates in thecontext of Hamiltonian mechanics, and generalization to infinite-dimensional case, that thedenominators appearing in these formulas are exactly the “1-loop determinants” of a would-besemiclassical approximation to the partition function. More aspects of the equivariant theoryin contact with symplectic geometry can be found in [22]. hapter 4Supergeometry and supersymmetry We give some definitions concerning graded spaces and super -spaces, that are useful for manyapplications of the localization theorems in physics. In particular, we will see how to translatethe problem of integration of differential forms in the context of supergeometry, and how this isuseful to prove the ABBV localization formula for circle actions. Also, in the next chapter wewill apply this theorem to path integrals in QM and QFT, where the coordinates over whichone integrates are those of a “field space” over a given manifold. To construct a suitable Cartanmodel over this kind of spaces, it is necessary to introduce a graded structure, that physicallymeans to distinguish between bosonic and fermionic fields, and some operation that acts as a“Cartan differential” transforming one type of field into the other. These structures arise inthe context of supersymmetric field theories , or in the context of topological field theories , andthe differentials here are called supersymmetry transformations or
BRST transformations. Theprecise mathematics behind this is a great subject and we do not seek to be complete here, wejust give some of the basic concepts that are necessary to understand what follows. For a moreextensive review of the subject, we suggest for example [38].
To understand the concept of a supermanifold, we need first to recall the linearized case. Wealready introduced a graded module or graded algebra V over a ring R , that is a collection of R -modules { V n } n ∈ Z such that V = (cid:76) n ∈ Z V n . If V is an algebra, it must also satisfy V n V m ⊆ V n + m .An element a ∈ V n for some n is called homogeneous of degree deg( a ) ≡ | a | := n . We now canspecialize to the case of super - vector spaces (or modules) and super -algebras. Definition 4.1.1. A super vector space is a Z -graded vector space V = V ⊕ V where V , V are vector spaces. Its dimension as a super vector space is defined as dim V := (dim V | dim V ).A superalgebra is a super vector space V with the product satisfying V V ⊆ V ; V V ⊆ V ; V V ⊆ V . A Lie superalgebra is a superalgebra where the product [ · , · ] : V × V → V , called Lie super-bracket , satisfies also[ a, b ] = − ( − | a || b | [ b, a ] (supercommutativity) , ( − | a || c | [ a, [ b, c ]] + ( − | b || c | [ c, [ a, b ]] + ( − | a || b | [ b, [ c, a ]] = 0 (super Jacobi identity) .
456 4.1. Gradings and superspaces
Definition 4.1.2.
The k-shift of a graded vector space V is the graded vector space V [ k ] suchthat ( V [ k ]) n = V n + k ∀ n ∈ Z .A few remarks are in order. First, it is clear that every graded vector space has naturallyalso the structure a super vector space, if we split its grading according to “parity”: V even := (cid:77) n ∈ Z V n , V odd := (cid:77) n ∈ Z +1 V n . (4.1)In physics, the Z -grading occurs on the “field space” as the so-called ghost number , whilethe Z -grading with respect to parity distinguish between bosonic and fermionic coordinates.Second, we notice that every vector space V can be considered as a (trivial) super vector space,if we think of it as V = V ⊕ V [1] = 0 ⊕ V in odd degree. Notice that theeven/odd parts of a super vector space can be considered as eigenspaces of an automorphism P : V → V such that P = id V . In this sense, a super vector space is a pair ( V, P ) made by avector space and the given automorphism P .Morphisms of graded vector spaces are graded linear maps, i.e. grading preserving maps: Definition 4.1.3. A graded linear map f between graded vector spaces V and W is a collectionof linear maps { f k : V k → W k } k ∈ Z . A linear map of k-degree is a graded linear map f : V → W [ k ].Now we can turn to the non-linear case and consider supermanifolds . Locally, they can bethought as extensions of a manifold via “anticommuting coordinates”: if we take an open set U ⊂ R n and a set of coordinates { x µ : U → R } µ =1 , ··· ,m , we can consider a set of additionalcoordinates { θ i } i =1 , ··· ,n with the algebraic properties x µ θ i = θ i x µ , θ i θ j = − θ j θ i . (4.2)The anticommuting { θ i } can be thought as generators of (cid:86) ( V ∗ ) for some vector space V , andthe product between them and coordinates of C ∞ ( U ) is then interpreted as a tensor productin C ∞ ( U ) ⊗ (cid:86) ( V ∗ ) =: C ∞ ( U × V [1]). If we then patch together different open sets we getglobally a manifold structure, with a modified atlas made by a graded ring of local functions C ∞ ( U × V [1]). More formally, we define: Definition 4.1.4.
A (smooth) supermanifold SM of dimension ( m | n ) is a pair ( M, A ), where M is a C ∞ -manifold of dimension m , and A is a sheaf of R -superalgebras that makes SM locally isomorphic to (cid:16) U, C ∞ ( U ) ⊗ (cid:94) ( V ∗ ) (cid:17) for some U ⊆ R m open and some vector space V of finite dimension dim V = n . M is calledthe body of SM and A is called the structure sheaf (or, sometimes, “soul”) of SM . Historically two (apparently) different concepts of supermanifolds and graded manifolds were firstly devel-oped. They both aimed to generalizing the mathematics of manifolds to a non-commutative setting, followingdifferent approaches. Eventually it was proven in [39] that their definitions are equivalent. Being generators of an exterior algebra, θ i are called “Grassmann-odd” coordinates, while x µ are called“Grassmann-even” consequently. This terminology is commonly inherited by every graded object (vector fields,forms, etc.) on the supermanifold. Note that also a regular d-dimensional smooth manifold can be viewed as a pair ( M, O M ) composed by atopological space M (Hausdorff and paracompact) with a structure sheaf of local functions O M : O M ( U ) = C ∞ ( U ) for every U ⊆ M open, such that locally every U is isomorphic to a subset of R d . hapter 4. Supergeometry and supersymmetry 47We just associated to a real manifold M a graded-commutative algebra C ∞ ( SM ) of func-tions over SM . Locally in a patch U ⊆ M , this matches the idea above of having coordinatesystems as tuples ( x µ , θ i ) µ =1 , ··· ,m i =1 , ··· ,n with the property (4.2). In particular, any local functionΦ ∈ A ( U ) can be trivialized with respect to the graded basis of (cid:86) ( V ∗ ):Φ( x, θ ) = Φ (0) ( x ) + Φ (1) i ( x ) θ i + Φ (2) ij θ i ∧ θ j + · · · + Φ ( n ) i , ··· ,i n ε i ··· i n θ ∧ · · · ∧ θ n (4.3)where Φ ( l ) i ··· i l ∈ C ∞ ( U ) ∀ l ∈ { , · · · , n } . The restriction to the zero-th degree (cid:15) : A → C ∞ M suchthat (cid:15) (Φ) := Φ (0) is usually called the evaluation map . Example 4.1.1.
To every super vector space V = V ⊕ V we can associate the supermanifoldˆ V = ( V , C ∞ ( V ) ⊗ Λ( V ∗ )) ∼ = R dim( V ) | dim( V ) . More generally, to every vector bundle E → M with sections Γ( E ) we can associate the oddvector bundle , denoted Π E or E [1], that is the supermanifold with body M and structure sheaf A = Γ ( (cid:86) E ∗ ). The odd tangent bundle Π T M is the supermanifold with A = Γ ( (cid:86) T ∗ M ), i.e. globally the functions here are the differential forms on M , C ∞ (Π T M ) = Ω( M ). Coordinateson Π T M are just ( x µ , dx µ ) µ =1 ··· ,m , exactly as the coordinates on the tangent bundle T M , butnow we consider them as generators of a graded algebra.Morphisms of supermanifolds can be given in terms of local morphisms of superalgebras, thatrespect compatibility between different patches. In particular, a morphism ( f, f ) : SM → SN is a pair such that f : M → N is a diffeomorphism, and for every U ⊆ M there is a morphismof superalgebras f U : A M ( U ) → A N ( f ( U )) that respects f V ◦ res U,V = res f ( U ) ,f ( V ) ◦ f U ,where res U,V is the restriction to a subset V ⊆ U . In less fancy words, if dim SM = ( m | p )and dim SN = ( n | q ), a local coordinate system ( x, θ ) in SM is mapped through n functions y ν = y ν ( x, θ ) and q functions ϕ j = ϕ j ( x, θ ) to a coordinate system ( y, ϕ ) of SN .A vector field X on a supermanifold SM , or supervector field , is a derivation on C ∞ ( SM ).Locally, considering U ⊆ M open and A ( U ) = C ∞ ( U ) ⊗ (cid:86) ( V ∗ ), it can be expressed withrespect to a coordinate system ( x, θ ) as X = X µ ( x, θ ) ∂∂x µ + X i ( x, θ ) ∂∂θ i , (4.4)where ( ∂/∂x µ ) acts as the corresponding vector field in Γ( T M ) on the C ∞ ( U ) components andacts trivially on the odd coordinates θ i ; ( ∂/∂θ i ) acts trivially on C ∞ ( U ), and as an interiormultiplication by the dual basis vector u i ∈ V : ∂∂θ i θ j := θ j ( u i ) = δ ji . X µ , X i are local sections in A ( U ). Supervectors on SM form the tangent bundle T SM . Notice that, in particular ( ∂/∂x µ )preserves the grading of an homogeneous function, i.e. it is a derivation of degree 0, while( ∂/∂θ i ) shifts the grading by -1. Definition 4.1.5. A graded vector field of degree k on SM is a graded linear map X : C ∞ ( SM ) → C ∞ ( SM )[ k ] that satisfies the graded Leibniz rule: X ( φψ ) = X ( φ ) ψ + ( − k | φ | φX ( ψ )for any φ, ψ ∈ A of pure degree. The graded commutator between graded vector fields X, Y isdefined as [
X, Y ] := X ◦ Y − ( − | X || Y | Y ◦ X. ∂/∂x µ ) with respect to even coordinatescommute between each other, while ( ∂/∂θ i ) anticommute, being respectively graded vectorfields of degree 0 and -1. Then from (4.4) and (4.3) we see that any supervector field can bedecomposed with respect to the Z -grading given by the parity, as the sum X = X (0) + X (1) of an even ( bosonic ) and an odd ( fermionic ) vector field. This makes (Γ( T SM ) , [ · , · ]) into aLie superalgebra. The value at a point p ∈ M of a supervector field X ∈ Γ( T SM ) is definedthrough the evaluation map: X p (Φ) := (cid:15) p ( X (Φ)) = (cid:20) X µ ( x, θ ) ∂ Φ ∂x µ + X i ( x, θ ) ∂ Φ ∂θ i (cid:21) x = x ( p ) θ =0 . (4.5)Clearly a super vector field X is not determined by its values at points, since the evaluationmap throws away all the dependence on the Grassmann-odd coordinates θ i in the coefficientfunctions X µ , X i . This means that at every point p ∈ M , T p SM is a super vector spacegenerated by the symbols ∂/∂x µ , ∂/∂θ i of opposite degrees, with real coefficients. We collectthis result in the following proposition. Proposition 4.1.1.
Let SM = ( M, A ) be a supermanifold such that for every chart U ⊆ M A = C ∞ ( U ) ⊗ Λ( V ∗ ). Then at every point p ∈ M , T p SM ∼ = T p M ⊕ V [1] as real super vectorspaces. For an odd vector bundle Π E this specializes as T p Π E ∼ = T p M ⊕ E p [1].Notice that one has always really both a Z and a Z grading of functions and supervectorfields, analogously to the remark (4.1). As already mentioned, in field theory and in particularin the BRST formalism, the first one is called ghost number , while the second one is thedistinction between bosonic and fermionic degrees of freedom in the theory. The physical ( i.e. gauge-invariant) combinations are those of ghost number zero.From the point of view of equivariant cohomology, it is very useful to relate the algebraicmodels we saw in Chapter 2 to these graded manifold structures. In particular, in field theorywe interpret the graded complex of fields as a Cartan model, with a suitable graded equivariantdifferential given in terms of supersymmetry or BRST transformations. Generically, a differ-ential in supergeometry can be interpreted as a special supervector field on a supermanifold: Definition 4.1.6. A cohomological vector field Q on a supermanifold SM is a graded super-vector field of degree +1 satisfying [ Q, Q ] = 0 . It is immediate that any cohomological vector field corresponds to a differential on thealgebra of functions C ∞ ( SM ), since being it of degree +1, Q ◦ Q = (1 / Q, Q ] = 0. Forexample, consider the de Rham differential d : Ω( M ) → Ω( M ) on a regular smooth manifold M . It corresponds to the cohomological vector field on the odd tangent bundle Π T M given inlocal coordinates by d = θ µ ∂∂x µ , (4.6)where now θ µ ≡ dx µ are odd coordinate functions on Π T M . Similarly, the interior multi-plication ι X : Ω( M ) → Ω( M ) with respect to some vector field X ∈ Γ( T M ) is a nilpotentsupervector field on Π
T M of degree -1, ι X = X µ ∂∂θ µ . (4.7)hapter 4. Supergeometry and supersymmetry 49 In the following we will use this graded machinery to translate the problem of integrationof differential forms Ω( M ) on a smooth manifold M , to an integration over the related oddtangent bundle Π T M . For this, we need to consider differential forms on a supermanifold SM .If SM has local coordinates ( x µ , θ i ), we can locally form an algebra generated by the 1-forms dx µ , dθ i , where now d is the de Rham differential on SM , acting as a cohomological vector fieldon Π T SM : d = dx µ ∂ x µ + dθ i ∂ θ i . (4.8)The odd tangent bundle Π T SM has thus coordinates ( x µ , θ i , dx µ , dθ i ), where now dx µ is oddwhereas dθ i is even. From this point of view, the de Rham differential acts as a “supersym-metry” transformation: d : (cid:40) x µ (cid:55)→ dx µ θ i (cid:55)→ dθ i (4.9)exchanging bosonic coordinates with fermionic coordinates. Since the odd 1-forms dθ i arecommuting elements, it is not possible to construct, at least in the usual sense, a form of “topdegree”on SM . We will thus interpret integration over the odd coordinates by the purelyalgebraic rules of Berezin integration for Grassmann variables: (cid:90) dθ i θ i = 1 , (cid:90) dθ i , (4.10)and such that Fubini’s theorem holds for multidimensional integrals. We see that symbolically (cid:90) dθ i ↔ ∂∂θ i , (4.11)and in particular (cid:82) dθ i ∂∂θ i Φ( θ i ) = 0 always holds. We will use the important property of theBerezin integral: (cid:90) d n θ e − θ i A ij θ j = Pf( A ) (4.12)where d n θ ≡ dθ dθ · · · dθ n , to be compared with the usual Gaussian integral for real variables (cid:90) d n x e − x i A ij x j = π n/ (cid:112) det( A ) . (4.13)Notice that under an homogeneous change of coordinates ϕ i = B ij θ j , the “measure” shiftsas (cid:82) d n θ → det B (cid:82) d n ϕ , such that the Gaussian integral (4.12) is invariant under similaritytransformations. For a mathematically refined theory of superintegration, we suggest lookingat [40].Concerning the integration of functions on the odd tangent bundle Π T M , we notice how,making use of the Berezin rules (4.10), this is nothing but a reinterpretation of the usual To be more precise, the algebra of functions locally generated by ( x µ , θ i , dx µ , dθ i ) on Π T SM has bi-grading , i.e. it inherits a Z grading from the original supermanifold SM and a Z grading from the action of the deRham differential (the form-degree). The coordinates have thus bi-degrees x µ : (even , , θ i : (odd , , dx µ : (even , , dθ i : (odd , , which result in a total even degree for dθ i and a total odd degree for dx µ . M . If dim M = d , the integral of the form ω = (cid:80) i ω ( i ) , where ω ( i ) ∈ Ω i ( M ) is (cid:90) M ω = (cid:90) M ω ( d ) = (cid:90) M d d x ω ( d ) ( x ) , (4.14)selecting the top-form of degree d . If we consider the same form ω ∈ C ∞ (Π T M ), its trivializa-tion in coordinates ( x, θ ) is ω ( x, θ ) = (cid:88) i ω ( i ) µ ··· µ i ( x ) θ µ · · · θ µ i . (4.15)Now the Berezin integration over d d θ selects just the term with the right number of θ ’s, giving (cid:90) Π T M d d xd d θ ω ( x, θ ) = (cid:90) M d d x ω ( d ) ( x ) (cid:90) d d θ θ d θ d − · · · θ = (cid:90) M d d x ω ( d ) ( x ) . (4.16) We give now a proof of the ABBV integration formula for a U (1)-action, starting from theexpression (3.14). The “localization 1-form” is chosen as β := g ( T , · ) (4.17)where g is a U (1)-invariant metric and T is the fundamental vector field corresponding to thegenerator T ∈ u (1). In local coordinates, the action of the Cartan differential d C = d + ι T onthis 1-form can be written as d C β = B µν ( x ) dx µ dx ν + g µν ( x ) T µ ( x ) T ν ( x ) B µν = ( ∇ µ T ) ν − ( ∇ ν T ) µ (4.18)where, again, we suppressed the S ( u (1) ∗ ) generator setting φ = − T M , identifying the odd coordinates θ µ ≡ dx µ : I [ α ] = (cid:90) M α = lim t →∞ (cid:90) M αe − td C β = lim t →∞ (cid:90) Π T M d d xd d θ α ( x, θ ) exp {− tB µν ( x ) θ µ θ ν − tg µν ( x ) T µ ( x ) T ν ( x ) } , (4.19)where the equivariantly closed form α is the sum of U (1)-invariant differential forms in Ω( M ) U (1) suppressing the S ( u (1) ∗ ) generator, α ( x, θ ) = (cid:88) i α ( i ) µ ··· µ i ( x ) θ µ · · · θ µ i , (4.20)such that d C α = ( θ µ ∂ x µ + T µ ∂ θ µ ) α = 0. Using the Gaussian integrals (4.12) and (4.13), we havethe following delta-function representations for Grassmann-even and Grassmann-odd variables δ ( n ) ( y ) = lim t →∞ (cid:18) tπ (cid:19) n/ √ det A e − ty µ A µν y ν δ ( n ) ( η ) = lim t →∞ t − n/ A e − tA µν η µ η ν (4.21)hapter 4. Supergeometry and supersymmetry 51where the limits are understood in the weak sense. Multiplying and dividing by ( t n/ ) in (4.19),and using the delta-representation we rewrite the integral as I [ α ] = π d/ (cid:90) Π T M d d xd d θ α ( x, θ ) Pf B ( x ) (cid:112) det g ( x ) δ ( d ) ( T ( x )) δ ( d ) ( θ ) . (4.22)The delta function on the odd coordinates simply puts θ µ = 0, that is analogous to selecting thetop-degree form in (4.19), so that it remains α ( x, ≡ α (0) ( x ) ∈ C ∞ ( M ). The delta functionon the even coordinates instead selects the values at which T µ ( x ) = 0, that corresponds to thefixed point set F (cid:44) → M of the U (1)-action. Suppose this fixed point set to be of dimension 0, i.e. composed by isolated points in M . If this is the case, we can simply separate the integral (cid:82) d d x in a sum of integrals, each of which domain D ( p ) : p ∈ F contains one and only one ofthose fixed points, and in each of them apply the delta function I [ α ] = π d/ (cid:88) p ∈ F (cid:90) D ( p ) d d x α (0) ( x ) Pf B ( x ) (cid:112) det g ( x ) δ ( d ) ( T ( x ))= π d/ (cid:88) p ∈ F α (0) ( p ) | det dT | ( p ) Pf B ( p ) (cid:112) det g ( p ) . (4.23)Here the factor | det dT | ( p ) is the Jacobian from the change of variable y µ := T µ ( x ). At anypoint p ∈ F we have ( ∇ µ T ) ν ( p ) = ∂ µ T ν ( p ) = ∂ µ T ρ ( p ) g ρµ ( p ) since T µ ( p ) = 0, so the pfaffian inthe numerator becomes Pf B ( p ) = √ d Pf dT ( p ) (cid:112) det g ( p ) , (4.24)and we get the result I [ α ] = (2 π ) d/ (cid:88) p ∈ F α (0) ( p )Pf dT ( p ) . (4.25)Notice that, at p ∈ F , the operator dT ( p ) = ∂ µ T ν θ µ ⊗ ∂ ν = − [ T , · ] coincide up to a signwith the infinitesimal action L T of T ∈ u (1) on the tangent space T p M , just because here T µ ( p ) = 0. If we consider a continuous fixed point set, so that F is a regular submanifold of M ,it is not possible to use the delta functions like in (4.23), but we can consider the decompositionas a disjoint union M = F (cid:116) N . Points far away from F give a zero contribution to I [ α ] inthe limit t → ∞ , so we can consider a neighborhood of F and split here the tangent bundleas T M ∼ = i ∗ T F ⊕ T N where
T N is the normal bundle to F in M ( i is the inclusion map).Consequently, in this neighborhood we can split the coordinates ( x µ , θ µ ) on the odd tangentbundle in tangent and normal to F , and rescale the normal components as 1 / √ t , x µ = x µ + x µ ⊥ √ t , θ µ = θ µ + θ µ ⊥ √ t . (4.26)The measure simply splits as d d xd d θ = d n x d ˆ n x ⊥ d n θ d ˆ n θ ⊥ , where n + ˆ n = d , thanks to theBerezin integration rules (4.10). Expanding B µν ( x ) , g µν ( x ) , T µ ( x ) around the x components,and taking the limit t → ∞ , the integral becomes [41] I [ α ] = (cid:90) d n x d n θ α ( x , θ ) (cid:90) d ˆ n x ⊥ d ˆ n θ ⊥ exp (cid:8) − B µσ ( x ) (cid:0) B σν ( x ) + R σνλρ ( x ) θ λ θ ρ (cid:1) x µ ⊥ x ν ⊥ − B µν ( x ) θ µ ⊥ θ ν ⊥ (cid:9) (4.27)2 4.3. Introduction to Poincar´e-supersymmetrywhere R σνλρ ( x ) is the curvature relative to the metric g . The integrals over the normal coordi-nates are Gaussian, giving the exact “saddle point” contribution1Pf N (cid:0) R + B π (cid:1) ( x ) = 1 e T ( R ) | N , (4.28)where e T ( R ) | N is the U (1)- equivariant Euler class of the normal bundle to F . We notice thatthis matches the definition in Appendix B.1, since B := ∇ T , seen as an element of the adjointbundle Ω ( M ; gl ( n )), is a moment map for the Riemannian curvature R satisfying (as can bechecked by direct computation) ∇ B σν = − ι T R σν = R σν ( · , T ) , (4.29)where the covariant derivative acts as in the adjoint bundle with respect to the Levi-Civitaconnection, ∇ = d + [Γ , · ]. As the final piece, the Berezin integration selects the componentof α of degree dim F evaluated on F , that is just the pull-back i ∗ α along the inclusion map.Summarizing, we are left with I [ α ] = (cid:90) F i ∗ αe T ( R ) | N (4.30)as presented in Section 3.2, for the case of a circle action.Notice that in Section 3.2 we called the moment map B σν ( p ) = µ T ( p ) σµ ∈ End( T p M ) at afixed point p . If the fixed point is isolated, the tangent space T p M splits, as in (3.22), as thedirect sum of the weight spaces of the U (1)-representation, and the vector field T acts as arotation in any subspace. On a suitable coordinate basis T = (cid:88) i v i (cid:18) x i ∂∂y i − y i ∂∂x i (cid:19) , (4.31)so that the moment map block-diagonalizes as B σν = ∂ ν T σ = − v · · · v · · · − v v . (4.32)Taking the pfaffian then one gets exactly the product of the weights (or the “exponents”) v i ofthe circle action, so that the Euler class results e T ( R ) = (2 π ) − dim( M ) / (cid:89) i v i . (4.33)This recovers (3.23) for the case of a circle action. See [6] for the result in presence of a torusaction. We have seen in the last section how it is useful to translate the integration problem of adifferential form on the manifold M , into an integration over the supermanifold Π T M . Here thedifferential forms Ω( M ) are seen as the graded ring of functions over Π T M , and the differential d C = d + ι T hapter 4. Supergeometry and supersymmetry 53being the sum of two graded derivations of degree ±
1, can be viewed as an infinitesimal supersymmetry transformation mapping odd-degree ( fermionic ) forms to even-degree ( bosonic )forms. In field theory, we already mentioned that the presence of a supersymmetry on therelevant complex of fields often arises in two different ways: • The differential d C is represented in the physical model by a BRST-like supercharge,introduced because of some gauge freedom. In this case, the complex of fields (analogouslyto the graded ring of functions Ω( M )) is the BRST complex, and the grading is referredto as ghost number . Physical states of the quantum field theory are then created byfields of 0-degree, i.e. the functions on M . We could refer to this type of supersymmetryas a “hidden” one, coming from the original internal gauge symmetry of the model. InHamiltonian systems, as we shall see in the next chapter, this gauge freedom can besimply associated to the Hamiltonian flow. • The original theory could also be explicitly endowed with a supersymmetry. In this casethe base space has a supermanifold structure, and the grading of the field complex follows.This is the case of QFT with Poincar´e supersymmetry, where the action of a supercharge as generator of the super-Poincar´e algebra can be interpreted as an equivariant differential.In this and the next sections we will review the geometric setup of Poincar´e supersymmetryand its generalization to curved spacetimes, interesting case for practical applications of thelocalization technique in QFT.As QFT (with global Poincar´e symmetry) is formulated on Minkowski spacetime R d ∼ = ISO ( R d ) /O ( d ) , (4.34)we can formulate a Poincar´e-supersymmetric theory on a super-extension of this space, comingfrom a given super-extension of the Poincar´e group ISO ( R d ). We then first introduce thesuper-Poincar´e groups starting from the super-extension of their algebras. Definition 4.3.1.
A super-Poincar´e algebra is the extension of the Poincar´e algebra iso ( d ) ∼ = R d ⊕ so ( d ) as a Lie superalgebra, via a given real spin representation space S of Spin ( d ) takenin odd-degree: siso S ( d ) ∼ = iso ( d ) ⊕ S [1] . (4.35)The super Lie bracket are extended on S [1] through the symmetric and Spin-equivariant bilinearform Γ : S × S → R d : [Ψ , Φ] := 2Γ(Ψ , Φ) = 2(Ψ γ µ Φ) P µ = 2(Ψ T Cγ µ Φ) P µ (4.36)where P µ are generators of the translation algebra R d , C is the charge conjugation matrix,Ψ ∈ S ∗ is the Dirac adjoint of Ψ, and γ µ are the generators of the Clifford algebra acting on Both d and ι T are nilpotent supervector fields on Π T M . Without suppressing the degree-2 generator φ of S ( u ( ) ∗ ), it is apparent that their sum d C is a cohomological vector field, i.e. a good differential of degree +1,on the subspace of U (1)-invariant forms. We are really interested in both the Lorentzian and the Euclidean case, so we will express both the Minkowskiand Euclidean spaces as R d without stressing of the signature in the notation. The choice of metric will be clearfrom the context. In Lorentzian signature Ψ = Ψ † β with β ab ≡ ( γ ) ab , in Euclidean signature Ψ = Ψ † . This is due tohermitianity of the generators of the Euclidean algebra, unlike the Lorentzian case. S . In a real representation, the Majorana condition Ψ T C ! = Ψ is satisfied. The other bracketsinvolving S are defined by the natural action of so ( d ) on it, and by the trivial action on R d : λ ∈ so ( d ) : [ λ, Ψ] := i λ µν Σ µν (Ψ) , (4.37)[Ψ , λ ] = − [ λ, Ψ] , (4.38) a ∈ R d : [ a, Ψ] := 0 , (4.39)where Σ µν = i γ [ µ γ ν ] are the generators of the rotation algebra in the Spin representation S . Ifthe charge conjugation matrix is symmetric in the given representation, we can further enlargethis superalgebra via a “central extension”, considering siso S ( d ) ⊕ R with the extended bracketsΨ , Φ ∈ S [1] : [Ψ , Φ] := Γ(Ψ , Φ) + (Ψ T C Φ) , (4.40) x ∈ R , A ∈ siso S ( d ) : [ x, A ] := 0 . (4.41)The super Jacobi identity is satisfied thanks to the Spin-equivariance of the spinor bilinearform: Γ (cid:16) e R ( s ) Ψ , e R ( s ) Φ (cid:17) = e R ( v ) Γ(Ψ , Φ) (4.42)where R ( s ) , R ( v ) are the same element R ∈ so ( d ) in the spin and vector representations, respec-tively.Often the bracket structure of this superalgebra is given in terms of the generators. Re-garding the odd part and picking a basis { Q a } of S [1], their brackets are [ Q a , Q b ] = 2( γ µ ) ab P µ + C ab . (4.43)The generators { Q a } are referred to as supercharges . If the real spin representation on S is irreducible as a representation of the corresponding Clifford algebra, we have the minimalamount of supersymmetry and we refer to siso S ( d ) as to an N = 1 supersymmetry algebra.If instead the representation is reducible, then S = (cid:76) N I =1 S ( I ) and we can split the basis ofsupercharges as (cid:8) Q Ia (cid:9) . This case is referred to as extended supersymmetry. In this basis thegamma matrices block-diagonalize as γ µ ⊗ I , with γ µ the (minimal) gamma matrices in every S ( I ) , and the central extension part separates as C ⊗ Z , with C being the (minimal) chargeconjugation matrix in every S ( I ) and Z a matrix of so-called central charges . The odd part ofthe superalgebra then looks like[ Q Ia , Q Jb ] = 2( γ µ ) ab δ IJ P µ + C ab Z IJ . (4.44)The matrix Z must be (anti)symmetric if C is (anti)symmetric. Definition 4.3.2.
The subspace st S ( d ) := R d ⊕ S [1] is a Lie superalgebra itself (if there is nocentral extension), and can be referred to as the super-translation algebra . We conventionally raise and lower spinor indices with the charge conjugation matrix:( γ µ ) ab := C ac ( γ µ ) cb , ( γ µ ) ab = ( γ µ ) ba . Notice that the matrix C represents an inner product on S , while the Clifford algebra generators { γ µ } act on S as endomorphisms, so the index structure follows. A review of classification of Clifford algebras, Spin groupsand Majorana spinors can be found in [42]. hapter 4. Supergeometry and supersymmetry 55Even if st S ( d ) it is not Abelian, it has the property[ a, [ b, c ]] = 0 ∀ a, b, c ∈ st S ( d ) (4.45)that can be easily checked by the definition. This means that the elements of the corresponding super-translation group can be computed exactly using the exponential map and the BHCformula. This space can be identified with the super-spacetime . Definition 4.3.3.
We can define the full super-Poncar´e group as
SISO S ( d ) = exp ( st S ( d )) (cid:111) Spin ( d ) , (4.46)so that we can identify the superspacetime with respect to the spin representation S , analo-gously to (4.34), as S R dS = SISO S ( d ) /Spin ( d ) ∼ = exp ( st S ( d )) . (4.47)As a supermanifold of dimension ( d | dim( S )), S R dS is characterized by its sheaf of functions, A = C ∞ ( R d ) ⊗ (cid:94) ( S ∗ ) . (4.48)So, S R dS is the odd vector bundle associated to the spinor bundle over R d of typical fiber S . In particular, on S R dS we have respectively even and odd coordinates ( x µ , θ a ), with µ =1 , · · · , d and a = 1 , · · · , dim( S ). As a Lie group, we can get the group operation (the sum bysupertranslation) from the exponentiation of its Lie superalgebra. Technically, to use the BHCformula e A e B = e A + B + [ A,B ]+ ··· (4.49)we would like to deal with a Lie algebra , so we consider the group operation on coordinatesfunctions instead of points on S R dS , taking the space( A ⊗ st S ( d )) (0) = (cid:0) A (0) ⊗ R d (cid:1) ⊕ (cid:0) A (1) ⊗ S [1] (cid:1) . (4.50)The Lie brackets on this space are inherited from those on st S ( d ) and the (graded) multiplicationof functions in A . The only non-zero ones come from couples of elements of A (1) ⊗ S [1]:[ f ⊗ (cid:15) , f ⊗ (cid:15) ] = − (cid:15) , (cid:15) ) f f (4.51)where the sign rule has been used since both f , f and (cid:15) , (cid:15) are odd. We will use the combi-nation (suppressing tensor products)[ θ a Q a , ϕ b Q b ] = − Q a , Q b ) θ a ϕ b = − γ µ ) ab θ a ϕ b P µ ≡ − θγ µ ϕ ) P µ . This makes (
A ⊗ st S ( d )) (0) into a Lie algebra, by the antisymmetry of the product betweenodd functions. Representing via the exponential map ( x, θ ) as exp { i ( xP + θQ ) } , where we We are being a little informal here, but this can be made more rigorous with the help of a constructioncalled functor of points . The important thing for us is that in this approach one can work with coordinatefunctions x, θ instead of some would-be “points” on the supermanifold (a misleading concept since we knowfrom the last section that a supermanifold is not a set). See [43] for more details. x, θ ) ( y, ϕ ) = exp { i ( xP + θQ ) } exp { i ( yP + ϕQ ) } = exp (cid:26) i (cid:18) xP + θQ + yP + ϕQ + i θQ, ϕQ ] (cid:19)(cid:27) = exp { i ( x + y + iθγϕ ) P + i ( θ + ϕ ) Q } = ( x + y − iθγϕ, θ + ϕ ) . (4.52)If the odd dimension dim ( S ) is zero, this reduces to a standard translation in R d . The in-finitesimal action of the superalgebra st S ( d ) is defined as a Lie derivative with respect to thefundamental vector field representing a given element of the supertranslation algebra. ForΦ ∈ A , (cid:15) = (cid:15) a Q a ∈ ( A (1) ⊗ S [1]) δ (cid:15) Φ( x, θ ) := L (cid:15) (Φ( x, θ )) = (cid:15) (Φ) = (cid:15) a Q a (Φ( x, θ )) , (4.53)where the odd vector field Q a is associated to the supercharge Q a through the left translation(4.52): Q a Φ( x, θ ) = ∂∂ϕ a (cid:0) e − iϕQ (cid:1) ∗ Φ( x, θ ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ =0 = ∂∂ϕ a Φ (cid:0) (0 , − ϕ )( x, θ ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ϕ =0 == ∂ µ Φ( x, θ )( iγ µ θ ) a + ∂ b Φ( x, θ ) δ ba . (4.54)We recognize then Q a = − ∂∂θ a + i ( θγ µ ) a ∂∂x µ , (4.55)and one can check that [ Q a , Q b ] = 2( γ µ ) ab P µ with P µ = − i∂/∂x µ is associated to the momen-tum generator.The rest of the super-Poincar´e group and its algebra acts naturally on superspace followingthe same type of arguments. In particular, the spin generators act in the vector and spinrepresentations on the even and odd sectors, respectively. It is useful to introduce also thefundamental vector fields with respect to right supertranslations on S R dS . These are called superderivatives and are easily obtained from the law (4.52) as D a := ∂∂θ a + i ( θγ µ ) a ∂∂x µ . (4.56)One can check that indeed they satisfy [ Q a , D b ] = 0 and [ D a , D b ] = − γ µ ) ab P µ , since rightinvariant and left invariant vector fields form anti-isomorphic algebras. It is customary to construct supersymmetric field theories starting not from a real spinrepresentation, but from a complex
Dirac representation S endowed with a real structure , i.e. an antilinear map J : S → S which is an involution ( J = id S ). In this case, diagonalizing J the representation splits as S ∼ = S R ⊗ C ∼ = S (+) ⊕ S ( − ) Remember that left and right actions correspond to opposite signs at the exponent in the definition of thefundamental vector fields. J is the generalization of the “complex conjugation” operation on a C -vector space. hapter 4. Supergeometry and supersymmetry 57where S R ∼ = S (+) ∼ = iS ( − ) are a real vector spaces. Choosing some basis, the matrix representingthe real structure is J = C ( γ ) T in Lorenzian signature, or J = C in Euclidean signature.The Majorana spinors are the elements of S R , that satisfy J (Ψ) = Ψ, or in matrix notationΨ = Ψ T C . The subspace of Majorana spinors, taken as a real representation, would then givethe super-extension of the last paragraph.If instead we are interested in working with the whole complex representation S , we areforced to introduce complexified supersymmetry algebra and superspace, and then impose con-straints on the resulting objects to properly reduce their degrees of freedom a posteriori. Inparticular, complex spinors from S generating the supertranslations are taken satisfying theMajorana condition. This is always the case in QFT. As a paradigmatic example, we can take N = 1 supersymmetry in (3+1)-dimensions, where Ψ is a Dirac spinor in S = C ∼ = R ⊗ C .On the complex representation, we can chose a to work in the chiral basis { Q a , ˜ Q ˙ a } , splittedin left- and right-handed Weyl spinors. The dotted and undotted indices now run between 1 , S × S → C is non-zero only on S ( L/R ) × S ( R/L ) , and and therestriction to the symmetrized subspace Γ : S L (cid:12) S R → C is actually an isomorphism, so therelevant non-zero brackets are [ Q a , ˜ Q ˙ b ] = 2( γ µ ) a ˙ b P µ . (4.57)On the chiral basis, γ µ = (cid:18) σ µ ¯ σ µ (cid:19) , C = (cid:18) ε − ε (cid:19) , σ µ = ( , σ i ) , σ µ = ( , − σ i ) , where ( σ i ) i =1 , , are the Pauli matrices and ε ab = ε ˙ a ˙ b is the totally antisymmetric tensor. Thesupercharges are represented by the odd vector fields Q a = − ∂∂θ a + i ˜ θ ˙ b ( γ µ ) ˙ ba ∂∂x µ , ˜ Q ˙ a = − ∂∂ ˜ θ ˙ a + iθ b ( γ µ ) b ˙ a ∂∂x µ , (4.58)and the supersymmetry action on a superfield is δ (cid:15) Φ = ( (cid:15) a Q a + ˜ (cid:15) ˙ a ˜ Q ˙ a )Φ (4.59)where (cid:15) a = ˜ (cid:15) ˙ a , being a Majorana spinor. The superderivatives are D a = ∂∂θ a + i ˜ θ ˙ b ( γ µ ) ˙ ba ∂∂x µ , ˜ D ˙ a = ∂∂ ˜ θ ˙ a + iθ b ( γ µ ) b ˙ a ∂∂x µ . (4.60)Here we split the odd coordinates θ a , ˜ θ ˙ a according to the split of the supercharges. The realityconstraint on the coordinates then reads θ a = ˜ θ ˙ a , x µ = x µ . (4.61)Since we are operating over C , the subspaces of the supertranslation algebra st ( L/R ) := C ⊕ S ( L/R ) [1] (4.62)are both Lie superalgebras over C , and determines the corresponding complex Lie supergroups S C ( L/R ) . Moreover these subalgebras are Abelian , since Γ vanishes on S ( L/R ) × S ( L/R ) . Definition 4.3.4. S C ( L ) is called chiral superspace and S C ( R ) anti-chiral superspace.8 4.3. Introduction to Poincar´e-supersymmetryWe can write the complexified superspace as S C S ∼ = S C ( L ) × C S C ( R ) (4.63)where the × C here denotes the fiber product with respect to the base C . The chiral and anti-chiral superspaces are those identified by the flows of the corresponding superderivatives, sincethey generates the chiral and anti-chiral part of the supertranslation algebra on Γ(
T S C S ). We can easily find sets of “holomorphic-like” coordinates y µ ( ± ) := x µ ± i ˜ θ ˙ a ( γ µ ) ˙ ab θ b , ϕ a := θ a , ˜ ϕ ˙ a := ˜ θ ˙ a , (4.64)where the superderivatives simplify as D a = ∂∂ϕ a ∂∂ϕ a + 2 i ( ˜ ϕγ µ ) a ∂∂y µ ( − ) , ˜ D ˙ a = ∂∂ ˜ ϕ ˙ a + 2 i ( ϕγ µ ) ˙ a ∂∂y µ (+) ∂∂ ˜ ϕ ˙ a . (4.65)Complexifying the superspace we doubled its real-dimension, and that leads to a sort of“reducibility” of the relevant physical objects, i.e. the fields on superspace, or superfields . Thesimplest kind of superfields in the complexified setting are complex even maps Φ : S C → C ,that is sections of a trivial C -line bundle over S C . We can impose now some constraints onthem in order to restore the correct number of degrees of freedom. This is usually done in twodifferent ways: asking the superfields to depend only on the chiral (or anti-chiral) sector of thecomplexified superspace, or imposing a reality condition. Definition 4.3.5. (i) A chiral (anti-chiral) superfield is a superfield Φ such that˜ D ˙ a Φ = 0 ( D a Φ = 0) . (ii) A vector superfield is a superfield V such that V = V † . Notice how the first one is a sort of (anti)holomorphicity condition with respect to thechiral/anti-chiral sectors of S C S , spanned by the coordinates ϕ a , ˜ ϕ ˙ a . Moreover, we can see thatthe complex conjugate Φ † of a chiral superfield Φ is antichiral. Next we will see how theseconditions reflect on the various component fields of the coordinate expansions of Φ and V .It is important to stress that in this particular case of N = 1 in (3+1)-dimensions, thecomplex representation S ∼ = C allows for a real structure and the presence of Majorana spinors,and also for a chiral decomposition into left- and right-handed parts. It does not exist thougha common basis for the two decompositions, i.e. S ( ± ) (cid:54) = S ( L/R ) . In other words, it is notpossible to require both the chirality and the Majorana conditions on spinors in 4-dimension,since Majorana spinors contain both left- and right- handed components. This means that in atheory with some supersymmetry in 4-dimensions there will be the same number of left-handedand right-handed degrees of freedom. In d = 2mod8 dimensions (in Lorentzian signature), This is analogous to the pull-back bundle, not to be confused with an homotopy quotient. Here Γ denotes the space of sections on the tangent bundle, i.e. the vector fields on S C S , not the spinorpairing. hapter 4. Supergeometry and supersymmetry 59instead, the minimal complexified spin representation S can be decomposed into Majorana-Weyl subrepresentations S = S (+) R ⊕ S ( − ) R , so that one can choose to work only with realleft-handed spinors. In the case of extended supersymmetry, we can thus have in general adifferent number of left-handed and right-handed real supercharges, and S = ( S R ) N + ⊕ ( S R ) N − .This is denoted with N = ( N + , N − ). When N + or N − is zero, the supersymmetry is called chiral . For a review on spinors in different dimensions, Majorana and chirality conditions werefer to [42]. Actions for supersymmetric field theories are constructed integrating over superspace combi-nations of superfields and their derivatives. For this purpose, it is useful to write the superfieldsin a so-called “component expansion”, with respect to the generators of (cid:86) ( S ∗ ). In this para-graph we continue with the example of N = 1 in (3+1)-dimensions, but the construction isimmediatly generalizable to other cases. Consider a complex superfield Φ : S C → C , itstrivialization on the set of coordinates ( x, θ, ˜ θ ) beingΦ( x, θ, ˜ θ ) = φ ( x ) + v µ ( x )(˜ θγ µ θ ) + ψ a ( x ) θ a + ˜ ψ ˙ a ˜ θ ˙ a + F ( x ) θ (2) + ˜ F ( x )˜ θ (2) ++ ξ a ( x ) θ a ˜ θ (2) + ˜ ξ ˙ a ( x )˜ θ ˙ a θ (2) + D ( x ) θ (2) ˜ θ (2) (4.66)where the the wedge product between the θ ’s has been suppressed, and we agree they areanticommuting, θ (2) := θ θ and ˜ θ (2) := ˜ θ ˙2 ˜ θ ˙1 . We used the isomorphism Γ : S L (cid:12) S R → C to represent the component of degree (1,1) as v a ˙ a (cid:55)→ v µ ( γ µ ) a ˙ a , the reason for this will becomeclear shortly. The expansion stops at top-degree (here 4) for the anticommuting property ofthe exterior product. In order for Φ to be an even (scalar) field, we must take the functions φ, v µ , F, ˜ F , D to be even (commuting), and the functions ψ a , ˜ ψ ˙ a , ξ a , ˜ ξ ˙ a carrying a spinor indexto be odd (anticommuting). These are called component fields of the superfield Φ. If we impose the chirality condition ˜ D ˙ a Φ = 0, when expressed in the coordinates ( y ( − ) , ϕ, ˜ ϕ )this simply requires the independence on ˜ ϕ , so on these coordinates a chiral superfield can bewritten as Φ( y ( − ) , ϕ, ˜ ϕ ) = φ ( y ( − ) ) + ψ a ( y ( − ) ) ϕ a + F ( y ( − ) ) ϕ (2) . (4.67)Taylor-expanding in the old coordinates, this is equivalent toΦ( x, θ, ˜ θ ) = φ ( x )+ ψ a ( x ) θ a + F ( x ) θ (2) − i∂ µ φ ( x )(˜ θγ µ θ )+ i (˜ θγ µ ∂ µ ψ ( x )) θ (2) − ∂ φ ( x ) θ (2) ˜ θ (2) (4.68)where higher order terms are again automatically zero for degree reasons. The “irreducible”chiral superfield has three non-zero field components: a complex scalar field φ , a left-handedWeyl spinor field ψ and another complex scalar field F . One can work out the supersymmetrytransformations of these component fields from the general rule (4.59), and the result is δ (cid:15) φ = (cid:15)ψδ (cid:15) ψ = 2 i (˜ (cid:15)γ µ ) ∂ µ φ + (cid:15)Fδ (cid:15) F = − i ˜ (cid:15)γ µ ∂ µ ψ (4.69) The possibility of writing down an expansion similar to (4.66) with these properties could again be justifiedmore rigorously thanks to the concept of functor of point . Here we used θ a θ b = ε ab θ (2) , and (˜ θγ µ θ )(˜ θγ µ θ ) = 2 η µν θ (2) ˜ θ (2) with a “mostly plus” signature. φ , that being a scalar field is ( d − / d = 4 dimensions. Since ψ is a spinor, it has dimension ( d − / /
2, so the odd coordinateshave always dimension -1/2. The highest component of any superfield has thus dimensiontwo more than the superfield. This means that to construct a Lagrangian we have to take aquadratic expression in Φ. Since the action should be real, the simplest choice is Φ † Φ. Itstop-degree component is (cid:90) d θd ˜ θ Φ † Φ = 4 | ∂φ | + i Ψ /∂ Ψ + | F | + ∂ µ ( · · · ) µ (4.70)where Ψ is a Majorana 4-spinor, whose left-handed component is ψ . Up to a total derivative,this is the Lagrangian of the free, massless Wess-Zumino model : S W Z,free [Φ] = (cid:90) S R C d xd θd ˜ θ Φ † Φ = (cid:90) R d x (cid:0) | ∂φ | + i Ψ /∂ Ψ + | F | (cid:1) . (4.71)Since the field F appears without derivatives, its equation of motion is an algebraic equation.For this reason, it is called an auxiliary field , and it is customary to substitute its on-shell valuein the action. For this simple model, this means putting F = 0. This procedure makes ingeneral the action to be supersymmetric only if the equations of motion (EoM) are imposed,and is often called on-shell supersymmetry. The physical field content of an N = 1 chiralsuperfield is thus the supersymmetry doublet ( φ, ψ ). By CPT invariance, the theory mustcontain both the chiral field Φ and its antichiral conjugate Φ † , so that the physical field contentof a meaningful theory constructed from it is made by two real scalar fields Re( φ ) , Im( φ ) andthe Majorana spinor Ψ.If we now start from a vector superfield V , satisfying the reality condition V = V † , we reacha different physical field content and supersymmetric Lagrangian. The component expansionin a chart ( x, θ, ˜ θ ) is the following: V ( x, θ, ˜ θ ) = C ( x ) + ξ a ( x ) θ a + ξ † ˙ a ˜ θ ˙ a + v µ ( x )(˜ θγ µ θ ) + G ( x ) θ (2) + G † ( x )˜ θ (2) ++ η a ( x ) θ a ˜ θ (2) + η † ˙ a ( x )˜ θ ˙ a θ (2) + E ( x ) θ (2) ˜ θ (2) (4.72)where C, v µ , E are real fields. It is clear that this is the right type of superfield needed todescribe (Abelian) gauge boson fields, represented here by v µ . This is why V is called vector superfield.Notice that the real part of a chiral superfield is a special kind of vector superfield. Inparticular, its vector component is a derivative: if Λ is chiral, from (4.68)Λ + Λ † ⊃ i∂ µ ( φ − φ † )(˜ θγ µ θ ) . (4.73)This suggest to interpret the transformation V (cid:55)→ V + (Λ + Λ † ) (4.74)as the action of a U (1) internal gauge symmetry on superfields. In terms of component fieldsthis gauge transformation reads C (cid:55)→ C + ( φ + φ † ) ξ a (cid:55)→ ξ a + ψ a G (cid:55)→ G + Fv µ (cid:55)→ v µ − i∂ µ ( φ − φ † ) η a (cid:55)→ η a − i ( ∂ µ ψ † γ µ ) a E (cid:55)→ E − ∂ ( φ + φ † ) . (4.75)hapter 4. Supergeometry and supersymmetry 61We can notice two main things. The first is that the combinations λ a := η a + i ( γ µ ∂ µ ξ † ) a D := E + ∂ C (4.76)are gauge invariants. The second is that, since C, G, ξ a transform as shifts, we can chose a specialgauge in which they vanish. This is called the Wess-Zumino (WZ) gauge . Chosing a gauge ofcourse breaks explicitly supersymmetry, but it is convenient for most of the calculations. Inthe WZ gauge, the vector superfield looks like V = v µ (˜ θγ µ θ ) + λ a θ a ˜ θ (2) + λ † ˙ a ˜ θ ˙ a θ (2) + Dθ (2) ˜ θ (2) . (4.77)Gauge transformations with immaginary scalar component ( φ + φ † = 0) preserve the Wess-Zumino gauge and moreover induce on v µ the usual U (1) transformation of Abelian vectorbosons. Indeed, if α := 2Im( φ ), v µ (cid:55)→ v µ + ∂ µ α. (4.78)As for F in the case of chiral superfields, D is the top component field of the vector superfield V . It will have a purely algebraic equation of motion, so it can be considered as an auxiliaryfield. The physical field content of an N = 1 vector superfield is thus the supersymmetry doublet ( v µ , λ ) composed by an Abelian gauge boson and a Majorana spinor, called the gaugino .A gauge-invariant supersymmetric action for the Abelian vector superfield can be given interms of the spinorial superfields defined as W a := 12 ˜ D D a V, ˜ W ˙ a := 12 D ˜ D ˙ a V. (4.79) W a ( ˜ W ˙ a ) is both chiral (antichiral) and gauge invariant, and moreover it satisfies the “reality”condition D a W a = ˜ D ˙ a ˜ W ˙ a . Expanding the component fields in coordinates ( y ( ± ) , θ, ˜ θ ), we have W a = λ a + if µν ( γ µ γ ν ) ab θ b + Dε ab θ b + 2 i ( γ µ ∂ µ λ † ) a θ (2) ˜ W ˙ a = λ † ˙ a − if µν ( γ µ γ ν ) ˙ a ˙ b ˜ θ ˙ b + Dε ˙ a ˙ b ˜ θ ˙ b − i ( γ µ ∂ µ λ ) ˙ a ˜ θ (2) (4.80)where f µν := ( ∂ µ v ν − ∂ ν v ν ) is the gauge invariant Abelian field-strength of v µ . A gauge invariantaction is then S [ V ] = (cid:90) d x (cid:18)(cid:90) d θ W a W a + (cid:90) d ˜ θ ˜ W ˙ a ˜ W ˙ a (cid:19) = (cid:90) d x (cid:8) f µν f µν + iλγ µ ∂ µ λ † + 2 D (cid:9) (4.81)that is an N = 1 supersymmetric extension of the Abelian Yang-Mills theory in 4-dimensions.The supersymmetry transformations of the component field, under which (4.81) is invariantcan be obtained applying the supertranslation on V in WZ gauge. The result will not be inthis gauge anymore, but can be translated back in WZ gauge applying an appropriate gaugetransformation as (4.75). The result is δ (cid:15) v µ = 12 (cid:0) ˜ (cid:15)γ µ λ − λ † γ µ (cid:15) (cid:1) δ (cid:15) λ = i f µν (cid:15)γ µ γ ν + D(cid:15)δ (cid:15) D = − i (cid:0) ˜ (cid:15)γ µ ∂ µ λ + (cid:15)γ µ ∂ µ λ † (cid:1) . (4.82)2 4.3. Introduction to Poincar´e-supersymmetryNotice that in this case the Super Yang-Mills (SYM) action remains supersymmetric even ifwe impose the EoM on the auxiliary field D , setting D = 0 in both (4.81) and (4.82). This isa special result, that holds in 4, 6 and 10 dimensions [44].In a generic gauge theory with gauge group G , we consider a G − valued chiral multiplet Φwhich transforms under a gauge transformations asΦ (cid:55)→ e Λ Φ (4.83)where Λ is a g -valued chiral superfield. Now the combination Φ † Φ is not gauge invariant, so weintroduce a g − valued vector superfield V , transforming as e V (cid:55)→ e Λ † e V e Λ (4.84)that reduces to the previous case (4.74) for Abelian G = U (1). The exponential of a superfieldcan be defined through its component field expansion, that stops at finite order for degreereasons: e V = 1 + v µ (˜ θγ µ θ ) + λ a θ a ˜ θ (2) + λ † ˙ a ˜ θ ˙ a θ (2) + ( D + 2 v µ v µ ) θ (2) ˜ θ (2) . (4.85)The kinetic term for the chiral superfield can be rewritten as a gauge invariant combination: (cid:90) d xd θd ˜ θ Φ † e V Φ . (4.86)Generalizing the supersymmetric field-strength W a to the non-Abelian case as W a = 12 ˜ D e − V D a e V (4.87)we can write the full matter-coupled gauge theory action: S [ V, Φ] = (cid:90) d x (cid:26)(cid:90) d θd ˜ θ Φ † e V Φ + (cid:20)(cid:90) d θ (cid:18)
14 Tr W a W a + W (Φ) (cid:19) + c.c. (cid:21)(cid:27) (4.88)where W (Φ) is a holomorphic function of Φ called superpotential . The expansion in terms ofcomponent fields and the supersymmetry variations can be calculated with the same procedurewe did in the other cases. Notice that whenever the center of the Lie algebra g is non-trivial, i.e. when there is a U (1) factor in G , we could add another supersymmetric and gauge-invariant term to the action(4.88). This is the so-called Fayet-Iliopoulos term : (cid:90) d xd θd ˜ θ ξ ( V ) = (cid:90) d x ξ A D A (4.89)where ξ = ξ A ˜ T A is a constant element in the dual of the center of g . The subgroup of (outer) automorphisms of the supersymmetry group which fixes the under-lying Poincar´e (Euclidean) group is called
R-symmetry group . At the level of the algebra, theseare linear transformations that act only on the spin representation S , leaving the brackets of A more detailed treatment can be found in [45], or [46]. hapter 4. Supergeometry and supersymmetry 63two spinors unchanged. In the complexified case, when different chiral sectors are present, theR-symmetry acts differently on any sector.For example, in the case of N = 1 in (3+1)-dimensions, there is a U (1) R R-symmetry groupacting as Q a (cid:55)→ e − iα Q a , ˜ Q ˙ a (cid:55)→ e iα ˜ Q ˙ a , (4.90)with α ∈ R . This clearly leaves the brackets [ Q a , ˜ Q ˙ a ] invariant. The odd coordinates θ a onsuperspacetime, being elements of S ∗ transform as θ a (cid:55)→ e iα θ a ˜ θ ˙ a (cid:55)→ e − iα ˜ θ ˙ a d θ (cid:55)→ e − iα d θ d ˜ θ (cid:55)→ e iα d ˜ θ, (4.91)so that the volume element d θ = d θd ˜ θ is invariant under R-symmetry. This fixes the R-charge of the superpotential W (Φ) to be 2, if we want the action to be invariant under R-symmetry: W (Φ) (cid:55)→ e iα W (Φ) . (4.92)In principle we can chose the chiral superfield Φ to have any R-charge r , since the combinationΦ † Φ is R-invariant. This, combined with (4.91) means that the different field components inthe chiral multiplet transform differently with respect to R-symmetry: φ (cid:55)→ e irα φ, ψ (cid:55)→ e i ( r − α ψ, F (cid:55)→ e i ( r − α F. (4.93)The vector superfield, being real is acted upon trivially by U (1) R . Its component fields are thenforced to transform as v µ (cid:55)→ v µ , λ (cid:55)→ e iα λ, D (cid:55)→ D, (4.94)thus the gauge-invariant supersymmetric field-strength W a has R-charge 1.In general, if the spin representation is reducible and we have extended supersymmetry,the R-group is always compact. For S = ( S ) N , where S is a real representation, it is of thetype U ( N ), while for S = ( S (+) ) N + ⊕ ( S ( − ) ) N − , where S ( ± ) are the two real representations ofdifferent chirality, it is of the type U ( N + ) × U ( N − ) [43]. Notice the isomorphism U ( n ) ∼ = ( SU ( n ) × U (1)) / Z n (4.95) i.e. U ( n ) is an n-fold cover of SU ( n ) × U (1). In particular, their Lie algebras are isomorphic. Interms of infinitesimal transformations then, the R-symmetry generators can be decomposed inone R-charge plus N − Q Ia (cid:55)→ e − iα U IJ Q Ja , ˜ Q I ˙ a (cid:55)→ e iα ( U † ) IJ ˜ Q J ˙ a . (4.96)In QFT, R-symmetry may or may not be present as a symmetry of the theory, and in manycases part of this symmetry may be broken by anomaly at quantum level. A geometric analysis as the one carried out in the last subsections allows one to find thephysical field content of a supersymmetric theory in every dimensions and for any degree ofreducibility of the spin representation S that is used to extend the Poincar´e algebra. Anothersystematic way to obtain the same result, from a more algebraic point of view, is to study therepresentation of the supersymmetry algebra siso S ( d ), in analogy with the Wigner analysis of4 4.3. Introduction to Poincar´e-supersymmetrymassive and massless representations of the Poincar´e algebra. As the cases encountered above,this study leads to the presence of different supersymmetry multiplets for different choices ofspin and N . We will not present this here but refer for example to [47] for a comprehensivereview, and list here some results for the multiplets at various N .For 1 ≤ N ≤ G be the gauge group, and g its Lie algebra. We are interested mainly in two types of multiplets. The first is the (massless) vector or gauge multiplet , which transforms under the adjoint representation of g . For N = 3 , N = 3 su-persymmetries coincide with those with N = 4 in view of CPT invariance, thus we shall limitour discussion to the N = 4 theories. For N = 1 ,
2, we also have (possibly massive) mattermultiplets: for N = 1, this is the chiral multiplet , and for N = 2 this is the hypermultiplet , bothof which may transform under an arbitrary (unitary, and possibly reducible) representation of G . In (3+1)-dimensions, the on-shell field content of these multiplets is: • N = 1 gauge multiplet ( A µ , λ ): a gauge boson and a Majorana fermion, the gaugino. • N = 1 chiral multiplet ( φ, ψ ): a complex scalar and a left-handed Weyl fermion. • N = 2 gauge multiplet ( A µ , λ ± , φ ): λ ± form a Dirac spinor, and φ is a complex gaugescalar . Under the SU (2) R symmetry, A µ and φ are singlets, while λ + , λ − transform as adoublet. • N = 2 hypermultiplet ( ψ + , H, ψ − ): ψ ± form a Dirac spinor and H ± are complex scalars.Under the SU (2) R symmetry, ψ + and ψ − transform as singlets, while H + , H − transformas a doublet. • N = 4 gauge multiplet ( A µ , λ i , Φ A ): λ i , i = 1 , , , A , A = 1 , · · · , SU (4) R symmetry the gauge field A µ is a singlet, the fermions λ i transform in the fundamental representation , the scalars Φ A transform in the rank-twoantisymmetric representation .Even though in this thesis we do not work explicitly with gravity theories, we will see in thenext section that the introduction of off-shell supergravity is necessary in a possible approach toconstruct globally supersymmetric theories on curved base-spaces. For this purpose, it is usefulto remind also the content of massless supersymmetry particle representations with helicitybetween 1 and 2. These are the gravitino multiplet and the graviton multiplet (or supergravitymultiplet , or metric multiplet ). In general the gravitino multiplet contains degrees of freedomwith helicity less or equal than 3/2. Since in a theory without gravity one cannot acceptparticles with helicity greater than one, that multiplet cannot appear in a supersymmetric To be more precise, it is possible to construct theories with genuine N = 3 supersymmetry, but they lackof a Lagrangian description in terms of component fields. The R-symmetry group is actually SU (2) × SU (2) × U (1), as we will see in a practical application in thefollowing. This comes from the so-called
Weinberg-Witten theorem [48]. hapter 4. Supergeometry and supersymmetry 65theory if also a graviton, with helicity 2, does not appear. In (3+1)-dimensions, the field contentof the relevant multiplets are: • N = 1 gravitino multiplet (Φ µ , B µ ): a helicity 3/2 fermionic particle and a vector boson. • N = 1 graviton multiplet ( h µν , Ψ µ ): the graviton , with helicity 2, and its supersymmetricpartner the gravitino , of helicity 3/2. • N = 2 gravitino multiplet : a spin 3/2 particle, two vectors and one Weyl fermion. • N = 2 graviton multiplet : graviton, two gravitinos and a vector boson.For N > N > N = 2 supersymmetric gauge theories As an example, which will be used in some applications of the localization principle in thenext chapter, we can look at N = 2 Euclidean supersymmetry in 3-dimensions. First, noticethat the rotation algebra for 3d Euclidean space is so (3). The corresponding spin group isthus SU (2), whose fundamental representation does not admit a real structure. In fact herethe charge conjugation can be taken as the totally antisymmetric symbol C ab = ε ab , and theMajorana condition would be inconsistent: ψ T C = ψ † ⇔ ψ = 0 . (4.97)Thus we cannot construct an N = 1 Euclidean supersymmetry algebra in 3 dimensions, in thesense of definition (4.3.1). The problem can be cured considering a reducible spin representation S , where the spinors and the charge conjugation matrix can be split asΨ = ( ψ aI ) a =1 , I =1 , ··· , N , C = (Ω IJ C ab ) a,b =1 , I,J =1 , ··· , N , (4.98)and the same reality condition Ψ † = Ψ T C now is consistent if also the matrix Ω squares to − and is anti-orthogonal: Ω = − Ω T = − Ω − . (4.99)If we now fix N = 2, the resulting spinor representation is analogous to the one of N = 1in 4-dimensions, but now the two Weyl sectors are independent since they generates the twosupersymmetries. To see this corrispondence, we can change basis of S = (1) ⊕ (2) from thenatural one in terms of the generators { Q a , Q a } to Q a := 1 √ Q a + iQ a ) , ˜ Q a := 1 √ Q a − iQ a ) . (4.100)In this basis, using (4.44) the super Lie brackets become[ Q a , ˜ Q b ] = 2( γ µ ) ab P µ + Zε ab [ Q a , Q b ] = 0 [ ˜ Q a , ˜ Q b ] = 0 (4.101)where Z is a constant central charge, and the gamma matrices in this representation can bechosen to be the Pauli matrices γ µ = σ µ for µ = 1 , , Note that the 4-dimensional
Spin (3 , There is also another inequivalent representation of the Clifford algebra, as in any odd dimensions, in which γ = − σ . We chose the former one. SU (2) × SU (2) R , where Spin (3) ∼ = SU (2) is the 3-dimensional Lorentzgroup, and the remaining SU (2) R is an R-symmetry acting on the N = 2 algebra. Thegenerators Q a and ˜ Q a are represented in superspace by odd vector fields whose expressions areformally the same as in (4.58), and the N = 2 supersymmetry variation of a superfield Φ is δ (cid:15),η Φ = ( (cid:15) a Q a + η a ˜ Q a )Φ (4.102)where now, as said before, (cid:15) and η are two independent complex spinors.If we want to construct a supersymmetric gauge theory in 3-dimensions, we consider thevector superfield, now expressed in WZ gauge as V ( x, θ, ˜ θ ) = A µ (˜ θγ µ θ ) + iσθ ˜ θ + λ a θ a ˜ θ (2) + λ † a ˜ θ a θ (2) + Dθ (2) ˜ θ (2) . (4.103)The off-shell N = 2 gauge multiplet is then composed by a gauge field A µ , two real scalars σ, D and a 2-component complex spinor λ . Notice that this is just the dimensional reduction of the N = 1 multiplet in 4-dimensions, with σ coming from the zero-th component of thegauge field in higher dimensions. The only difference with the 4-dimensional vector multipletis that this zero-th component has been considered purely immaginary, i.e. A = iσ with real σ . This ensures the kinetic term for σ to be positive definite and the path integral to converge,matching the would-be dimensional reduction from an Euclidean 4-dimensional theory. If thegauge group is G , all fields are valued in its Lie algebra g .For what we are going to discuss in the next chapter, we now adopt the convention of [49,50] for the supersymmetry variations of the vector superfield and the supersymmetric actions.Under a proper rescaling of the component fields and of the supercharges, one can work themout in an analogous way to which we did in the last sections, and get δ (cid:15),η A µ = i η † γ µ λ − λ † γ µ (cid:15) ) δ (cid:15),η σ = 12 ( η † λ − λ † (cid:15) ) δ (cid:15),η D = i (cid:0) η † γ µ D µ λ − ( D µ λ † ) γ µ (cid:15) (cid:1) − i (cid:0) η † [ λ, σ ] − [ λ † , σ ] (cid:15) (cid:1) δ (cid:15),η λ = (cid:18) − γ µν F µν − D + iγ µ D µ σ (cid:19) (cid:15)δ (cid:15),η λ † = η † (cid:18) − γ µν F µν + D − iγ µ D µ σ (cid:19) (4.104)where D µ = ∂ µ + [ A µ , · ] is the gauge-covariant derivative and γ µν := [ γ µ , γ ν ]. Up to someprefactors, they can be seen as a dimensional reduction of (4.82).We can consider two types of gauge supersymmetric actions constructed from the vectormultiplet in 3 Euclidean dimensions: the Super Yang-Mills theory, that is a reduction of (4.88),and the Super Chern-Simons (SCS) theory. In superspace, the former one is constructed in thesame way as the 4-dimensional case from the spinorial superfield W a , while the SCS term isconstructed as S CS = (cid:90) d xd θd ˜ θ k π (cid:18)(cid:90) dt Tr (cid:110) V ˜ D a e − tV D a e tV (cid:111)(cid:19) . (4.105)hapter 4. Supergeometry and supersymmetry 67Integrating out the odd coordinates in superspace, these are given by [49]: S Y M = (cid:90) d x Tr (cid:26) i λ † γ µ D µ λ + 14 F µν F µν + 12 D µ σD µ σ + i λ † [ σ, λ ] + 12 D (cid:27) , (4.106) S CS = k π (cid:90) d x Tr (cid:26) ε µνρ (cid:18) A µ ∂ ν A ρ + 2 i A µ A ν A ρ (cid:19) − λ † λ + 2 σD (cid:27) . (4.107) N = 4 , , ∗ supersymmetric gauge theories We describe here another example that will be useful in the next chapter, when we willapply the localization principle to supersymmetric QFT. The N = 4 SYM theory on flat spacecan be derived via dimensional reduction of N = 1 SYM in (9 + 1) dimensions. The N = 2and N = 2 ∗ theories can be derived as modification of the N = 4 theory, as we will see later.We start recalling the structure of the 10-dimensional Clifford algebra following the con-ventions of [11]. This is independent from the choice of the signature, Cl (9 , ∼ = Cl (1 , ∼ =Mat ( R ), it is real, and generated by the gamma matrices ( γ M ) M =0 , , ··· , such that { γ M , γ N } = 2 η MN where η is the 10-dimensional Minkowski metric, that we take with signature ( − , + , · · · , +).The fundamental representation of the spin group Spin (9 , (cid:44) → Cl (9 ,
1) is then Majorana,and it is moreover reducible under chirality [42] γ c := − iγ γ · · · γ as Spin (9 ,
1) = S + ⊕ S − ∼ = Mat ( R ) ⊕ Mat ( R ). Thus fundamental spinors are Majorana-Weyl, and have 16 realcomponents. In the chiral basis we denote γ M = (cid:18) M Γ M (cid:19) ˜Γ M , Γ M : S ± → S ∓ γ MN = (cid:18) ˜Γ [ M Γ N ]
00 Γ [ M ˜Γ N ] (cid:19) =: (cid:18) Γ MN
00 ˜Γ MN (cid:19) (4.108)where Γ M , ˜Γ M act on the Majorana-Weyl subspaces, exchanging chirality, and are taken to besymmetric. Let the gauge group G be a compact Lie group, and g its Lie algebra. The (on-shell)component field content of the gauge multiplet in 10 dimensions is of a gauge field, locallyrepresented as A ∈ Ω ( R , , g ), and a gaugino, a Mayorana-Weyl spinor Ψ : R , → S + ⊗ g with values in the Lie algebra g . The field strength of the gauge field is locally represented by F = dA + [ A, A ], and the associated gauge-covariant derivative on R , is D M = ∂ M + A M . Thesupersymmetry variations under the action of the 10-dimensional super-Poincar´e algebra are δ (cid:15) A M = (cid:15) Γ M Ψ δ (cid:15) Ψ = 12 Γ MN F MN (cid:15) (4.109) For convergence of the partition function, it would be nicer to start from the (10 ,
0) Euclidean signature.We follow the convention of [11] and start from the (9 ,
1) one, Wick rotating a posteriori the path integral whenneeded, to match the would-be reduction from the (10 ,
0) theory. In the Euclidean signature, we would use Γ ME = { Γ , · · · , Γ , i Γ } . (cid:15) is a Majorana-Weyl spinor, analogously to the on-shell version of (4.82) up to thechirality projection and conventional prefactors. The action functional for the N = 1 10-dimensional theory is S d = (cid:82) d x L , with Lagrangian L = 1 g Y M Tr (cid:18) F MN F MN − ΨΓ M D M Ψ (cid:19) (4.110)where Tr denotes a symmetric bilinear pairing in g , and g Y M is the Yang-Mills couplingconstant. As we remarked in Section 4.3.3, this action is exactly supersymmetric under (4.109)without the addition of auxiliary fields.To get the Euclidean 4-dimensional theory, we perform dimensional reduction along thedirections x , x , · · · , x , assuming independence of the fields on these coordinates. The fieldssplit as A M → (( A µ ) µ =1 , ··· , , (Φ A ) A =5 , ··· , , )Ψ → (cid:0) ψ L χ R ψ R χ L (cid:1) T (4.111)where ψ L/R , χ
L/R are four-component real chiral spinors. The spacetime symmetry group
Spin (9 ,
1) is broken to
Spin (4) × Spin (5 , R (cid:44) → Spin (9 , Spin (4) ∼ = SU (2) L × SU (2) R acts on the x , · · · , x directions, and the R-symmetry group Spin (5 , R rotates the other ones.It is often convenient to further break the R-symmetry group to Spin (4) R × SO (1 , R (cid:44) → Spin (5 , R , where the first piece Spin (4) R ∼ = SU (2) R L × SU (2) R R rotates the x , · · · , x direc-tions, and SO (1 , R acts on the x , x ones. We thus consider the symmetry group SU (2) L × SU (2) R × SU (2) R L × SU (2) R R × SO (1 , R (4.112)under which the fields behave as • A µ : vector of SU (2) L × SU (2) R , scalar under R-symmetry; • (Φ I ) I =4 , ··· , : 4 scalars under SU (2) L × SU (2) R , vector of SU (2) R L × SU (2) R R , scalars under SO (1 , R ; • Φ , Φ : scalars under SU (2) L × SU (2) R × SU (2) R L × SU (2) R R , vector of SO (1 , R ; • ψ L/R : (cid:0) , (cid:1) / (cid:0) , (cid:1) of SU (2) L × SU (2) R , (cid:0) , (cid:1) of SU (2) R L × SU (2) R R , + / − of SO (1 , R ; • χ L/R : (cid:0) , (cid:1) / (cid:0) , (cid:1) of SU (2) L × SU (2) R , (cid:0) , (cid:1) of SU (2) R L × SU (2) R R , − / + of SO (1 , R ;here we denoted + , − the inequivalent Majorana-Weyl representations of SO (1 , R , seen as asubgroup of Cl (1 , ∼ = Mat ( R ).The above decomposition of Spin (9 ,
1) into four subrepresentations of
Spin (4), rotated intoeach other by the R-symmetry group, gives the N = 4 supersymmetry algebra on R . Thesupersymmetry variations of the reduced component fields are given by (4.109), read in termsof the splitting (4.111), δ (cid:15) A µ = (cid:15) Γ µ Ψ δ (cid:15) Φ A = (cid:15) Γ A Ψ δ (cid:15) Ψ = 12 (cid:0) Γ µν F µν + Γ AB [Φ A , Φ B ] + Γ µA D µ Φ A (cid:1) (cid:15). (4.113) For semisimple g , this is the Killing form as usual. hapter 4. Supergeometry and supersymmetry 69The action of the 4d N = 4 SYM theory is S N =4 = (cid:82) d x L with the Lagrangian obtained bythe reduction of (4.110). More explicitly, S N =4 = (cid:90) d x g Y M Tr (cid:18) F µν F µν + ( D µ Φ A ) − ΨΓ µ D µ Ψ + 12 [Φ A , Φ B ] − ΨΓ A [Φ A , Ψ] (cid:19) . (4.114)Notice that, since contractions of A, B indices are done with a reduced Minkowski metric, upondimensional reduction from the Lorentzian theory the scalar Φ has a negative kinetic term.Analogously to the last section, we consider it to be purely immaginary, i.e. Φ =: i Φ E withΦ E real. This makes the path integral match with the would-be reduction from the Euclidean(10 , N = 4 algebra closes on-shell . In fact, it can be obtained from (4.113) that δ (cid:15) = 12 [ δ (cid:15) , δ (cid:15) ] = −L v − G Φ (4.115)up to the imposition of the EoM for Ψ, Γ M D M Ψ = 0. Here v M := (cid:15) Γ M (cid:15) , L v is the Lie derivative(the action of the translation algebra) with respect to v ∼ v µ ∂ µ , and G Φ is an infinitesimal gaugetransformation with respect to Φ := A M v M . A famous non-renormalization theorem by Seiberg[51] states that the N = 4 theory is actually superconformal , i.e. it has a larger supersymmetryalgebra that squares to the conformal algebra , whose generators are the Poincar´e generators plusthe generators of dilatations and special conformal transformations. In fact, one can see that S N =4 is classically invariant under supersymmetry variations with respect to the non-constantspinor (cid:15) = ˆ (cid:15) s + x µ Γ µ ˆ (cid:15) c (4.116)where ˆ (cid:15) s , ˆ (cid:15) c are constant spinors parametrizing supertranslations and superconformal transfor-mations. This enlarged supersymmetry algebra closes now on the superconformal algebra, δ (cid:15) = −L v − G Φ − R − Ω (4.117)where R is a Spin (5 , R rotation, acting on scalars as ( R · Φ) A = R BA Φ B , and on spinors as( R · Ψ) = R AB Γ AB Ψ, where R AB = 2 (cid:15) ˜Γ AB ˜ (cid:15) . Ω is an infinitesimal dilatation with respect tothe parameter 2(˜ (cid:15)(cid:15) ), acting on the gauge field trivially, on scalars as Ω · Φ = − (cid:15)(cid:15) )Φ and onspinors as Ω · Ψ = − (cid:15)(cid:15) )Ψ. This new bosonic transformations are clearly symmetries of S N =4 .Now we can restrict the attention to an N = 2 subalgebra, considering the variations withrespect to Majorana-Weyl spinors of the form (cid:15) = (cid:0) (cid:15) L (cid:15) R (cid:1) T (4.118)so in the subrepresentation (cid:0)(cid:0) , (cid:1) ⊕ (cid:0) , (cid:1)(cid:1) ⊕ (cid:0) , (cid:1) R ⊕ (+ ⊕ − ) R , the eigenspace of Γ witheigenvalue +1. With respect to these supersymmetry variations, the gauge multiplet furthersplits in • ( A µ , Φ , Φ , ψ L , ψ R ): the N = 2 vector multiplet; • (Φ I , χ L , χ R ): the N = 2 hypermultiplet, with value in the adjoint representation of G . More precisely, the theorem states that the beta function of g Y M is zero non-perturbatively. This meansthat the theory is fully scale invariant at quantum level. g Y M → Thesame Lagrangian thus equivalently describes an N = 2 matter-coupled gauge theory. It is alsopossible to insert a mass for the hypermultiplet, breaking explicitly the conformal invariance,and obtain the so-called N = 2 ∗ theory. Since the fields of the vector multiplet are all scalarsunder SU (2) R R , and the hypermultiplet fields are all in the representation, these mass termscan at most rotate the hypermultiplet content with an SU (2) R R transformation. Thus replacing D Φ I (cid:55)→ [Φ , Φ I ] + M JI Φ J and D Ψ (cid:55)→ [Φ , Ψ] + M IJ Γ IJ Ψ, where ( M IJ ) represents an SU (2) R R rotation in the vector representation, one obtains the mass terms for the Φ I and χ fields. Noticethat δ (cid:15) gets a contribution from the Lie derivative with respect to v ∂ ∼ = 0 (cid:55)→ v M , so that inthe 2 ∗ theory δ (cid:15) Φ I (cid:55)→ ( δ (cid:15) Φ I ) N =2 − v M JI Φ J δ (cid:15) χ (cid:55)→ ( δ (cid:15) χ ) N =2 − v M IJ Γ IJ χ. (4.119)In the limits of infinite or zero mass, the pure N = 2 or N = 4 theory is recovered. Notice that,since we argued that Φ should be integrated over purely immaginary values for the convergenceof the path integral, also ( M IJ ) should be taken purely immaginary. Recently, localization theory has been extensively used in the framework of quantum fieldtheories with rigid super-Poincar´e symmetry, to compute exactly partition functions or expec-tation values of certain supersymmetric observables, when the theory is formulated on a curved compact manifold. This cures the corresponding partition functions from infrared divergencesmaking the path integral better defined, and is consistent with the requirement of periodicboundary conditions on the fields, that allows to generalize properly the Cartan model on theinfinite dimensional field space. We will come back to this last point in the next chapter, whenwe will study circle localization of path integrals, while we close this chapter reviewing the ideabehind some common approaches used to formulate rigid supersymmetry on curved space.Following the approach of the last section, we would have to understand what does it mean tohave supersymmetry on a generic metric manifold ( M , g ) (Riemannian or pseudo-Riemannian)of dimension dim M = d from a geometric point of view. The supersymmetry of flat spacewas constructed as a super-extension of Minkowski (or Euclidean) space R d , starting from asuper-extension of the Lie algebra of its isometry group, the Poincar´e group. Now in generalthe Poincar´e group is not an isometry group for M , so the super Poincar´e algebra siso S ( d )with respect to some (real or Majorana) spin representation S cannot be fully interpreted asa “supersymmetry” algebra for the space at hand. We can nonetheless associate in some waythis algebra to a suitable super-extension of M , and then ask what part of it can be preservedas a supersymmetry of this supermanifold. We follow [52] for this geometric introduction.Since we want to work with spinors, we assume that M admits a spin-structure. In particu-lar, it exists a (real) spinor bundle S → M associated to the spin-structure, with structure sheaf S : S ( U ) = Γ( U, S ) , ∀ U ⊂ M open. Analogously to the flat superspace of the last section, wemake now a super-extension of M through this spinor bundle considering the odd spinor bundle S M S := Π S , with body M and structure sheaf (cid:86) S ∗ : (cid:86) S ∗ ( U ) = C ∞ ( U ) ⊗ (cid:86) ( S ∗ ) , ∀ U ⊂ M Working out the restricted supersymmetry variations taking into account the splitting of the gaugino, somenon-linear term, coupling the fermionic sectors of the two multiplets, survives because of the gauge interaction.In the free theory limit, after the rescaling A M (cid:55)→ g Y M A M , Ψ (cid:55)→ g Y M
Ψ, these terms go to zero. hapter 4. Supergeometry and supersymmetry 71open, where S is the typical fiber of S . From proposition 4.1.1, for any p ∈ M , there is anisomorphism of Z -graded vector spaces T p S M S ∼ = T p M ⊕ S p [1].Now, the vector bundle V := T M ⊕ S over M carries the canonical spin-connection inducedby the Levi-Civita connection of the manifold ( M , g ). Assume that we can pick a parallelnon-degenerate Spin ( d )-invariant bilinear form β on S with respect to this connection. Wecan think of the
Spin ( d )-invariant bilinear form ˜ g = g + β as a (pseudo-)Riemannian metricon the supermanifold S M S . Moreover, associated to the bilinear form β we have the mapΓ : S → T M , that is a point-wise generalization of the usual symmetric and Spin-equivariantbilinear form for a Spin representation S ∼ = S p , ∀ p ∈ M . This means that we can consider thebundle p ( V ) := spin ( d ) ⊕ V (4.120)as a bundle of super Poincar´e algebras over M , with the bracket structure extended throughΓ. Having found how to (point-wise) set up the super Poincar´e algebra on top of the superman-ifold S M S constructed from ( M , g ), we wish to establish which section of the super Poincar´ebundle p ( V ) produces a suitable generalization of “super-isometry” for S M S . In particular,we pay attention to which sections of S , as the odd subbundle of p ( V ), generates “supersym-metries” of the generalized metric ˜ g . This problem was analyzed in [52], and connected to theproblem of finding solution to the so called Killing spinor equation for a section ψ of S → M . Definition 4.4.1.
A section ψ of the spinor bundle S → M is called a twistor spinor (or conformal Killing spinor ) if it exists another section φ such that, for any vector field X ∈ Γ( T M ), ∇ X ψ = X · φ (4.121)where X · φ = X µ γ µ φ is the Clifford multiplication . If in particular φ = λψ , for some constant λ , the spinor ψ is called Killing spinor .The equation (4.121) is called twistor or Killing spinor equation . Note that (4.121) directlyimplies φ = ± (1 / dim( M )) / ∇ ψ , where / ∇ := γ µ ∇ µ is the Dirac operator, and the sign dependson conventions. The twistor spinor equation is thus equivalently written as ∇ X ψ = ± M ) X · / ∇ ψ. (4.122)This characterizes the Killing spinors as those twistor spinors that satisfies also the Diracequation / ∇ ψ = mψ for some constant m . The main result proved in [52] is stated in thefollowing theorem. Theorem 4.4.1.
Consider the supermanifold S M S with the bilinear form ˜ g = g + β , and asection ψ of S . The odd vector field X ψ associated to ψ is a Killing vector field of ( S M S , ˜ g ) ifand only if ψ is a twistor spinor.Here the Killing vector condition on the supermanifold is a conceptually straightforwardgeneralization of the usual concept of Killing vector fields on a smooth manifold. It can benatually stated in terms of superframe fields. We refer to the above cited article for the details.Notice that, in particular, Killing spinors generate infinitesimal isometries of the supermanifold S M S , and thus are good candidates to describe the “preserved” supersymmetries of the odd This is always true if M is simply-connected. M in the way we saw above. See also [53] for a review on Killing spinors in (pseudo-)Riemanniangeometry.From the QFT point of view, it is possible to derive a (generalized) Killing spinor equation,describing the preserved supercharges on the curved space, from a dynamical approach. Thisidea is based on a procedure also valid in the non-supersymmetric setting, when one aims todeform a certain QFT to redefine it on a generic curved manifold. In this case, one couplesthe theory to background gravity, letting the metric fluctuate. Then the gravitational sectoris decoupled from the rest of the theory taking the gravitational constant G N →
0, whilethe metric is linearized around the chosen off-shell configuration g = η + h and the higherorder corrections disappear in the limit of weak gravitational interaction. It is importantthat we do not constrain the gravitational field to satisfy the equation of motion, since it isconsidered as a background (classical) field. The same idea applies when the theory is definedin the supersymmetric setting: in this case to preserve supersymmetry one has to couple tobackground supergravity (SUGRA) . The resulting field theory will contain then more fieldsbelonging to the so-called supergravity multiplet . This time, taking the limit G N → preserved on this background.This procedure was systematically introduced in [54], then many cases and classification weremade in different dimensions and with different degree of supersymmetriy (see for example[55–58]). Suppose we have a supersymmetric field theory formulated on flat space specified by itsLagrangian L (0) , whose variation under supersymmetry is a total derivative: δ L (0) = ∗ d ∗ ( · · · ) = ∂ µ ( · · · ) µ (4.123)We can introduce supergravity by requiring the action of the super-Poincar´e group to be local ,employing the usual gauge principle and minimal coupling or Noether procedure.In the non-supersymmetric setting, this would mean to introduce a gauge symmetry underlocal coordinate transformations, realized via diffeomorphisms on M . The Noether current as-sociated to such infinitesimal transformations is the energy-momentum tensor T µν , that we take Since the metric can fluctuate and the field theory is defined locally, there is no harm in principle inconsidering different topologies of the base manifold, like requiring it to be compact. For simplicity, we consider now the formulation on the even space R d or M , at the level of componentfields of the given supersymmetry multiplets. The supersymmetry variation of these component fields are thosecoming from the action of the odd supertranslations in superspace. hapter 4. Supergeometry and supersymmetry 73to be symmetric. The minimal coupling procedure then requires to modify the Lagrangian, L (cid:48) = L (0) + h µν T µν + O ( h ) (4.125)where h µν is regarded as a variation of the metric from the flat space values η µν , and O ( h ) areseagull non-linear terms that can be fixed requiring the gauge invariance of L (cid:48) . The resultingnon-linear coupling is obtainable substituting in the original theory d (cid:55)→ ∇ , η (cid:55)→ g = η + h, (4.126)where ∇ is the gauge-covariant derivative with respect to a connection Γ, that we take as theLevi-Civita connection. The theory is now coupled to a gravitational (classical) background . Ifwe want to make the graviton field h dynamical, we can add an Hilbert-Einstein term to L (0) , L HE = − (cid:112) | det g | κ Ric g (4.127)where Ric g is the Ricci scalar associated to g , and κ := 1 /M p = √ πG N .In the supersymmetric setting, gauging the super-Poincar´e group leads to the introductionof more fields into the theory, since as we know they can be interpreted as components ofsuperfields in superspace, and thus belong to supersymmetry multiplets. In particular, we haveto introduce the graviton multiplet composed by the metric g , the gravitino Ψ and other (maybeauxiliary) fields. The particular field content depends on the number N of supersymmetries,the dimensionality of the theory, the presence or absence of an R-symmetry and whether thetheory is or not superconformal. Consequently, also the energy-momentum tensor will belongto a multiplet, the so-called supercurrent multiplet , composed by T , a supercurrent J associatedto the local invariance under (odd) supertranslations, and other fields. We can schematicallyperform the first steps of the Noether procedure to see how the components of these multipletsarise naturally.We start from the odd part of the super-Poincar´e algebra iso ( R d ) ⊕ S [1], writing the in-finitesimal variation of the Lagrangian in terms of the supercurrent: δ (cid:15) L (0) ≡ (cid:15) · L (0) = ( ∂ µ J µ ) (cid:15) (4.128)where (cid:15) is now a Majorana spinor field, i.e. a section of the spinor bundle with fiber S . Thesupercurrent J is an S -valued vector field, and the spinor contraction is done via the usualcharge conjugation matrix. We couple this current to a gauge field Ψ, to be identified with thegravitino, that is locally an S -valued 1-form such that, at linearized level, δ (cid:15) Ψ µ = 1 κ ∂ µ (cid:15) (4.129) In general this will not be a symmetric tensor, but there always exists a suitable modification that makesit symmetric, and moreover equivalent to the Hilbert definition of energy momentum tensor as a source ofgravitational field. This is the
Belinfante–Rosenfeld tensor ˜ T µν := T µν + 12 ∇ λ ( S µνλ + S νµλ − S λνµ ) (4.124)where S µνλ is the spin part of the Lorentz generators in a given spin representation satisfying ∇ µ S µνλ = T νµ − T µν ,and ∇ is an appropriate torsion-free spin-covariant derivative induced from the metric (see for example [59]). κ is introduced for dimensional reasons. Then we add a term to theLagrangian: L (cid:48) = L (0) + κ Ψ µ J µ . (4.130)Now the variation of L (cid:48) is proportional to the variation of the current δ (cid:15) J . Since the supercurrentis a supersymmetry variation of the original Lagrangian, its variation will be proportional tothe action of the translation generators P µ : δ (cid:15) L (0) = (cid:15) · ( (cid:15) · L (0) ) = ∂ µ ( δ (cid:15) J µ ) (cid:15) = (1 / (cid:15), (cid:15) ] · L (0) = (cid:15)γ ν (cid:15) ( P ν · L (0) ) = ( (cid:15)γ ν (cid:15) ) ∂ µ T µν ⇒ δ (cid:15) J µ = (cid:15)γ ν T νµ (4.131)where in the second line we wrote the variation under the action of the translation generatorsin terms of the energy-momentum tensor T . We try to restore the gauge-invariance of theLagrangian by minimally coupling this new current to a new gauge field h , that we identify asa metric variation, the graviton L (cid:48)(cid:48) = L (0) + κ Ψ µ J µ + h µν T µν , (4.132)and naturally requiring the supersymmetry variation of the graviton to be δ (cid:15) h µν = κ(cid:15)γ ( µ Ψ ν ) , (4.133)making it the superpartner of the gravitino Ψ.The Lagrangian L (cid:48)(cid:48) is again not supersymmetric, since the variation δ (cid:15) T µν (cid:54) = 0 in general, sothe Noether procedure is not terminated yet. It is not easy to complete this procedure in thisway, but in principle repeating this passages we would introduce more linearly coupled currentsand gauge fields that, motivated by supersymmetry, we expect to come from the SUGRAsupermultiplets mentioned above. To ensure the supersymmetry of the full Lagrangian atnon-linear level, as in non supersymmetric gauge theories, non-linear couplings could have tobe introduced as well as non-linear terms in the supersymmetry variations. Summarizing, weexpect the fully coupled Lagrangian to be schematically of the form L = L (0) + κJ µ Ψ µ + h µν T µν + (cid:88) i B i · J i + (seagull terms) (4.134)where B is the multiplet of background gauge fields ( h, Ψ , · · · ), J the supercurrent multiplet( T, J, · · · ), and we referred to possible higher-order terms in the background fields as seagullterms. As already said, the particular field content of these multiplets is not unique, so weremain generic for the moment and refer to the next subsections for some examples. Wecan absorb the terms proportional to h as in the non-supersymmetric case, by making thesubstitutions d (cid:55)→ ∇ and η (cid:55)→ g = η + h . If we want to have a full gravitational theory, wecan add a kinetic term for the source fields, and complete their supersymmetry variations withpossible non-linear terms to ensure gauge invariance. Regarding the metric and the gravitino, If we canonically take mass dimensions of scalars to be ( d − / d − /
2, and sinceschematically δ (cid:15) ( boson ) = ( f ermion ) (cid:15) than [ (cid:15) ] = − /
2, so κ must be a dimensionfull parameter of mass dimension [ κ ] = (2 − d ) /d . We can so identifythis constant as the gravitational constant previously defined. hapter 4. Supergeometry and supersymmetry 75the kinetic terms are given by the Hilbert-Einstein action (4.127) and the Rarita-Schwinger action L RS = − (cid:112) | det g | µ γ µνρ ( ∇ ν Ψ) ρ , (4.135)where γ µνρ := γ [ µ γ ν γ ρ ] , and ∇ acts on spinors via the spin connection, ( ∇ µ Ψ) ν = ∂ µ Ψ ν + ω abµ γ ab Ψ ν − Γ ρµν Ψ ρ . The supersymmetry variations will be generically δ (cid:15) h µν = κ(cid:15) (cid:8) γ ( µ Ψ ν ) + ( · · · ) F (cid:9) δ (cid:15) Ψ µ = 1 κ (cid:8) ∇ µ + ( · · · ) Bµ (cid:9) (cid:15) + O ( κ Ψ (cid:15) ) (4.136)where we stressed that non-linear higher-oreder terms for the gravitino are κ -suppressed, andthe ellipses in both cases collect contributions from the other (fermionic or bosonic, respectively)fields of the supergravity multiplet. Notice that also the supersymmetry variations of the fieldcontent of the original L (0) get modified with respect to their flat-space version. Once one hasthe full supergravity theory, their transformation rules follow from the corresponding formulasin the appropriate matter-coupled off-shell supergravity.We now consider the rigid limit G N → κ →
0, or M P → ∞ ) together with the choiceof a given background gravitational multiplet B compatible with the original request ( M , g ). Since we think at this classical configuration as a VEV, we require all the fermion fields in thesupergravity multiplet to vanish on this background. We also look for those supergravity trans-formations that leave this background invariant. These requirements produce the followingeffects: • Fermionic gravitational fields as well as the kinetic term for the bosonic gravitationalsector do not contribute to the lagrangian: L = L (0) (cid:12)(cid:12) d →∇ η → g + (cid:88) i B iB · J iB + (seagull terms) B (4.137)At the same time, supersymmetry variations of the bosonic gravitational fields automat-ically vanish. • Requiring the supersymmetry of the background is then equivalent to δ (cid:15) B F = 0 . (4.138)In particular, this condition on the gravitino generates the generalized Killing spinorequation δ (cid:15) Ψ µ = 0 ⇔ ∇ µ (cid:15) = ( · · · ) µ (cid:15) (4.139)where, again, ellipses stand for terms proportional to bosonic fields in the graviton multi-plet. The solutions to this equation determine which sections of the spinor bundle S → M generates the preserved supersymmetry transformations on M . We stress that a rigid supersymmetric background is characterized by a full set of supergravity backgroundfields, i.e. specifying only the metric does not determine the background. In particular, there are distinctbackgrounds that have the same metric but lead to different partition functions. This can be interpreted as a superisometry requirement with respect to the graviton multiplet.
As we wrote before, the field content of supercurrent multiplets depends on the generalproperties of the theory at hand. In [60] it was given a definition from basic general requirementsstarting from superfields in superspace, that allows a classification by specializing to the variousparticular cases. It is shown that the most general supercurrent is a real superfield S a ˙ a satisfying˜ D ˙ a S a ˙ a = χ a + Y a ˜ D ˙ a χ a = 0 ˜ D ˙ a χ † ˙ a = D a χ a D ( a Y b ) = 0 ˜ D Y a = 0 . (4.140)Every supersymmetric theory has such an S -multiplet, containing the stress energy tensor T and the supercurrent S . They are the only component fields with spin larger than one, sincethey couple to the graviton and the gravitino in the metric multiplet, that respectively arethe only component fields with spin higher than one in this multiplet. We report here specialexamples in 4 and 3 dimensions, that can be derived solving the constraints (4.140) in caseswhere additional conditions on the superfields χ a and Y a hold.For N = 1 in 4-dimensions we have three possible interesting special cases:1. The majority of theories admit a reduction of the S -multiplet into the so-called Ferrara-Zumino (FZ) multiplet J F Zµ → ( j µ , ( S µ ) a , x, T µν ) (4.141)where j µ is a vector field and x a complex scalar field.2. If the theory has a U (1) R symmetry, the S -multiplet reduces to the so-called R -multiplet ,whose lower degree component is the conserved R-current j ( R ) µ : R µ → (cid:0) j ( R ) µ , ( S µ ) a , T µν , C µν (cid:1) (4.142)where C µν are the components of a conserved 2-form current, the so-called brane current .
3. For a superconformal theory, the S -multiplet decomposes into the smaller supercurrent J µ → (cid:0) j ( R ) µ , ( S µ ) a , T µν (cid:1) (4.143)where j ( R ) µ is a conserved superconformal U (1) R -current.Both the FZ multiplet and the R -multiplet contain 12+12 real degrees of freedom out ofthe initial 16+16 of the general S -multiplet, while the superconformal multiplet is reduced to8+8 real degrees of freedom. The FZ multiplet can be coupled to the so-called “old minimalsupergravity multiplet” [63]: H µ → ( b µ , (Ψ µ ) a , M, h µν ) (4.144) In curved space and in presence of topological defects as strings (1-brane) or domain walls (2-brane), thesupersymmetry algebra (4.57) can be modified by the presence of brane charges , (cid:104) Q a , ˜ Q ˙ b (cid:105) = 2( γ µ ) a ˙ b ( P µ + Z µ )[ Q a , Q b ] = ( γ µν ) ab ˜ Z µν where Z µ , ˜ Z µν are non-zero for strings and domain walls, respectively. The corresponding tensor currents arethe brane currents C µν , ˜ C µνρ , that are topologically conserved. See [60–62] for more details. hapter 4. Supergeometry and supersymmetry 77where b µ is a genuine 1-form field ( i.e. non gauge), M is a complex scalar, (Ψ µ ) a is the gravitinoand h µν is the graviton. The variation of the gravitino in this case is given by [54] δ (cid:15) Ψ µ = − ∇ µ (cid:15) + i M γ µ + 2 b µ + 2 b ν γ µν ) (cid:15) (4.145)that implies a generalized Killing spinor equation of the form ∇ µ (cid:15) = i M γ µ + 2 b µ + 2 b ν γ µν ) (cid:15) (4.146)in the Majorana spinor (cid:15) , given a background multiplet. In theories with an R-symmetry, onecan couple the R -multiplet to the “new minimal supergravity multiplet”[64]: H ( new ) µ → (cid:0) A ( R ) µ , (Ψ µ ) a , h µν , B µν (cid:1) (4.147)where A ( R ) µ is the Abelian gauge field associated to the U (1) R symmetry, and B µν is a 2-formgauge field that is often treated through its Hodge dual V µ := i ( (cid:63)B ) µ = ( i/ ε µνρσ ∂ ν B ρσ . Thevariation of the gravitino in this case gives rise to the following Killing spinor equation, that in2-component notation is [54] (cid:0) ∇ µ − iA ( R ) µ (cid:1) (cid:15) a = − iV µ (cid:15) a − iV ν ( γ µν (cid:15) ) a (cid:0) ∇ µ + iA ( R ) µ (cid:1) ˜ (cid:15) ˙ a = iV µ ˜ (cid:15) ˙ a + iV ν ( γ µν ˜ (cid:15) ) ˙ a (4.148)where in the parenthesis on the LHS there is a gauge-covariant derivative with respect to the U (1) R symmetry, that acts with opposite charge on the two chiral sector of the spin represen-tation, see (4.90).The N = 2 case in 3 Euclidean dimensions can be derived in superspace by dimensionalreduction from the four dimensional case: the supercurrent is reduced to a three dimensional S -multiplet with 12+12 real DoF, plus a real scalar superfield ˆ J = S ≡ S a ˙ a ( σ ) a ˙ a , thatcontains 4+4 real DoF. Again, there are special cases analogue of those above: a FZ multiplet,an R -multiplet, and a superconformal multiplet. For example, the N = 2 R -multiplet in 3dimensions has the field content R µ → (cid:16) j ( R ) µ , j ( Z ) µ , J, ( S µ ) a , ( ˜ S µ ) a , T µν (cid:17) (4.149)where j ( R ) µ is the conserved R-current, j ( Z ) µ is the conserved central charge current and J is ascalar operator, that with the conserved supercurrents and enery-momentum tensor sum up to8+8 real DoF. This multiplet couples to the tree dimensional N = 2 new minimal supergravitymultiplet H ( new ) µ → (cid:16) A ( R ) µ , C µ , H, ( ψ µ ) a , ( ˜ ψ µ ) a , h µν (cid:17) (4.150)with the graviton, two gravitini, two gauge 1-forms A ( R ) and C , and a scalar H . The 1-form C is often treated in terms of the vector field V µ := i ( (cid:63)dC ) µ = iε µνρ ∂ ν C ρ that is Hodge dualto its field strength. Putting to zero the gravitini and their variations leads to the generalizedKilling spinor equations [56] (cid:0) ∇ µ − iA ( R ) µ (cid:1) (cid:15) = − (cid:18) Hγ µ + iV µ + 12 ε µνρ V ν γ ρ (cid:19) (cid:15) (cid:0) ∇ µ + iA ( R ) µ (cid:1) ˜ (cid:15) = − (cid:18) Hγ µ − iV µ − ε µνρ V ν γ ρ (cid:19) ˜ (cid:15) (4.151)where in this case the two spinors (cid:15), ˜ (cid:15) have to be treated as independent. Notice that bothequations (4.148) and (4.151) are linear in the 4 spinor components, so their solutions (if exist)span a vector space of dimension less or equal than 4.8 4.4. From flat to curved space N = 2 gauge theories on the round 3-sphere It was shown that, in general, solutions of the Killing condition (4.148) in four dimensionsexist if ( M , g ) is an Hermitian manifold, i.e. M has an integrable complex structure and g is a compatible Hermitian metric. Analogously, the existence of a solution to (4.151) in threedimensions was shown to be equivalent to the manifold admitting a transversally holomorphicfibration . If one is interested in the case of maximal number of Killing spinor solutions, asuitable integrability condition (see [56]) gives H = const , d ( A ( R ) − V ) = 0 , g ( V, V ) = const , ( ∇ µ V ) ν = − iHε µνρ V ρ ,R µν = − V µ V ν + g µν ( g ( V, V ) + 2 H ) . (4.152)In particular, if we take A ( R ) = V = 0, then M is of Einstein type and so it has constantsectional curvature. H is then interpreted as a cosmological constant, and M can be S , T or H if H is purely immaginary, zero or real. All of them are examples of maximally super-symmetric backgrounds in N = 2, thus we have 2 solutions for (cid:15) and 2 solutions for ˜ (cid:15) to theKilling equations ∇ µ (cid:15) = − H γ µ (cid:15) ; ∇ µ ˜ (cid:15) = − H γ µ ˜ (cid:15). (4.153)In particular, if we take H = − ( i/l ), this solutions are consistent with the S round metric g = l (cid:0) dϕ ⊗ dϕ + sin ϕ dϕ ⊗ dϕ + sin ϕ sin ϕ dϕ ⊗ dϕ (cid:1) . (4.154)The action of the supersymmetry algebra on the curved manifold can be derived by takingthe rigid limit of the appropriate algebra of supergravity transformations. In the 3-dimensionalcase, it can be derived by a “twisted” reduction of the N = 1 supergravity in 4 dimensions.The 4-dimensional supersymmetry algebra realizes on the curved manifold as[ δ (cid:15) , δ (cid:15) ] φ ( r ) = [ (cid:15), (cid:15) ] · φ ( r ) = 2( (cid:15)γ µ (cid:15) ) P µ · φ ( r ) (4.155)where (cid:15) is a Majorana Killing spinor, φ ( r ) is a generic field of R-charge r , and the local actionof the momentum operator is through the fully covariant derivative P µ → − (cid:0) i ∇ µ + rA ( R ) µ (cid:1) , (4.156)so that (4.155) can be written as[ δ (cid:15) , δ (cid:15) ] φ ( r ) = − i (cid:0) L v − irA ( R ) ( v ) (cid:1) φ ( r ) (4.157)where v µ := ( (cid:15)γ µ (cid:15) ) is a Killing vector field thanks to (cid:15) being a Killing spinor field. This isreduced to the 3-dimensional case, taking (cid:15), ˜ (cid:15) now as independent 2-component Killing spinorsand v µ := ˜ (cid:15)γ µ (cid:15) in 3 dimensions, as (see again [56])[ δ ˜ (cid:15) , δ (cid:15) ] φ ( r,z ) = − i (cid:20) L v − iv µ (cid:18) r ( A ( R ) µ − V µ ) + zC µ (cid:19) + ˜ (cid:15)(cid:15) ( z − rH ) (cid:21) φ ( r,z ) , [ δ ˜ (cid:15) , δ ˜ (cid:15) ] φ ( r,z ) = 0 , [ δ (cid:15) , δ (cid:15) ] φ ( r,z ) = 0 , (4.158) This is an odd-dimensional analogue to a complex structure. It means, roughly speaking, that M is locallyisomorphic to R × C , and its transition functions are holomorphic in the C -sector. hapter 4. Supergeometry and supersymmetry 79where z is the charge associated to the action of the central charge Z in (4.101). For the3-sphere of radius l = 1, this is simplified to[ δ ˜ (cid:15) , δ (cid:15) ] φ ( r,z ) = − i [ L v + ˜ (cid:15)(cid:15) ( z + ir )] φ ( r,z ) , [ δ ˜ (cid:15) , δ ˜ (cid:15) ] φ ( r,z ) = 0 , [ δ (cid:15) , δ (cid:15) ] φ ( r,z ) = 0 . (4.159)We report the resulting supersymmetry variation for the 3-dimensional N = 2 vector mul-tiplet ( A µ , σ , λ a , ˜ λ a , D ), with respect to two Killing spinors ˜ (cid:15), (cid:15) . This multiplet is unchargedunder the action of R-symmetry and of the central charge Z . Following conventions of [50] and[49], δA µ = i (cid:15)γ µ λ − ˜ λγ µ (cid:15) ) δσ = 12 (˜ (cid:15)λ − ˜ λ(cid:15) ) δλ = (cid:18) − F µν γ µν − D + i ( D µ σ ) γ µ + 2 i σγ µ D µ (cid:19) (cid:15)δ ˜ λ = (cid:18) − F µν γ µν + D − i ( D µ σ ) γ µ − i σγ µ D µ (cid:19) ˜ (cid:15)δD = − i (cid:16) ˜ (cid:15)γ µ D µ λ − ( D µ ˜ λ ) γ µ (cid:15) (cid:17) + i (cid:16) [˜ (cid:15)λ, σ ] − [˜ λ(cid:15), σ ] (cid:17) − i (cid:16) ˜ λγ µ D µ (cid:15) + ( D µ ˜ (cid:15) ) γ µ λ (cid:17) (4.160)where now D µ = ∇ µ − iA µ is the gauge-covariant derivative. On the 3-sphere, the actions in(4.106) and (4.107) acquire a factor √ g in the measure, and the Super Yang-Mills Lagrangiangets modified to L Y M = Tr (cid:26) i λγ µ D µ λ + 14 F µν F µν + 12 D µ σD µ σ + i λ [ σ, λ ] + 12 (cid:16) D + σl (cid:17) − l ˜ λλ (cid:27) (4.161)where we reinstated the radius l , to see that indeed in the limit l → ∞ this becomes the standardEuclidean SYM theory in 3 dimensions. We note an important feature of this Lagrangian, thatwill be important for the application of the localization principle: this can be written as asupersymmetry variation, i.e. ˜ (cid:15)(cid:15) L Y M = δ ˜ (cid:15) δ (cid:15) Tr (cid:26)
12 ˜ λλ − Dσ (cid:27) . (4.162)The SCS Lagrangian does not get modified on curved space, since the term depending on thegauge field is topological, and the other ones do not contain derivatives.It is important to remark that, in general, unbroken supersymmetry is consistent only withAnti-de Sitter geometry (or, in Euclidean signature, hyperbolic geometry) [54]. An exceptionto this is given by those theories that possess a larger group of symmetries, the superconformal group. This is an extension of the super-Poincar´e group, to include also conformal transfor-mations of the metric. In this case, supersymmetry can be consistent also on conformally flatbackgrounds with positive scalar curvature, of which the n -spheres are an example. The N = 2SCS theory of above is an example of superconformal theory. The pure Chern-Simons term (cid:0) A ∧ dA + i A (cid:1) is actually unmodified, being already a 3-form. N = 4 , , ∗ gauge theories on the round 4-sphere We continue also the example of the N = 4 4-dimensional theory, understanding how it canbe realized on a different background compatible with S , and what part of the supersymmetryalgebra can be preserved on this background. As in Section 4.3.7, the N = 2 and N = 2 ∗ casesfollow from modifications of the N = 4 theory.Using stereographic coordinates x , · · · , x , such that the North pole is located at x µ = 0,the round metric of the 4-sphere of radius r looks explicitly as a conformal transformation ofthe flat Euclidean metric, g ( x ) µν = δ µν e x ) , where e x ) = 1 (cid:0) x r (cid:1) (4.163)where x = (cid:80) µ =1 ( x µ ) . As remarked at the end of the last section, the conformal flatness of S allows us to deform the superconformal YM theory on it, provided we preserve the conformalsymmetry. In order to do this, we modify the kinetic term of the scalars (Φ A ) A =5 , ··· , , addinga conformal coupling to the curvature:( ∂ Φ A ) → ( ∂ Φ A ) + R A ) where R = r is the scalar curvature of the metric g . This ensures conformal invariance ofthe action on the 4-sphere, S N =4 S = (cid:90) S d x √ g g Y M Tr (cid:18) F MN F MN − ΨΓ M D M Ψ + 2 r Φ A Φ A (cid:19) (4.164)where the derivatives have been promoted to covariant derivatives with respect to the Levi-Civita connection of g .Now we have to understand which supersymmetries of the N = 4 algebra can be preservedon the new curved background. From theorem 4.4.1, we know that a necessary condition for asection (cid:15) of the Majorana-Weyl spinor bundle on S , to produce a superisometry for the newbackground, is that it satisfies the twistor spinor equation , or conformal Killing equation ∇ µ (cid:15) = ˜Γ µ ˜ (cid:15) (4.165)for some other section ˜ (cid:15) . It can be checked that, to ensure supersymmetry of (4.164), ˜ (cid:15) mustbe also a twistor spinor satisfying ∇ µ ˜ (cid:15) = − r Γ µ (cid:15), (4.166)and the variations (4.109) have to be modified as the superconformal transformations δ (cid:15) A M = (cid:15) Γ M Ψ δ (cid:15) Ψ = 12 Γ MN F MN (cid:15) + 12 Γ µA Ψ A ∇ µ (cid:15). (4.167)Since S is conformally flat, the number of solutions to (4.165) is maximal and equal to2 dim ( S ± ) = 32 [53], so the whole N = 4 superconformal algebra is preserved. If one restricts In d -dimensions, the conformal coupling to the curvature scalar is made adding a term ξR (Φ) for the scalarfield of canonical mass dimension d − , with ξ = ( d − / d −
1) (see [65], Appendix D). The scalar curvatureof the d -sphere of radius r is R = d ( d − /r . The number of generators of the N = 4 super-Euclidean algebra is dim ( S ± ) = 16. The other 16 are theremaining generators of the superconformal algebra. hapter 4. Supergeometry and supersymmetry 81the attention to the N = 2 subalgebra, then half of the generators are preserved. If insteadthe N = 2 ∗ theory is considered, the conformal symmetry is broken and only 8 superchargesare preserved on S . With the above modifications, the N = 4 superconformal algebra closesagain only on-shell: imposing the EoM for Ψ, one gets (see Appendix of [11] for the details ofthe computation) δ (cid:15) = −L v − G Φ − R − Ω (4.168)as (4.117).In order to prepare the ground for the exploitation of the localization principle on supersym-metric gauge theories, we remark that, if we want to define correctly an equivariant structurewith respect to (at least a U (1) subgroup of) the Poincar´e group, we need at least an N = 1supersymmetry subalgebra to close properly ( i.e. off-shell). If this is the case, we can use thecorresponding variation δ (cid:15) as a Cartan differential with respect to this equivariant cohomology(we are going to justify better this in the next chapter). It is not possible to close off-shell thefull N = 2 algebra on the hypermultiplet, but fixing a conformal Killing spinor (cid:15) satisfying(4.165) and (4.166), it is possible to close the subalgebra generated by δ (cid:15) only. To do this,one has to add auxiliary fields to match the number of off-shell bosonic/fermionic degrees offreedom of the theory [66], analogously to what happens for example to the N = 1 vector mul-tiplet in 4-dimensions. In 10-dimensions, we have 16 real fermionic components, and (10 − K i ) i =1 , ··· , . Themodified action S N =4 S = (cid:90) S d x √ g g Y M Tr (cid:32) F MN F MN − ΨΓ M D M Ψ + 2 r Φ A Φ A − (cid:88) i =1 K i K i (cid:33) (4.169)is supersymmetric under the modified N = 4 superconformal transformations δ (cid:15) A M = (cid:15) Γ M Ψ δ (cid:15) Ψ = 12 Γ MN F MN (cid:15) + 12 Γ µA Ψ A ∇ µ (cid:15) + (cid:88) i =1 K i ν i δ (cid:15) K i = − ν i Γ M D M Ψ . (4.170)Here (cid:15) is a fixed conformal Killing spinor, and ( ν i ) are seven spinors satisfying (cid:15) Γ M ν i = 0( (cid:15) Γ M (cid:15) )˜Γ Mab = 2 (cid:32)(cid:88) i ( ν i ) a ( ν i ) b + (cid:15) a (cid:15) b (cid:33) ν i Γ M ν j = δ ij (cid:15) Γ M (cid:15). (4.171)To ensure convergence of the path integral, as we did with the scalar field Φ , we path integratethe new auxiliary scalars on purely immaginary values, i.e. K j =: iK Ej with K Ej real. For everyfixed non-zero (cid:15) , there exist seven linearly independent ν i satisfying these constraints, up toan SO (7) internal rotation, ensuring the closure (4.117) off-shell. Although, if we want δ (cid:15) todescribe the equivariant cohomology of a subgroup of the Poincar´e group (not the conformalone), we should restrict to those (cid:15) that generates only translations and R-symmetries at most(up to unphysical gauge transformations). Thus the dilatation term in (4.117) must vanish,imposing the condition ( (cid:15) ˜ (cid:15) ) = 0 on the conformal Killing spinors.2 4.4. From flat to curved spaceWe describe now which modifications to the above discussion have to be made in orderto describe the N = 2 and N = 2 ∗ theories. The pure N = 2 is classically obtained byrestricting to the N = 2 supersymmetry algebra generated by (4.118) and putting all the fieldsof the hypermultiplet to zero. At quantum level, this theory breaks in general the conformalinvariance, so it is equivalent to consider the N = 2 ∗ with hypermultiplet masses introducedas at the end of Section 4.3.7, by D (cid:55)→ D + M where M is an SU (2) R R mass matrix. The mass terms for the fermions break the SO (1 , R R-symmetry, so we must restrict the superconformal algebra further to those (cid:15) for which thecorresponding piece of the R-symmetry in (4.117) vanish. This imposes (˜ (cid:15) Γ (cid:15) ) = 0. Moreover,this deformed theory is not invariant under the N = 2 supersymmetry, because of the non-triviality of the conformal Killing spinor. In fact, using the conformal Killing equations itresults that δ (cid:15) (cid:18) F MN F MN − ΨΓ M D M Ψ + 2 r Φ A Φ A (cid:19) = − i ˜Γ ˜ (cid:15)M ji Φ j (4.172)where i, j = 5 , · · · ,
8, up to a total derivative. If (cid:15), ˜ (cid:15) are restricted to the the +1 eigenspaceof Γ , we write ˜ (cid:15) = r Λ (cid:15) , where Λ is a generator of SU (2) R L . Explicitly Λ = Γ ij R ij ,with components ( R ij ) normalized such that R ij R ij = 4. Then, after some gamma matrixtechnology, (4.172) gives δ (cid:15) ( · · · ) = 12 r (ΨΓ i (cid:15) ) R ik M jk Φ j = 12 r ( δ (cid:15) Φ i ) R ik M jk Φ j . (4.173)Hence, we can modify further the mass-deformed action to get invariance with respect to thissubalgebra of the original superconformal algebra on S , adding the new term − r ( R ki M jk )Φ i Φ j . (4.174)Finally, the action S N =2 ∗ S = (cid:90) S d x √ g g Y M Tr (cid:32) F MN F MN − ΨΓ M D M Ψ + 2 r Φ A Φ A −− r ( R ki M jk )Φ i Φ j − (cid:88) i =1 K i K i (cid:33) (4.175)where D Φ i (cid:55)→ [Φ , Φ i ]+ M ji Φ j and D Ψ (cid:55)→ [Φ , Ψ]+ M ij Γ ij Ψ, is invariant under the subalgebragenerated by a fixed conformal Killing spinor satisfying the conditionsΓ (cid:15) = (cid:15), ∇ µ (cid:15) = 18 r Γ µ Γ ij R ij (cid:15). (4.176) Another method that was extensively used in the physics literature to promote a supersym-metric theory on curved spaces is based on a trial and error procedure [67]. Suppose to have asupersymmetric QFT formulated in terms of component fields on flat Minkowski (or Euclidean)space R d , specified by the Lagrangian density L (0) invariant under the supersymmetry varia-tion δ (0) . The starting point of this approach is to simply “covariantize” the original theory,hapter 4. Supergeometry and supersymmetry 83replacing the flat metric η to the desired metric g defined on M and every derivative ∂ µ withthe appropriate Levi-Civita or spin covariant derivative ∇ µ corresponding to g . The problemis that in general this does define the theory on the curved space, but it is not guaranteed thatthe supersymmetry survives: (cid:2) δ (0) L (0) (cid:3) ( η,d ) → ( g, ∇ ) (cid:54) = ∇ µ ( · · · ) µ . (4.177)The idea then is to correct the action of the supersymmetry and the Lagrangian with anexpansion in powers of 1 /r , where r is a characteristic length of M , δ = δ (0) + (cid:88) i ≥ r i δ ( i ) L = L (0) + (cid:88) i ≥ r i L ( i ) (4.178)requiring order by order the symmetry of the Lagrangian and the closure of the super-algebra.This “trial and error” terminates if one is able to ensure both conditions at some finite order in1 /r , even though a priori the series contains an infinite number of terms. This procedure hasthe quality to be simple and operational in principle, but can be very cumbersome in practiceto apply. In gauge theories, BRST cohomology is a useful device to provide an algebraic description ofthe path integral quantization procedure, and the renormalizability of non-Abelian Yang-Millstheory in 4 dimensions. This formalism makes use of Lie algebra cohomology, while the BRSTmodel of Section 2.5 describes equivariant cohomology, that is what we use in topological orsupersymmetric field theories. It is natural to ask whether there is a relation between thesetwo cohomology theories, and in fact there is. It turns out that equivariant cohomology of aLie algebra g is the same as a “supersymmetrized” Lie algebra cohomology of a correspondinggraded Lie algebra g [ (cid:15) ] := g ⊗ (cid:86) (cid:15) [20].Let us first see how the Weil model W ( g ) = S ( g ∗ ) ⊗ (cid:94) ( g ∗ ) (4.179)for the equivariant cohomology of g can be seen in more supergeometric terms. Notice that thespace S ( g ∗ ) may be identified with the (commutative) algebra of functions on the Lie algebra g , and thus us we can see the Weil algebra W ( g ∗ ) as the space of functions on a supermanifoldbuilt from the tangent bundle of g , that is exactly the odd tangent bundle Π T g ≡ Π g . Denoting { ˜ c i } and { c i } the generators of S ( g ∗ ) and (cid:86) ( g ∗ ) respectively, indeed a function onthis superspace is trivialized asΦ = Φ (0) (˜ c ) + Φ (1) j (˜ c ) c j + Φ (2) jk (˜ c ) c j c k + · · · (4.180) Being M compact, we can take it as an embedding in R n for some n , and scale the metric according tosome characteristic length r . Notice that since g is a vector space, T ∗ g ∼ = g ∗ . { ˜ b i } and { b i } of g [1] and g , such that b i ( c j ) := c j ( b i ) = δ ji , ˜ b i (˜ c j ) := c j ( b i ) = δ ji , (4.181)the Weil differential (2.29) can be written as d W = ˜ c i b i + f ijk c j ˜ c k b i − f ijk c j c k ˜ b i , (4.182)that is very reminiscent of the form of a “BRST operator”.In Lie algebra cohomology, the Chevalley-Eilenberg differential on (cid:86) ( g ∗ ) is defined on 1-forms α ∈ g ∗ as δα = − α ([ · , · ]) = α i (cid:18) − f ijk c j c k (cid:19) , (4.183)and then extended as an antiderivation on the whole complex. If we consider a g -module V , such as the target space of a given field theory of gauge group G , with a representation ρ : g → End( V ), then the CE differential is extended to (cid:86) ( g ∗ ) ⊗ V as δv ( X ) := ρ ( X ) v ∀ v ∈ V, X ∈ g δ ( α ⊗ v ) = δα ⊗ v + ( − k α ⊗ δv ∀ v ∈ V, α ∈ (cid:94) k ( g ∗ ) . (4.184)This, expressed with respect to a basis { b i } of g , coincide with the BRST operator δ = c i ρ ( b i ) − f ijk c j c k b i (4.185)that satisfies δ = 0. The c i are ghosts , while the b i are anti-ghosts . The zero-th cohomologygroup of the complex (cid:86) ( g ∗ ) ⊗ V with respect to the differential (4.185) contains those statesthat have ghost number g -invariant, H ( g , V ) ∼ = V g (4.186)so the interesting “physical” states.We see that there is a difference between the BRST operator (4.185) and the Weil differential(4.182), but we can connect these differentials as follows. To the Lie algebra g we can associatethe differential graded Lie algebra g [ (cid:15) ] := g ⊗ (cid:86) (cid:15) . Here (cid:15) is a single generator taken in odddegree, deg( (cid:15) ) := −
1, while deg( g ) := 0. A differential ∂ : g [ (cid:15) ] → g [ (cid:15) ] is defined as ∂(cid:15) := 1 ∈ g , ∂X := 0 ∀ X ∈ g . (4.187)This superalgebra has generators b i := b i ⊗ b i := b i ⊗ (cid:15) , and the superbracket structurecoming from the Lie brackets on g and the (trivial) wedge product on (cid:86) (cid:15) :[ b i , b j ] = f kij b k [ b i , ˜ b j ] = f kij ˜ b k [˜ b i , ˜ b j ] = 0 (4.188) Sometimes the action of this generators is denoted as a graded bracket structure, like [ b i , c j ] = [˜ b i , ˜ c j ] + = δ ji . If V is a field space C ∞ ( M ) over some (super)manifold M , g acts as usual as a Lie derivative with respectto the fundamental vector field, ρ ( X ) = L X . hapter 4. Supergeometry and supersymmetry 85making it into a Lie superalgebra. The differential on the generators is rewritten as ∂b i = 0 , ∂ ˜ b i = b i . (4.189)To this “supersymmetrized” algebra we can associate the Lie algebra cohomology with respectto the complex (cid:86) ( g [ (cid:15) ] ∗ ), that is generated by { c i , ˜ c i } of degrees deg( c i ) = 1 , deg(˜ c i ) = 2, suchthat c i ( b j ) = ˜ c i (˜ b j ) = δ ij . (4.190)The BRST differential for the g [ (cid:15) ]-Lie algebra cohomology is naturally defined analogously tobefore as Qα := − α ([ · , · ]) (4.191)on 1-forms α ∈ g [ (cid:15) ] ∗ . But now, because of the Lie algebra extension (4.188), its expression interms of the generators results Q = − f kij c i c j b k + f kij c i ˜ c j ˜ b k . (4.192)Moreover, the dual ∂ ∗ acts as ∂ ∗ = ˜ c i b i (4.193)with b i acting as in (4.181). Then the total differential on this complex coincides with the Weildifferential, d W ∼ = ∂ ∗ + Q (4.194)and we identify an isomorphism of dg algebras (cid:16)(cid:94) ( g [ (cid:15) ] ∗ ) , ∂ ∗ + Q (cid:17) ∼ = ( W ( g ) , d W ) . (4.195)If we bring into the game the g -module V as before, we can work with Ω( V ) as a g [ (cid:15) ]-dgalgebra, defining the g [ (cid:15) ] action as ( X ⊗ → L X , ( X ⊗ (cid:15) ) → ι X . (4.196)On the complex (cid:86) ( g [ (cid:15) ] ∗ ) ⊗ Ω( V ), the total differential inherited from the Weil differential andthe g [ (cid:15) ]-action is d B := c k ⊗ L k + ˜ c k ⊗ ι k + Q ⊗ ∂ ∗ ⊗ ⊗ d (4.197)and it coincides with the one of the BRST model of equivariant cohomology (2.51)!This demonstrates how the BRST quantization formalism and equivariant cohomology areintimately related, and suggests that indeed BRST symmetry operators are good candidates torepresent equivariant differentials in QFT, that can be used to employ the localization principlein these kind of physical systems. Again, if V = C ∞ ( M ), then Ω( M ) is the space we considered when we constructed the equivariantcohomology of a G -manifold. hapter 5Localization for circle actions insupersymmetric QFT In this chapter we describe how the equivariant localization principle can be carried out inthe infinite dimensional case of path integrals in QM or QFT. In this setting, the first objectof interest is the partition function Z = (cid:90) F Dφ e iS [ φ ] (5.1)where F = Γ( M, E ) is the space of fields, i.e. sections of some fiber bundle E → M with typicalfiber (the target space ) V over the (Lorentzian) n -dimensional spacetime M , and S ∈ C ∞ ( F ) isthe action functional. The fields are supposed to satisfy some prescribed boundary conditionson ∂M . In the Riemannian case, the corresponding object has the form Z = (cid:90) F Dφ e − S E [ φ ] (5.2)where we denoted S E the Euclidean action. If the spacetime has the form M = R t × Σ, this lastexpression can be reached from the Lorentzian theory via
Wick rotation of the time direction t (cid:55)→ τ := it . If the τ direction is compactified to a circle of length T , we can interpret theEuclidean path integral as the canonical ensemble partition function describing the originalQFT at a finite temperature 1 /T . If needed, we are always free to set the length T of thecircle to be very large, and find the zero temperature limit when T → ∞ . Given an observable O ∈ C ∞ ( M ), its expectation value is given by (cid:104)O(cid:105) = 1 Z (cid:90) F Dφ O [ φ ] e iS [ φ ] or (cid:104)O(cid:105) E = 1 Z (cid:90) F Dφ O [ φ ] e − S E [ φ ] . (5.3)The path integral measure Dφ on the infinite dimensional space F is not rigorously defined, but it is usually introduced as (cid:90) Dφ = N (cid:89) x ∈ M (cid:90) V dφ ( x ) (5.4) In the (common) case of a trivial bundle, this is equivalent to considering F = C ∞ ( M, V ), i.e. V -valuedfunctions over M . Often V is a vector space, otherwise the theory describes a so-called non-linear σ -model .In supersymmetric field theories, V = (cid:86) ( S ∗ ) for some vector space S , and the field space acquires a naturalgraded structure. In fact, it does not exist in general. N is some (possibly infinite) multiplicative factor, and at every point x ∈ M we havea standard integral over the fiber V . Notice that the infinite factors N cancel in ratios inthe computations of expectation values, so we can still make sense of such objects and for-mally manipulate them to obtain physical information. Another convergence issue comes withthe prescription of boundary conditions in computing the action S [ φ ]. If M is non compact,this often requires a specific regularization, while taking compact spacetimes ensure betterconvergence properties.Very few quantum systems have an exactly solvable path integral. When this functionalintegral method was introduced, the only examples where (5.1) could be directly evaluated werethe free particle and the harmonic oscillator. Both these theories are quadratic in the fieldsand their derivatives, thus the partition function can be computed using the formal functionalanalog of the classical Gaussian integration formula (cid:90) ∞−∞ d n x e − x k M kl x l + A k x k = (2 π ) n/ √ det M e A i ( M − ) ij A j (5.5)where M = [ M ij ] is an n × n non singular matrix. The analogue in field theory has n → ∞ and a functional determinant at the denominator, that must be properly regularized in orderto be a meaningful convergent quantity (see any standard QFT book, like [69, 70]).In perturbative QFT, one almost never has to explicitly perform such a functional integra-tion. Suppose that the action has the generic form S = S + S int (5.6)where S is the free term containing up to quadratic powers of the fields and their derivatives,and the rest is collected in S int . Then the expectation value of an observable O , expressibleas a combination of local fields, is computed expanding the exponential of the interacting partin Taylor series, and exploiting Wick’s theorem for the vacuum expectation values in the freetheory: (cid:104)O(cid:105) = (cid:88) k k ! (cid:10) ( iS int ) k O (cid:11) . (5.7)Another perturbative approach, especially useful to compute the effective action in a giventheory, is the so-called background field method , where the action S is expanded around aclassical “background”element φ ∈ F , S [ φ + η ] = S [ φ ] + (cid:90) M d n x (cid:18) δSδφ ( x ) (cid:19) φ η ( x ) + 12 (cid:90) M d n xd n y (cid:18) δ (2) Sδφ ( x ) δφ ( y ) (cid:19) φ η ( x ) η ( y ) + · · · (5.8)If we chose φ to be a solution of the classical equation of motion (cid:16) δSδφ ( x ) (cid:17) φ = 0, the first orderterm disappears from the expansion. If we also neglect the terms of order higher than quadraticin η , and substitute the resulting expression in (5.1) or (5.2), we get the equivalent of the saddlepoint approximation, or “one-loop approximation” of the partition function Z ≈ e − S [ φ ] Z − loop [ φ ] , (5.9) For example, one can first assume that the spacetime just extends up to some large but finite typical lenght r , and then send this value to infinity at the end of the calculations. Later on, ad hoc methods for other particular systems were developed, like the solution of the Hydrogenatom by Duru and Kleinert [68], and others. Again, see any standard QFT book. hapter 5. Localization for circle actions in supersymmetric QFT 89where Z − loop [ φ ] := (cid:90) F Dη e − η · ∆[ φ ] · η ≡ [det (∆[ φ ])] − / ∆[ φ ]( x, y ) := (cid:18) δ (2) Sδφ ( x ) δφ ( y ) (cid:19) φ , (5.10)and we denoted convolution products over M with ( · ) for brevity.We are interested in those cases in which such a “semiclassical” approximation of the par-tition function turns out to give the exact result for the path integral in the full quantumtheory. This is possible if some symmetry of the field theory, i.e. acting on the space F , allowsus to formally employ the equivariant localization principle and reduce the path integrationdomain from F to a (possibly finite-dimensional) subspace. In the next part of the chapterwe will see some cases in which it is possible to interpret F , or a suitable extension of it, asa Cartan model with some (super)symmetry operator acting as the Cartan differential. As wealready anticipated, this is possible if F has a graded structure that can both arise from thesupersymmetry of the underlying spacetime, or can be introduced via an extension analogousto what happens in the BRST formalism.We will first describe the application of the Duistermaat-Heckman theorem in the caseof Hamiltonian QM, where the equivariant structure can be constructed from the symplecticstructure of the underlying theory. Then we will be concerned with more general applications ofthe localization principle in supersymmetric QFT, where the super-Poincar´e group action allowsfor an equivariant cohomological interpretation. In both frameworks, we present examples oflocalization under the action of a single supersymmetry, whose “square” generates a bosonic U (1) symmetry. Supersymmetric localization was recently applied to many cases of QFT oncurved spacetimes, so we must consider those curved background that preserve at least onesupersymmetry, as discussed in Section 4.4. We consider now, following [19] and refernces therein, the path integral quantization of anHamiltonian system (
M, ω, H ), of the 2 n -dimensional phase space M with symplectic form ω ,and an Hamiltonian function H ∈ C ∞ ( M ). This is simply QM viewed as a (0+1)-dimensionalQFT over the base “spacetime” R or S , that now is only “time”, and with target space M ,that physically represents the phase space of the system. The fields of the theory are thepaths γ : R ( S ) → M , that means we consider a trivial total space E = R ( S ) × M . Inprinciple the time axis can be chosen to be the real line (or an interval with some prescribedboundary conditions), or the circle (that corresponds to periodic boundary conditions), but wewill soon see that it is much convenient technically to chose the latter possibility, so considerthe “loop space” F = C ∞ ( S , M ). A field for us is so a closed curve γ : [0 , T ] → M such that γ (0) = γ ( T ). Since we make the periodicity explicit in t , we interpret this parameter as an“Euclidean” Wick-rotated time, so that T is the inverse temperature of the canonical ensemble.We set up now some differential geometric concept on the loop space that we are going touse in the following. If { x µ } are coordinates on M , on the loop space we can choose an infiniteset of coordinates { x µ ( t ) } for µ ∈ , · · · , n and t ∈ [0 , T ], such that for any γ ∈ F x µ ( t )[ γ ] := x µ ( γ ( t )) . T F ) can be thus expressedlocally with respect to these coordinates as X = (cid:90) T dt X µ ( t ) δδx µ ( t ) (5.11)where X µ ( t ) are functions over F , and ( δ/δx µ ( t )) γ is a basis element of the tangent space T γ F at γ . Many other geometric objects can be lifted from M to F following this philosophy. Forexample, for any function in C ∞ ( M ) as the Hamiltonian H , we can define H ( t ) ∈ C ∞ ( F ) at agiven time t , such that H ( t )[ γ ] := H ( γ ( t )). The action functional instead is the function overthe loop space such that S [ γ ] = (cid:90) T dt [ ˙ q a ( t ) p a ( t ) − H ( p ( t ) , q ( t ))]= (cid:90) T dt (cid:2) θ γ ( t ) ( ˙ γ ) − H ( γ ( t )) (cid:3) (5.12)where in the first line we expressed γ through its trivialization in Darboux coordinates { q a , p a } with a ∈ { , · · · , n } , and in the second line we expressed the same thing more covariantly usingthe (local) symplectic potential θ of ω and the velocity vector field ˙ γ of the curve. Concerningdifferential forms, if we consider the basis set { η µ ( t ) := dx µ ( t ) } , a k -degree element of Ω( F )can be expressed locally as α = (cid:90) T dt · · · (cid:90) T dt k k ! α µ ··· µ k ( t , · · · , t k ) η µ ( t ) ∧ · · · ∧ η µ k ( t k ) (5.13)and we recall that Ω( F ) can be considered as the space of functions over the super-loop space Π T F , of coordinates { x µ ( t ) , η µ ( t ) } . The de Rham differential on the loop space can be expressedas the cohomological vector field on Π T F d F = (cid:90) T dt η µ ( t ) δδx µ ( t ) . (5.14)Finally, it is natural to lift on the loop space the symplectic structure of M , as well as a choiceof Riemannian metric, as Ω = (cid:90) T dt ω µν ( t ) η µ ( t ) ∧ η ν ( t ) G = (cid:90) T dt g µν ( t ) η µ ( t ) ⊗ η ν ( t ) , (5.15) i.e. Ω µν ( t, t (cid:48) ) := ω µν ( t ) δ ( t − t (cid:48) ) and G µν := g µν ( t ) δ ( t − t (cid:48) ). Ω is closed under the loop spacedifferential d F . Considering the standard Liouville measure on Mω n n ! = d n x Pf( ω ( x )) = d n qd n p, (5.16) Strictly speaking, the 2-form Ω should be called “pre-symplectic”, since although it is certainly closed, it isnot necessarily non-degenerate on the loop space. hapter 5. Localization for circle actions in supersymmetric QFT 91we can write now the path integral measure for QM on the loop space as an infinite product ofthe Liouville one for any time t ∈ [0 , T ], and get (cid:90) F Ω n n ! = (cid:90) F D n x Pf(Ω[ x ]) = (cid:90) Π T F D n xD n η Ω n n ! . (5.17)Here in the last equality we rewrote the integral over F as an integral over the super-loop space,analogously to (4.16). The path integral for the quantum partition function is thus Z ( T ) = (cid:90) F D n x Pf(Ω[ x ]) e − S [ x ] = (cid:90) Π T F D n xD n η Ω n [ x, η ] n ! e − S [ x ] = (cid:90) Π T F D n xD n η e − ( S [ x ]+Ω[ x,η ]) (5.18)where in the last line we exponentiated the loop space symplectic form, making explicit theformal analogy with the Duistermaat-Heckman setup. In particular, we associated to the“classical” Hamiltonian system ( M, ω, H ) an Hamiltonian system ( F , Ω , S ) on the loop space.Here the loop space Hamiltonian function S generates an Hamiltonian U (1)-action on F , thatcan be used to formally apply the same equivariant localization principle as in the finite-dimensional case. As for the proof of the ABBV formula, we introduced a graded structure onfield space formally rewriting the path integration on the super-loop space Π T F . This gradedstructure is now simply given by the form-degree on the extended field space Ω( F ).Let now X S be the Hamiltonian vector field associated to S ∈ C ∞ ( F ), such that d F S = − ι X S Ω, or equivalently X S = Ω( · , d F S ). Explicitly, in coordinates { x µ ( t ) } X µS ( t ) = (cid:90) T dt (cid:48) Ω µν ( t, t (cid:48) ) δSδx ν ( t )= ω µν ( x ( t )) (cid:0) ω νρ ( x ( t )) ˙ x ρ ( t ) − ∂ ν H ( x ( t )) (cid:1) = ˙ x µ ( t ) − X µH ( x ( t )) (5.19)where ˙ x ( t ) is the vector field on F with components such that ˙ x µ ( t )[ γ ] := ( x µ ◦ γ ) (cid:48) ( t ) ≡ ˙ γ µ ( t ).The flow of X S defines the Hamiltonian U (1)-action on F and the infinitesimal action of theLie algebra u (1) on Ω( F ) through the Lie derivative L X S . The Cartan model for the U (1)-equivariant cohomology of F is then defined by the space of equivariant differential formsΩ S ( F ) := ( R [ φ ] ⊗ Ω( F )) U (1) ∼ = Ω( F ) U (1) [ φ ] (5.20)and the equivariant differential Q S := ⊗ d F − φ ⊗ ι X S ≡ d F + ι X S = (cid:90) T dt (cid:0) η µ ( t ) + ˙ x µ ( t ) − X µH ( x ( t )) (cid:1) δδx µ ( t ) , (5.21)where as usual we localized algebraically setting φ = − Q S = (cid:90) T dt (cid:18) ddt − L X H | x ( t ) (cid:19) (5.22)2 5.1. Localization principle in Hamiltonian QMwhere the second term is the Lie derivative on M with respect to X H , lifted on F at everyvalue of t . The first term, when evaluated on a field, gives only contributions from the values at t = 0 , T , and so it vanishes thanks to the fact that we chose periodic boundary conditions! Thismeans that, with this choice, the Cartan model on field space is completely determined by thelift of the U (1)-invariant forms on M , for which L X H α = 0. Consistently, if we restrict to thissubspace of Ω( M ) where the energy is preserved, Q S ≡ Q ˙ x = d F + ι ˙ x acts as the supersymmetry operator generating time-translations on the base S : Q x = 12 [ Q ˙ x , Q ˙ x ] = (cid:90) T dt ddt (5.23)resembling the supersymmetry algebra (4.44) for N = 1 and 1-dimensional spacetime. We willsee in the next section that this is not just a coincidence, but we can relate this model to asupersymmetric version of QM. This restricted differential acts on coordinates of the super-loopspace as Q ˙ x x µ ( t ) = η µ ( t ) , Q ˙ x η µ ( t ) = ˙ x µ ( t ) , (5.24)while the full equivariant differential acts as Q S x µ ( t ) = η µ ( t ) , Q S η µ ( t ) = X µS ( t ) , (5.25)both exchanging “bosonic” with “fermionic” degrees of freedom.We remark that we started from a standard (non supersymmetric) Hamiltonian theory on F , and from this we constructed a supersymmetric theory on Π T F , whose supersymmetry isencoded in the Hamiltonian symmetry (so in the symplectic structure) of the original theory.This “hidden” supersymmetry is thus interpretable, in the spirit of Topological Field Theory,as a BRST symmetry, and the differential Q S as a “BRST charge” under which the augmentedaction ( S + Ω) is supersymmetric: Q S ( S + Ω) = d F S + d F Ω + ι X S Ω = d F S + 0 − d F S = 0 . (5.26)In other words, ( S + Ω) is an equivariantly closed extension of the symplectic 2-form Ω, anal-ogously to the finite-dimensional Hamiltonian geometry discussed in Chapter 3.3. The sameargument cannot be straightforwardly applied to any QFT, since in general we do not have asymplectic structure on the field space, but in the presence of a gauge symmetry we know thata BRST supersymmetry can be used to define the equivariant cohomology on the field spaceand exploit the localization principle. We will expand this a bit in the next chapter.It is now possible to mimic the procedure of Section 4.2 to localize the supersymmetric pathintegral Z ( T ) = (cid:90) Π T F D n xD n η e − ( S [ x ]+Ω[ x,η ]) (5.27)seen as an integral of an equivariantly closed form. We modify the integral introducing the“localizing action” S loc [ x, η ] := Q S Ψ[ x, η ], with localization 1-form Ψ ∈ Ω ( F ) U (1) , the so-called“gauge-fixing fermion”: Z T ( λ ) = (cid:90) Π T F D n xD n η e − ( S [ x ]+Ω[ x,η ] − λQ S Ψ[ x,η ]) (5.28) A symplectic structure can be induced from the action principle on the subspace of solutions of the classicalEoM, but it does not lift on the whole field space in general. hapter 5. Localization for circle actions in supersymmetric QFT 93where λ ∈ R is a parameter. The resulting integrand is again explicitly equivariantly closed,and we can check that this path integral is formally independent on the parameter λ . Indeed,after a shift λ (cid:55)→ λ + δλ , (5.28) becomes Z T ( λ + δλ ) = (cid:90) Π T F D n xD n η e − ( S [ x ]+Ω[ x,η ] − λQ S Ψ[ x,η ] − δλQ S Ψ[ x,η ]) , (5.29)and we can make a change of integration variables to absorb the resulting shift at the expo-nential. Since the exponential is supersymmetric, we change variables using a supersymmetrytransformation: x µ ( t ) (cid:55)→ x (cid:48) µ ( t ) := x µ ( t ) + δλ Ψ Q S x µ ( t ) = x µ ( t ) + δλη µ ( t ) η µ ( t ) (cid:55)→ η (cid:48) µ ( t ) := η µ ( t ) + δλ Ψ Q S η µ ( t ) = η µ ( t ) + δλX µS ( t ) (5.30)so that the only change in the integral comes from the integration measure, D n xD n η (cid:55)→ D n x (cid:48) D n η (cid:48) = Sdet (cid:20) ∂x (cid:48) /∂x ∂x (cid:48) /∂η∂η (cid:48) /∂x ∂η (cid:48) /∂η (cid:21) D n xD n η = e − δλQ S Ψ D n xD n η. (5.31)Putting all together, Z T ( λ + δλ ) = (cid:90) Π T F D n x (cid:48) D n η (cid:48) e − ( S [ x (cid:48) ]+Ω[ x (cid:48) ,η (cid:48) ] − λQ S Ψ[ x (cid:48) ,η (cid:48) ] − δλQ S Ψ[ x (cid:48) ,η (cid:48) ]) = (cid:90) Π T F D n xD n η e − ( S [ x ]+Ω[ x,η ] − λQ S Ψ[ x,η ]) = Z T ( λ ) . (5.32)The same property can be seen less rigorously by exploiting some sort of (arguable) infinite-dimensional version of Stokes’ theorem: ddλ Z T ( λ ) = (cid:90) Π T F D n xD n η ( Q S Ψ) e − ( S [ x ]+Ω[ x,η ] − λQ S Ψ[ x,η ]) = (cid:90) Π T F D n xD n η (cid:0) Q S Ψ e − ( S [ x ]+Ω[ x,η ] − λQ S Ψ[ x,η ]) (cid:1) = 0 , (5.33)that holds if we assume the path integration measure to be non-anomalous under Q S .Since the path integral (5.28) is independent on the parameter, we can take the limit λ → ∞ and obtain the localization formula Z ( T ) = lim λ →∞ (cid:90) Π T F D n xD n η e − ( S [ x ]+Ω[ x,η ] − λQ S Ψ[ x,η ]) (5.34)that “localizes” Z ( T ) onto the zero locus of S loc . Of course different choices of gauge-fixingfermion induce different final localization formulas for the path integral, but at the end theyshould all give the same result. We now present two different localization formulas derived from(5.34) with different choices of localizing term.The fist canonical choice of gauge fixing fermion we can make mimics the same procedurewe used in the finite-dimensional case. Under the same assumptions we made in Section 3.1,we pick a U (1) H -invariant metric g on M , and lift it to F using (5.15), so that the resulting G is U (1) S -invariant: L S G = Q S G = 0. Then the localization 1-form is taken to beΨ[ x, η ] := G ( X S , · ) = (cid:90) T dt g µν ( x ( t )) (cid:16) ˙ x µ ( t ) − X µH ( x ( t )) (cid:17) η ν ( t ) , (5.35)4 5.1. Localization principle in Hamiltonian QMso that the localization locus is the subspace where X S = 0, i.e. the moduli space of classicaltrajectories [41]: F S = (cid:40) γ ∈ F : (cid:18) δSδx µ ( t ) (cid:19) γ = 0 (cid:41) . (5.36)If this space consists of isolated, non-degenerate trajectories, we can apply the non-degenerateversion of the ABBV formula for a circle action, and get Z ( T ) = (cid:88) γ ∈F S Pf [ ω ( γ ( t ))] (cid:112) det [ dX S [ γ ] / π ] e − S [ γ ] (5.37)where the pfaffian and the determinant are understood in the functional sense, spanning boththe phase space indices µ ∈ { , · · · , n } and the time continuous index t ∈ [0 , T ]. In general,for non isolated classical trajectories we have to decompose any γ ∈ F near to the fixed pointset, splitting the classical component and normal fluctuations, as we did in Section 4.2. Then,rescaling the normal fluctuations by √ λ and thanks to the Berezin integration rules on thesuper-loop space, the same argument of the finite-dimensional case leads to Z ( T ) = (cid:90) F S D n x Pf [ ω ( x ( t ))]Pf [ δ µν ∂ t − ( B + R ) µν ( x ( t ))] | N F S e − S [ x ] (5.38)where B µν = g µρ ( ∇ [ ρ X H ) ν ] , while ∇ and R are the connection and curvature of the metric g on M , evaluated on F time-wise as usual. Notice that this localization scheme makes thecontribution from the classical configurations explicit, resembling the exactness of the saddlepoint approximation (5.9), with 1-loop determinant given by the pfaffian at the denominator.However, even if the integration domain has been reduced, one must still perform a difficultinfinite-dimensional path integration to get the final answer, whose T -dependence for examplelooks definitely non-trivial from (5.38).We can consider another choice of localizing term to simplify the final result, setting thegauge-fixing fermion to Ψ[ x, η ] := G ( ˙ x, · ) = (cid:90) T dt g µν ( x ( t )) ˙ x µ ( t ) η ν ( t ) . (5.39)With this choice, the gauge-fixed action reads S [ x ] + Ω[ x, η ] + λQ S Ψ[ x, η ] == (cid:90) T dt (cid:18) ˙ x µ θ µ − H + 12 ω µν η µ η n u + λ ( g µν,σ ˙ x µ η σ η ν + η µ ∂ t ( g µν η ν ) + g µν ˙ x µ ˙ x ν − g µν ˙ x µ X νH ) (cid:19) = (cid:90) T dt (cid:18) λg µν ˙ x µ ˙ x ν + λη µ ∇ t η ν + ˙ x µ θ µ + 12 ω µν η µ η ν − H − λg µν ˙ x µ X νH (cid:19) (5.40)where in the second line the time-covariant derivative acts as ∇ t η ν = ∂ t η ν + Γ νρσ ˙ x ρ η σ , and thelocalization locus is the subset of constant loops F := { γ ∈ F : ˙ γ = 0 } ∼ = M (5.41)that is, points in M . Splitting again the loops near this subspace in constant modes plusfluctuations, and rescaling the latter as we did before, the path integral is reduced to a finite-dimensional integral over M , the Niemi-Tirkkonen localization formula [71] Z ( T ) = (cid:90) Π T M √ gd n xd n η e − T ( H ( x ) − ω ( x,η )) (cid:112) det (cid:48) [ g µν ∂ t − ( B µν + R µν )] (5.42)hapter 5. Localization for circle actions in supersymmetric QFT 95where the prime on the determinant means it is taken over the normal fluctuation modes, ex-cluding the constant ones, giving exactly the equivariant Euler form of the normal bundle to F . This formula is much more appealing since it contains no refernce to T -dependent subman-ifolds of F , and the evaluation of the action on constant modes simply gives the Hamiltonianat those points multiplied by T . The functional determinant at the denominator requires aspecific regularization: using the ζ -function method, it can be simplified to1 (cid:112) det (cid:48) [ g µν ∂ t − ( B µν + R µν )] = (cid:118)(cid:117)(cid:117)(cid:116) det (cid:34) T ( B + R ) µν sinh (cid:0) T ( B + R ) µν (cid:1) (cid:35) = ˆ A H ( T R ) (5.43)where in the last equality we rewrote, by definition, the determinant as the U (1) H -equivariantDirac ˆ A -genus of the curvature R up to a constant T , that is the Dirac ˆ A -genus of the equiv-ariant extension of the curvature, R + B [72]. Note that the exponential can be rewritten asthe U (1) H -equivariant Chern character of the symplectic form, e − H + ω = ch H ( ω ) (5.44)and so the partition function can be nicely rewritten as Z ( T ) = (cid:90) M ch H ( T ω ) ∧ ˆ A H ( T R ) (5.45)in terms of equivariant characteristic classes of the phase space M with respect to the U (1) H Hamiltonian group action, that are determined by the initial classical system. The only remnantof the quantum theory is in the, now very explicit, dependence on the inverse temperature T .This form of the partition function emphasizes the fact that if we put H = 0, we end up witha topological theory. In this case there are no propagating physical degrees of freedom, and thepartition function only describes topological properties of the underlying phase space. In thenext section we will report a non-trivial example of this kind. A famous and important application of the localization principle to loop space path integralsis the supersymmetric derivation of the
Atiyah-Singer index theorem [73] [74] [75]. The theoremrelates the analytical index of an elliptic differential operator on a compact manifold to atopological invariant, connecting the local data associated to solutions of partial differentialequations to global properties of the manifold. Supersymmetric localization allowed to provethis statement, in a new way with respect to the original proof, for different examples of classicaldifferential operators. We describe here the application to the index of the Dirac operatoracting on the spinor bundle S on an even-dimensional compact manifold M , in presence of agravitational and electromagnetic background, specified by the metric g and a U (1) connection1-form A on a C -line bundle L C over M . The Dirac operator is defined as the fiber-wise Clifford action of the covariant derivative onthe twisted spinor bundle
T M ⊗ S ⊗ L C over M : i / ∇ = iγ µ (cid:18) ∂ µ + 18 ω µij [ γ i , γ j ] + iA µ (cid:19) (5.46) This can be extended to non-Abelian gauge groups as well, but for simplicity we report the Abelian case. { γ µ } are the gamma-matrices generating the Clifford algebra in the given spin represen-tation, satisfying the anticommutation relation { γ µ , γ ν } = 2 g µν , (5.47)and ω is the spin connection related to the metric. The “curved” and “flat” indices are relatedthrough the vielbein e iµ ( x ), g µν ( x ) = e iµ ( x ) e jν ( x ) η ij , γ i = e iµ ( x ) γ µ ( x ) , (5.48)with η the flat metric. The analytical index of the Dirac operator is defined as [76]index( i / ∇ ) := dim Ker( i / ∇ ) − dim coKer( i / ∇ ) = dim Ker( i / ∇ ) − dim Ker( i / ∇ † ) . (5.49)We have thus to find the number of zero-energy solutions of the Dirac equation i / ∇ Ψ = E Ψ , (5.50)where Ψ is a Dirac spinor. In even dimensions, we can decompose the problem in the chiralbasis of the spin representation, where γ i = (cid:18) σ i σ i (cid:19) , γ c = (cid:18) − (cid:19) , i / ∇ = (cid:18) DD † (cid:19) , Ψ = (cid:18) ψ − ψ + (cid:19) , (5.51)and the chirality matrix is denoted by γ c . In this representation, we see that the index countsthe number of zero-energy modes with positive chirality minus the number of zero-energy modesof negative chirality, index( i / ∇ ) = dim Ker( D ) − dim Ker( D † ) . (5.52)It is possible to give a path integral representation of this index, a key ingredient to applythe localization principle. In order to do that, we first prove that it can be rewritten as a Witten index , index( i / ∇ ) = Tr H (cid:0) γ c e − T ∆ (cid:1) (5.53)where ∆ := ( i / ∇ ) is the Shr¨oedinger operator (the covariant Laplacian) and the parameter T > H of Dirac spinors, sections ofthe twisted spinor bundle over M . Proof of (5.53) . First, we notice that ( i / ∇ ) is symmetric and elliptic, and since M is compactit is also essentially self-adjoint [77]. Thus, it has a well defined spectrum { Ψ E } that forms abasis of the function space at hand. The same modes diagonalize also the Schr¨odinger operator,∆Ψ E = E Ψ E (5.54)so we can shift the attention to solutions of the Schr¨odinger equation with eigenvalue satisfying E = 0. It is easy to see that the Dirac operator anticommutes with the chirality matrix γ c , and so the Schr¨odinger operator commutes with it. Thus we can split the field space incomplementary subspaces S E ± := { Ψ ∈ H : ∆Ψ = E Ψ , γ c Ψ = ± Ψ } , for every eigenvalue E and chirality ( ± ). For every non-zero energy, we establish an isomorphism S E + ∼ = S E − : using the See Appendix A.2. hapter 5. Localization for circle actions in supersymmetric QFT 97fact that ( i / ∇ ) and γ c anticommute, starting from a solution with eigenvalue E and definitechirality Ψ ± we can construct another one with opposite chirality ( i / ∇ Ψ ± ),∆( i / ∇ Ψ ± ) = i / ∇ i / ∇ i / ∇ Ψ ± = E ( i / ∇ Ψ ± ) γ c ( i / ∇ Ψ ± ) = − i / ∇ γ c Ψ ± = − ( ± )( i / ∇ Ψ ± ) . (5.55)Thus the maps φ ± : S E ± → S E ∓ such that φ ± (Ψ) := i / ∇| E | Ψ (5.56)are both well defined and are right and left inverse of each other, giving the bijective corre-spondence for every | E | (cid:54) = 0. This is the well known fact that particles and antiparticles arecreated in pairs with opposite energy and chirality.If we take the trace over the field space H , this is splitted into the sum of the traces overevery subspace S E of definite energy squared. In every one of them the restricted Witten indexgives Tr S E (cid:0) γ c e − T ∆ (cid:1) = e − T E (cid:16) Tr S E + ( ) − Tr S E − ( ) (cid:17) = 0 (5.57)for every E (cid:54) = 0. So the whole trace is formally independent on T , resolving on the subspaceof zero-energy, where the number of chirality + and - eigenstates is different in general:Tr H (cid:0) γ c e − T ∆ (cid:1) = Tr E =0 ( γ c )= E =0 (chirality (+) modes) − E =0 (chirality ( − ) modes) . (5.58) (cid:4) The Witten index representation (5.53) and the chirality decomposition of the operatorsof interest (5.51) permit to see the current problem as a N = 1 supersymmetric QM onthe manifold M , identifying chirality +(-) spinors with bosonic(fermionic) states. Here, thesupersymmetry algebra (4.57) is simply (choosing an appropriate normalization)[ Q, Q ] = 2 H, (5.59)and corresponds to the Schr¨odinger operator above, if we make the following identifications: i / ∇ ↔ Q ∆ = ( i / ∇ ) ↔ H = Q . (5.60)The chirality matrix γ c is identified with the operator ( − F , where F is the fermion numberoperator, that assigns eigenvalue +1 to bosonic states and − i / ∇ ) = Tr (cid:0) ( − F e − T H (cid:1) = n E =0 (bosons) − n E =0 (fermions) . (5.61)The proof above, translated in terms of the quantum system, shows that in supersymmetricQM eigenstates of the Hamiltonian have non-negative energy, and are present in fermion-boson Often this is called N = 1 / N = 1 for the complexified algebra withgenerators Q, ˜ Q , and imposition of Majorana condition. Q is a positive-definite Hermitian operator, the zeromodes | (cid:105) of H are supersymmetric, Q | (cid:105) = 0 (they do not have supersymmetric partners).Thus the non-vanishing of the Witten index (5.61) is a sufficient condition to ensure that thereis at least one supersymmetric vacuum state available, whereas its vanishing is a necessarycondition for spontaneous braking of supersymmetry by the vacuum.The Witten index has a super-loop space path integral representation [78], so that we canrewrite (5.61) as index( i / ∇ ) = (cid:90) D n φD n ψ e − T S [ φ,ψ ] (5.62)where S is the Euclidean action corresponding to the Hamiltonian H and the fields are definedon the unit circle. The appropriate supersymmetric theory which describes a spinning particleon a gravitational background is the 1-dimensional supersymmetric non-linear σ model . Thesuperspace formulation of this model considers the base space as a super extension of the1-dimensional spacetime with coordinates ( t, θ ), and M as the target space with covariantderivative given by the Dirac operator. A superfield, trivialized with respect to coordinates( t, θ ) and ( x µ ) on M is thenΦ µ ( t, θ ) ≡ ( x µ ◦ Φ)( t, θ ) = φ µ ( t ) + ψ µ ( t ) θ, (5.63)and supersymmetry transformations are given by the action of the odd vector field Q = ∂ θ + θ∂ t ,that in terms of component fields reads δφ µ = ψ µ , δψ µ = ˙ φ µ . (5.64)Denoting the superderivative as D = − ∂ θ + θ∂ t , the action of the non-linear σ model coupledto the gauge field A can be given as S [Φ] = (cid:90) dt (cid:90) dθ g Φ( t,θ ) (cid:16) D Φ , ˙Φ (cid:17) + A ( D Φ) (5.65)where D Φ and ˙Φ are thought as (super)vector fields on M such that, for any function f ∈ C ∞ ( M ), D Φ( f ) := D ( f ◦ Φ) and ˙Φ( f ) := ∂ t ( f ◦ Φ). Inserting the trivialization for the metriccomponents g µν (Φ( t, θ )) = g µν ( φ ) + θψ σ ( t ) g µν,σ ( φ ), and the component expansion for Φ, theaction is simplified to S [ φ, ψ ] = (cid:90) dt (cid:18) g µν ( φ ) ˙ φ µ ˙ φ ν + 12 g µν ( φ ) ψ µ ( ∇ t ψ ) ν + A µ ( φ ) ˙ φ µ − ψ µ F µν ψ ν (cid:19) (5.66)where we suppressed the t -dependence, F µν = ∂ [ µ A ν ] are the components of the electromagneticfield strength, and the time-covariant derivative ∇ t acts as( ∇ t V ) σ ( φ ( t )) = ˙ V σ ( φ ( t )) + Γ σµν ˙ φ µ ( t ) V ν ( φ ( t )) . The action (5.66) is formally equivalent to the model-independent action (5.40) of the lastsection, with T behaving like the localization parameter λ , if we identify θ with the electro-magnetic potential A and ω with the field strength F , and if we set the Hamiltonian and itsassociated vector field H, X H to zero. This means that we can give to it an equivariant coho-mological interpretation on the super-loop space Π T F over M , with coordinates identified with( φ µ , ψ µ ), as was pointed out first by Atiyah and Witten [8]. Moreover, since the Hamiltonianvanishes, this action describes no propagating physical degrees of freedom, and thus the modelhapter 5. Localization for circle actions in supersymmetric QFT 99is topological. Indeed, its value has to give the index of the Dirac operator, expected to bea topological quantity. To emphasize the equivariant cohomological nature of the model, wenotice that the action functional can be split in the loop space (pre-)symplectic 2-formΩ[ φ, ψ ] := (cid:90) dt ψ µ ( g µν ∇ t − F µν ) ψ ν (5.67)and the loop space Hamiltonian H = (cid:90) dt (cid:18) g µν ˙ φ µ ˙ φ ν + A µ ˙ φ µ (cid:19) . (5.68)They satisfy d F H = − ι ˙ φ Ω, so the supersymmetry transformation (5.64) is rewritten in termsof the Cartan differential Q ˙ φ = d F + ι ˙ φ , and Q ˙ φ S = Q ˙ φ ( H + Ω) = 0. Moreover, we can find aloop space symplectic potential Σ such that S is equivariantly exact: S [ φ, ψ ] = Q ˙ x Σ[ φ, ψ ]where Σ[ φ, ψ ] := (cid:90) dt (cid:16) g µν ( φ ) ˙ φ µ + A ν ( φ ) (cid:17) ψ ν . (5.69)Notice that, since the Hamiltonian H vanishes, the localizing U (1) symmetry is the one gener-ated by time-translation with respect to the base space S , an intrinsic property of the geometricstructure that underlies the model.We can now apply the Niemi-Tirkkonen formula, localizing the path integral (5.62) into themoduli space of constant loops, i.e. as an integral over M . The result is the same as equation(5.45), but now since the Hamiltonian vanishes, the Chern class and the Dirac ˆ A -genus (seeAppendix B.1) are not equivariantly extended by the presence of an Hamiltonian vector field,giving the topological formula index( i / ∇ ) = (cid:90) M ch( F ) ∧ ˆ A ( R ) . (5.70)This is the result of the Atiyah-Singer index theorem for the Dirac operator on the twisted spinorbundle over M . Similar applications of the localization principle to variations of the non-linear σ model give correct results for other classical complexes as well (de Rham, Dolbeault forexample), in terms of different topological invariants [74]. In the last section we saw how to give an equivariant cohomological interpretation to a modelexhibiting Poincar´e-supersymmetry, in terms of the super-loop space symplectic structure in-troduced before. In the following we are interested in applying the same kind of supersymmetriclocalization principle to higher dimensional QFT on a, possibly curved, n -dimensional space-time M , where there is some preserved supersymmetry.In [9][10] it is argued that any generic quantum field theory with at least an N = 1 Poincar´esupersymmetry admits a field space Hamiltonian (symplectic) structure and a corresponding U (1)-equivariant cohomology responsible for localization of the supersymmetric path integral.The key feature is an appropriate off-shell component field redefinition which defines a splitting00 5.3. Equivariant structure of supersymmetric QFTof the fields into loop space “coordinates” and their associated “differentials”. In general, unlikethe simplest case of the last section where bosonic fields were identified with coordinates andfermionic fields with 1-forms, loop space coordinates and 1-forms involve both bosonic andfermionic fields. It is proven that, within this field redefinition on the super-loop space, anysupersymmetry charge Q can be identified with a Cartan differential Q = d F + ι X + (5.71)analogously to (5.21), whose square generates translations in a given “light-cone” direction x + , Q = L X + ∼ (cid:90) M ∂∂x + (5.72)that corresponds to the U (1) symmetry that can be used to exploit the localization principle.Taking the base spacetime to be compact in the light-cone direction, the periodic boundaryconditions ensure Q = 0, analogously to the loop space assumption of the one dimensionalcase. Also, it is argued that the supersymmetric action can be generally split into the sum ofa loop space scalar function and a (pre-)symplectic 2-form, S susy = H + Ω (5.73)related by d F H = − ι X + Ω. Thus, the supersymmetry of the action can be seen in general asthe U (1)-equivariant closeness required to the application of the localization principle, and thepath integral localizes onto the locus of constant loops, i.e. zero-modes of the fields.Even without entering in the details of this construction in terms of auxiliary fields redefi-nition, we feel now allowed to translate in full generality the circle localization principle in theframework of Poincar´e-supersymmetric QFT. In the component-field description, we considera rigid supersymmetric background over the given compact spacetime M and a graded fieldspace F that plays the role of the super-loop space over M , whose even-degree forms are bosonicfields and the odd-degree forms are fermionic fields. The (infinitesimal) supersymmetry actionof a preserved supercharge Q plays the role of the Cartan differential d F , squaring to a bosonicsymmetry that corresponds to the (infinitesimal) action of a U (1) symmetry group, Q ∼ L X with X an even vector field. The Cartan model for the U (1)-equivariant cohomology of F isdefined by the subcomplex of Q -invariant (or supersymmetric, or “BPS”) observables, wherethe supercharge squares to zero.We consider a supersymmetric model specified by the (Euclidean) action functional S ∈ C ∞ ( F ) such that δ Q S = 0, and a BPS observable O such that δ Q O = 0. Now the partitionfunction (5.2) and the expectation value (5.3) are seen as integrations of equivariantly closedforms with respect to the differential Q . The supersymmetric localization principle then tellsus that we can modify the respective integrals adding an equivariantly exact localizing term tothe action, λS loc [Φ] := λδ Q V [Φ] (5.74)where V ∈ Ω ( F ) U (1) is a Q -invariant fermionic functional, the “gauge-fixing fermion” ofSection 5.1, and λ ∈ R is a parameter. Assuming the supersymmetry δ Q to be non anomalous,the partition function and the expectation value are not changed by this modification, i.e. the This ensures the loop space interpretation of above. hapter 5. Localization for circle actions in supersymmetric QFT 101the new integrand lies in the same equivariant cohomology class, ddλ Z ( λ ) = (cid:90) F D Φ ( − δ Q V [Φ]) e − ( S + λδ Q V )[Φ] = − (cid:90) F D Φ δ Q (cid:0) V [Φ] e − ( S + λδ Q V )[Φ] (cid:1) = 0 ddλ (cid:104)O(cid:105) λ = − Z ( λ ) (cid:18) ddλ Z ( λ ) (cid:19) (cid:104)O(cid:105) λ + 1 Z ( λ ) (cid:90) F D Φ δ Q (cid:0) V [Φ] O [Φ] e − ( S + λδ Q V )[Φ] (cid:1) = 0 (5.75)by the same argument of Section 5.1. Assuming the bosonic part of δ Q V to be positive-semidefinite, and using the λ -independence of the path integral, we can evaluate the partitionfunction or the expectation value in the limit λ → + ∞ , getting the localization formulas Z = lim λ →∞ (cid:90) F D Φ e − ( S + λS loc )[Φ] , (cid:104)O(cid:105) = 1 Z lim λ →∞ (cid:90) F D Φ O [Φ] e − ( S + λS loc )[Φ] . (5.76)The path integrals localize then onto the locus F of saddle points of S loc . Following againthe same argument of Section 4.2 we can in fact expand the fields about these saddle pointconfigurations, rescale the normal fluctuations asΦ = Φ + 1 √ λ ˜Φ , (5.77)and the augmented action functional as( S + λS loc )[Φ] = S [Φ ] + 12 (cid:90) M d n x (cid:90) M d n y (cid:18) δ S loc δ Φ( x ) δ Φ( y ) (cid:19) Φ ˜Φ( x ) ˜Φ( y ) + O ( λ − / ) . (5.78)The functional measure on the normal sector D ˜Φ is not affected by the rescaling, since thesupersymmetric model contains the same number of bosonic and fermionic physical componentfields, and the corresponding Jacobians cancel by Berezin integration rules. The integral overthis fluctuations is Gaussian and can be performed, giving the “1-loop determinant” analogousto the equivariant Euler class that appeared in Theorem 3.2.2. The leftover integral correspondsto the saddle point formula (5.9), but as an exact equality: Z = (cid:90) F D Φ e − S [Φ ] Z − loop [Φ ] (cid:104)O(cid:105) = 1 Z (cid:90) F D Φ O [Φ ] e − S [Φ ] Z − loop [Φ ] (5.79)where Z − loop [Φ ] := (cid:18) Sdet (cid:20) δ S loc δ Φ( x ) δ Φ( y ) [Φ ] (cid:21)(cid:19) − (5.80)and the super-determinant denotes collectively the result of Gaussian intergrations over bosonicor fermionic fields.Although the choice of localizing term V is arbitrary, and different choices give in principledifferent localization loci, the final result must be the same for every choice. At the endof Section 4.4 we remarked that the N = 2 supersymmetric Yang-Mills Lagrangian on a 3-dimensional maximally supersymmetric background is Q -exact, and thus can be used as alocalizing term for supersymmetric gauge theories on this type of 3-dimensional spacetimes. It Note that this has to be true off-shell , i.e. without imposing any EoM.
02 5.3. Equivariant structure of supersymmetric QFTturns out that also the N = 2 matter (chiral) Lagrangian is Q -exact in three dimensions [79].A canonical choice of localizing action can be, schematically [67] S loc [Φ] := (cid:90) M δ Q (cid:88) f (cid:16) ( δ Q Φ f ) † Φ f + Φ † f ( δ Q Φ † f ) † (cid:17) (5.81)where the sum runs over the fermionic fields of the theory. Its bosonic part is S loc [Φ] | bos = (cid:88) f (cid:16) | δ Q Φ f | + | δ Q Φ † f | (cid:17) , (5.82)that is indeed positive semidefinite. The corresponding localization locus is the subcomplex ofBPS configurations, [fermions] = 0 , δ Q [fermions] = 0 . (5.83)Concluding this general and schematic discussion, there are a couple of remarks we wish topoint out. Firstly, in the above formulas we always considered generic BPS observables that areexpressed through (local or non-local) combinations of the fields. In other words, their quantumexpectation values are defined as insertions in the path integral of corresponding (classical)functionals on the field space. Examples of local objects of this kind are correlation functionsof fundamental fields. A famous class of non-local quantum operators that are expressible asclassical functionals are the so-called Wilson loops . In gauge theory with gauge group G andlocal gauge field A , a Wilson loop in the representation R of Lie ( G ) over the closed curve C : S → M is defined by W R ( C ) := 1dim R Tr R (cid:18) P exp i (cid:73) S C ∗ ( A ) (cid:19) (5.84)where the trace Tr R is taken in the given representation. This is gauge invariant, and repre-sents physically the phase acquired by a charged probe particle in the representation R aftera tour on the curve C , in presence of the gauge potential A . Mathematically, if R is the ad-joint representation, the Wilson loop represents the parallel transport map between the fibersof the principal G -bundle defining the gauge theory. These operators have many interestingapplications in physics: depending on the chosen curve C their expectation value can be inter-preted as an order parameter for the confinement/deconfinement phase transitions in QCD orthe Bremsstrahlung function for an accelerated particle [80–82]. In the case of 3-dimensionalChern-Simons theory, they can be used to study topological invariants in knot theory [83]. Insupersymmetric theories, they are particularly relevant for tests of the AdS/CFT correspon-dence [84]. In the next sections we will review some interesting cases in which expectationvalues of this type of operators can be evaluated exactly using the supersymmetric localizationprinciple. There exists another class of interesting operators in the quantum theory, that can-not be expressed as classical functionals on the field space. These are the so-called disorderoperators , and their expectation values are defined by a restriction of the path integral to thosefield configurations which have prescribed boundary conditions around some artificial singular-ity introduced in spacetime. An example of these are the ’t Hooft operators , which introduce aDirac monopole singularity along a path in a 4-dimensional space [85]. These kind of operators In the adjoint representation, this denotes an invariant inner product in g , for example the Killing form fora semisimple Lie algebra. hapter 5. Localization for circle actions in supersymmetric QFT 103can be also studied non-perturbatively with the help of localization techniques [86]. For a greatreview of different examples of localization computations in supersymmetric QFT, see [12].The second remark we wish to make is that, in presence of a gauge symmetry, the actionfunctionals in the above formulas have to be understood as the quantum ( i.e. gauge-fixed)action in order to give meaning to the corresponding partition function. That is, one has tointroduce Faddeev-Popov ghost fields in the theory and the associated BRST transformations δ BRST . We have seen in Section 4.5 that it is always possible to see the BRST complex in termsof equivariant cohomology on the field space, so this supersymmetry transformation have tobe incorporated in the equivariant structure of the supersymmetric theory. In this case, thefield space acquires a Z -grading corresponding to the ghost number, on top of the Z one fromsupersymmetry, and the appropriate Cartan differential with respect to which the equivariantcohomological structure is defined is then the total supersymmetry variation Q = δ susy + δ BRST . N = 4 , , ∗ gauge theory on the 4-sphere In this section we review, following the seminal work of Pestun [11], how to exploit thesupersymmetric localization principle in N = 4 Euclidean Super Yang-Mills theory on thefour-sphere S . The N = 2 and N = 2 ∗ theories can be also treated with the same technique.In particular, it was possible to solve exactly the partition function of the theory and theexpectation value of the Wilson loop defined by W R ( C ) := 1dim R Tr R (cid:18) P exp i (cid:73) C ( S ) ( A µ ˙ C µ + | ˙ C | Φ ) dt (cid:19) (5.85)where C is a closed equatorial curve on S of tangent vector ˙ C , and the scalar field Φ isrequired by supersymmetry, as will become clear later. The localization procedure makes thepath integral reduce to a finite-dimensional integral over the Lie algebra of the gauge group, aso-called “matrix model”. We will consider the theories revisited in Sections 4.3.7 and 4.4.4. We report the action ofthe N = 2 ∗ theory on the 4-sphere, S N =2 ∗ S = (cid:90) S d x √ g g Y M Tr (cid:0) F MN F MN − ΨΓ M D M Ψ + 2 r Φ A Φ A −− r ( R ki M jk )Φ i Φ j − (cid:88) i =1 K i K i (cid:33) (5.86)where D Φ i (cid:55)→ [Φ , Φ i ] + M ji Φ j and D Ψ (cid:55)→ [Φ , Ψ] + M ij Γ ij Ψ, i, j = 5 , · · · ,
8. In the limitof zero mass M we get the N = 4 YM theory, while in the limit of infinite mass the N = 2hypermultiplet decouples and the pure N = 2 YM theory is recovered. This is invariant under04 5.4. Localization of N=4,2,2* gauge theory on the 4-spherethe superconformal transformations (4.170) that we report here, δ (cid:15) A M = (cid:15) Γ M Ψ δ (cid:15) Ψ = 12 Γ MN F MN (cid:15) + 12 Γ µA Ψ A ∇ µ (cid:15) + (cid:88) i =1 K i ν i δ (cid:15) K i = − ν i Γ M D M Ψ (5.87)with ( ν i ) i =1 , ··· , satisfying (4.171), and (cid:15) being a conformal Killing spinor satisfying (4.165) and(4.166). When the mass is non-zero, the Killing condition is restricted to (4.176), or equivalently˜ (cid:15) = 12 r Λ (cid:15) (5.88)where Λ is an SU (2) RL generator. The superconformal algebra closes schematically as δ (cid:15) = −L v − G Φ − ( R + M ) − Ω . (5.89)To obtain a Poincar´e-equivariant differential interpretation of this variation, we want δ (cid:15) togenerate rigid supersymmetry, i.e. square only to the Poincar´e algebra (plus R-symmetry, upto gauge transformations). Thus, to eliminate the dilatation contribution, we impose also thecondition (cid:15) ˜ (cid:15) = 0 . (5.90)If the mass is non-zero, the SU (1 , R is broken, so also its contribution should be eliminated,imposing further the condition ˜ (cid:15) Γ (cid:15) = 0 . (5.91)Solutions to (4.165) and (4.166) are easy to compute in the flat space limit r → ∞ : here ∇ µ = ∂ µ , and ∂ µ ˜ (cid:15) = 0 imposes ˜ (cid:15) ( x ) = ˆ (cid:15) c constant. Thus, the conformal Killing spinor in flatspace is just the one considered in (4.116), (cid:15) ( x ) = ˆ (cid:15) s + x µ Γ µ ˆ (cid:15) c (5.92)where the first constant term generates supertranslations, while the term linear in x generatessuperconformal transformations. The constant spinors ˆ (cid:15) s , ˆ (cid:15) c parametrize in general the space ofsolutions of the conformal Killing spinor equation. For a finite radius r , using stereographic co-ordinates and the round metric (4.163), the covariant derivative acts as ∇ µ (cid:15) = (cid:0) ∂ µ + ω ijµ Γ ij (cid:1) (cid:15) ,where ω is the spin connection ω ijµ = (cid:0) e iµ e νj − e jµ e iν (cid:1) ∂ ν Ω (5.93)and e is the vielbein corresponding to the metric. The general solution in this coordinatesystem is (cid:15) ( x ) = 1 (cid:113) x r (ˆ (cid:15) s + x µ Γ µ ˆ (cid:15) c ) ˜ (cid:15) ( x ) = 1 (cid:113) x r (cid:18) ˆ (cid:15) c − x µ Γ µ r ˆ (cid:15) s (cid:19) (5.94)that indeed simplifies to (5.92) in the limit of infinite radius. The conditions (5.90), (5.91),(5.88) are rewritten in terms of the constant spinors asˆ (cid:15) s ˆ (cid:15) c = ˆ (cid:15) s Γ ˆ (cid:15) c = 0 ˆ (cid:15) s Γ µ ˆ (cid:15) s = 14 r ˆ (cid:15) c Γ M ˆ (cid:15) c ˆ (cid:15) c = 12 r Λˆ (cid:15) s . (5.95) Here we use latin indices as “flat” indices and greek indices as “curved” indices, so that as g µν = e iµ e jν δ ij . hapter 5. Localization for circle actions in supersymmetric QFT 105The second condition is solved if the two constant spinors are taken to be chiral with respect tothe 4-dimensional chirality operator Γ , so that both terms vanish automatically. In Pestun’sconventions, they are chosen to have the same definite chirality and orthogonal to each other(to satisfy the first condition), so that (cid:15) is chiral only at the North and South poles, where x = 0 , ∞ . The Wilson loop under consideration is of the type considered in [87, 88], W R ( C ) := 1dim R Tr R (cid:18) P exp i (cid:73) C dt ( A µ ˙ C µ + | ˙ C | Φ ) (cid:19) (5.96)where C : [0 , → S is an equatorial closed curve, parametrized in stereographic coordinatesas ( x ◦ C )( t ) = 2 r (cos ( t ) , sin ( t ) , , r . Its tangent vector is˙ C ( t ) = 2 r ( − sin( t ) , cos( t ) , , | ˙ C | = 2 r in front of Φ is needed for thereparametrization invariance of the line integral. We argue now that this Wilson loop preservessome supersymmetry under the action of δ (cid:15) . In fact, its variation is proportional to δ (cid:15) W R ( C ) ∝ (cid:15) (cid:16) Γ µ ˙ C µ + 2 r Γ (cid:17) Ψ (5.97)and for this to vanish for every value of the gaugino Ψ, it must be that0 = (cid:15) (cid:16) Γ µ ˙ C µ + 2 r Γ (cid:17) ∝ (ˆ (cid:15) s + C µ Γ µ ˆ (cid:15) c ) (cid:16) Γ µ ˙ C µ + 2 r Γ (cid:17) ⇔ t ) ( − ˆ (cid:15) s Γ + 2 r ˆ (cid:15) c Γ Γ ) + cos( t ) (ˆ (cid:15) s Γ + 2 r ˆ (cid:15) c Γ Γ ) + (2 r ˆ (cid:15) c Γ Γ + ˆ (cid:15) s Γ ) (5.98)where we inserted the values for C µ , ˙ C µ and simplified some trivial terms. For this to vanishat all t , the three parentheses have to vanish separately, giving the conditionˆ (cid:15) c = 12 r Γ Γ Γ ˆ (cid:15) s . (5.99)This condition halves the number of spinors that preserve the Wilson loop under supersymme-try, so this is called a operator. In the N = 4 case, it preserves 16 supercharges.If the mass of the hypermultiplet is turned on, the third condition in (5.95) has a non-zerosolution for ˆ (cid:15) s if det(Λ − Γ Γ Γ ) = 0, that fixes Λ up to a sign. Without considering the unphysical redundancy in field space given by the gauge symmetryof the theory, we can give a quick argument for the localization of the N = 4 SYM, using theprocedure outlined in Section 5.3. We consider the U (1)-equivariant cohomology generated by There cannot be chiral spinor fields on S without zeros, because a chiral spinor defines an almost complexstructure at each point, but S has no almost complex structure. Since S has constant scalar curvature, it canbe proved that the conformal Killing condition on (cid:15) actually implies that (cid:15) is also a Killing spinor, ∇ µ (cid:15) = µ Γ µ (cid:15) for some constant µ . This condition implies that the spinor is never zero, since it has constant norm. Thus (cid:15) cannot be chiral. [55] This is just common terminology, that does not refer to any BPS condition between mass and central chargesin the supersymmetry algebra (see [47]). It only means that the observable under consideration preserves halfof the supercharges.
06 5.4. Localization of N=4,2,2* gauge theory on the 4-spherethe action of a fixed supersymmetry δ (cid:15) . Since δ (cid:15) S N =4 S = 0 is equivariantly closed (off-shell)with respect to the variations (5.87), we can perform the usual trick and add the localizingterm λS loc := λδ (cid:15) V where V := Tr (cid:0) Ψ δ (cid:15) Ψ (cid:1) (5.100)where δ (cid:15) Ψ is defined by complex conjugation in the Euclidean signature, δ (cid:15) Ψ = 12 ˜Γ MN F MN (cid:15) + 12 Γ µA Ψ A ∇ µ (cid:15) − (cid:88) i =1 K i ν i . (5.101)The bosonic part of the localizing action is S loc | bos = Tr (cid:0) δ (cid:15) Ψ δ (cid:15) Ψ (cid:1) (5.102)that is positive semi-definite. The localization locus is then the subspace of fields such that[fermions] = 0; δ (cid:15) [fermions] s.t. S loc | bos = 0 . (5.103)The solution to (5.103) is found by inserting the relevant supersymmetry variation in S loc | bos ,collecting the terms as a sum of positive semi-definite contributions and requiring them to vanishseparately. Under the assumption of smooth gauge field, this is given by the field configurationssuch that, up to a gauge transformation (see [11] for the details) A µ = 0 µ = 1 , · · · , i = 0 i = 5 , · · · , E = a ∈ g constant K Ei = − ν i ˜ (cid:15) ) a i = 5 , , K I = 0 I = 1 , · · · , . (5.104)So the physical sector of the theory localizes onto the zero-modes of Φ . If also singulargauge field configurations are allowed, (5.103) receives contributions from instanton solutions,where F µν = 0 everywhere except from the North or the South pole. These configurationscan contribute non-trivially to the partition function. Computing the action on the smoothsolutions one gets S N =4 S [ a ] = 1 g Y M (cid:90) S d x √ g Tr (cid:18) r (Φ E ) + ( K Ei ) (cid:19) = 1 g Y M vol( S ) 3 r Tr (cid:0) a (cid:1) = 8 π r g Y M
Tr( a )(5.105)where we used vol( S ) = π r and ( ν i ˜ (cid:15) ) = r . This last equation can be derived fromthe conditions (4.171) and the form of the conformal Killing spinor (cid:15) . The action is givenby constant field contributions, so the path integral is expected to be reduced to a finite-dimensional integral over the Lie algebra g , of the form Z ∼ (cid:90) g da e − π r g Y M
Tr( a ) | Z inst [ a ] | Z − loop [ a ] . (5.106) δ (cid:15) squares to the Poincar´e algebra up to an gauge transformation, so we are really considering an ( U (1) (cid:111) G )-equivariant cohomology, because δ (cid:15) -closed equivariant forms are supersymmetric and gauge invariant observ-ables. Ignoring the gauge fixing procedure, we are really not considering the complete field space, since theBRST procedure teaches us that in presence of a gauge symmetry this is automatically extended to includeghosts, that may contribute to the localization locus. It turns out that ghosts contribution is trivial, so therough argument already gives the correct localization locus. Here we continue with this simplified procedure,and in the next section we are going to argue the above claim. hapter 5. Localization for circle actions in supersymmetric QFT 107Here Z inst is the instanton partition function, coming from the singular gauge field contributionsof above. Since G is often considered to be a matrix group, this partition function is said todescribe a matrix model . The Wilson loop (5.96), evaluated on this locus is given by W R ( C ) = 1dim R Tr R e πra . (5.107)This type of matrix models can be approached by reducing the integration over g to an in-tegration over its Cartan subalgebra h (we will discuss this better later, in Section 5.4.5). Assuming the zero-mode a ∈ h , we can conveniently rewrite the trace in the representation R as the sum of all the weights ρ ( a ) of a in R , W R ( C ) = 1dim R (cid:88) ρ ∈ Ω( R ) n ( ρ ) e πrρ ( a ) , (5.108)where n ( ρ ) is the multiplicity of the weight ρ , and Ω( R ) is the set of all the weights in therepresentation R .We finally notice that the same result for the localization locus works also for the N = 2and the N = 2 ∗ theories, and both theories localize to the same matrix model. If the massterm for the hypermultiplet is considered, this can of course give a non-trivial contribution tothe 1-loop determinant. As remarked at the end of Section 5.3, in presence of a gauge symmetry the path integralhas to be defined with respect to a gauge-fixed action. To do so, one has to enlarge the fieldspace to include the appropriate Faddeev-Popov ghosts in a BRST complex with the differential δ B . We consider then the total differential Q = δ (cid:15) + δ B (5.109)where (cid:15) is a fixed conformal Killing spinor that closes off-shell the superconformal algebra,so that the gauge-invariant SYM action is Q -closed. From the equivariant cohomology pointof view, this operator is an equivariant differential with respect to the U (1) (cid:15) (cid:111) G symmetrygroup acting on the enlarged field space. To gauge fix the path integral, following the BRSTprocedure with respect to the differential Q , the action has to be extended as S phys [ A, Ψ , K, ghosts ] = S SY M [ A, Ψ , K ] + Q O g.f. [ A, ghosts ] (5.110)with a gauge-fixing fermion O [ A, ghosts ]. Upon path integration over ghosts, this new termhas to give the gauge-fixing action and Fadee-Popov determinant. The localization principle isthen exploited augmenting again the action with a Q -exact term, λQ V with again V := Tr(Ψ δ (cid:15) Ψ) . (5.111) For finite-dimensional semi-simple complex Lie algebras, this is the maximal Abelian subalgebra. In generalit is the maximal Lie subalgebra such that there exists a basis extension h ⊕ span ( e α ) ∼ = g , and it holds theeigenvalue equation [ h, e α ] = ρ α ( h ) e α for any h ∈ h and a certain eigenvalue ρ α ( h ). ρ α : h → C are called roots of g . Analogously to the definition of root in the adjoint representation of g , if the Lie algebra acts on therepresentation R , the weight space R ρ of weight ρ : h → C is defined as the subspace of elements A ∈ R suchthat h · A = ρ ( h ) A .
08 5.4. Localization of N=4,2,2* gauge theory on the 4-sphereThis effectively gives the same localization term of the previous paragraph, since V is gauge-invariant.The BRST-like complex considered in [11] is given by the following ghost and auxiliaryfield extension. The ghost c , anti-ghost ˜ c and standard Lagrange multiplier for the R ξ -gauges b (“Nakanishi-Lautrup” field) are introduced, respectively odd, odd and even with respect toGrassmann parity. Since the path integral is expected to localize on zero-modes, constant fields c , ˜ c (odd) and a , ˜ a , b (even) are also introduced. On the original fields of the SYM theory,the BRST differential acts as a gauge transformation parametrized by c . On the gauge field A µ δ B A µ = − [ c, D µ ] . (5.112)On ghosts and zero-modes the BRST transformation is defined by δ B c = − a − [ c, c ] δ B ˜ c = b δ B ˜ a = ˜ c δ B b = c δ B a = 0 δ B b = [ a , ˜ c ] δ B ˜ c = [ a , ˜ a ] δ B c = [ a , b ] (5.113)and its square generates a gauge transformation with respect to the (bosonic) constant field a , δ B = [ a , · ] . (5.114)The supersymmetry complex, constituted by the original fields, is reparametrized withrespect to the basis { Γ M (cid:15), ν i } , with M = 1 , · · · , i = 1 , · · · ,
7, of the 10-dimensional Majorana-Weyl bundle over S . Expanding Ψ over such a basis we haveΨ = (cid:88) M =1 Ψ M (Γ M (cid:15) ) + (cid:88) i =1 Υ i ν i (5.115)and the superconformal transformations are rewritten as δ (cid:15) A M = Ψ M δ (cid:15) Ψ M = − ( L v + R + G Φ ) A M δ (cid:15) Υ i = H i δ (cid:15) H i = − ( L v + R + G Φ )Υ i , (5.116)where H i := K i + 2( ν i ˜ (cid:15) )Φ + 12 F MN ν i Γ MN (cid:15) + 12 Φ A ν i Γ µA ∇ µ (cid:15). (5.117)With this field redefinition, we see that the supersymmetry transformations can be schematizedin the form δ (cid:15) X = X (cid:48) δ (cid:15) X (cid:48) = [ φ + (cid:15), X ] ( δ (cid:15) φ = 0) (5.118)where φ := − Φ = v M A M , [ φ, X (cid:48) ] := − G Φ X (cid:48) denotes a gauge transformation, [ (cid:15), X (cid:48) ] := − ( L v + R ) X (cid:48) denotes a Lorentz transformation. Here X = ( A M ( x ) , Υ i ( x )), X (cid:48) = (Ψ M ( x ) , H i ( x )) arethe coordinates in the super-loop space interpretation of the supersymmetric model, of oppositestatistics. As we pointed out in Section 5.3, we espect every Poincar´e-supersymmetric theoryto have such super-loop equivariant structure, and this is an example of the fact that in higherdimensional QFT the reparametrization of the fields necessary to make this apparent can benon-trivial. Indeed, the loop space coordinates X and the corresponding 1-forms X (cid:48) mix thebosonic/fermionic field components of the original parametrization!hapter 5. Localization for circle actions in supersymmetric QFT 109Combining the two complexes, and giving supersymmetry transformation properties to theghost sector, the equivariant differential Q is taken to act as QX = X (cid:48) − [ c, X ] Qc = φ − a − [ c, c ] Q ˜ c = bQX (cid:48) = [ φ + (cid:15), X ] − [ c, X (cid:48) ] Qφ = − [ c, φ + (cid:15) ] Qb = [ a + (cid:15), ˜ c ] Q ˜ a = ˜ c Qb = c Q ˜ c = [ a , ˜ c ] Qc = [ a , b ] . (5.119)Moreover, Qa = Q(cid:15) = 0. This differential squares to a constant gauge transformation gener-ated by a and the Lorentz transformation generated by (cid:15) , Q = [ a + (cid:15), · ] . (5.120)Notice that to make explicit the super-loop structure when the combined complex is taken intoaccount, one needs another non-trivial reparametrization of the fields,˜ X (cid:48) := X (cid:48) − [ c, X ] ˜ φ := φ − a −
12 [ c, c ] . (5.121)This makes the tranformations look like Q (field) = field (cid:48) Q (field (cid:48) ) = [ a + (cid:15), field] (5.122)and the new pairs of coordinate/1-form in the extended super-loop space are ( c, ˜ φ ), (˜ c, b ),(˜ a , ˜ c ), ( b , c ).The gauge-fixing term considered for the extended quantum action is, schematically S g.f. = (cid:90) S Q (cid:18) ˜ c (cid:18) ∇ µ A µ + ξ b + b (cid:19) − c (cid:18) ˜ a − ξ a (cid:19)(cid:19) (5.123)where the bilinear product in g is suppressed in the notation, assuming contraction of Liealgebra indices. Upon integration of the auxiliary field, this term produces the usual gaugefixing term for the Lorentz gauge ∇ µ A µ = 0, and the ghost term of the action. Moreover,the path integral is independent of the parameters ξ , ξ (we refer to [11] for the proof). Wefinally claim that the localization principle for the gauge-fixed theory remains the same, with theadditional condition of vanishing ghosts in the localization locus, and identifying the zero-modeof Φ with a . In fact from the gauge-fixing term, S g.f. ⊃ − (cid:90) S (cid:18) φ − a −
12 [ c, c ] (cid:19) ˜ a (5.124)and integrating over ˜ a , we have the condition φ = a + [ c, c ], that in the localization locuswhere c = 0 and φ = − v M A M = Φ , becomes precisely Φ = a . We stated that the path integral localizes (apart from instanton corrections) to the zero-modes of the bosonic constant a ∈ g , that correspond to the zero-modes of Φ . For the samereason of the scalar field corresponding to the reduced time-direction of the (9,1)-theory, weintegrate over immaginary a = ia E , where a E is real. The application of the localization prin-ciple is now straightforward in principle, although very cumbersome in practice. In particular,10 5.4. Localization of N=4,2,2* gauge theory on the 4-sphereintegrating out the Gaussian fluctuations around the localization locus, the arising one-loopdeterminant in the partition function results of the form Z − loop = (cid:18) det K f det K b (cid:19) / (5.125)where K f , K b are the kinetic operators acting on the fermionic and bosonic fluctuation modesafter the usual expansion of Q V . This factor requires in general a regularization, and it hasbeen computed for the N = 2 , N = 2 ∗ and N = 4 theory, using an appropriate generalizationof the Atiyah-Singer theorem seen in Section 5.2 applied to transversally elliptic operators. Theinstanton partition functions have been also simplified for the theories under consideration. Werefer to [11, 89, 90] for the explicit form of instanton contributions in the cases of N = 2 , ∗ .For the maximally supersymmetric N = 4 SYM theory, the results for the 1-loop determi-nant and the instanton partition function are of the very simple form Z N =41 − loop = 1 , Z N =4 inst = 1 , (5.126)so that the resulting localization formulas for the partition function and the expectation valueof the supersymmetric Wilson loop presented before become Z S = 1vol( G ) (cid:90) g da e − π r g Y M
Tr( a ) , (cid:104) W R ( C ) (cid:105) = 1dim R Z vol( G ) (cid:90) g da e − π r g Y M
Tr( a ) Tr R (cid:0) e πra (cid:1) . (5.127)This result proved a previous conjecture, based on a perturbative analysis by Erickson-Semenoff-Zarembo [91]. Their calculation for (cid:104) W R ( C ) (cid:105) with G = U ( N ) showed that the Feynmandiagrams with internal vertices cancel up to order g N , and that the sum of all ladder diagrams(planar diagrams with no internal vertices) exponentiate to a matrix model. The result of thisexponentiation gives an expectation value that coincides with the strong-coupling prediction ofthe AdS/CFT correspondence for N = 4 SYM, thus they conjectured that the diagrams withvertices have to vanish at all orders. Later this conjecture was supported by Drukker-Gross[93], and finally proven with the exact localization technique described above.We quote now the results for the 1-loop determinants in the N = 2 , ∗ theories. For this, itis useful to introduce the notationdet R f ( a ) := (cid:89) ρ f ( ρ ( a )) H ( z ) := e − (1+ γ ) z ∞ (cid:89) n =1 (cid:18) − z n (cid:19) n ∞ (cid:89) n =1 e z /n (5.128)with ρ running over the weights of R (if R = Ad , the weights are the roots of g ), γ beingthe Euler-Mascheroni constant. Let also be m := M ij M ij , and recall that m (as well as a ) This “correspondence” conjectures a duality between the N = 4 SYM in 4 dimensions and type IIBsuperstring theory in an AdS × S background. In particular, when the parameters of the gauge theory aretaken to be such that N → ∞ and g Y M N → ∞ (namely, in the planar and strong ’t Hooft coupling limit), N = 4 SYM is dual to classical type IIB supergravity on AdS × S and the computation of the Wilson loop inthis limit is mapped to the evaluation of a minimal surface in this space [80, 92]. hapter 5. Localization for circle actions in supersymmetric QFT 111should take immaginary values. From [11] we have Z N =2 ∗ − loop [ a ; M ] = exp (cid:32) − r m (cid:32) (1 + γ ) − ∞ (cid:88) n =1 n (cid:33)(cid:33) det Ad (cid:34) H ( ra )[ H ( r ( a + m )) H ( r ( a − m ))] − / (cid:35) , (5.129) Z N =2 ,pure − loop [ a ] = det Ad H ( ra ) , (5.130) Z N =2 ,W − loop [ a ] = det Ad H ( ra )det W H ( ra ) , (5.131)where the first result is for the massive N = 2 ∗ theory, the second one is derived putting m = 0 in the first line, and describes the pure N = 2 SYM, the third one is for the matter-coupled theory to a massles hypermultiplet in the representation W . Notice that the exponentialprefactor in the first line diverges, but is independent of a , and thus simplifies in ratios duringthe computation of expectation values. Also, the third line holds literally if the ( a -independent)divergent factors are the same for the vector and the hypermultiplet. N = 4 SYM
As an example, we include here an explicit computation for the Gaussian matrix model(5.127) in the case of N = 4 SYM [85, 93, 94]. We will take in particular the case of thecompact matrix group G = U ( N ) with the Wilson loop in the fundamental representation R = N , but first we analyze generically how to simplify such an integration over the Liealgebra g . We normalize the invariant volume element da on g such that (cid:90) g da e − ξ Tr( a ) = (cid:18) ξ π (cid:19) dim( G ) / (5.132)for any parameter ξ . In the U ( N ) case, g = u ( N ) = { Hermitian N × N matrices } , so thismeans taking da = 2 N ( N − / N (cid:89) i =1 da ii (cid:89) ≤ j
12 5.4. Localization of N=4,2,2* gauge theory on the 4-sphere group of G , that we call W . Taking this into account, after the gauge-fixing we can performthe integral over the orbits obtaining a volume factorvol( G/T ) |W| . (5.136)The gauge-fixing can be done with the usual Faddeev-Popov (FP) procedure, that is insertingthe unity decomposition 1 = (cid:90) dg ∆ ( X ) δ ( F ( a ( g ) )) , (5.137)where the delta-function fixes the condition (5.135), and the FP determinant is given by∆( X ) = (cid:89) α | α ( X ) | = (cid:89) α> α ( X ) , (5.138)where α : h → C are the roots of g , and in the second equality we used that roots come in pairs( α, − α ). We can rewrite the expectation value of the circular Wilson loop as (cid:104) W R ( C ) (cid:105) = (cid:18) ξ π (cid:19) dim( G ) / vol( G/T ) |W| dim R (cid:90) h dX ∆( X ) e − ξ Tr( X ) Tr R (cid:0) e πrX (cid:1) = (cid:18) ξ π (cid:19) dim( G ) / vol( G/T ) |W| dim R (cid:88) ρ ∈ Ω( R ) n ( ρ ) (cid:90) h dX ∆( X ) e − ξ Tr( X ) e πrρ ( X ) (5.139)where in the second line we used (5.108).We specialize now to the case G = U ( N ), h = { X = diag( λ , · · · , λ N ) | λ i ∈ R } , and wetake r = 1. The adjoint action of U ( N ) is the conjugation a (cid:55)→ gag † , so the FP determinant isdefined by 1 = (cid:90) dg ∆( X ) (cid:89) ij δ (( gXg † ) ij ) , (5.140)that imposes the off-diagonal terms to vanish in the given gauge. Expressing g = e M with M ∈ u ( N ),∆( X ) = (cid:89) ij det kl (cid:12)(cid:12)(cid:12)(cid:12) δ ( e M Xe − M ) ij δM kl (cid:12)(cid:12)(cid:12)(cid:12) = (cid:89) ij det kl | δ ki δ lj ( λ j − λ i ) | = (cid:89) i>j ( λ i − λ j ) (5.141)is the so called Vandermonde determinant , that can be related to the following matrix∆( λ ) = det || λ j − i || = det λ λ · · · λ N − λ λ · · · λ N − ...1 λ N λ N · · · λ N − N . (5.142)The partition function can thus be expressed as Z S = 1 N ! 1(2 π ) N (cid:90) (cid:32)(cid:89) i dλ i (cid:33) (cid:32)(cid:89) i>j ( λ i − λ j ) (cid:33) e − ξ (cid:80) i λ i , (5.143)hapter 5. Localization for circle actions in supersymmetric QFT 113where N ! is the order of the Weyl group W = S N and (2 π ) N is the volume of the N -torus U (1) N , while the Wilson loop in the fundamental representation inserts in the path integral afactor 1 N N (cid:88) j =1 e πλ j . (5.144)There are two main approaches to the evaluation of this matrix model and the computation ofthe Wilson loop expectation value, at least in the limit N → ∞ .1 st method: saddle-point The first method that we present is based on a suitable saddle-point approximation in the large- N limit. To see the possibility for this interpretation, werewrite the partition function as Z S = 1 N ! (cid:90) (cid:89) i dλ i π e − N S eff ( λ ) with S eff ( λ ) := 8 π tN N (cid:88) i =1 λ i − N (cid:88) i>j log | λ i − λ j | , (5.145)where t := g Y M N is the ’t Hooft coupling constant. This can be viewed as an effective action ofa zero-dimensional QFT describing N sites (the eigenvalues λ i ), where the first piece is a “one-body” harmonic potential, and the second one is a repulsive “two-body” interaction. Noticethat every sum is roughly of order ∼ N , so S eff ∼ O (1) in N . The limit N → ∞ , with t fixed,can be regarded as a semi-classical approximation (we could compare it to “1 / √ (cid:126) → ∞ ”), andin that limit we can solve the integral using a saddle-point approximation. The saddle pointsare those values of λ i that solve the classical EoM0 = δS eff δλ i ⇒ π tN λ i − N (cid:88) j (cid:54) = i λ i − λ j . (5.146)In the large- N limit we can study this equation in the continuum approximation, assuming theeigenvalues λ i to take values in a compact interval I = [ a, b ], so that the (normalized) eigenvaluedistribution ρ ( λ ) = 1 N N (cid:88) i =1 δ ( λ − λ i ) (5.147)is regarded as a continuous function of compact support on I . Then every sum can be replacedby an integration over the reals, 1 N N (cid:88) i =1 f ( λ i ) → (cid:90) dλ f ( λ ) ρ ( λ ) , (5.148)and (5.146) becomes 8 π t λ = P (cid:90) ρ ( λ (cid:48) ) dλ (cid:48) λ − λ (cid:48) , (5.149)where we took the principal value of the integral to avoid the pole at λ i = λ j . This is an integralequation in ρ ( λ ), whose solution gives the distribution of the eigenvalues at the saddle-pointlocus of the partition function.14 5.4. Localization of N=4,2,2* gauge theory on the 4-sphereIt is useful to introduce an auxiliary function on the complex plane, the “resolvent” ω ( z ) := (cid:90) ρ ( λ ) dλz − λ , (5.150)that has three important properties for our purposes:(i) it is analytic on C \ I , since there are poles for z = λ when z ∈ I ;(ii) thanks to the normalization of ρ , asymptotically for | z | → ∞ it goes as ω ( z ) ∼ z ;(iii) using the residue theorem and the delta-function representation (cid:15)z + (cid:15) (cid:15) → + −−−→ πδ ( z ) , (5.151)it relates to the eigenvalue distribution by the discontinuity equation ρ ( λ ) = − πi lim (cid:15) → + [ ω ( λ + i(cid:15) ) − ω ( λ − i(cid:15) )] . (5.152)Knowing the resolvent we can easily compute the eigenvalue distribution by this last property,so we rewrite the saddle-point equation in terms of it. To compute ω , we can start again from(5.149), multiply by 1 / ( λ − z ) and integrate over λ with the usual measure ρ ( λ ) dλ :8 π t (cid:90) dλ ρ ( λ ) λλ − z = (cid:90) dλ ρ ( λ ) λ − z P (cid:90) dλ (cid:48) ρ ( λ (cid:48) ) λ − λ (cid:48) . (5.153)We can add ± z at the numerator of the LHS, and use the formula ( Sokhotski–Plemelj theorem ) P (cid:90) f ( z ) z dz = lim (cid:15) → + (cid:18)(cid:90) f ( z ) z + i(cid:15) dz + (cid:90) f ( z ) z − i(cid:15) dz (cid:19) (5.154)to break the principal value on the RHS. Inserting the definition of the resolvent and using theresidue theorem, this gives 8 π t − π t λω ( λ ) = − ω ( λ ) , (5.155)that is solved for ω ( λ ) = 8 π t (cid:32) λ ± (cid:114) λ − t π (cid:33) . (5.156)In order to match the right asymptotic behavior ω ( z → ∞ ) ∼ /z , we have to chose theminus sign. With this choice, we can compute the saddle-point eigenvalue distribution usingthe discontinuity equation (5.152), ρ ( λ ) = − πi π t lim (cid:15) → + [ ω ( λ + i(cid:15) ) − ω ( λ − i(cid:15) )]= 4 πit lim (cid:15) → + (cid:34)(cid:114) λ − t π + 2 i(cid:15)λ − (cid:114) λ − t π − i(cid:15)λ (cid:35) = 4 πit (cid:32) (cid:114) λ − t π (cid:33) = 8 πt (cid:114) t π − λ (5.157)hapter 5. Localization for circle actions in supersymmetric QFT 115where we used that the principal square root has a branch cut on the real line. This functionis called Wigner semi-circle distribution , it has support on the interval I = [ −√ t/ π, √ t/ π ],and here it is correctly normalized to 1.Now that we have the saddle-point locus in terms of the eigenvalue distribution, we cancompute the expectation value for the circular Wilson loop in the fundamental representation.Since the exponential factor (5.144) is of order ∼ N , this does not contribute to the saddle-point equation in the N → ∞ limit. We can thus still use the Wigner distribution at zero-orderin 1 /N , and insert in the path integral the trace in the continuum limit, (cid:104) W N ( C ) (cid:105) = (cid:90) dλ (cid:104) ρ ( λ ) (cid:105) e πλ = 8 πt (cid:90) √ t/ π −√ t/ π dλ e πλ (cid:114) t π − λ + O (cid:0) /N (cid:1) = 2 √ t I (cid:16) √ t (cid:17) + O (cid:0) /N (cid:1) (5.158)where I ( z ) is a modified Bessel function of the first kind. In the weak and strong couplinglimits t (cid:29) , (cid:28) t (cid:28) (cid:104) W N ( C ) (cid:105) ∼ t t
192 + · · · (5.159) t (cid:29) (cid:104) W N ( C ) (cid:105) ∼ (cid:114) π t − / e √ t , (5.160)so it explodes in the strong coupling limit, with an essential singularity. nd method: orthogonal polynomials Another technique to solve matrix models involvethe use of orthogonal polynomials [93]. Our starting point is again the partition function, Z = 1 N ! (cid:90) N (cid:89) i =1 (cid:18) dλ i π e − π Nt λ i (cid:19) ∆( λ ) . (5.161)Introducing the L ( R ) measure dµ ( x ) := dx e − π Nt x , (5.162)we can write the partition function as Z = 1 N ! (cid:90) N (cid:89) i =1 dµ ( λ i )∆( λ ) . (5.163)Recalling that the Vandermonde determinant is evaluated from the matrix (5.142), expressed interms of the polynomials { , x, x , · · · } , we notice that we can equivalently express it in termsof another set of monic polynomials, p k ( x ) = x k + k − (cid:88) j =0 a ( k ) j x j (5.164) Interestingly, the strong coupling limit can be checked independently using holography, where Wilson loopsare given by minimal surfaces in AdS [80, 92].
16 5.4. Localization of N=4,2,2* gauge theory on the 4-spheresince by elementary row operations∆( λ ) = det || λ j − i || = det || p j − ( λ i ) || . (5.165)It is useful to chose the set { p k } k ≥ to be orthogonal with respect to the matrix model measure, (cid:90) dµ ( λ ) p n ( λ ) p m ( λ ) = h n δ nm (5.166)since the knowledge of this set, and in particular of the normalization constants h n , allows tocompute the partition function. Writing the determinant as∆( λ ) = (cid:88) σ ∈ S N ( − ( σ ) N (cid:89) k =1 p σ ( k ) − ( λ k ) , then (5.163) reduces to Z = N − (cid:89) k =0 h k . (5.167)In our case the matrix model is Gaussian, and the corresponding set of orthogonal polyno-mials are the Hermite polynomials , H n ( x ) := e x (cid:18) − ddx (cid:19) n e − x , (cid:90) + ∞−∞ dx e − x H n ( x ) H m ( x ) = δ nm n n ! √ π (5.168)so, normalizing h n = 1 and inserting the correct prefactors, we consider the set of orthonormal polynomials with respect to the measure dµ ( λ ) P n ( λ ) := (cid:115)(cid:114) πNt n n ! H n (cid:32) √ π Nt λ (cid:33) . (5.169)The expectation value of any observable of the type Tr( f ( X )) = (cid:80) k f ( λ k ) can be simplifiedas (cid:104) Tr f ( X ) (cid:105) = 1 N ! Z (cid:90) (cid:32) N (cid:89) i =1 dµ ( λ i ) (cid:33) ∆( λ ) N (cid:88) k =1 f ( λ k )= 1 N ! (cid:88) k (cid:88) σ ∈ S N (cid:90) dµ ( λ ) P σ (1) − ( λ ) · · · (cid:90) dµ ( λ k ) P σ ( k ) − ( λ k ) f ( λ k ) · · ·· · · (cid:90) dµ ( λ N ) P σ ( N ) − ( λ N ) = N − (cid:88) j =0 (cid:90) dµ ( λ ) P j ( λ ) f ( λ ) . (5.170)Applying this formula to the expectation value of the circular Wilson loop in the fundamentalrepresentation we have (cid:104) W N ( C ) (cid:105) = 1 N (cid:104) Tr exp(2 πX ) (cid:105) = 1 N N − (cid:88) j =0 (cid:90) dλ P j ( λ ) e − π Nt λ +2 πλ . (5.171)hapter 5. Localization for circle actions in supersymmetric QFT 117A useful formula to simplify this integral is (cid:90) + ∞−∞ dx H n ( x ) e − ( x − c ) = 2 n n ! √ πL n ( − c ) (5.172)where c is a constant and L n ( x ) are the Laguerre polynomials , satisfying the properties L ( m ) n ( x ) = 1 n ! e x x m (cid:18) ddx (cid:19) n (cid:0) e − x x n + m (cid:1) , (5.173) L n ( x ) ≡ L (0) n ( x ) , (5.174) L ( m +1) n ( x ) = n (cid:88) j =0 L ( m ) j ( x ) , (5.175) L ( m ) n ( x ) = n (cid:88) k =0 (cid:18) n + mn − k (cid:19) ( − x ) k k ! . (5.176)Substituting (5.172) in (5.171), and expanding in series we have (cid:104) W N ( C ) (cid:105) = 1 N e c L (1) N − ( − c ) with c := (cid:114) t N = 1 N ∞ (cid:88) k =0 k ! (cid:18) t N (cid:19) k N − (cid:88) j =0 N !( j + 1)!( N − − j )! 1 j ! (cid:18) t N (cid:19) j −− N N − (cid:88) j =1 j !( j + 1)! (cid:18) t (cid:19) j j ( j + 1)2 + 1 N N − (cid:88) j =0 j !( j + 1)! (cid:18) t (cid:19) j
12 + O (1 /N )= N − (cid:88) j =0 j !( j + 1)! (cid:18) t (cid:19) j + O (cid:0) /N (cid:1) (5.177)where we expanded the first terms with respect to powers of 1 /N , and already noticed that for N (cid:29) N limit, and inserting thedefinition of the modified Bessel function I n (2 x ) = (cid:80) ∞ k =0 x n +2 k k !( n + k )! the expectation value gives (cid:104) W N ( C ) (cid:105) = 2 √ t I (cid:16) √ t (cid:17) + O (1 /N ) , (5.178)matching the result obtained with the saddle point technique in (5.158). In general, the ex-pansion is in powers of 1 /N rather than 1 /N , as expected from the analogy “ N ↔ / (cid:126) ” thatwe noticed in (5.145). Solutions to the matrix model for higher representations have also beenfound, see [85, 95, 96]. N = 2 Chern-Simons theory on the3-sphere
In this section we review another example of supersymmetric localization applied to thecomputation of Wilson loop expectation values, in an N = 2 matter-coupled Euclidean SuperChern-Simons (SCS) theory on the 3-sphere S . We follow the derivation of Kapustin-Willet-Yaakov [49], and Mari˜no [50], inspired in part by the work discussed in the previous section.We consider a generic compact Lie group G as the gauge group, with Lie algebra g .18 5.5. Localization of N=2 Chern-Simons theory on the 3-sphere N = 2 Euclidean SCS theory on S The case of N = 2 Euclidean supersymmetry on S was discussed as an example in Sections4.3.6 and 4.4.3 for the gauge sector. We report the action for the SCS theory S CS = k π (cid:90) S d x √ g Tr (cid:26) ε µνρ √ g (cid:18) A µ ∂ ν A ρ + 2 i A µ A ν A ρ (cid:19) − ˜ λλ + 2 σD (cid:27) (5.179)and the supersymmetry variations, already considered in curved space δA µ = i (cid:15)γ µ λ − ˜ λγ µ (cid:15) ) δσ = 12 (˜ (cid:15)λ − ˜ λ(cid:15) ) δλ = (cid:18) − F µν γ µν − D + i ( D µ σ ) γ µ + 2 i σγ µ D µ (cid:19) (cid:15)δ ˜ λ = (cid:18) − F µν γ µν + D − i ( D µ σ ) γ µ − i σγ µ D µ (cid:19) ˜ (cid:15)δD = − i (cid:16) ˜ (cid:15)γ µ D µ λ − ( D µ ˜ λ ) γ µ (cid:15) (cid:17) + i (cid:16) [˜ (cid:15)λ, σ ] − [˜ λ(cid:15), σ ] (cid:17) − i (cid:16) ˜ λγ µ D µ (cid:15) + ( D µ ˜ (cid:15) ) γ µ λ (cid:17) . (5.180)where the D µ are gauge-covariant derivatives with respect to the metric and spin connectioninduced by the round metric (4.154), that in stereographic coordinates x µ =1 , , is given by g µν = e x ) δ µν e x ) = (cid:18) x r (cid:19) − (5.181)with r being the radius of the embedding S (cid:44) → R . We remark again that this supersymmetricaction is actually superconformal, thus can preserve supersymmetry on this conformally flatbackground, even with positive scalar curvature. The new background preserves all the original N = 2 algebra, generated by conformal Killing spinors (cid:15), ˜ (cid:15) , taken to satisfy ∇ µ (cid:15) = i r γ µ (cid:15), ∇ µ ˜ (cid:15) = i r γ µ , ˜ (cid:15) (5.182)where every equation has two possible solutions.We consider also coupling the theory to matter fields, adding them in chiral multipletsin a representation R of the gauge group, to preserve supersymmetry. The 3-dimensional N = 2 chiral multiplet (or hypermultiplet) is, as for the gauge multiplet, given by dimensionalreduction of the N = 1 chiral multiplet in 4 dimensions: a complex scalar φ , a 2-componentDirac spinor ψ and an auxiliary complex scalar F . Every field comes with its complexconjugate from the corresponding anti-chiral multiplet. The supersymmetric action for thematter multiplet coupled to the gauge multiplet is given by S m = (cid:90) S d x √ g (cid:18) D µ ˜ φD µ φ + 34 r ˜ φφ + i ˜ ψ /Dψ + ˜ F F + ˜ φσ φ + i ˜ φDφ + i ˜ ψσψ + i ˜ φ ˜ λψ − i ˜ ψλφ (cid:19) (5.183)where the g -valued fields in the gauge multiplets act on the chiral multiplet in the represen-tation R . This is the “covariantization” of the flat space action for the matter multiplet (see Recall that
Spin (3) = SU (2) has no Majorana spinors. We consider the reduced 4-dimensional Majoranaspinor ψ as a 3-dimensional Dirac (complex) spinor, since they have the same number of real components. hapter 5. Localization for circle actions in supersymmetric QFT 119for example [97]), with the addition of the conformal coupling of the scalar field to the curva-ture, r ˜ φφ . The supersymmetry transformations for the chiral multiplet, with respect to theconformal Killing spinors (cid:15), ˜ (cid:15) , are δφ = ˜ (cid:15)ψ δ ˜ φ = ˜ ψ(cid:15)δψ = ( − iγ µ D µ φ − iσφ ) (cid:15) − i γ µ ( ∇ µ (cid:15) ) φ + ˜ (cid:15)Fδ ˜ ψ = ˜ (cid:15) ( iγ µ D µ ˜ φ + iσ ˜ φ ) + i ∇ µ ˜ (cid:15) ) γ µ ˜ φ + (cid:15) ˜ FδF = (cid:15) ( − iγ µ D µ ψ + iλφ + iσψ ) δ ˜ F = ( iD µ ˜ ψγ µ − i ˜ λ ˜ φ + iσ ˜ ψ )˜ (cid:15). (5.184)The above variations generates a superconformal algebra that closes off-shell:[ δ (cid:15) , δ ˜ (cid:15) ] = − i ( L v + G Λ + R α + Ω f ) (5.185)where L v is the Lie derivative (translation) along the Killing vector field v = (˜ (cid:15)γ µ (cid:15) ) ∂ µ , actingon one forms as L v ( A ) µ = v ν ∂ ν A µ + A ν ∂ µ v ν , and on spinors as L v ψ = ∇ ν ψ − ( ∇ µ v ν ) γ µν ψ . G Λ is a gauge transformation with respect to the parameter Λ := A ( v ) + σ (˜ (cid:15)(cid:15) ). R α is a U (1) R R-symmetry transformation, and Ω f is a dilatation [50]. The matter coupled action S CS + S m is known to be superconformal at quantum level, but one could also add a superpotential forthe matter multiplet. This choice is restricted by the condition of unbroken superconformalsymmetry both at classical and at quantum level, since the localization principle works only ifthe supersymmetry algebra closes off-shell. It turns out that the localization locus is at trivialconfigurations of the matter sector, thus the precise choice of superpotential does not influencethe computation. The Wilson loop under consideration, in the representation R of the gauge group, is definedas [97] W R ( C ) = 1dim R Tr R (cid:18) P exp (cid:73) C dt ( iA µ ˙ C µ + σ ) (cid:19) (5.186)with C : S → S a closed curve of tangent vector ˙ C , normalized such that | ˙ C | = 1. In order tolocalize its expectation value, we have to consider those curves such that this operator preservessome supersymmetry on the 3-sphere. Its variation under (5.180) is proportional to δW R ( C ) ∝ − ˜ (cid:15) ( γ µ ˙ C µ + 1) λ + ˜ λ ( γ µ ˙ C µ − (cid:15). (5.187)Imposing the vanishing of this expression for all gauginos, we get the following conditions onthe conformal Killing spinors,˜ (cid:15) ( γ µ ˙ C µ + 1) = 0 , ( γ µ ˙ C µ − (cid:15) = 0 . (5.188)We have two more conditions on the conformal Killing spinors, thus the maximum number ofsolutions is reduced by half. The Wilson loop can at most be invariant under two of the fourpossible supersymmetry variations, and for that it is called .We can find explicitly one family of supersymmetric Wilson loops and one supersymmetryvariation with respect to which we are going to perform the localization procedure. In order20 5.5. Localization of N=2 Chern-Simons theory on the 3-sphereto solve the conformal Killing equations and the conditions (5.188), we chose explicitly anorthonormal basis and a corresponding vielbein on S . Since as a manifold S ∼ = SU (2), wecan use Lie theory to describe the geometry on the 3-sphere. In particular, the vielbein can bechosen proportional to the Maureer-Cartan form Θ ∈ T ∗ ( SU (2)) ⊗ su (2), e iµ := r e i (Θ( ∂ µ )) (5.189)where { e i } is a basis of su (2) ∗ , dual to a basis { T i } of su (2). One can check that this vielbeinis consistent with the round metric, giving g µν = e iµ e jν δ ij (see [50]). Using this orthonormalbasis, the spin connection components are( ω µ ) ij = 1 r e kµ ε ijk (5.190)where ε ijk is the Levi-Civita symbol. In this basis the conformal Killing spinor equation for (cid:15) looks particularly simple, (cid:18) ∂ µ + 18 ( ω µ ) ij [ γ i , γ j ] (cid:19) (cid:15) = i r γ µ (cid:15) ⇔ ∂ µ (cid:15) = 0 (5.191)where we used the commutator [ γ i , γ j ] = 2 i ε ij k γ k . We see that the components of (cid:15) areconstants. The corresponding condition for the supersymmetry of the Wilson loop then requires γ µ ˙ C µ to be constant too, as the components of the vector field ˙ C i in the orthonormal frame.This means that the Wilson loop has to describe grat circles on S . Following [49], we take ˙ C parallel to one of the e i , say e , and the conformal Killing spinor to satisfy( γ − (cid:15) = 0 . (5.192)We will consider the one dimensional subalgebra generated by such restricted spinor, and put˜ (cid:15) = 0. We focus now on the localization of the Chern-Simons path integral, without coupling tothe matter multiplet. Ignoring the issue of gauge fixing, we would add to the action the local-izing term tS loc = tδ V , with t ∈ R + a parameter, δ being the supersymmetry transformationgenerated by the conformal Killing spinor (cid:15) described in the last section, and V some fermionicfunctional whose bosonic part is positive semi-definite. At the end of Section 4.4, we pointedout that the Super Yang-Mills Lagrangian is an example of δ -exact term, so we put S loc := 2 S Y M = (cid:90) S d x √ g Tr (cid:18) i ˜ λγ µ D µ λ + 12 F µν F µν + D µ σD µ σ + i ˜ λ [ σ, λ ]++ (cid:16) D + σr (cid:17) − r ˜ λλ (cid:19) (5.193)whose bosonic part is indeed positive semi-definite. This localizing term can be derived alsofrom the functional [49] V = (cid:90) S d x √ g Tr (cid:16) ( δ ˜ λ ) λ (cid:17) (5.194) Again, we use Roman letters as “flat” indices, and Greek letters as “curved” indices. Say, the standard basis given by the Pauli matrices, T i := σ i / √ hapter 5. Localization for circle actions in supersymmetric QFT 121analogously to the one used in the previous chapter for the gauge multiplet. S Y M being super-symmetric means that δ = 0 on V , making the localization principle applicable. As usual, thelimit t → ∞ localizes the path integral on the configurations that make this term vanish: theterms involving bosonic fields are separately non-negative, while the gaugino and its conjugatehave to vanish identically. Summarizing, the localization locus is given by λ = ˜ λ = 0 F = 0 ⇒ A = 0 (up to a gauge transformation) σ = a ∈ g (constant) D = − r a (5.195)Keeping into account the gauge-fixing procedure (as we should), the ghost c , anti-ghost ˜ c and Lagrange multiplier b are added to the theory, taking value in the Lie algebra g , togetherwith the BRST differential δ B that acts as δ B X = − [ c, X ] δ B c = −
12 [ c, c ] δ B ˜ c = b δ B b = 0 (5.196)where X is any field in the original theory, acted by a gauge transformation parametrized by c . The BRST differential is nilpotent, δ B = 0. The total differential Q := δ (cid:15) + δ B (5.197)acts now as the equivariant differential for the ( U (1) (cid:111) G )-equivariant cohomology in the BRST-augmented field space. The original CS action is automatically Q -closed since it is gaugeinvariant, so we can combine the localization principle with the gauge-fixing procedure addingto the Lagrangian the term Q (cid:18) ( δ ˜ λ ) λ − ˜ c (cid:18) ξ b − ∇ µ A µ (cid:19)(cid:19) (5.198)where we suppressed the Lie algebra bilinear Tr for notational convenience. Since the firstterm is gauge invariant, δ B (cid:16) ( δ ˜ λ ) λ (cid:17) = 0, this gives the same localization term as before. If δ [ghosts] = 0 on the gauge-fixing subcomplex, the second term gives Q (cid:18) ˜ c (cid:18) ξ b − ∇ µ A µ (cid:19)(cid:19) = ξ b − b ∇ µ A µ + ˜ c ∇ µ D µ c + ˜ c ∇ µ δA µ . (5.199)The first two terms give, upon path integration over b , the usual gauge-fixing Lagrangian inthe R ξ -gauge; the third term is the ghost Lagrangian. The fourth term ∝ (cid:16) ˜ c ∇ µ ˜ λγ µ (cid:17) does notchange the partition function: if we see this term as a perturbation of the gauge-fixed action, alldiagrams with insertion of (cid:16) ˜ c ∇ µ ˜ λγ µ (cid:17) will vanish, since ˜ c is coupled only to c via the propagatorbut there are no vertices containing c . In other words, the fermionic determinant arising fromthe path integration over ghosts is not changed by this term. The modified localizing term(5.199) is Q -closed: the old localizing term because of gauge invariance and supersymmetry,while the gauge-fixing and ghost terms follows by Q A µ = 0 that is easy to check. After pathintegration over the auxiliary b the limit t → ∞ finally localizes the theory to the same locus(5.195), with ghosts put to zero.22 5.5. Localization of N=2 Chern-Simons theory on the 3-sphereEvaluating the classical action at the saddle point configuration, we get S CS [ a ] = k π (cid:90) S d x √ g Tr (cid:18) − r a (cid:19) = − kπr Tr( a ) (5.200)where we used vol( S ) = 2 π r . The supersymmetric Wilson loop observable (5.186) localizesto W R ( C ) = 1dim R Tr R (cid:0) e πra (cid:1) (5.201)since the curve C is a great circle of radius r . Integrating as usual the rescaled fluctuationsabove the localization configuration, and taking the limit t → ∞ as in (5.76), the partitionfunction and the Wilson loop expectation value are thus given by a finite-dimensional integralover g with Gaussian measure, the “matrix model” Z = (cid:90) g da e − kπr Tr( a ) Z g − loop [ a ] (cid:104) W R ( C ) (cid:105) = 1 Z dim R (cid:90) g da e − kπr Tr( a ) Z g − loop [ a ]Tr R (cid:0) e πra (cid:1) . (5.202)As we pointed out in the last section, the integration over the Lie algebra g can be reducedover its Cartan subalgebra h , exploiting the gauge invariance of the matrix model under theadjoint action of g itself. This for example means, in the case of a matrix gauge group, thatwe integrate over the diagonalized matrices “fixing the gauge” of the matrix model. Thecorresponding Faddeev-Popov determinant is also called Vandermonde determinant , (cid:89) α ( ρ α ( a )) (5.203)where the product runs over the roots of g . There is left an overcounting given by the possiblepermutations of the roots, the action of the Weyl group W of g , cured dividing by its order |W| . The path integrals are thus rewritten as Z = 1 |W| (cid:90) h da (cid:89) α ( ρ α ( a )) e − kπr Tr( a ) Z g − loop [ a ] (cid:104) W R ( C ) (cid:105) = 1 Z |W| dim R (cid:90) h da (cid:89) α ( ρ α ( a )) e − kπr Tr( a ) Z g − loop [ a ]Tr R (cid:0) e πra (cid:1) . (5.204)Here we summarize the computation of the 1-loop determinant from [49]. For convenience,we put r = 1 and ξ = 1. Inserting the contribution of ghosts, the Lagrangian for the localizingterm is given by (suppressing the Tr) L loc = 12 F µν F µν + D µ σD µ σ +( D + σ ) + i ˜ λ /Dλ + i [˜ λ, σ ] λ −
12 ˜ λλ + ∂ µ ˜ cD µ c − b + b ∇ µ A µ . (5.205)Considering the limit t → ∞ , we rescale as usual the fields around the configuration (5.195): σ = a + σ (cid:48) / √ t, D = − a + D (cid:48) / √ t, X = X (cid:48) / √ t, (5.206)where X are all the fields without zero modes, and then rename σ (cid:48) → σ , D (cid:48) → D , X (cid:48) → X . Inthe limit, only quadratic terms in the fluctuations survive, L loc ∼ ∂ [ µ A ν ] ∂ [ µ A ν ] − [ A µ , a ] + ( ∂σ ) + ( D + σ ) + i ˜ λ / ∇ λ + i [˜ λ, a ] λ −
12 ˜ λλ + | ∂ ˜ c | − b + b ∇ µ A µ . (5.207)hapter 5. Localization for circle actions in supersymmetric QFT 123The resulting theory is free, and we can integrate it giving the corresponding 1-loop determinant.We will neglect all overall normalization constant from the Gaussian integrations. The integralover the auxiliary field b gives the gauge fixing term − ( ∇ µ A µ ) . The contribution from D ispurely Gaussian and can be integrated out removing the corresponding term. The integrationover σ gives a determinant det ( ∇ ) − / , and the (Grassman) integral over the ghosts givesdet ( ∇ ). It is useful to separate the gauge field as (Helmolz-Hodge decomposition) A µ = B µ + ∂ µ φ with φ scalar and B µ divergenceless, ∇ µ B µ = 0. With this decomposition, the Lorentz gaugecondition becomes ∇ φ = 0, and we can integrate φ giving a determinant det ( ∇ ) − / , thatcancels the above two other contributions. We are left with − B µ ∆ B µ − [ a, B µ ] + i ˜ λ / ∇ λ + i [˜ λ, a ] λ −
12 ˜ λλ (5.208)where ∆ is the vector Laplacian. Now we use the fact that the path integral can be reducedover the Cartan subalgebra of g , considering a ∈ h , and B µ = B ( h ) µ + B αµ e α (5.209)where B ( h ) µ is the component of B µ along h , and similarly for the gaugino. This component doesnot enter in the Lie brackets with a , so its contribution to the path integral is independent of a , and we drop it. The remaining interesting terms are (cid:88) α (cid:18) B − αµ ( − ∆ + ρ α ( a ) ) B αµ + ˜ λ − α (cid:18) i / ∇ + iρ α ( a ) − (cid:19) λ α (cid:19) (5.210)where the a -dependent kinetic terms are clearly identified, and the component fields appearingare real or complex valued scalars and spinors. The Gaussian integration over these fields leadto the determinant factors Z g − loop [ a ] = (cid:89) α det (cid:0) i / ∇ + iρ α ( a ) − (cid:1) det ( − ∆ + ρ α ( a ) ) / . (5.211)Now, using the fact that the eigenvalues of the Laplacian on divergenceless vectors are( l + 1) with degeneracy 2 l ( l + 2), and the eigenvalues of i / ∇ are ± (cid:0) l + (cid:1) with degeneracy l ( l + 1), where l ∈ Z + , the corresponding determinants can be written as infinite products (cid:89) α ∞ (cid:89) l =1 ( l + iρ α ( a )) l ( l +1) ( − l − iρ α ( a )) l ( l +1) (( l + 1) + ρ α ( a ) ) l ( l +2) = (cid:89) α ∞ (cid:89) l =1 ( l + iρ α ( a )) ( l +1) ( l − iρ α ( a )) ( l − (5.212)where the equality follows after some simplifications. Since roots come in pairs ( ρ α , − ρ α ), takingthe square of this one gets (cid:0) Z g − loop [ a ] (cid:1) = (cid:89) α ∞ (cid:89) l =1 ( l + ρ α ( a ) ) ( l +1) ( l + ρ α ( a ) ) ( l − = (cid:89) α ∞ (cid:89) l =1 (cid:0) l + ρ α ( a ) (cid:1) . (5.213)Collecting a factor l the product splits in the factorization formula for the hyperbolic sine,sinh( πz ) πz = ∞ (cid:89) l =1 (cid:18) z l (cid:19) (5.214)24 5.5. Localization of N=2 Chern-Simons theory on the 3-sphereand an a -independent divergent part that can be regularized with the zeta-function method, ∞ (cid:89) l =1 l = e (cid:80) ∞ l =1 log( l ) = e − ζ (cid:48) (0) = e π ) . (5.215)Up to an overall normalization constant, the a -dependence of the 1-loop determinant is finallygiven by Z g − loop [ a ] = (cid:89) α (cid:18) πρ α ( a )) πρ α ( a ) (cid:19) (5.216)where we see cancellation between the denominator and the Vandermonde determinant (5.203).Collecting the above results, the localization formulas for the partition function and theexpectation value of the supersymmetric Wilson loop in the pure CS theory are Z ∼ |W| (cid:90) h da e − kπ Tr( a ) (cid:89) α (2 sinh( πρ α ( a ))) (cid:104) W R ( C ) (cid:105) = 1 Z |W| dim R (cid:90) h da e − kπ Tr( a ) Tr R (cid:0) e πa (cid:1) (cid:89) α (2 sinh( πρ α ( a )))= 1dim R (cid:82) h da e − kπ Tr( a ) Tr R ( e πa ) (cid:81) α (2 sinh( πρ α ( a ))) (cid:82) h da e − kπ Tr( a ) (cid:81) α (2 sinh( πρ α ( a ))) . (5.217)These general localization formulas can be tested comparing their results for specific choicesof G to perturbative calculations, for example. In the case of U ( N ) gauge group, the integralover the Cartan subalgebra is an integral over diagonal matrices a = diag( λ , · · · , λ N ), and theroots are given by ρ ij ( a ) = λ i − λ j for i (cid:54) = j . The Weyl group is S N , thus |W| = N !. If we takethe Wilson loop in the fundamental representation, from (5.217) we get Z ∼ N ! (cid:90) (cid:32)(cid:89) i dλ i e − kπλ i (cid:33) (cid:89) i (cid:54) = j π ( λ i − λ j )) , (cid:104) W N ( C ) (cid:105) = 1 ZN ! N (cid:90) (cid:32)(cid:89) i dλ i e − kπλ i (cid:33) (cid:0) e πλ + · · · + e πλ N (cid:1) (cid:89) i (cid:54) = j π ( λ i − λ j )) , (5.218)that are sums of Gaussian integrals, and can be computed exactly. The result for the Wilsonloop expectation value is (cid:104) W N ( C ) (cid:105) = 1 N e − Niπ/k sin (cid:0) πNk (cid:1) sin (cid:0) πk (cid:1) , (5.219)which is known as the exact result [83], up to the overall phase factor e − Niπ/k . This kind ofphase factors arise in perturbative calculations in the so-called framing of the Wilson loop. Aperturbative calculation of the Wilson loop involves computations of correlators of the type (cid:104) A µ ( x ) A µ ( x ) · · · (cid:105) , where x , x , · · · are coordinates of points on the image of the curve C .This contribution diverges when x = x , so it is necessary to choose some regularization schemeto perform the computations. For example, considering the 2-point function (cid:104) A µ ( x ) A µ ( y ) (cid:105) ,this clashing of points can be avoided requiring that y is integrated over a shifted curve C f such that C µf ( τ ) = C µ ( τ ) + α n µ ( τ ) (5.220)hapter 5. Localization for circle actions in supersymmetric QFT 125where n is orthogonal to ˙ C . The choice of such an orthogonal component (frame) at everypoint on the curve is called framing. Even if at the end of the calculation one takes α →
0, thisprocedure leaves a deformation-dependent term, that in pure U ( N ) CS is e iπNk χ ( C,C f ) (5.221)where χ ( C, C f ) is a topological invariant that takes integer values corresponding to the numberof times the path C f winds around C . We see that localization produces an expectation valueat framing -1 (see also [98] for a detailed discussion about framing). We turn now to the result for the localization of the matter-coupled theory. This is ofcourse gauge invariant, so the equivariant differential acts effectively as Q ∼ δ , since the ghostsector has been already considered in the previous paragraph. This means that, following thelocalization principle, we have to extend the matter action with a δ -exact term. We are freeto consider the canonical choice (5.81) as in [49], or using the fact that [50] the matter action(5.183) is actually given by a supersymmetry variation, as the case of the YM action. Thismeans that we can consider the localizing terms tS m + tS Y M + ghosts or, schematically t (cid:90) δ (cid:16) ( δψ ) † ψ + ˜ ψ ( δ ˜ ψ ) † (cid:17) + tS Y M + ghosts that have positive semi-definite bosonic parts. The second term in both choices is the oneanalyzed in the previous paragraph, and gives the same localization locus for the gauge andghost sector, while both the first terms vanishes for the field configurations ψ = 0 , φ = 0 , F = 0 . (5.222)This means that the classical action of the matter sector does not contribute to the partitionfunction, but only in the 1-loop determinant. Expanding the fields around this configurationand scaling the fluctuations with the usual 1 / √ t factor, we see that there are no couplings tothe gauge sector fluctuations that survive in the t → ∞ limit, but only to the zero mode a of σ . Thus the determinant factorizes as Z − loop [ a ] = Z g − loop [ a ] Z m − loop [ a ] . (5.223)If matter is present in different copies of chiral multiplets, in maybe different representationsof the gauge group, the determinant factorizes in the same way for each multiplet.The determinant for the matter sector can be computed diagonalizing the the kinetic op-erators acting on the scalar and the fermion field, after having integrated out the auxiliary F ,and considering the path integration over the Cartan subalgebra with a ∈ h . In particular, therelevant kinetic operators that have to be diagonalized are K ( ρ ) b = (cid:18) −∇ + ρ ( a ) − iρ ( a ) + 34 (cid:19) , K ( ρ ) f = (cid:0) i / ∇ + iρ ( a ) (cid:1) , (5.224)for the (complex) bosonic and fermionic parts, where a is regarded as acting on the repre-sentation R with weights { ρ } . The eigenvalues of −∇ are 4 j ( j + 1) with j = 0 , , · · · with26 5.5. Localization of N=2 Chern-Simons theory on the 3-spheredegeneracy (2 j + 1) , that we can rewrite as l ( l + 2) with degeneracy ( l + 1) and l = 0 , , · · · .The eigenvalues of i / ∇ are ± (cid:0) l + (cid:1) with degeneracy l ( l + 1), with l = 1 , , · · · . Thus the oneloop determinant results, after a change of dummy index and some simplifications Z m − loop [ a ] = (cid:89) ρ det( K f )det( K b )= (cid:89) ρ ∞ (cid:89) l =1 (cid:0) l + + iρ ( a ) (cid:1) l ( l +1) (cid:0) l + − iρ ( a ) (cid:1) l ( l +1) (cid:0) l + + iρ ( a ) (cid:1) l (cid:0) l − − iρ ( a ) (cid:1) l = (cid:89) ρ ∞ (cid:89) l =1 (cid:18) l + + iρ ( a ) l − − iρ ( a ) (cid:19) l (5.225)This product can be regularized using the zeta-function. We refer to [50] for the details of thecomputation, and report here the result in the case the fields take value in a self-conjugaterepresentation R of the gauge group: Z m − loop [ a ] = (cid:89) ρ (2 cosh ( πρ ( a ))) − / (5.226)where now a ∈ R ( h ) and ρ ( a ) is the weight of the Cartan element in the representation R .Summarizing, we have seen that the application of the supersymmetric localization principleto the matter-coupled SCS theory on S reduces the path integral to a finite-dimensional integraldescribing a matrix model over the Lie algebra of the theory. Using the notation (5.128), thelocalization formulas for the partition function and the supersymmetric Wilson loop expectationvalue, with matter multiplets coming in self-conjugate representations R ⊕ R ∗ , R ⊕ R ∗ , · · · are Z = 1 |W| (cid:90) h da e − kπ Tr( a ) det ad πa )(det R πa )) (det R πa )) · · ·(cid:104) W R ( C ) (cid:105) = 1 Z |W| dim R (cid:90) h da e − kπ Tr( a ) Tr R (cid:0) e πa (cid:1) det ad πa )(det R πa )) (det R πa )) · · · (5.227) ABJM theory is a special type of matter-coupled SCS theory in 3-dimensions constructedin [99], that has the interesting property to be dual under the AdS/CFT conjecture to a certainorbifold background in M-theory. It consists of two copies of N = 2 SCS theory, each onewith gauge group U ( N ), and opposite levels k, − k . In addition, the are four matter (chiraland anti-chiral) supermultiplets Φ i , ˜Φ i , with i = 1 ,
2, in the bi-fundamental representation of U ( N ) × U ( N ), ( N , ¯ N ) and ( ¯ N , N ). This field content can be represented as the quiver in Fig.5.1. The superpotential for the matter part is given by W = 4 πk (Φ ˜Φ Φ ˜Φ − Φ ˜Φ Φ ˜Φ ) , (5.228) For example, if R = S ⊕ S ∗ . hapter 5. Localization for circle actions in supersymmetric QFT 127Figure 5.1: The quiver for ABJM theory. The two nodes represent the gauge multiplets, withthe convention of specifying the level of the CS term. The oriented links represent the mattermultiplets in the bi-fundamental and anti-bi-fundamental representations.and this structure actually enhance the supersymmetry of the resulting theory to N = 6. Ifnow a = diag( λ , · · · , λ N , ˆ λ , · · · , ˆ λ N ), the weights in the bi-fundamental representations are ρ ( N, ¯ N ) i,j ( a ) = λ i − ˆ λ j , ρ ( ¯ N,N ) i,j ( a ) = ˆ λ j − λ i . (5.229)Plugging this information into (5.227), the partition function in this case localizes to thefollowing matrix model, Z ∼ N ! N ! (cid:90) (cid:32)(cid:89) i dλ i d ˆ λ i e − kπ ( λ i − ˆ λ i ) (cid:33) (cid:81) i (cid:54) = j (cid:16) π ( λ i − λ j ))2 sinh( π (ˆ λ i − ˆ λ j )) (cid:17)(cid:81) i,j (cid:16) π ( λ i − ˆ λ j )) (cid:17) . (5.230)The circular Wilson loop under consideration can be called now with respect to theenhanced supersymmetry of the model. Its expectation value in the fundamental representationis obtained by plugging a factor (1 /N ) (cid:80) i e πλ i as before. This matrix model cannot be solvedexactly as in the case of the pure CS discussed above, but can be studied in the N → ∞ limitwith the saddle-point technique showed in Section 5.4.5 [50, 94]. We also mention that, in thisparticular theory with enhanced N = 6 supersymmetry, it was possible to construct a Wilson loop (so invariant under half of the N = 6 supersymmetry algebra). The latter can besolved applying the same localization scheme that brings to the matrix model describing the1/6 BPS Wilson loop presented above [100, 101]. A compact review introducing the state ofthe art on recent results about supersymmetric Wilson loops in ABJM and related theories canbe found in [84]. This is not apparent from the original action, but can be realized noticing that the superpotential has an SU (2) × SU (2) symmetry that rotates separately the Φ i and the ˜Φ i . This, combined with the original SU (2) R symmetry of the theory, gives an SU (4) ∼ = Spin (6) symmetry that acts non-trivially on the supercharges. Thusthe final theory has to have an enhanced N = 6 supersymmetry. hapter 6Non-Abelian localization and 2d YMtheory In this chapter we are going to summarize the result obtained mainly in [15] by Witten. Thiswas the first attempt in the physics literature of extending the equivariant localization formalismto possibly non-Abelian group actions. In that work, a modified definition of equivariantintegration was defined, and this allowed for an extension of the same procedure discussed inChapter 3 to show the localization property of integrals computed over spaces with genericsymmetry group G . This new formalism was applied to the study of 2-dimensional Yang-Mills(YM) theory over a Riemann surface, a relatively simple model from the physical point ofview, but with a very rich underlying mathematical structure. In the following, we are goingfirst to review the geometry of this special model, in connection with the symplectic geometryintroduced in Section 3.3, as a motivation for the more mathematical discussion about theWitten’s equivariant integration and non-Abelian localization principle that will follow. Next,we will review the ideas underlying the application of this new localization principle to theYM theory, and how this application results in a “mapping” between this model and a suitabletopological theory, establishing the topological nature of the YM theory in the weak couplinglimit. In the final section, we will summarize the interpretation given by the localizationframework to the already existing solution for the partition function of this model.As we pointed out in the Introduction, other generalizations of the Duistermaat-Heckmantheorem to non-Abelian Hamiltonian systems also appeared in the mathematical literature, asthe result obtained by Jeffrey and Kirwan in [16]. Other applications of this extended formalismfollowed, and Witten’s approach was used for example more recently to describe Chern-Simonstheories over a special class of 3-manifolds in [102]. In the next section we are going to review Witten’s extension of the equivariant localizationprinciple to possibly non-Abelian group actions, and a generalization of the DH formula in thisdirection. In [15] this was applied to reinterpret the weak coupling limit of pure YM theory ona Riemann surface. This theory is exactly solvable, in the sense that its partition function canbe expressed in closed form, and its zero-coupling limit is known to describe a topological fieldtheory. These features make 2-dimensional YM theory very appealing from the mathematicalstructure it carries, and make it possible to compare results or interpretations obtained via this“new” localization method with already existing solutions of the problem.12930 6.1. Prelude: moment maps and YM theoryWe are going to discuss more about the topological interpretation of 2d YM theory later,while in this section we review some results introduced by Atiyah and Bott [103] about thesymplectic structure underlying this special QFT. This can be useful to contextualize the genericdiscussion of the next section, and it prepares the ground for the formal application of the non-Abelian localization principle.We start by considering the partition function of YM theory on a compact orientable Rie-mannian manifold Σ of arbitrary dimension, Z ( (cid:15) ) = 1vol( G ( P )) (cid:18) π(cid:15) (cid:19) dim( G ) / (cid:90) A ( P ) DA e − S [ A ] ,S [ A ] = − (cid:15) (cid:90) Σ Tr( F A ∧ (cid:63)F A ) . (6.1)Here (cid:15) := g Y M is the square of the YM coupling constant. To describe the rest of the ingredients,let us recall the geometry underlying the gauge theory (to fill some of the details, see AppendixA.1). The dynamical field here is the connection A ∈ Ω( P ; g ) on a principal G -bundle P π −→ Σ,where G is a compact connected Lie group with Lie algebra g . The path integral is thus takenover the space A ( P ) of G -equivariant vertical 1-forms with values in g , that is naturally an affine space modeled on the infinite-dimensional vector space a of G -equivariant horizontal 1-forms with values in g . This gives to A ( P ) the structure of an infinite-dimensional manifold,whose tangent spaces are T A A ( P ) ∼ = a ∼ = Ω (Σ; ad( P )), where we identified horizontal formsover P with forms over the base Σ. In other words, any vector field α ∈ Γ( T A ( P )) can beexpanded locally as α = α aµ T a ⊗ dx µ , α aµ ∈ C ∞ (Σ × A ( P )) , (6.2)with coefficients that depend on the point A ∈ A ( P ) and p ∈ Σ. The curvature F A = dA + [ A ∧ ,A ] of the connection A is a horizontal 2-form over P , so we can identify it as a 2-form on theadjoint bundle without loss of information, F A ∈ Ω (Σ; ad( P )). As such, it can be integratedas a differential form over Σ. In the action S [ A ], “Tr” represents a (negative definite) invariantinner product on g , and (cid:63) is the Hodge dual operation, that is identified by the presence of ametric on Σ. G ( P ) ∼ = Ω (Σ; Ad( P ) is the group of gauge transformations, that is locally equivalent tothe space of G -valued functions over Σ, and acts naturally on A ( P ). If φ ∈ Lie ( G ( P )) ∼ =Ω (Σ; ad( P )) is an element of the Lie algebra of infinitesimal gauge transformations, its asso-ciated fundamental vector field at the point A ∈ A ( P ) is φ A ≡ δ φ A = ∇ A φ = dφ + [ A, φ ] . (6.3)The path integral measure DA can be defined formally as the Riemannian measure inducedby a metric on the affine space A ( P ). The latter can be induced by the metrics on Σ and on g , and defined pointwise in A ( P ) as( α, β ) A := − (cid:90) Σ Tr( α A ∧ (cid:63)β A ) (6.4) Recall that horizontality means essentially to have components only in the “directions” of the base space,and the G -equivariance ensures the right transformation behavior as forms valued in the adjoint bundle ad( P ),the associated bundle to P that has g as typical fiber. Thus a ∼ = Ω (Σ; ad( P )). The definition of the Hodge star is, implicitly, α ∧ (cid:63)β = g − ( α, β ) ω for any α, β ∈ Ω k (Σ). Here g − is the “inverse” metric on Σ, that extends multi-linearly its action on every tangent space as g − ( α, β ) = g µ ν · · · g µ k ν k α µ ··· µ k β ν ··· ν k . ω is a volume form (that can be induced by g , for example). The Hodge starsatisfies the property (cid:63) α = ( − k (dim(Σ) − k ) α . hapter 6. Non-Abelian localization and 2d YM theory 131for every α A , β A ∈ Ω (Σ; ad( P )). With this definition, the YM action can be rewritten as S [ A ] = 12 (cid:15) ( F, F ) A . (6.5)We can now specialize the discussion to the case in which dim(Σ) = 2, i.e. the base space isa Riemann surface. It is a well-known fact in geometry that any Riemann surface is a K¨ahlermanifold : it admits a Riemannian metric g , a symplectic form ω (that can be a choice of volumeform), and a complex structure J such that the compatibility condition g ( · , · ) = ω ( · , J ( · )) issatisfied. This special property holds also for A ( P ), since in addition to the metric (6.4) wecan define the symplectic form Ω ∈ Ω ( A ( P )) such thatΩ A ( α, β ) := − (cid:90) Σ Tr( α A ∧ β A ) , (6.6)and the complex structure on T A ( P ) is provided by the Hodge duality, (cid:63) : Ω (Σ; ad( P )) → Ω (Σ; ad( P )) such that (cid:63) = −
1. Then the compatibility condition is immediately satisfied,since ( · , · ) = Ω( · , (cid:63) ( · )). The fact that Ω is symplectic can be seen by noticing that, in any basis,it has constant components ( i.e. independent from A ∈ A ( P )):Ω µνab ( A ) = Ω A ( T a ⊗ dx µ , T b ⊗ dx ν ) = − Tr( T a T b ) ε µν (cid:18)(cid:90) Σ dx dx (cid:19) ∈ R . (6.7)The non-degeneracy follows from the non-degeneracy of Tr and of (cid:82) Σ , and the skew-symmetryis obvious from the definition. Thus A ( P ) is K¨ahler.For our applications, we focus on the fact that A ( P ) has now a canonical symplectic struc-ture. It is natural to wonder if it possible to extend all the machinery that we introduced inSection 3.3 also to this case, and in particular if the G ( P )-action on A ( P ) results to be sym-plectic or Hamiltonian with respect to Ω. The answer was given by in [103], and we state it inthe following theorem. Theorem 6.1.1 (Atiyah-Bott) . In 2-dimensions, the group G ( P ) of gauge transformations actsin an Hamiltonian way on A ( P ), with a moment map identified by the curvature F . Proof.
To see this, let us introduce the moment map as µ : Lie ( G ( P )) → C ∞ ( A ( P )) such that µ φ ( A ) := (cid:104) F A , φ (cid:105) = − (cid:90) Σ Tr( F A φ ) , (6.8)and check that the Hamiltonian property is satisfied. For every α ∈ Γ( T A ( P )) and φ ∈ Lie ( G ( P )), we compute( ι φ Ω A )( α ) = Ω A ( φ, α ) = − (cid:90) Σ Tr( ∇ A φ ∧ α A ) = (cid:90) Σ Tr( φ ∇ A α A ) , (6.9) δµ φ | A ( α ) = − (cid:90) Σ Tr (cid:0) F A + α φ − F A φ (cid:1) = − (cid:90) Σ Tr (cid:0) φ ∇ A α A (cid:1) , (6.10) A complex structure on a vector space V is an isomorphism J : V → V such that J = − id V . It intuitivelyplays the role of “multiplication by i ” when one considers the complexified V C := V ⊗ C , allowing for adecomposition of V C in a holomorphic subspace (generated by the eigenvectors with eigenvalue + i ) and anti-holomorphic subspace (generated by the eigenvectors with eigenvalue − i ). A manifold M has almost complexstructure if there is a tensor J ∈ Γ( T M ) that acts as a complex structure in every tangent space. If theholomorphic decomposition can be extended on an entire neighborhood of every point by a suitable choice ofcoordinates, M has complex structure , and admits an atlas of holomorphic coordinates. Riemann surfaces canthus be thought as 2-dimensional real manifolds, or 1-dimensional complex manifolds.
32 6.1. Prelude: moment maps and YM theorywhere δ is the de Rham differential on A ( P ), that acts in the usual sense of variational calculus.We see that ι φ Ω = − δµ φ , thus µ provides a correct moment map for the G ( P )-action. If weidentify Lie ( G ( P )) with Lie ( G ( P )) ∗ through the pairing (cid:104)· , ·(cid:105) introduced above, and regard thecurvature F : A ( P ) → Ω (Σ; ad( P )) as an element of C ∞ ( A ( P )) ⊗ Lie ( G ( P )) ∗ , we can simplywrite that µ ≡ F . (cid:4) Another corollary of A ( P ) being K¨ahler is that the path integral measure DA is formallyequivalent to the Liouville measure induced from Ω, since by compatibility of the structuresthe latter is equivalent to the Riemannian measure induced by ( · , · ). Since we are working onan infinite-dimensional space, we can write this measure formally as DA = exp(Ω) , (6.11)as we did in (3.43) but with n = ∞ . With this identification, we see that the path integral ofthe 2-dimensional YM theory acquires the very suggestive form Z ( (cid:15) ) ∝ (cid:90) A ( P ) exp (cid:18) Ω − (cid:15) ( µ, µ ) (cid:19) . (6.12)This path integral resembles very much an infinite-dimensional version of the type of integralswe treated when discussing the Duistermaat-Heckman localization formula in Section 3.3, butwith the fundamental difference that now the exponent of the integrand is not the momentmap, but its square. We will return to this point in the next section.Here we notice that in the weak coupling limit (cid:15) → S = (cid:15) ( µ, µ ), that is the space of solutions of theclassical equations of motion ∇ A (cid:63) F A = 0. Every one of these contributions brings roughly aterm that decays as ∼ exp ( − /(cid:15) ) to the partition function, the main one being determined bythe absolute minimum at µ = 0, the subspace of flat connections µ − (0) ⊂ A ( P ). Eliminatingthe redundancy from the gauge freedom of the theory, the most interesting piece of the physical field space, especially in the weak coupling limit, is thus determined by the quotient A := µ − (0) (cid:30) G ( P ) , (6.13)or in other words, when computing the path integral one is interested in the G ( P )-equivariantcohomology of µ − (0), H ∗G ( µ − (0)) ∼ = H ∗ ( A ). The quotient A is the moduli space of flatconnections . It turns out that this space has a nice interpretation in symplectic geometry interms of symplectic reduction . A theorem by Marsden-Weinstein-Meyer (MWM) [34, 104, 105]in fact states that, in a generic Hamiltonian G -space ( M, ω, G, µ ), if the zero-section of themoment map µ − (0) ⊂ M is acted on freely by G , then the base space M := µ − (0) /G of theprincipal G -bundle µ − (0) π −→ M is a symplectic manifold, with symplectic form ω ∈ Ω ( M )satisfying ω | µ − (0) = π ∗ ( ω ) . (6.14)In other words, the restriction of ω to µ − (0) is a basic form, completely determined by asymplectic form ω on the base space. The space ( M , ω ) is called Marsden–Weinstein quotient,symplectic quotient or symplectic reduction of M by G . Returning to the case of YM theory, Symplectic reduction in classical mechanics on M = R n occurs when one of the momenta is an in-tegral of motion, 0 = ˙ p n = − ∂ n H . In that case, one can solve the system in the reduced coordinates( q , · · · , q n − , p , · · · , p n − ) and then solve for the n th coordinate separately. The MWM theorem essentiallygeneralizes this process in a fully covariant setting. hapter 6. Non-Abelian localization and 2d YM theory 133this means that in the limit (cid:15) →
0, when the path integral is reduced to A by gauge fixing,the symplectic form can be reduced without loss of information on this base space. In the nextsection we will see that, applying localization, this is extended to the whole exponential. In the last section we found an Hamiltonian interpretation of the system ( A ( P ) , Ω , G ( P ) , µ ≡ F ) for YM theory on a 2-dimensional Riemann surface Σ. Here we would like to make contactwith the DH formula, that we described for analogous systems in finite-dimensional geometry.We notice that the main differences with the case treated in Section 3.3 are essentially two: G ( P ) is non-Abelian in general for non-Abelian gauge groups G , and the path integral is notin the form of an oscillatory integral of the DH type. Indeed, schematically we haveDH → (cid:90) exp ( ω + iµ ) , YM → (cid:90) exp (cid:18) Ω − | µ | (cid:19) . (6.15)In the following, we will describe the solution proposed in [15] to generalize the DH formulato the non-Abelian case starting from the first integral in (6.15), and how this procedure canbe used to recover the second one, of the YM type. We consider a generic Hamiltonian system( M, ω, G, µ ) with compact semisimple Lie group G of dimension dim( G ) = s , and the associatedCartan model defined by the space Ω G ( M ) = ( S ( g ∗ ) ⊗ Ω( M )) G of equivariant forms, on whichwe defined the action of the extended operators d C = 1 ⊗ d − iφ a ⊗ ι a , L a ≡ ⊗ L a + L a ⊗ L a φ b = f bac φ c ,ι a ≡ ⊗ ι a . (6.16)An element α ∈ Ω G ( M ) is an invariant polynomial in the generators φ a of S ( g ∗ ), with differentialforms on M as coefficients. This means that integration over M provides a map in equivariantcohomology of the type (cid:90) M : H ∗ G ( M ) → S ( g ∗ ) G , (6.17)or in other words that the integral of an equivariant form is in general a polynomial in the φ a .This is not quite satisfactory, as we would like an integration that generalizes the standard deRham case, giving a map H ∗ G ( M ) → R (or C ). In the case of G = U (1) we often solved thisproblem by setting the unique generator φ = − φ = i in this conventions), thus constructinga map in the localized cohomology, (cid:82) M : H ∗ U (1) ( M ) φ → R . Here in the non-Abelian case, thetrick of algebraic localization is not so trivial in practice, and we avoid it.An alternative and fruitful idea to saturate the φ -dependence is to make them dynamicalvariables, and integrate over them too. Since φ a can be regarded as an Euclidean coordinateover g , this means defining an integration over M × g . As a vector space, the Lie algebra has anatural measure d s φ that is unique up to a multiplicative factor. We fix that factor by choosing We adopt Witten’s conventions and substitute φ a (cid:55)→ iφ a in the definition of the Cartan differential, analo-gously to the DH case of Section 3.3.
34 6.2. A localization formula for non-Abelian actionsa (positive-definite) inner product ( · , · ) on g ∗ and setting (cid:90) g d s φ e − (cid:15) ( φ,φ ) = (cid:18) π(cid:15) (cid:19) s/ , (6.18)essentially as we did in (5.132). Since our goal is to integrate equivariant forms that havepolynomial dependence on the φ a , or at most expressions of the form exp( φ a ⊗ µ a ) that haveexponential dependence, integrating over g with the bare measure d s φ would produce possibledivergences. To ensure convergence of these class of functions, the equivariant integration isdefined as [15] (cid:90) M × g α := 1vol( G ) (cid:90) M (cid:90) g d s φ (2 π ) s e − (cid:15) ( φ,φ ) α (6.19)where (cid:15) is inserted as a regulator. Notice that in general the limit (cid:15) → G ( M ), we can applythe equivariant localization principle to the present case. Let α ∈ Ω G ( M ) be an equivariantlyclosed form, so that d C α = 0, and choose an equivariant 1-form β ∈ Ω G ( M ) = Ω ( M ) G .The latter is independent on φ a , and plays the role of the localization 1-form. By the samearguments of Section 3.1, α and αe d C β are representatives of the same equivariant cohomologyclass in H ∗ G ( M ), and we can deform the integral of α as I [ α ; (cid:15) ] := (cid:90) M × g α = (cid:90) M × g αe td C β ∀ t ∈ R . (6.20)In particular, taking the limit t → ∞ , this integral localizes on the critical point set of thelocalization 1-form β . This can be seen simply by expanding the definition of equivariantintegration from (6.20), I [ α ; (cid:15) ] = 1vol( G ) (cid:90) M (cid:90) g d s φ (2 π ) s α exp (cid:16) − (cid:15) φ, φ ) − itφ a ( ι a β ) + tdβ (cid:17) = 1vol( G ) (cid:90) M (cid:90) g d s φ (2 π ) s α exp (cid:32) − (cid:15) φ, φ ) − t (cid:15) (cid:88) a ( ι a β ) + tdβ (cid:33) , (6.21)where in the second line we completed the square and shifted variable in the φ -integral. Sincethe term tdβ gives a polynomial dependence on t (by degree reasons, it is expanded up to afinite order), the limit t → ∞ converges and makes the integral localize on the critical points of ι a β . This shows the localization property of equivariant integrals in the non-Abelian setting. Iffor example we suppose α to be independent on the φ a , we can perform the Gaussian integrationto further simplify I [ α ; (cid:15) ], I [ α ; (cid:15) ] = 1vol( G )(2 π(cid:15) ) s/ (cid:90) M α exp (cid:32) − t (cid:15) (cid:88) a ( ι a β ) + tdβ (cid:33) . (6.22) The inner product on g ∗ is induced from an inner product on g , and when we write ( φ, φ ) we really mean (cid:80) a ( φ a , φ a ). This can be stated more formally defining φ := φ a ⊗ T a ∈ g ∗ ⊗ g , and then letting act the inner-product on the T a ’s, normalized in order to produce a Kronecker delta. We avoid this cumbersome notation,since the action of the various inner products is always clear from the context. hapter 6. Non-Abelian localization and 2d YM theory 135We now apply the above non-Abelian localization principle to the special case in which α = exp( ω − iφ a ⊗ µ a ), i.e. generalizing the DH formula of Section 3.3. We will suppress tensorproducts in the following, for notational convenience. First of all, it is straightforward to seethat this form is equivariantly closed, d C e ω − iφ a µ a ∝ d C ( ω − iφ a µ a ) = − iφ a ι a ω − iφ a dµ a = − iφ a ι a ω + iφ a ι a ω = 0 . (6.23)The DH oscillatory integral becomes, following the same steps of (6.20) and (6.21), Z ( (cid:15) ) = (cid:90) M × g ω n n ! e − iφ a µ a = 1vol( G ) (cid:90) M (cid:90) g d s φ (2 π ) s exp (cid:16) ω − iφ a µ a − (cid:15) φ, φ ) + t ( dβ − iφ a ( ι a β )) (cid:17) = 1vol( G )(2 π(cid:15) ) s/ (cid:90) M exp (cid:32) ω − (cid:15) ( µ, µ ) − t (cid:15) (cid:88) a ( ι a β ) + it(cid:15) (cid:88) a µ a ( ι a β ) (cid:33) , (6.24)and it is independent of t . Specializing to the case t = 0, we get (cid:90) M × g ω n n ! e − iφ a µ a = 1vol( G )(2 π(cid:15) ) s/ (cid:90) M exp (cid:18) ω − (cid:15) ( µ, µ ) (cid:19) , (6.25)that shows the equivalence of the YM type partition function and the equivariant integral of theDH type! If instead we take the limit t → ∞ , we see that the integral localizes on the criticalpoints of ( ι a β ). With a smart choice of localization 1-form, we can show that this localizationlocus coincides with the critical point set of the function S := ( µ, µ ). Proof.
Since (
M, ω ) is symplectic, it admits an almost complex structure J ∈ Γ( T M ) and aRiemannian metric G ∈ Γ( T M ) such that ω ( · , J ( · )) = G ( · , · ) (see [34], proposition 12.6). Wepick the localization 1-form β := dS ◦ J = J σν ∂ σ S dx ν = J σν (cid:88) a µ a ( ∂ σ µ a ) dx ν , and the localization condition ι a β = 0. Now we use the compatible metric G , that has com-ponents G µν = ω ( ∂ µ , J ( ∂ n u )) = ω µσ J σν . We consider its “inverse” G − acting on T ∗ M withcomponents G µν = J µσ ω νσ , where ω νσ are the components of the “inverse” symplectic form, andcompute the norm of the 1-form dS , G − ( dS, dS ) = ( ∂ µ SJ µσ ) ω νσ ∂ ν S = β σ (cid:88) a µ a ( ω νσ ∂ ν µ a ) = − (cid:88) a µ a β σ ( T a ) σ = − (cid:88) a µ a ( ι a β ) = 0 , where we used the Hamiltonian equation dµ a = − ι a ω and the localization condition ι a β = 0.By the non-degeneracy of G , this condition is equivalent to dS = 0, that precisely identifies thecritical points of S . (cid:4) Rephrasing the above result in the language of the last section, we just showed in generalterms that the 2-dimensional YM partition function localizes on the moduli space of solutions ofthe EoM, meaning that this theory is essentially classical. We remark again that this localization In the case of 2-dimensional YM theory, we recall that J ≡ (cid:63) is the Hodge duality operator.
36 6.2. A localization formula for non-Abelian actionslocus consists of two qualitatively different types of points: those that minimize absolutely S ,that is µ − (0) ⊂ M , and the higher extrema with µ (cid:54) = 0. The former ones in the gauge theoryare the flat connections, and they give the dominant contribution to the partition function.The latter ones decays exponentially in the limit (cid:15) → ∼ exp( − S/(cid:15) ). In general thus thepartition function can be written as a sum of terms coming from all these disconnected regionsof M , Z ( (cid:15) ) = (cid:88) n Z n ( (cid:15) ) . (6.26)Let us consider the dominant piece Z ( (cid:15) ) coming from µ − (0), that we interpret in the gaugetheory as the rough answer in the weak coupling limit, and that we can select by restrictingthe integration over M to a suitable neighborhood N of µ − (0), Z ( (cid:15) ) = 1vol( G ) (cid:90) N (cid:90) g d s φ (2 π ) s exp (cid:16) ω − iφ a µ a − (cid:15) φ, φ ) + td C ( dS ◦ J ) (cid:17) , (6.27)where we inserted the localization 1-form such that the t → ∞ limit identifies the critical locus µ − (0). Cohomological arguments show that, if G acts freely on µ − (0), this integral retractson the symplectic quotient M := µ − (0) /G , giving Z ( (cid:15) ) = (cid:90) M exp ( ω + (cid:15) Θ) (6.28)for some 4-form Θ ∈ Ω ( M ). In particular, we see that in the weak coupling limit (cid:15) → Z (0)gives the volume of the symplectic quotient M . Argument for (6.28) . The precise proof is technical and it can be found in [15], we only sketchthe main instructive ideas here. The neighborhood N is chosen small enough to be preservedby the G -action, and represents the split with respect to the normal bundle we used in Section4.2. Thus it retracts equivariantly onto µ − (0), meaning that it is homotopic to µ − (0) andthat the homotopy commutes with the G -action.First of all we recall what we noticed at the end of the last section: if G acts freely on µ − (0) the MWM theorem tells us that the symplectic form retracts on the symplectic quotient M := µ − (0) /G , so it does not contribute to the integration over the “normal directions” to M in N . Here we are not considering a simple integration over N , but an equivariant integration,that provides a map H ∗ G ( N ) → C . So in this case we consider the equivariantly closed extension˜ ω = ω − iφ a µ a as representative of a cohomology class [˜ ω ] ∈ H G ( M ). When restricted over N ,this class is the pull-back of a cohomology class [ ω ] ∈ H ( M ), since H ∗ G ( N ) ∼ = H ∗ G ( µ − (0)) ∼ = H ∗ ( M ), the first equivalence following from the retraction of N onto µ − (0) and the secondfrom the fact that the G -action is free on µ − (0) (these properties were explained in Chapter2). Thus we can substitute in the integral ˜ ω (cid:55)→ ω without changing the final result.The same kind of argument works for the term ( φ, φ ). It is easy to check that this is both G -invariant and equivariantly closed, so it represents an element (cid:2) ( φ, φ ) (cid:3) ∈ H G ( M ). Whenwe restrict it to N , as above, this class is the pull-back of some class [Θ] ∈ H ( M ), and wecan make the substitution ( φ, φ ) (cid:55)→ Θ in the integral without changing the final result.Since both ω and Θ are standard differential forms over M and thus independent of φ , theintegration over g goes along only with the remaining term d C ( dS ◦ J ). We already know thaton µ − (0) ⊂ N this term is zero, so one has to show that its integral over the normal directionsto µ − (0) in N produces a trivial factor of 1. In [15] it is proven that1vol( G ) (cid:90) F (cid:90) g d s φ (2 π ) s exp ( td C ( dS ◦ J )) = 1 , hapter 6. Non-Abelian localization and 2d YM theory 137where F is any fiber of the normal bundle to µ − (0) in N . From this (6.28) follows. (cid:4) Example 6.2.1 (The height function on the 2-sphere, again) . Beside the main application ofWitten’s localization principle to non-Abelian gauge theories, we try now to apply this newformalism to the old and simple example of the height function on the 2-sphere, to compareit with the results obtained in Chapter 3. Setting G = U (1), M = S , µ = cos( θ ) and ω = d cos( θ ) ∧ dϕ , the equivariant integration (6.19) of the DH oscillatory integral gives Z ( (cid:15) ) = (cid:90) S × g ωe − iφµ = 12 π (cid:90) π dϕ (cid:90) +1 − d cos θ (cid:90) + ∞−∞ dφ π exp (cid:16) − iφ cos θ − (cid:15) φ (cid:17) = 1 √ π(cid:15) (cid:90) +1 − dx e − x (cid:15) = 1 − I ( (cid:15) ) , where I ( (cid:15) ) := (cid:82) ∞ dx √ π(cid:15) exp( − x / (cid:15) ) is a trascendental error function. The three terms in thefinal result for Z ( (cid:15) ) (two of which are equal to − I ( (cid:15) )) correspond to the contributions of theextrema of (cos θ ) : the two maxima at θ = 0 , π contribute with − I ( (cid:15) ) and the minimum at θ = π/ (cid:15) →
0, since I (0) = 0.We see that in general the modified equivariant integration of this new formalism gives anincredibly complicated answer, when compared to the simple result of Example 3.2.1 obtainedvia the usual equivariant localization principle. Remark.
We notice that we could have expressed equivalently the whole dissertation above insupergeometric language, since we discussed in Section 4.1 that integration over M is equivalentto integration over Π T M . In these terms, maybe more common in QFT, we can introducecoordinates ( x µ , ψ µ , φ a ) over Π T M × g , where ψ µ := dx µ are Grassmann-odd, and interpretelements of Ω G ( M ) as elements of C ∞ (Π T M × g ) G . An equivariant form α is thus (locally)a G -invariant function of ( x, ψ, φ ). For example, the Cartan differential and the definition ofequivariant integration become d C = ψ µ ∂∂x µ − iφ a T µa ∂∂ψ µ , (cid:90) M × g α := 1vol( G ) (cid:90) d n xd n ψ d s φ (2 π ) s α ( x, ψ, φ ) e − (cid:15) ( φ,φ ) . (6.29) In this section we are going to review the relation between 2-dimensional YM theory thatwe described in Section 6.1 and a topological field theory (TFT) that can be viewed as its“cohomological” counterpart. We can translate almost verbatim the general principles that wediscussed in the last section, setting M (cid:55)→ A ( P ) , ω (cid:55)→ Ω , G (cid:55)→ G ( P ) , µ (cid:55)→ F, (6.30)while we regard G as a compact connected Lie group that acts as the gauge group on theprincipal bundle P → Σ over a Riemann surface Σ. The moment map is formally equivalentto the curvature F if we identify Lie ( G ( P )) ∼ = Lie ( G ( P )) ∗ through an inner product on g , asin (6.8).38 6.3. “Cohomological” and “physical” YM theoryThe non-Abelian localization principle of the last section, if used in reverse, already showedthat an equivalent way to express the standard YM theory is through a “first-order formulation” S [ A, φ ] = − (cid:90) Σ Tr (cid:16) iφF A + (cid:15) φ (cid:63) φ (cid:17) (6.31)where we consider φ ∈ Ω (Σ; ad( P )) ∼ = Lie ( G ( P )), and F A ∈ Ω (Σ; ad( P )) is the curvature of A . This is essentially what is written in (6.25), where on the LHS we have the first-order action(the (cid:15) dependence is contained in the equivariant integration), and on the RHS we have thestandard YM action S [ A ] = (cid:15) ( F, F ) A . The first-order formulation has the quality of showingvery clearly the weak coupling limit behavior when (cid:15) →
0, that is less obvious in the standardformulation. In this limit, the theory becomes topological, in the sense that the action doesnot depend on the metric anymore (the metric appears in the Hodge duality (cid:63) ), S (cid:15) → [ A, φ ] = − (cid:90) Σ Tr( iφF A ) . (6.32)This theory is called “BF model”, and it is the prototype of a TFT of Schwarz-type. The YMtheory can thus be seen as a “regulated version” of a truly topological field theory. At leastclassically, it is intuitive from the EoM with respect to φ that the only contribution to theclassical solutions comes from the moduli space of flat connections, where F A = 0 up to gaugetransformations. It is not trivial, though, to infer that this is all the theory has to offer also atthe quantum level, that is essentially the result we showed in general terms in the last section,via the localization principle applied to the path integral Z ( (cid:15) → (cid:15) (cid:54) = 0 will not follow precisely this behavior, as we already know that higher extrema of theYM action contribute to the partition function Z ( (cid:15) ), but these will have a nice interpretationin terms of the moduli space. Intermezzo: TFT
This is a good moment to explain briefly in more general terms what one usually means byTFT, and how localization enters in this subject. Traditionally, TFT borrows the language ofBRST formalism for quantization of gauge theories, as many examples of topological theories More precisely, we should say that A aµ and φ a are coordinates functions on A ( P ) × Lie ( G ( P )), so theyeffectively are elements of C ∞ ( A ( P )) ⊗ Lie ( G ( P )) ∗ . This caveat will be logically important in the following,and it goes along with the functor of points approach we used in the supergeometric discussion of Chapter 4. Notice that, although YM theory is clearly dependent on the metric of Σ, in 2 dimensions it shows avery “weak” dependence to it. In fact, in this dimensionality the action can be simplified as (suppressing theconstants and the Lie algebra inner product) (cid:90) F ∧ (cid:63)F = (cid:90) g µρ g νσ F µν F ρσ (cid:112) | g | d x = (cid:90) ( F ) g µρ g νσ ε µν ε ρσ (cid:112) | g | d x == (cid:90) ( F ) | g − | (cid:112) | g | d x = (cid:90) ( F ) | g | − / d x, so the metric does not appear through its components, but only in the invariant quantity det( g ). hapter 6. Non-Abelian localization and 2d YM theory 139arise from that context. Recall that the standard BRST quantization procedure is based onthe definition of a differential, the “BRST charge” Q , that acts on an extended graded fieldspace, whose grading counts the “ghost number” (in other words, Q is an operator of degreegh( Q ) = +1). The BRST charge represents an infinitesimal supersymmetry transformation,that squares to zero in the gauge-fixed theory. On the Hilbert space, physical states are thoseof ghost number zero, and that are annihilated by the BRST charge, Q | phys (cid:105) = 0 , gh | phys (cid:105) = 0 . (6.33)The action of Q on the field space is often denoted with a Poisson bracket-like notation, Φ (cid:55)→ δ Q Φ := −{ Q, Φ } for any field Φ. By gauge invariance of the vacuum, Q | (cid:105) = 0, and so for anyoperator O one has that (cid:104) |{ Q, O}| (cid:105) = 0.For a QFT being “topological”, in physics one usually means that all its quantum proper-ties are independent from a choice of a metric on the base space M . This is rephrased in therequirement that the partition function of the theory should be metric-independent. Assum-ing the path integral measure to be metric-independent and Q -invariant (so that the BRSTsymmetry is non anomalous), the variation with respect to the metric of the partition functionis δ g Z ∝ (cid:90) F [ D Φ] e − S [Φ] δ g S [Φ] = (cid:90) F [ D Φ] e − S [Φ] (cid:18)(cid:90) M d n x (cid:112) | g | δg µν T µν (cid:19) ∝ (cid:104) | T µν | (cid:105) , (6.34)so a suitable definition of TFT is the one that requires the energy-momentum tensor T µν to bea BRST variation, T µν = { Q, V µν } for some operator V µν . This would ensure δ g Z = 0 for whatwe said above.Collecting the above remarks, we can give the following “working definition” [106]. A Topological Field Theory is a QFT defined over a Z -graded field space F , with a nilpotentoperator Q ( i.e. a cohomological vector field on F ), and a Q -exact energy-momentum tensor T µν = { Q, V µν } , for some V µν ∈ C ∞ ( F ). Physical states are defined to be elements of the Q -cohomology of F in degree zero, | phys (cid:105) ∈ H ( F , Q ). Remark. • Q is called “BRST charge” or “operator”, but in general it can be every su-persymmetry charge (as it is a cohomological vector field). We saw examples in the lastchapter where Q = δ susy + δ BRST , where δ BRST is the actual gauge-supersymmetry, and δ susy is a Poincar´e-supersymmetry. • The one above is a good “working definition” for most examples, but it is not completelyadequate in all cases. Indeed, there are examples of QFT where T µν fails to be BRST-exact, but nonetheless one can still establish the topological nature of the model. We donot need to treat any example of this kind, so we refer to [106] for more details. Here the word “topological” is somewhat overused. Mathematically, a topological space consists of a set M and a topology O M , that is roughly the set of all “open neighborhoods” in M . In QFT one almost alwaysworks on base spaces that have the structure of a manifold of some kind (smooth, complex, ...), so that it allowsfor the presence of an atlas A M of charts that identifies it locally as R n for some (constant) n . A manifold isthus a triple ( M, O M , A M ), and the choice of a metric is only on top of this structure. So metric-independencedoes not generically mean that the QFT describes only the topology of M , but it can depend on the choice ofa smooth (or complex, ...) structure on M . We stress that, if we see F as a graded extension of an original field space F acted upon by gauge trans-formations, the Q -cohomology of F is exactly the analogous of the gauge-equivariant cohomology H ∗ gauge ( F ),computed in the Cartan model with Cartan differential Q .
40 6.3. “Cohomological” and “physical” YM theory • We already encountered an example of TFT in Section 5.2, i.e. supersymmetric QM.There, we called it topological because its partition function was determined by topolog-ical invariants of the base space, here we point out that it indeed fits into this generaldiscussion. In fact, in (5.69) we saw that its action can be expressed as S = { Q, Σ } ,where Q ≡ Q ˙ x was the U (1)-Cartan differential on the loop space, and Σ some other loopspace functional. The Q -exactness of the energy-momentum tensor follows from this. • If the energy-momentum tensor is Q -exact, in particular the Hamiltonian satisfies (cid:104) phys | H | phys (cid:105) = (cid:104) phys | T | phys (cid:105) = 0 . (6.35)This means that the energy of any physical state is zero, and thus the TFT does notcontain propagating degrees of freedom. It can only describe “topological” properties ofthe base space.TFTs fall in two main broad categories. The first one is constituted by the so-called TFT of Witten-type (or also cohomological TFT ). Their defining property is to have a Q -exact quantumaction, S q = { Q, V } (6.36)for some operator V . As the case of supersymmetric QM, the energy-momentum tensor isautomatically Q -exact too, T µν = 2 (cid:112) | g | { Q, δV /δg µν } . (6.37)From the equivariant point of view, these theories have a very simple localization property. Theaction is representative of the trivial Q -equivariant cohomology class, so the partition functioncan be in fact written as Z ∝ (cid:90) F [ D Φ] e − tS q [Φ] (6.38)for any t ∈ R , by the standard argument of supersymmetric localization. In this case F isnon-compact and so simply taking t = 0 is not really allowed, but the limit t → ∞ is perfectlydefined, producing the path integral localization onto the space of solutions of the classical EoMof S q . This means that all TFT of Witten-type are completely determined by their semiclassicalapproximation!The second main class of TFT is called of Schwarz-type (or also quantum TFT ). In thiscase one starts with a classical action S that is metric-independent, so that the classical energy-momentum tensor is zero. The usual BRST quantization of this action produces a quantumaction of the type S q = S + { Q, V } for some V , and again T µν = 2 (cid:112) | g | { Q, δV /δg µν } , (6.39)since the classical piece does not contribute. Chern-Simons theory and BF theory are of thistype. Regarding the second remark above, we mention that some Schwarz-type TFTs fail torespect the Q -exactness property of T µν , for example in the case of non-Abelian BF theoriesin dimension d > Q has to be defined in a metric-dependent way).hapter 6. Non-Abelian localization and 2d YM theory 141As we anticipated before we are interested in BF theories, an example of Schwarz-type TFT,and in particular in their 2-dimensional realization. On a base space Σ of generic dimension d ,the classical action of the BF theory is S BF [ A, B ] := − (cid:90) Σ Tr( B ∧ F A ) , (6.40)where F A is the curvature 2-form of a connection A , and B is a ( d − B ≡ iφ ∈ Ω (Σ; ad( P )), and the naive BRST quantization of this theory has no problems(as we mentioned, in more than 3 dimensions the definition of Q have to be modified and thingscomplicate). The quantum action is S q [ A, B, π, b, c ] = − (cid:90) Σ Tr (cid:0) BF A + π ( ∇ A (cid:63) A ) + b ( ∇ A (cid:63) ∇ A ) c (cid:1) = S BF [ A, B ] + { Q, V } , with V := − (cid:90) Σ Tr (cid:0) b ( ∇ A (cid:63) A ) (cid:1) , (6.41)where we introduced a ghost c , an anti-ghost b and an auxiliary field π with the BRST trans-formations properties δ Q A = ∇ A c, δ Q c = −
12 [ c, c ] , δ Q b = π, δ Q π = 0 , δ Q B = − [ B, c ] . (6.42)Even though this theory is not of Witten-type, so it has no direct localization onto the subspaceof classical solutions, it turns out that this is still the case. Traditionally, this is shown finding asuitable redefinition of coordinates in field space that trivializes the bosonic sector of the action(meaning that there are no derivatives acting on bosonic fields), and whose Jacobian cancelsin the path integral with the 1-loop determinant obtained by integrating out the fermions (inthis case, the ghosts fields). This is called Nicolai map , and for the 2-dimensional BF model isgiven by the redefinitions [106, 107] ξ ( A ) := F A , η ( A ) := ∇ A c (cid:63) A q , (6.43)where A := A c + A q is the expansion of the gauge field around a classical (on-shell) solution A c . Assuming the fermions being integrated out, the path integrals over B and π exactlyidentify the space of zeros of ξ, η , that is the moduli space of solutions to F A = 0 up to gaugetransformations, A . We do not pursue this direction further, but summarize the approachtaken in [15], more related to localization. Cohomological approach
Thinking in equivariant cohomological terms, one can expect to show the localization of theBF model (and of its “regulated” version, i.e.
YM theory) finding a suitable “localization 1-form” V (cid:48) . This has to be such that the deformation of the action given by t { Q, V (cid:48) } induces thepath integral to localize in this subspace when t → ∞ , analogously to what we did for examplein Section 5.1, and also to the discussion of the last section in the finite-dimensional case. Thisis exactly what is shown in [15], where the partition function of the model of interest is found42 6.3. “Cohomological” and “physical” YM theoryas an expectation value inside a cohomological TFT, proving automatically its localizationbehavior.This cohomological TFT is constructed in a way such that the action of the BRST op-erator Q coincides with the Cartan differential d C arising in the symplectic formulation of2-dimensional YM theory. Using the more common supergeometric language of Sections 5.1,5.2 and recalled in (6.29) (in finite dimensions), we move from the field space A ( P ) × Lie ( G ( P ))to F := Π T A ( P ) × Lie ( G ( P )), introducing the graded coordinates ( A aµ , ψ aµ , φ a ) and regarding d C as a supersymmetry (BRST) transformation, d C ≡ −{ Q, ·} = (cid:90) Σ d Σ (cid:18) ψ aµ δδA aµ − i ( φ a ) µ δδψ aµ (cid:19) , (6.44)where we recall from (6.3) that the fundamental vector field associated to the action of a Liealgebra element φ ∈ Lie ( G ( P )) is φ = ∇ φ . The BRST transformation of every field follows therule δ Q Φ = −{ Q, Φ } ≡ d C Φ, and on the coordinates we have δ Q A = ψ, δ Q ψ = − i ∇ φ, δ Q φ = 0 . (6.45)The ghost numbers of the elementary fields are gh( A, ψ, φ ) = (0 , , S c = { Q, V } , (6.46)for some operator V , that ensures the localization onto the moduli space A of flat connections.Then the mapping to the physical theory is done by the common equivariant localizationprocedure. The TFT is deformed adding a cohomologically trivial localizing action, S ( t ) = S c + t { Q, V (cid:48) } = { Q, V + tV (cid:48) } , (6.47)for some gauge invariant operator V (cid:48) that forces only the interesting YM multiplet ( A, ψ, φ )to survive. Since the field space here is non-compact, some additional care must be taken inclaiming the t -independence of the deformed theory. In particular, the new term must notintroduce new fixed points of the Q -symmetry, that would contribute to the localization locusof the resulting theory. These fixed points, if presents in the theory at t (cid:54) = 0, can be interpretedas “flowing from infinity” in the moduli space (since this is, as just remarked, non-compact). If this is not the case, one can infer properties of the “physical” theory at t = ∞ by makingcomputations in the cohomological one at t = 0. The cohomological theory
The cohomological theory considered in [15] makes use of theadditional multiplets ( λ, η ) and ( χ, H ) with the transformation properties δ Q λ = η, δ Q η = − i [ φ, λ ] , δ Q χ = − iH, δ Q H = [ φ, χ ] . (6.48) This will be exactly the case in going to the YM theory with (cid:15) (cid:54) = 0, where the new fixed points are just thehigher extrema of the action S = ( F, F ). Notice that this is logically the opposite of what one does usually in using localization. Here the “easy”theory is the cohomological one at t = 0, while the more difficult (but more interesting) is the one at t (cid:54) = 0. hapter 6. Non-Abelian localization and 2d YM theory 143The extended field space and ghost numbers are F = Π T A ( P ) × Lie ( G ( P )) (cid:124) (cid:123)(cid:122) (cid:125) × (cid:0) Ω (Σ; ad( P )) (cid:1) (cid:124) (cid:123)(cid:122) (cid:125) × (cid:0) Ω (Σ; ad( P )) (cid:1) (cid:124) (cid:123)(cid:122) (cid:125) ( A, ψ, φ ) ( λ, η ) ( χ, H )gh = (0 , ,
2) ( − , −
1) ( − , . (6.49)Of course the definition of the BRST operator will be extended from (6.44), but we do not needit explicitly. The operator V defining the TFT is chosen to be V = 1 h (cid:90) Σ d Σ Tr (cid:18) χ ( H − (cid:63)F A )) + g µν ( ∇ Aµ λ ) ψ ν (cid:19) , (6.50)where h ∈ R is a parameter from which the theory is completely independent (it is analogousto 1 /t appearing in (6.38)), that can be interpreted as the “coupling constant” of the TFT.Computing the cohomological action S c one sees that the H field plays an auxiliary role, and canbe eliminated setting H = (cid:63)F . Analyzing then the theory in the h → δ Q χ = 0 ( ⇒ F = 0) , δ Q ψ = 0 ( ⇒ ∇ φ = 0) . (6.51)This is analogous to the usual situation in Poincar´e-supersymmetric theories, as discussedin Chapter 5, where the localization locus was always identified by the subcomplex of BPSconfigurations, given by the vanishing of the variation of the fermions. Here the “fermions”are the fields with odd ghost number. This means that the final moduli space contains A ,plus maybe some contributions from the zero-modes of the other bosonic (even ghost number)fields, λ and φ . t (cid:54) = 0 deformation The deformation (6.47) can be made in order to reduce effectively thefield content of the theory to the YM multiplet only, (
A, ψ, φ ). In particular, to eliminatethe non-trivial presence of the field λ from the contributions to the localization locus one canconsider V (cid:48) = − h (cid:90) Σ d Σ Tr ( χλ ) . (6.52)Computing the deformed action S ( t ), one sees that for t (cid:54) = 0 all the additional fields H, χ, λ, η can be integrated out (again, some more details on the technical passages can be found in [15]).In particular, the EoM for H and λ are H = 0 , λ = − t ( (cid:63)F ) . (6.53)This is already the sign that the localizing term V (cid:48) qualitatively changed the localizationproperty of the theory. In fact, we see that for t = 0 we do not have an algebraic equation for λ , but (6.53) reduces to the solution F = 0 , H = 0 of the cohomological theory. The theoriesdefined by S ( t ) and S c may be thus different, but the failure of their equivalence can only comefrom new components of the moduli space that flow in from infinity for t (cid:54) = 0; the contributionof the “old” component must be independent of t . Taking the limit t (cid:29)
1, the dominantcontribution to the deformed action is (suppressing the A -dependence) S ( t (cid:29)
1) = − t (cid:26) Q, (cid:90) Σ d Σ Tr ( ψ µ ∇ µ ( (cid:63)F )) (cid:27) = 1 t (cid:90) Σ d Σ Tr (cid:16) i ∇ µ φ ∇ µ ( (cid:63)F ) − ( (cid:63)F )[ ψ µ , ψ µ ] + ∇ µ ψ µ (cid:15) νσ ∇ ν ψ σ (cid:17) . (6.54)44 6.3. “Cohomological” and “physical” YM theoryThe main point is that the φ -EoM is actually equivalent to the YM equation ∇ (cid:63) F = 0. Thismeans that the moduli space of the deformed theory contains the moduli space of the standardYM theory, and indeed includes all the higher extrema corresponding to non-flat connections.These solutions with F (cid:54) = 0 have λ ∼ − /t , and thus their contribution to the path integralgoes roughly as exp( − /t ), as expected. When t = 0, the cohomological theory is recoveredand the only contribution to the moduli space is given by the flat connections. Connection with 2-dimensional YM theory
We already argued that the deformed TFTgained all the YM spectrum “flowing from infinity in the moduli space”, but it remains tosee how one can get practically the YM (and BF) partition function from the theory definedby S ( t ). This is simply obtained by another deformation of the exponential in the partitionfunction: we notice that the YM action is gauge invariant, so it is meaningful to compute theexpectation value of e S Y M in the TFT. Thus we consider an exponential operator of the formexp ( ω + (cid:15) Θ)with ω := (cid:90) Σ Tr (cid:18) iφF + 12 ψ ∧ ψ (cid:19) , Θ := 12 (cid:90) Σ Tr( φ (cid:63) φ ) . (6.55)Since the quantity (cid:104) exp ( ω + (cid:15) Θ) (cid:105) t ∝ (cid:90) Π T A ( P ) × Lie ( G ( P )) DADψDφ exp ( ω + (cid:15) Θ − S ( t )) (6.56)is well defined for t → ∞ , we can actually take t = ∞ and drop S ( ∞ ) = 0 (recall that the pathintegral is independent on the actual value of t ), getting exactly the YM partition function (cid:104) exp ( ω + (cid:15) Θ) (cid:105) t ∝ (cid:90) Π T A ( P ) × Lie ( G ( P )) DADψDφ exp ( ω + (cid:15) Θ) ∝ Z ( (cid:15) ) , (6.57)up to some normalization constant.In the limit (cid:15) →
0, when only the BF model survives, the path integral over φ producesthe constraint δ ( F ), so localizing the expectation value onto the space of flat connections. Thismeans that, although we started from different theories S c ∼ = S (0) and S ( t ), this particularexpectation value satisfies (cid:104) exp ( ω ) (cid:105) t = (cid:104) exp ( ω ) (cid:105) t =0 (6.58)and the BF model is recovered as an expectation value in the cohomological theory. This givesanother interpretation to the topological behavior of the BF model, and a measure of the failureof 2-dimensional YM theory in being topological.Concluding, we only point out that the the operators ω and Θ are precisely the infinite-dimensional realization in this example of the general expressions in (6.28). In fact, the sym-plectic 2-form on A ( P ) Ω = (cid:90) Σ Tr( ψ ∧ ψ ) (6.59)only serves to have a formal interpretation of the measure DADψe Ω , since the field ψ is reallya spectator in the action S [ A, ψ, φ ] = − (cid:90) Σ Tr (cid:18) iφF + (cid:15) φ (cid:63) φ + 12 ψ ∧ ψ (cid:19) . (6.60)hapter 6. Non-Abelian localization and 2d YM theory 145 As we said at the beginning of the chapter, 2-dimensional YM theory is an exactly solv-able theory, whose partition function can be expressed in closed form, for example by groupcharacters expansion methods [15, 20]. This makes it possible to compare results from the lo-calization formalism, and obtain a new geometric interpretation of the already present solutionof the theory. In general, its partition function on a Riemann surface Σ of genus g , with asimply-connected gauge group G , is given as Z ( (cid:15) ) = (vol( G )) g − (cid:88) R R ) g − e − (cid:15) ˜ C ( R ) , (6.61)where the sum runs over the representations R of G , and ˜ C ( R ) is related to the quadraticCasimir C ( R ) := (cid:80) a Tr R ( T a T a ) of the representation R by some normalization constant. Fornot simply-connected G this formula has to be slightly modified (see [15]). We are notinterested in reviewing the proof of (6.61) in general, but we present a quick argument for asimple example, that already contains the logic behind it.
Example 6.4.1 (YM theory with genus g = 1) . We quickly motivate the result for the YMpartition function on a genus 1 surface. We can think of this surface as a disk with boundary,that is homeomorphic to a sphere with one hole. The radial direction is identified with theinterval [0 , T ], and the angular coordinate as [0 , L ] with the edges identified.It is more natural to compute the partition function of the theory in the Hamiltonianformulation, Z = Tr H P exp (cid:18) − (cid:90) T dt H ( t ) (cid:19) (6.62)up to possible normalization factors, where H is the Hilbert space of the system. To this end,let us consider the canonical quantization of the YM action. To make sense of the partitionfunction we must fix a gauge, and we do this setting A t = 0 (temporal gauge). In this gaugethe action simplifies as S [ A ] = − (cid:15) (cid:90) dxdt Tr( F ) = 12 (cid:15) (cid:90) dxdt ( ∂ t A ax ) , (6.63)where we expanded A µ = A aµ T a with respect to the generators of g , and suppressed the innerproduct implicitly summing over the Lie algebra indices. We see that the only non-zero canon-ical momentum is Π ax = (1 /(cid:15) ) ∂ t A ax , acting on the Hilbert space as Π ax ( t, x ) (cid:55)→ δδA ax ( t,x ) . Thecanonical Hamiltonian in temporal gauge is thus H ( t ) = (cid:15) (cid:90) L dx (Π ax ( t, x )) (cid:55)→ (cid:15) (cid:90) L dx δδA ax ( t, x ) δδA ax ( t, x ) . (6.64)The Hilbert space H can be considered to consist of gauge invariant functions Ψ( A ). Theonly gauge invariant data obtained from the gauge field at any point p ∈ Σ is its holonomy, U p [ A ] := P exp (cid:18)(cid:73) C ( p ) A (cid:19) ∈ G, (6.65) Simply-connectedness implies that the principal G -bundle P → Σ has to be trivial. When one drops thiscondition, the triviality is not ensured and contributions to the formula appear due to singular points in Σ forthe connection.
46 6.4. Localization of 2-dimensional YM theorywhere C ( p ) is a loop about p . In the case of the one holed-sphere, all loops are homotopic tothe one on the boundary, so Ψ ∈ H must be an invariant function of U = P exp (cid:90) L dxA , (6.66)and independent of t ∈ [0 , T ]. Any invariant function must be expandible in characters ofrepresentations of G , so Ψ( U ) = (cid:80) R Ψ R Tr R ( U ), where Ψ R ∈ C and Tr R ( U ) is the Wilson loopin the representation R of G . We notice that the basis functions χ R ( U ) := Tr R ( U ) diagonalizethe Hamiltonian, since Hχ R ( U ) = (cid:15) Tr R (cid:90) dx T a T a P exp (cid:90) L dxA = (cid:15)LC ( R ) χ R ( U ) , (6.67)where C ( R ) := Tr R ( T a T a ) is the quadratic Casimir in the representation R , a time-independenteigenvalue of H . Via this diagonalization the partition function is easily computed, Z = (cid:88) R e − T L(cid:15)c ( R ) , (6.68)matching (6.61) for g = 1. All the geometric information about Σ that enters in Z is itstotal area T L , and any other local property. Notice that in the topological limit (cid:15) →
0, theHamiltonian vanishes (as the theory has no propagating degrees of freedom) and the partitionfunction simplifies further.From the localization formalism discussed in the last sections, we expect the partitionfunction to be of the type Z ( (cid:15) ) = Z ( (cid:15) ) + (cid:88) n Z n ( (cid:15) ) , (6.69)with Z ( (cid:15) ) representing the contribution from the moduli space A of flat connections, suchthat Z (0) ∼ vol( A ), and the other Z n ( (cid:15) ) coming from contributions of the higher extremaof the YM action, such that Z n ( (cid:15) ) ∼ exp( − /(cid:15) ) in the weak coupling limit (cid:15) (cid:28)
1. Usingcohomological arguments, in [15] (also nicely reviewed in [102]) it was shown how to recoverthe general features of (6.61), and in particular how to interpret it in terms of an (cid:15) -expansion atweak coupling, in relation to the expected form (6.69). The detailed derivation is cumbersomeand requires some more technical background, so we refer to the article for it, but the logicis essentially the same as for the discussion at the end of Section 6.2. The strategy is thefollowing. Any solution to the YM EoM identifies a disconnected region S n ⊂ A ( P ). For everysuch region, one fixes a small neighborhood N n around S n that equivariantly retracts onto it.The technically difficult passage is to perform the integral over the “normal directions” to S n in N n , and then reduce it on the moduli space A n := S n / G ( P ). The main difficulty is that ingeneral the MWM theorem (or its equivariant counterpart) does not work, since the action of G ( P ) is not generally free on S n (also for n = 0), as we assumed in writing down (6.28) forthe S ≡ µ − (0) component. For the higher extrema, this is readily seen by the fact that theequation ∇ ( (cid:63)F ) = 0 with F (cid:54) = 0 (6.70)identifies a vacuum f := ( (cid:63)F ) as a preferred element of g (being it covariantly constant overΣ), and thus the gauge group is spontaneously broken to a subgroup G f ⊆ G . The actionof the whole gauge group thus cannot be free on this subspace, and the quotient S n / G ( P ) ishapter 6. Non-Abelian localization and 2d YM theory 147singular. Via a suitable choice of localization 1-form one is still able to extract information bythis integral over the normal directions, and in particular to compute the (cid:15) -dependence of thehigher extrema contributions.We limit ourselves now to the comparison of the exact result (6.61) applied to the case G = SU (2), with the expectation (6.69) obtained by cohomological arguments. For this gaugegroup, the character expansion of the partition function results Z ( (cid:15) ) = 1(2 π ) g − ∞ (cid:88) n =1 exp( − (cid:15)π n ) n g − . (6.71)Simply taking (cid:15) = 0, we see that this is finite and proportional to a Riemann zeta-function,but to explore better the (cid:15) -dependence it is convenient to consider ∂ g − Z∂(cid:15) g − = (cid:18) − (cid:19) g − ∞ (cid:88) n =1 exp (cid:0) − (cid:15)π n (cid:1) = ( − g − g (cid:32) − (cid:88) n ∈ Z exp (cid:0) − (cid:15)π n (cid:1)(cid:33) . (6.72)This is not quite in the expected form, since the exponentials in the sum go to zero for (cid:15) → − /(cid:15) ). We can bring this expression closer to the desired result using the Poissonsummation formula (cid:88) n ∈ Z f ( n ) = (cid:88) k ∈ Z ˆ f ( k ) with ˆ f ( k ) := (cid:90) + ∞−∞ f ( x ) e − πikx , (6.73)where f is a function and ˆ f its Fourier transform, and rewriting the sum of exponentials in(6.72) as ∂ g − Z∂(cid:15) g − = ( − g − g (cid:32) − (cid:114) π(cid:15) (cid:88) k ∈ Z exp (cid:18) − k (cid:15) (cid:19)(cid:33) . (6.74)This is exactly the result that could be obtained via integration over normal coordinates in thelocalization framework (see [102], eq. (4.102)), but fundamentally differs from our expectation,since for (cid:15) → k = 0) is singular for thepresence of the square root. This means that the partition function is not really a polynomialin (cid:15) for small couplings, but an expression of the form Z ( (cid:15) ) = g − (cid:88) m =0 a m (cid:15) m + a g − / (cid:15) g − / + exponentially small terms . (6.75)The singularity in Z ( (cid:15) →
0) arises because, for gauge group SU (2), the subspace µ − (0) issingular and the MWM theorem does not apply.A simpler situation would occur considering the gauge group SO (3) (which is not simplyconnected) and a non-trivial principal bundle over Σ. In this case, the character expansion ofthe partition function requires some modifications with respect to (6.61), the result being Z ( (cid:15) ) = 1(8 π ) g − ∞ (cid:88) n =1 ( − n +1 exp( − π (cid:15)n ) n g − . (6.76)Following the same idea as above, we look at the ( g − th derivative ∂ g − Z∂(cid:15) g − = ( − g g − ∞ (cid:88) n =1 ( − n exp( − π (cid:15)n ) = ( − g g − (cid:32) − (cid:88) n ∈ Z ( − n exp( − π (cid:15)n ) (cid:33) , (6.77)48 6.4. Localization of 2-dimensional YM theoryand we rewrite the sum using the Poisson summation formula, getting ∂ g − Z∂(cid:15) g − = ( − g · g − (cid:32) − (cid:114) π(cid:15) (cid:88) k ∈ Z exp (cid:32) − ( k + 1 / (cid:15) (cid:33)(cid:33) . (6.78)This time we see that the contribution for k = 0 from the moduli space of flat connections isfinite for (cid:15) →
0, and the whole ∂ g − Z/∂(cid:15) g − is constant up to exponentially small terms. Thismeans that the partition function at weak coupling Z ( (cid:15) →
0) is a regular polynomial of degree g − (cid:15) , up to exponentially decaying terms, Z ( (cid:15) ) = g − (cid:88) m =0 a m (cid:15) m + O ( (cid:15) g − ) , (6.79)and it reflects the fact that, for a non-trivial SO (3)-bundle, µ − (0) is smooth and acted onfreely by G . These two quick examples capture the way this localization framework can give avery geometric interpretation to the (cid:15) -expansion of the partition function, and its dependenceon the classical geometry of the moduli space.Finally, we point out that an analogous treatment was done more recently in [102] to analyzein this cohomological framework Chern-Simons theory on 3-dimensional Seifert manifolds . ASeifert manifold is a smooth object that can be described as an S -bundle over a 2-dimensionalorbifold, and this feature makes it possible to dimensionally reduce the Chern-Simons theoryalong the direction of the circle S to a 2-dimensional YM theory over a singular base space. Itturns out that the localization locus of the resulting theory receives contributions only from theflat connections over the total space. This is in accordance with the fact that Chern-Simonstheories are by themselves TFT (of Schwarz-type). 2-dimensional YM theories have been stud-ied extensively in the past years, and many interesting results were obtained thanks to theirnon-perturbative solvability. For example, exact results for Wilson loops expectation valuesand their relation with higher dimensional supersymmetric theories were studied in [108–110].A relation with certain topological string theories and supersymmetric black hole entropy com-putations were analyzed in [111]. A duality between higher-dimensional supersymmetric gaugetheories and deformations of 2-dimensional YM theory was revisited in [112, 113]. Localizationtechniques play an important role in all those cases. hapter 7Conclusion In this thesis we reviewed and summarized the main features of the formalism of equivariantcohomology, the powerful localization theorems first introduced by Atiyah-Bott and Berline-Vergne, and the principles that allow to formally apply these integration formulas to QFT.From the physical point of view, the equivariant (or supersymmetric) localization principlegives a systematic approach to understand when the “semiclassical” approximation of the pathintegral, describing the partition function or an expectation value in QFT, can give an exactresult for the full quantum dynamics. We discussed the applicability of these techniques inthe context of supersymmetric theories. These are characterized by a space of fields that isendowed with a graded structure and the presence of some symmetry operator whose “square”gives a standard “bosonic” symmetry of the action functional. This supersymmetry operator isinterpreted as a differential acting on the subspace of symmetric configurations in field space,and its cohomology describes the field theoretical analog of the G -equivariant cohomology of a G -manifold.After having introduced the general features of the mathematical theory of equivariantcohomology and equivariant localization, we reviewed the concepts in supergeometry that al-low for the construction of supersymmetric QFT, and that constitute the correct frameworkto translate the mathematical theory in the common physical language. Since many recentapplications of the localization principle aimed at the computations of path integrals in su-persymmetric QFT on curved spaces, we included a discussion of the main tools needed todefine supersymmetry in such instances. Then we collected some examples from the literatureof application of the supersymmetric localization principle to path integrals in QFT of diversedimensions. The common feature of these examples is that, via a suitable “cohomological”deformation of the action functional, it is possible to reduce the infinite-dimensional path in-tegral to a finite-dimensional one that represents its semiclassical limit, as stressed above. Wedescribed cases in which this reduction relates the partition function to topological invariantsof the geometric structure underlying the theory, namely the cases of supersymmetric QM(a 1-dimensional QFT) and the weak coupling limit of 2-dimensional Yang-Mills theory (its“topological” limit). We also reviewed the more recent applications to the computations ofthe expectation values of supersymmetric Wilson loops in 3- and 4-dimensional gauge theories,namely Supersymmetric Chern-Simons theory and Supersymmetric Yang-Mills theory definedon the 3- and the 4-sphere. In these cases, the path integral results to be equivalently describedby some 0-dimensional QFT with a Lie algebra as target space, called “matrix model”.In the last few decades, the literature concerning the applications of supersymmetric local-ization has grown exponentially, and many other advanced examples of its use in the physics14950context have been found. From the point of view of supersymmetric QFT, a consistent sliceof the state-of-the-art on the subject can be found in [12], including computations analogueto the one we showed for Wilson loop expectation values or topological invariants over morecomplicated geometries. From the point of view of Quantum (Super)Gravity, these techniqueshave found applications in the computations of the Black Hole quantum entropy [114, 115]. Inmany circumstances, localization allows for the analysis of properties of QFT at strong cou-pling, an otherwise prohibited region of study with conventional perturbative techniques. Thisfeature can be used also to test a class of conjectural dualities between some types of gauge the-ories and string theories, the so-called AdS/CFT correspondences [96]. Concerning the subjectof Wilson loops in 3-dimensional Chern-Simons theories and their relations to matrix modelsand holography, for which localization has played an important role, a recent reference thatconcisely reviews the state-of-the-art is [84]. ppendix ASome differential geometry A.1 Principal bundles, basic forms and connections
Here we recall some notions about principal bundles that can be useful to follow the dis-cussion, especially of the first chapters of this thesis. Principal bundles are the geometricconstruction behind the concepts of covariant derivatives and connections in gauge theory orGeneral Relativity, for example. If G is a Lie group, a principal G -bundle is a smooth bundle P π −→ M such that(i) P is a (right) G -manifold;(ii) the G -action on P is free ;(iii) as a bundle, P → M is isomorphic to P → P/G , where the projection map is canonicallydefined as p (cid:55)→ [ p ].Notice that since the G -action is free, a principal G -bundle is a fiber bundle with typical fiber G , and by the third property it is at least locally trivial , i.e. over every open set U ⊆ M it lookslike G × U . Morphisms of principal bundles are naturally defined as maps between bundles thatpreserve the G -structure, so G -equivariant maps. A principal bundle is trivial if it is isomorphicthrough a principal bundle isomorphism to the trivial product bundle G × M . A useful factis that the triviality of a principal bundle is completely captured by the existence of a global section σ : M → P such that π ◦ σ = id M . Since every principal bundle is locally trivial, thanlocal sections can always be chosen and they constitute a so-called local trivialization of thebundle .The main example of principal bundle that occurs in the geometric construction of spacetimeis the frame bundle LM over some n -dimensional smooth manifold M . At every point p ∈ M ,the elements of the fiber L p M are the frames at p , i.e. all the possible bases e = ( e , · · · , e n )for the tangent space T p M . LM has a natural GL ( n, R ) right action that corresponds to therotation of the basis, e · g := ( e k g k , · · · , e k g kn ) for g ∈ GL ( n, R ). In gauge theories, the structuregroup (or sometimes gauge group ) G of the theory is the Lie group acting on the right on aprincipal G -bundle.Since the fibers of the principal bundle are essentially the Lie group G , tangent vectors on P can come from its Lie algebra g . This leads to the following definition. Definition A.1.1.
The vertical sub-bundle
V P of the tangent bundle
T P is the disjoint union
V P := (cid:71) p ∈ P V p P, with V p P := Ker( π ∗ p ) = { X ∈ T p P | π ∗ ( X ) = 0 } ⊂ T p P. basic forms inside Ω( P ) are those forms ω ∈ Im( π ∗ ),so that it exists an α ∈ Ω( M ) such that ω = π ∗ α . The space of basic forms is denoted Ω( P ) bas .The vertical vectors in every V p P are in one to one correspondence with the Lie algebraelements in g , through the Lie algebra homomorphism X ∈ g (cid:55)→ X := ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 (cid:0) · e tX (cid:1) ∗ (A.1)that maps Lie algebra elements to the corresponding fundamental vector fields . Fundamentalvector fields satisfy the following properties:(i) [
X, Y ] = [
X, Y ];(ii) the integral curve of X through p ∈ M is γ p : R → Mt (cid:55)→ γ p ( t ) = p · e tX ; (A.2)(iii) denoting with r g the right action of g ∈ G ,( r g ) ∗ ( X p ) = (cid:16) Ad g − ∗ ( X ) (cid:17) r g ( p ) . (A.3)As for the vertical vector fields being encoded in the Lie algebra g , also the basic formscan be characterized in terms on the (infinitesimal) action of g on Ω( P ). This can be seenintroducing the following definitions. Definition A.1.2.
A differential form ω ∈ Ω( P ) is said to be G -invariant if it is preserved bythe G action: ω = ( r g ) ∗ ω ∀ g ∈ G. The space of G -invariant forms is commonly denoted Ω( P ) G . A differential form is called horizontal if it is annihilated by vertical vector fields, ι X ω = 0 ∀ X ∈ Γ( V P ) . The properties of being invariant and horizontal can be also stated infinitesimally withrespect to the action of the Lie algebra g . If we define L X := L X , ι X := ι X , ∀ X ∈ g , (A.4)then an invariant form is characterized by L X ω = 0 for every X ∈ g , and a horizontal formby ι X ω = 0 for every X ∈ g . This makes the concepts of invariant and horizontal elementsindependent from the principal bundle structure, so that they can be defined by this character-ization for every g -dg algebra, as in Section 2.3. Also basic forms can be defined for every g -dgalgebra, combining the definitions of invariant and horizontal forms, thanks to the followingtheorem: The choice of the sign at the exponential differs from the one in (2.19) because here we are considering aright action. ppendix A. Some differential geometry 153
Theorem A.1.1 (Characterization of basic forms) . Schematically,invariant + horizontal ⇔ basic . Proof.
For notational convenience only, let us consider 1-forms.( ⇐ ) If ω = π ∗ α is basic, then ( r g ) ∗ ω = ( r g ) ∗ π ∗ α = ( π ◦ r g ) ∗ α = ω, since the principal bundle is locally trivial. So ω is also invariant. For a vertical vector X ∈ V P , ι X ω = ( π ∗ α )( X ) = α ( π ∗ ( X )) = 0 . So ω is also horizontal.( ⇒ ) Let ω ∈ Ω ( P ) be horizontal and invariant. Since π is surjective, for every vector X ∈ T p P there exists Y = π ∗ ( X ) ∈ T π ( p ) M . We can define α ∈ Ω ( M ) such that, at every x ∈ Mα x ( Y ) := ω p ( X ) for p ∈ π − ( x ) , and thanks to the horizontality and invariance of ω we can check that this form is welldefined, i.e. independent from the choice of point p in the fiber π − ( x ) and from thechoice of vector X such that π ∗ ( X ) = Y . In fact, if X (cid:48) ∈ T p P is another vector such that π ∗ ( X (cid:48) ) = Y , then π ∗ ( X − X (cid:48) ) = 0 so ( X − X (cid:48) ) ∈ V p P . By horizontality, ω ( X − X (cid:48) ) = 0 ⇒ ω ( X ) = ω ( X (cid:48) ), so α is independent from the choice of vector. Moreover, if p (cid:48) ∈ π − ( x ) isanother point in the fiber, there exists a g ∈ G such that r g ( p ) = p (cid:48) , so by G -invariance ω p (cid:48) = ω p and thus α is independent from the choice of point in the fiber. (cid:4) Proposition A.1.1.
The differential d closes on the subspace Ω( P ) bas of basic forms, defininga proper subcomplex. This extends to any g -dg algebra. Proof.
Consider ω ∈ Ω( P ) bas , and its differential dω . We characterize basic forms by beinghorizontal and G -invariant. By Cartan’s magic formula the Lie derivative commutes with thedifferential, so for every X ∈ g , L X dω = d ( L X ω ) = 0. Thus dω is still G -invariant. Also, ι X dω = L X ω − dι X ω = 0. Thus dω is still horizontal, and so basic. (cid:4) We recall now the definition of connection and curvature on principal bundles, from whichone inherits covariant derivatives on associated vector bundles.
Definition A.1.3. An (Ehresmann) connection on a principal G -bundle P → M is an horizon-tal distribution HP , i.e. a smooth choice at every point p ∈ P of vector subspaces H p P ⊂ T p P such that(i) T p P = V p P ⊕ H p P ;(ii) ( r g ) ∗ ( H p P ) = H p · g P ( G -equivariance of the horizontal projection).Given a horizontal distribution HP , every vector X ∈ T p P decomposes into an horizontal anda vertical part, X = hor ( X ) + ver ( X ) . A connection 1-form on P → M is Lie algebra-valued 1-form A ∈ Ω ( P ) ⊗ g ) such that54 A.1. Principal bundles, basic forms and connections(i) for any X ∈ g , ι X A = A ( X ) = X ( vertical g ∈ G , ( r g ) ∗ A = ( Ad g − ∗ ◦ A ) ( G -equivariance).The choice of a horizontal distribution is equivalent to the choice of a connection 1-form on P ,since at every p ∈ P one can use A as a projection onto the vertical subspace V p P ∼ = g , and π ∗ asa projection onto the horizontal subspace, identifying V p P := Ker( π ∗ p ) and H p P := Ker( A p ).This choice is smooth and G -equivariant since A is, by definition. Notice that the splitting HP ⊕ V P induces a splitting Ω ( P ) = Ω hor ( P ) ⊕ Ω ver ( P ), and that we can identify the “spaceof connection 1-forms” as A ( P ) := { A ∈ (Ω ver ( P ) ⊗ g ) | A is G -equivariant } . (A.5)It is easy to see that for every A, A (cid:48) ∈ A ( P ), their difference is not a connection, and in fact itis an horizontal g -valued 1-form,( A − A (cid:48) ) ∈ a := { a ∈ (Ω hor ( P ) ⊗ g ) | a is G -equivariant } . (A.6)This means that every connection A can be written as another connection A (cid:48) plus a horizontalform, or in other words that ( A ( P ) , a ) can be seen as a natural affine space , modeled on theinfinite-dimensional vector space a . As for any affine space, one can think of the space ofconnections as an infinite-dimensional smooth manifold, with tangent spaces T A A ( P ) ∼ = a atevery A ∈ A ( P ). Definition A.1.4.
The covariant exterior derivative on Ω( P ) is D := d ◦ hor ∗ . The curvature of a connection 1-form A is F := DA = dA ( hor ( · ) , hor ( · )) ∈ Ω ( P ) ⊗ g . The curvature F satisfies the following properties:(i) by definition, F is horizontal: ι X F = 0 for every X ∈ g ;(ii) by G -equivariance of A , F is G -equivariant too;(iii) it obeys the structural equation F = dA + 12 [ A ∧ , A ] (A.7)where [ A ∧ , A ] = f abc A b ∧ A c ⊗ T a with respect to a basis { T a } of g and the structureconstants f abc ;(iv) it obeys the second Bianchi identity, DF = 0 or dF = [ F ∧ , A ] . (A.8)One can consider the very trivial construction of a principal G -bundle as G → pt , where P = G × pt ∼ = G . Here the right G -action is simply the diagonal action (trivial on pt , inducedby the natural action on G ). On this bundle there is a canonical choice of connection 1-form,the Maureer-Cartan (MC) form Θ ∈ Ω ( G ) ⊗ g . For every vector X ∈ T g G at some g ∈ G ,ppendix A. Some differential geometry 155there is a Lie algebra element A ∈ T e G ∼ = g such that X = l g ∗ ( A ), and the MC form is definedby Θ g ( X ) := l g − ∗ ( X ) = A. (A.9)One can check that this form is indeed G -equivariant, and it is obviously vertical, giving aconnection 1-form. Moreover it satisfies the Maurer-Cartan equation d Θ + 12 [Θ ∧ , Θ] = 0 , (A.10)so that by (A.7) we see that its curvature is zero. On the trivial principal G -bundle G × M → M one can always define a connection 1-form by pulling back the MC connection along theprojection π : G × M → G . In the general case, the principal bundle P is locally trivial, so inany local patch G × U α → U α one can pull back the MC connection and use a suitable partitionof unity to glue together the local pieces to a global connection 1-form on P . This shows thatany principal bundle allows for a connection. The curvature F of the chosen connection A measures, in a sense, the deviation of A from being the Maurer-Cartan connection.As said before, a connection on a principal G -bundle allows for the definition of a covariantderivative on associated vector bundles . An associated vector bundle to the principal G -bundle P π −→ M is a vector bundle constructed over M with some typical fiber V (a vector space)that has a (left) G -action compatible with the one on P . Precisely, the associated bundle is P V π V −→ M , where P V := ( P × V ) (cid:30) ∼ G with ( p, v ) ∼ G ( p · g, g − · v ) ∀ ( p, v ) ∈ P × V, g ∈ G,π V ([ p, v ]) := π ( p ) , (A.11)and it has indeed typical fiber V . In the case of the frame bundle P = LM , one can constructthe tangent bundle T M , the cotangent bundle T ∗ M and all the tensor bundles as associatedto LM . In fact, for the tangent bundle for example, the typical fiber is V := R n and the GL ( n, R )-action is ( g · v ) k = g kj v j . This encodes the change of basis rule if we see vectors aselements [ e, v ] ∈ LM R n ,[ e, v ] ≡ e k v k ∼ GL [ e · g, g − · v ] ≡ e j g jk ( g − ) kj v j = e k v k . (A.12)On the associated vector bundle, a field (in physics terms) is a (local, at least) section φ : M → P V , that can be always seen locally as a V -valued function on every U ⊆ M , ˜ φ : U → V , sothat φ ( x ) = [ p, ˜ φ ( x )] for some chosen p ∈ π − ( x ) , x ∈ U . Another example of this conceptthat came up in Chapter 2 is the homotopy quotient M G := ( M × EG ) /G of a G -manifold M .This is precisely the associated bundle with fiber V = M to the principal G -bundle EG → BG (however, this is not an associated vector bundle, since M is not a vector space in general).As we said at the beginning of this appendix, every principal bundle is locally trivial, sothat it exists a set of local trivializations { U α , ϕ α : π − ( U α ) → U α × G } , where { U α } covers M ,and ϕ α is G -equivariant. This means that ϕ α ( p ) = ( π ( p ) , g α ( p )) for some G -equivariant map g α : π − ( U α ) → G , that makes every fiber diffeomorphic to G . To this local trivialization, onecan canonically associate a family of local sections { σ α : U α → π − ( U α ) } , determined by themaps ϕ α so that for every m ∈ U α , ϕ α ( σ α ( m )) = ( m, e ), where e ∈ G is the identity element.In other words, g α ◦ σ α : U α → G is the constant function over the local patch U α ⊆ M that In this case G -equivariance means g α ( p · h ) = g α ( p ) · h .
56 A.1. Principal bundles, basic forms and connectionsmaps every point to the identity. Conversely, a local section σ α allows us to identify the fiberover m with G . Indeed, given any p ∈ π − ( m ), there is a unique group element g α ( p ) ∈ G such that p = σ α ( m ) · g α ( p ). Using these canonical local data, the connection A and thecurvature F can be pulled back on M giving the local gauge field A ( α ) := σ ∗ α ( A ) and fieldstrength F ( α ) := σ ∗ α ( F ). The covariant derivative along the tangent vector X ∈ T M of a local V -valued function ˜ φ : U α → V is defined as ∇ X ˜ φ := d ˜ φ ( X ) + A ( α ) ( X ) · ˜ φ, (A.13)where the second term denotes the action of the Lie algebra on V , that for matrix groupscoincides with the action of G . We denote schematically the covariant derivative as ∇ = d + A on a generic associated vector bundle. When V = g we have the so-called adjoint bundle , oftendenoted ad( P ), that is in one-to-one correspondence with the space a above, of horizontal and G -equivariant Lie algebra-valued forms on P . On this special associated bundle, the covariantderivative acts with the infinitesimal adjoint action of g , ∇ = d + [ A, · ] . (A.14)By the horizontal property of the curvature, we see that F can be regarded as a 2-form on M with values in ad( P ). Then the Bianchi identity can be rewritten in terms of the covariantderivative, ∇ F = dF + [ A, F ] = [
F, A ] + [
A, F ] = 0 . (A.15)As the last piece of information, we recall the meaning of gauge transformations from theperspective of the principal bundle. Locally, we can think of them as local actions of the gaugegroup G , so that a gauge transformation is a map that associates to every point x ∈ M anelement g ( x ) ∈ G , acting on the local field strength in the adjoint representation. At the levelof the principal bundle, this can be viewed more formally defining the group G ( P ) ⊂ Diff( P )of principal bundle maps of the type P PM. Ψ π π (A.16)We notice right-away that, from the local point of view, this can indeed be identified with thespace of sections Ω ( M ; Ad( P )) of the bundle Ad( P ), associated to P with typical fiber G and G -action defined by conjugation (the adjoint representation of G on itself). Proof of G ( P ) ∼ = Ω ( M ; Ad( P )) . We can see that associated to every element Ψ ∈ G ( P ) thereis a unique class of local sections { ψ α : U α → G } that transforms in the adjoint representation,and vice versa. Being horizontal means that pulling it back on the base space, we do not lose information on the 2-form.In fact, the local representation of the curvature F ( α ) = σ ∗ α ( F ) still transforms covariantly also as a g -valued 2-form over M . Strictly speaking, the covariant derivative on the adjoint bundle acts on this local representative.Notice that the gauge field A ( α ) instead looks only locally as an element of the adjoint bundle, but globally itdoes not respect the “right” transformation property, and indeed it comes from a global 1-form on P that isnot horizontal, but vertical. As a principal bundle map it is by definition G -equivariant, Ψ( p · g ) = Ψ( p ) · g , and it commutes with theprojection, π (Ψ( p )) = π ( p ), for every p ∈ P . Notice that this looks like P → Σ as a fiber bundle, since both are locally trivial with fiber G . It is onlythe G action that distinguishes them. On P we have a right action, on Ad( P ) we have the left action on thefibers g · f := gf g − . ppendix A. Some differential geometry 157( ⇒ ) In every local patch U α , let us define the map ˜ ψ α : π − ( U α ) → G such that˜ ψ α ( p ) := g α (Ψ( p )) g α ( p ) − , where g α is the trivialization map inside U α . By equivariance of Ψ and g α , ˜ ψ α is G -invariant, so it depends only on the base point π ( p ). Thus we can define ψ α : U α → G such that ψ α ( x ) := ˜ ψ α ( p ) for some p ∈ π − ( x ) . In changing local patch, this transforms in the adjoint representation. In fact, if x ∈ U α ∩ U β ψ β ( x ) = g β (Ψ( p )) g β ( p ) − = (cid:2) g β (Ψ( p )) g α (Ψ( p )) − (cid:3) g α (Ψ( p )) g α ( p ) (cid:2) g α ( p ) − g β ( p ) (cid:3) = g αβ ( x ) ψ α ( x ) g αβ ( x ) − , where in the last passage we recognized that g β ( p ) g α ( p ) − is G -invariant and thus can bewritten as a map g αβ : U α ∩ U β → G that depends only on the base point π ( p ), and weused that π ◦ Ψ = π .( ⇐ ) Starting from a class of local sections { ψ α } , we define the G -invariant maps ˜ ψ α := ψ α ◦ π .Then we can obtain Ψ by “inverting” the above definition in every patch and gluing themtogether, Ψ( p ) := σ α ( p ) · (cid:16) ˜ ψ α ( p ) g α ( p ) (cid:17) . (cid:4) The group of gauge transformations G ( P ) acts naturally on the space A ( P ) via pull-back,Ψ · A := Ψ ∗ ( A ) ∀ A ∈ A ( P ) , Ψ ∈ G ( P ) . (A.17)If we consider the local gauge field A ( α ) , one can prove that the trivialization of the gauge-transformed connection follows the usual rule(Ψ · A ) ( α ) = ψ − α A ( α ) ψ α + ψ − α ( dψ α ) . (A.18)From this local expression, it is easy to find the representation of Lie ( G ( P )) on T A ( P ). In fact,writing Ψ as exp( X ) for some X ∈ Lie ( G ( P )) ∼ = Ω ( M ; ad( P )), we can recognize the associatedfundamental vector field as X A = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 e − tX · A = dX + [ A, X ] = ∇ A X. (A.19)This make us see the usual “infintesimal variation” δ X A as a tangent vector δ X A ≡ X A ∈ T A A ( P ) at the point A ∈ A ( P ). A.2 Spinors in curved spacetime
In QFT, fermionic particles are described geometrically by spinors , i.e. fields that transformunder the Lorentz algebra in representations whose angular momentum is half-integer . At the58 A.2. Spinors in curved spacetimelevel of Lie groups, they transform thus in representations of the double-cover of the rotationgroup of spacetime, SO (1 , d −
1) (or SO ( d ) in the Euclidean case), SO (1 , d − ∼ = Spin (1 , d − (cid:30)Z . (A.20)For simplicity, let us denote the dimension by d for the rest of the section, since the discussionis valid both for the Euclidean and the Lorentzian signature. In Minkowski spacetime ( R d , η ),there exists a preferred class of global coordinate systems, the global “inertial frames”, wherethe metric is diagonal η µν = diag ( − , +1 , · · · , +1) (A.21)and that are preserved by the Lorentz transformations. Working only with such special typeof coordinate systems, one can introduce and work with spinors as living in double-valuedrepresentations of the Lorentz algebra, and transforming asvector fields : V µ (cid:55)→ Λ µν V ν , spinor fields : Ψ α (cid:55)→ S (Λ) βα Ψ β , (A.22)where, if Λ = exp( iω µν M µν ), S (Λ) = exp( iω µν Σ µν ). When we move on to the description ofa generically curved spacetime M , there is a priori no such choice of “preferred” coordinatesystems, and a general coordinate transformation (GCT) is generated by a diffeomorphism M → M , reflecting on the tangent spaces as GL ( d, R ) basis changes. SO ( d ) injects as asubgroup of the General Linear group, but Spin ( d ) does not, since it is a double cover, so it isnot clear a priori how GCTs act on spinor fields. Tensor fields are naturally present in the fullycovariant formalism as fields over the manifold M , but to define spinors one has to introducefurther structure.The solution to this puzzle is really to (try to) mimic the same idea applied the theMinkowski case, and employ the presence of a (pseudo-)Riemannian metric g on M . On themetric manifold ( M, g ) all the tangent bundles arise as associated bundles to the frame bundle LM , that is a principal GL ( d, R )-bundle over M . Using the presence of a metric on M , onecan restrict the frame bundle to a principal SO ( d )-bundle, by considering only those frames e = ( e , · · · , e d ) such that, at a given point g ( e i , e j ) = η ij , (A.23)where η is the “flat” Minkowski (or Euclidean) metric. This reduction defines the so called orthonormal frame bundle LM ( SO ) π −→ M . A section of this bundle is an orthonormal frame,or tetrad . It is customary to denote with Latin indices the expansion of every vector fieldwith respect to an orthonormal frame, and with Greek indices the expansion with respect to ageneric (for example chart-induced) frame: V = V i e i = V µ ∂∂x µ for V ∈ Γ( T M ) . (A.24)The choice of an orthonormal frame is encoded in the choice of a vielbein , or solder form on M , that is a linear identification of the tangent bundle with the typical fiber R d : E : T M → R d V (cid:55)→ E ( V ) := (˜ e i ( V )) i =1 , ··· ,d , (A.25) Latin indices are sometimes called “flat”, and Greek ones “curved”. If one needs to raise and lower indices,flat indices are understood to be multiplied by the diagonalized metric η ij , curved indices by g µν . ppendix A. Some differential geometry 159where (˜ e i ) is the dual frame to a chosen orthonormal frame ( e i ). Notice that the choice of ametric is in one to one correspondence with the choice of a vielbein, since g ( · , · ) = (cid:104) E ( · ) , E ( · ) (cid:105) , (A.26)where (cid:104)· , ·(cid:105) is the canonical inner product on R d with the chosen signature. In a chart-inducedbasis, g µν = e iµ e jν η ij , where we denoted the components of the vielbein ( E ( ∂ µ )) i ≡ e iµ . The“inverse vielbein” at any point is the matrix e µi such that e iµ e µj = δ ij .Once this orthonormal reduction is made, one can define spinor bundles as associated bun-dles to a principal Spin ( d )-bundle, that must be compatible with the orthonormal frame bundle.This is made precise by defining the presence of a spin-structure on M . Definition A.2.1. A spin-structure on ( M, g ) is a principal
Spin ( d )-bundle Spin ( M ) π S −→ M ,together with a principal bundle map Spin ( M ) LM ( SO ) M Φ π S π with respect to the double-cover map ϕ : Spin ( d ) → SO ( d ). This means that the equivariancecondition is Φ( s · g ) = Φ( s ) · ϕ ( g ) ∀ s ∈ Spin ( M ) , g ∈ Spin ( d ) . A section of
Spin ( M ) → M is called spin-frame .We notice that the equivariance condition in this definition is just the formal requirementthat spinors and tensors transform all together with compatible rotations by the action of therespective groups. Although the above restriction of the frame bundle to the orthonormal framebundle can always be done in presence of a metric on M , a spin-structure does not necessarilyexist, and if it does it is not necessarily unique. There can be topological obstructions to thisprocess that can be characterized in terms of the cohomology of M . By this construction, and from the canonical Levi-Civita covariant derivative ∇ on ( M, g ),we can induce a connection 1-form on the orthonormal frame bundle and on the spin-framebundle, and thus have a compatible covariant derivative on associated spinor bundles. Let usrecall that the Levi-Civita connection on (
M, g ) is the unique metric-compatible and torsionfree connection, i.e. ∇ X g = 0 ⇔ X ( g ( Y, Z )) = g ( ∇ X Y, Z ) + g ( Y, ∇ X Z ) ,T = 0 ⇔ ∇ X Y − ∇ Y X = [ X, Y ] . (A.27)This covariant derivative is associated to the gauge field Γ ∈ Ω ( M ) ⊗ gl ( n, R ) such thatΓ ρµν := ( ∇ µ ( ∂ ν )) ρ . Simply restricting to orthonormal frames, one can induce a connection 1-form on LM ( SO ) , ω ∈ Ω ( LM ( SO ) ) ⊗ so ( d ) such that in any trivialization induced by a localframe ( U ⊂ M, e : U → LM ( SO ) ) the gauge field has components ω ( X ) ij := ( ∇ X ( e i )) j = X µ e jν ( ∇ µ e i ) ν = X µ e jν (cid:0) ∂ µ e νi + Γ νµσ e σi (cid:1) or ω ( X ) ij = g ( ∇ X e i , e j ) , (A.28) Recall that a principal bundle map by definition commutes with the projections, π (Φ( p )) = π S ( p ). In particular, it turns out that a spin-structure exists if and only if the second Stiefel–Whitney class of M vanishes [27].
60 A.2. Spinors in curved spacetimeand it can be written as ω ( U ) := e ∗ ω = ω ij M ij , where M ij are the generators of so ( d ).Given a spin-structure as in the above definition, we can induce a compatible spin-connection ˜ ω ∈ Ω ( Spin ( M )) ⊗ so ( d ) by pulling back ω , ˜ ω := Φ ∗ ω . In a given patch U ⊂ M , if s : U → Spin ( M ) is a local spin-frame and e := Φ ◦ s is the associated tangent frame, the localgauge fields representing the spin-connection and the Levi-Civita connection coincide,˜ ω ( U ) := s ∗ ˜ ω = (Φ ◦ s ) ∗ ω = ω ( U ) , (A.29)so in particular the local components of the compatible spin-connection are defined as˜ ω ( X ) ij = ( ∇ X ( e i )) j . (A.30)The covariant derivative on an associated spinor bundle is defined as usual. Let V be thetypical fiber, acted upon by the representation ρ : Spin ( d ) → GL ( V ). Then for every local V -valued function ψ : U → V , ∇ X ψ = dψ ( X ) + 12 ω ( X ) ij ρ ( M ij ) · ψ. (A.31)If in particular we take the fundamental representation of Spin ( d ), i.e. ψ is a Dirac spinor , thegenerators are ρ ( M ij ) = Σ ij := [ γ i , γ j ], where γ i are the Dirac matrices. Thus, ∇ µ ψ = ∂ µ ψ + 18 ω µij [ γ i , γ j ] · ψ. (A.32)We quote the fact that, in general, one is not forced to consider a spin-connection thatis compatible with the Levi-Civita connection. However, in this work we always implicitlydefine covariant derivatives on spinors via a compatible spin-connections. A discussion aboutspinors in curved spacetime can be found also in [65]. Notice that
Lie ( Spin ( d )) ∼ = Lie ( SO ( d )) ∼ = so ( d ). For example in SUGRA it is sometimes convenient to work with torsion-full spin connections. ppendix BMathematical background onequivariant cohomology
B.1 Equivariant vector bundles and equivariant charac-teristic classes
We recall the definitions of characteristic classes on principal bundles [116] and then theirequivariant version when the bundle supports a G -action for some Lie group G . Consider aprincipal H -bundle P π −→ M with connection 1-form A , and curvature F . Both are forms on P with values in the Lie algebra h . A polynomial on h is an element f ∈ S ( h ∗ ), and it is called invariant polynomial if it is invariant with respect to the adjoint action of H on h , f ( Ad ∗ h X ) = f ( X ) ∀ X ∈ h , h ∈ H. (B.1)For example, if H is a matrix group, the adjoint action is simply Ad ∗ h X = hXh − . If f is an invariant polynomial of degree k , then f ( F ) is an element of Ω k ( P ). Explicitly, withrespect to a basis ( T a ) a =1 , ··· , dim h of h and the dual basis ( α a ) a =1 , ··· , dim h of h ∗ , if F = F a T a and f = f a ··· a k α a · · · α a k , then f ( F ) = f a ··· a k F a ∧ · · · ∧ F a k . (B.2)The above form has three remarkable properties:(i) f ( F ) is a basic form on P , i.e. it exists a 2 k -form Λ ∈ Ω k ( M ) such that f ( F ) = π ∗ Λ;(ii) d Λ = 0, or equivalently df ( F ) = 0;(iii) the cohomology class [Λ] ∈ H k ( M ) is independent on the connection F .The cohomology class [Λ] on M is called characteristic class of P associated to the invariantpolynomial f . Denoting Inv( h ) ⊆ S ( h ∗ ) the algebra of invariant polynomials on h , the map w : Inv( h ) → H ∗ ( M ) f (cid:55)→ [Λ] (B.3)is called Chern-Weil homomorphism .If one is considering a vector bundle E → M associated to the principal H -bundle P → M ,here the connection 1-form A and the curvature F are represented only locally via h -valued forms16162 B.1. Equivariant vector bundles and equivariant characteristic classeson M . Under a change of trivialization the local connection does not transform covariantly,but the local curvature does (by conjugation), so the invariant polynomial f ( F ) is independenton the frame and it defines a global form on M . The definition of characteristic classes couldbe thus given in terms of the local curvature of a vector bundle, without changing the result.We need mainly three examples of characteristic classes, associated to the invariant poly-nomials Tr , det and Pf, that corresponds for matrix groups to the standard trace, determinantand pfaffian. These are the Chern character ch( F ) := Tr (cid:0) e F (cid:1) , (B.4)the Euler class e ( F ) := Pf (cid:18) F π (cid:19) , (B.5)and the Dirac ˆ A -genus ˆ A ( F ) := (cid:118)(cid:117)(cid:117)(cid:116) det (cid:34) F sinh (cid:0) F (cid:1) (cid:35) . (B.6)Now we turn the discussion to the case of G -equivariant bundles [7, 19, 35]. Definition B.1.1. A G -equivariant vector bundle is a vector bundle E π −→ M , such that:(i) both E and M are G -spaces and π is G -equivariant;(ii) G acts linearly on the fibers.A principal H -bundle P π −→ M is G -equivariant if(i) both E and M are G -spaces and π is G -equivariant;(ii) the G -action commutes with the H -action on P .Usually, a connection A on a G -equivariant principal bundle is required to be G -invariant ,that is L X A = 0 for every X ∈ g . If G is compact, this choice is always possible by averaging anyconnection over G to obtain a G -invariant one [35]. Since the G - and the H -actions commute, aprincipal H -bundle P π −→ M induces another principal H -bundle P G π G −→ M G over the homotopyquotient M G . Topologically, the equivariant characteristic classes of P π −→ M are the ordinarycharacteristic classes of P G π G −→ M G , thus defining elements in the G -equivariant cohomology H ∗ G ( M ). From the differential geometric point of view, they can be derived as equivariantlyclosed extensions of the ordinary characteristic classes in the Cartan model. In particular, in [7,35] it was shown that the equivariant characteristic class associated to an invariant polynomial f is represented by f ( F g ), where F g = 1 ⊗ F + φ a ⊗ µ a (B.7)is the equivariant extension of the curvature F on the principal H -bundle. φ a =1 , ··· , dim g are thegenerators of S ( g ∗ ) in the Cartan model, and the map µ : g → Ω( P ; h ) such that µ X := − ι X A = − A ( X ) (B.8)is called moment map , with analogy to the symplectic case. We denoted µ a ≡ µ T a with T a =1 , ··· , dim g the basis of g dual to φ a . If we define ∇ = d + A the covariant derivative, thatppendix B. Mathematical background on equivariant cohomology 163in the adjoint bundle acts as ∇ ω = dω + [ A ∧ , ω ], we notice that we can obtain the aboveequivariant curvature in the Cartan model from the equivariant covariant derivative ∇ g := 1 ⊗ ∇ − φ a ⊗ ι a (B.9)that is completely analogous to the definition of the Cartan differential (2.46). With thisdefinition, the equivariant curvature can be expressed as F g = ( ∇ g ) + φ a ⊗ L a , (B.10)where the last piece takes care of the non-nilpotency of the Cartan differential on genericdifferential forms, and moreover it satisfies an equivariant variation of the Bianchi identity( ∇ g F g ) = 0 . (B.11)Notice that, if we assume the connection A to be G -invariant, the moment map µ indeedsatisfies a moment map equation with respect to the curvature F (see Section 3.3.2), ∇ µ X = − ι X F ∀ X ∈ g . (B.12)Once a suitable equivariant extension of the curvature F g is known, the particular equivari-ant characteristic classes are simply a modification of the old ones, so the equivariant versionof the above Chern character, Euler class and Dirac ˆ A -genus are given bych G ( F ) := Tr (cid:0) e F g (cid:1) , e G ( F ) := Pf (cid:18) F g π (cid:19) , ˆ A G ( F ) := (cid:118)(cid:117)(cid:117)(cid:116) det (cid:34) F g sinh (cid:0) F g (cid:1) (cid:35) , (B.13)respectively. B.2 Universal bundles and equivariant cohomology
In this section we motivate the well-definiteness of equivariant cohomology of Section 2.2,starting from the definition of the space EG . Proofs for the various propositions we are goingto state informally and/or without proof can be found for example in [18, 21, 24]. We shouldmention that the mathematically correct approach to this subject works considering only CWcomplexes . These are special types of topological spaces that can be constructed by “attachingdeformed disks” to each other [24]. We only quote that any smooth manifold can be given thestructure of a CW complex, so that in the smooth setting we do not need to bother with thissubtlety. Definition B.2.1.
A principal G -bundle π : EG → BG is called universal G-bundle if:(i) for any principal G -bundle P → X , there exists a map h : X → BG such that P ∼ = h ∗ ( EG ) (the pull-back bundle of EG through h );(ii) if h , h : X → BG are such that h ∗ ( EG ) ∼ = h ∗ ( EG ), then the two maps are homotopic.The base space BG is called classifying space . This is a result of Morse theory, see [18] and references therein.
64 B.2. Universal bundles and equivariant cohomologyThe classifying property (i) required of EG means that for every principal G -bundle thereis a copy of it sitting inside EG → BG . The important fact is that existence can be provenfor a large class of interesting cases, the argument going as follows. First recall that homotopicmaps pull back to isomorhic bundles , i.e. if E → B is a vector bundle, X a paracompact space,then g, h : X → B homotopic maps ⇒ g ∗ ( E ) ∼ = h ∗ ( E ) . (B.14)Then the property (ii) in the definition above states that if E → B is a universal bundle, ⇒ isreplaced by ⇔ . Now let, for any paracompact space X , P G ( X ) := { isomorphism classes of principal G-bundles over X } (B.15)and for some space BG (to be identified with the classifying space),[ X, BG ] := { homotopy classes of maps X → BG } . (B.16)Notice that the definition of P G ( X ) is totally independent from the notion of universal G -bundle.Considering then the map φ : [ X, BG ] → P G ( X )[ h : X → BG ] (cid:55)→ h ∗ ( EG ) , (B.17)by (B.14) we have that it is well-defined (independent from the representatives). The conditions(i) and (ii) are equivalent to surjectivity and injectivity of φ , so finally P G ( X ) ∼ = [ X, BG ]. Since P G ( X ) exists, this proves the existence of the classifying space BG and of the universal bundle EG → BG . We can now motivate the well-definiteness of the Borel construction for equivariant coho-mology. A fundamental result for this is that a principal G -bundle is a universal bundle if andonly if its total space is (weakly) contractible . The contractibility of EG makes its cohomologytrivial, so that, since ( M × EG ) ∼ M , we have H ∗ ( M × EG ) ∼ = H ∗ ( M ). When we take thehomotopy quotient, the product by EG acts as a “regulator” of the resulting cohomology. Infact, if the action of G on M is free, such that M → M/G is a principal G -bundle, one canprove that for any (weakly) contractible G -space E ( M × E ) (cid:30) G ∼ M (cid:30) G, (B.18)where ∼ here stands for “weakly homotopic”. In general, even if the G -action is not free,two homotopy quotients with respect to different (weakly) contractible G -spaces E and E (cid:48) are(weakly) homotopy equivalent, ( M × E ) (cid:30) G ∼ ( M × E (cid:48) ) (cid:30) G. (B.19)Another known fact is that weakly homotopic spaces have the same (co)homology groups, forall coefficients , generalizing (2.9). Putting together these properties, we have that H ∗ (cid:16) ( M × E ) (cid:30) G (cid:17) ∼ = H ∗ (cid:16) ( M × E (cid:48) ) (cid:30) G (cid:17) , (B.20) In the language of category theory, we could say that P G ( · ) is a contravariant functor, representable through[ · , BG ]. A weakly contractible space is a topological space whose homotopy groups are all trivial. Clearly anycontractible space is weakly contractible. It is a fact that every CW complex that is weakly contractible is alsocontractible [24], so for our purposes the two concepts coincide. Two spaces are weakly homotopic if they have the same homotopy groups. Again, homotopy equivalenceimplies weak homotopy equivalence, and for CW complexes these two concepts coincide. ppendix B. Mathematical background on equivariant cohomology 165so that the resulting cohomology is independent of the choice of contractible principal G -bundle.The homotopy quotient thus well-defines the G -equivariant cohomology of M , producing an“homotopically correct” version of its orbit space. As pointed out in Section 2.2, when the G -action is free on M this reproduces the naive definition of cohomology of the quotient space M/G . Every compact Lie group has a universal bundle
In Section 2.2 we gave the example of the universal bundle for the circle, EU (1) = S ∞ and BU (1) = C P ∞ . One can generalize this construction to concretely define a universal bundlefor any compact Lie group G . This is because any such Lie group embeds into U ( n ) or O ( n )(the maximal compact subgroups of GL ( n, C ) and GL ( n, R )), for some n , and for them one canconstruct universal bundles explicitly. As a subgroup, G will act freely on the given universalbundle. Then one can take this to be its universal bundle too.A class of principal O ( n )- or U ( n )-bundles is given by the so-called Stiefel manifolds . AStiefel manifold V k ( F n ) is the set of all orthonormal k -frames in F n , where F = R , C , and theorthonormality is defined with respect to the canonical Euclidean or sesquilinear inner products.A k -frame is an ordered set ( v , · · · , v k ) of k linearly independent vectors in F n . Notice thatwhen k = 1, V ( C n ) is the set of all unit vectors in C n ∼ = R n , i.e. the (2 n − U (1) by diagonal multiplication, and analogously the Stiefelmanifold V k ( C n ) is acted freely by U ( k ), that essentially rotates the vectors of the k -frames.Analogously, V k ( R n ) is acted freely by O ( k ). Thus we have the generalization of the sequence ofprincipal U ( k )- and O ( k )-bundles, that in the limit n → ∞ produces the contractible universalbundles EU ( k ) = V k ( C ∞ ) and EO ( k ) = V k ( R ∞ ). The Stiefel manifolds can thus be seen as a“higher dimensional versions” of the spheres, in the sense of the following consideration: V k ( R n ) ∼ = O ( n ) (cid:30) O ( n − k ) . (B.21)Comparing with Example 2.2.1, where we remarked that S n − ∼ = O ( n ) /O ( n − S n − ≡ V ( R n ). The ( n − R n , so orthonormal1-frames.The base spaces G k ( C n ) = V k ( C n ) /U ( k ) and G k ( R n ) = V k ( R n ) /O ( k ) are the sets of equiv-alence classes of k -frames, that identify k -hyperplanes through the origin inside C n or R n .These manifolds are called Grassmannians . The infinite Stiefel manifold V k ( F ∞ ) and the infi-nite Grassmannian G k ( F ∞ ) are thus the total space of the universal bundle and the classifyingspace for the unitary and orthogonal groups U ( k ) and O ( k ), and generalize the universal bundle S ∞ → C P ∞ of the circle.As recalled above, any compact Lie group G can be embedded as a closed subgroup of anorthogonal group (or a unitary group). This means that G also acts freely on V k ( F ∞ ) for some k , and in turn V k ( F ∞ ) → V k ( F ∞ ) /G is a principal G -bundle, whose total space is a contractiblespace. This gives the universal bundle for any compact Lie group G . Module structure of equivariant cohomology
We end this section with a more algebraic comment about the construction of equivariantcohomology. Notice first that pt G = ( pt × EG ) (cid:30) G ∼ = BG ⇒ H ∗ G ( pt ) ∼ = H ∗ ( BG ) , (B.22)66 B.2. Universal bundles and equivariant cohomologyso the equivariant cohomology of a point is the standard cohomology of the classifying space BG , generalizing Example 2.2.2. Thus the equivariant cohomology H ∗ G ( · ) inherits analogousfunctorial properties to the standard (singular) cohomology of the last section, with respectto the ring H ∗ ( BG ) instead of the coefficient ring A ∼ = H ∗ ( pt ; A ). To see this, let us firstnotice that a G -equivariant function f : M → N between the two G -spaces M, N induces awell-defined map between the two homotopy quotients, f G : M G → N G [ m, e ] (cid:55)→ [ f ( m ) , e ] . (B.23)This induced map inherits many properties from f :(i) if f is injective (surjective), then f G is injective (surjective);(ii) if id : M → M is the identity, then id G : M G → M G is the identity;(iii) ( h ◦ f ) G = h G ◦ f G ;(iv) if f : M → N is a fiber bundle with fiber F , then f G : M G → N G is also a fiber bundlewith fiber F .As pointed out in Section 2.1, a map between two topological spaces induces a map (in theopposite direction) between the associated singular cohomologies, so f ∗ G : ( H ∗ ( N G ) ≡ H ∗ G ( N )) → ( H ∗ ( M G ) ≡ H ∗ G ( M )) . (B.24)Defining thus a trivial map φ : M → pt , we see from (B.22) that the induced homomorphism φ ∗ G : H ∗ ( BG ) → H ∗ G ( M ) makes the equivariant cohomology H ∗ G ( M ) naturally into a H ∗ ( BG )-module! Also, in general f ∗ G : H ∗ G ( N ) → H ∗ G ( M ) is a H ∗ ( BG )-module homomorphism. Noticethat the cohomology of the classifying space BG is usually very simple, as we pointed out inSection 2.4 via its associated Weil model.There is a curious difference between standard cohomology and equivariant cohomologyregarding the associated coefficient rings. In the former case, it is clear from the variousexamples in Section 2.1 that the coefficient ring R ∼ = H ∗ ( pt ) always embeds into the cohomology H ∗ ( M ) (also for other commutative rings). In the case of equivariant cohomology, on the otherhand, the coefficient ring H ∗ G ( pt ) = H ∗ ( BG ) = S ( g ∗ ) G does not, since the map φ ∗ G above is notinjective in general, as it is clear also from the example of H ∗ U (1) ( S ) = R . It turns out thatthe condition for H ∗ ( BG ) to embed in H ∗ G ( M ) is that G acts on M with fixed points . We canargue briefly why this is the case. Let p ∈ M be a fixed point. The inclusion i : { p } → M is G -equivariant since the action on p is trivial, so there is a well-defined map i G : pt G = BG → M G .This is easily checked to be a section of the bundle M G π −→ BG , with respect to the projectionmap π ([ m, e ]) := [ e ] ∈ BG . The identity π ◦ i G = id M G lifts to the pull-backs in the oppositedirection: i ∗ G ◦ π ∗ = id on H ∗ G ( pt ) = H ∗ ( BG ). This means that the map π ∗ : H ∗ ( BG ) → H ∗ G ( M )has a left-inverse, and thus it is injective. This property can be seen in the example of the U (1)-equivariant cohomology of the 2-sphere. In this case there are two fixed points, andindeed H ∗ ( BU (1)) = R [ φ ] embeds in H ∗ U (1) ( S ) = R [ φ ] ⊕ R [ φ ] y , where y can be identified inthe Cartan model with the equivariantly closed extension of the volume form, y ≡ [˜ ω ]. Recall that singular cohomology has a ring structure. In category theory terminology, we could say that the Borel construction ( · ) G is a covariant functor fromthe category of G -spaces to Top (or
Man ), and H ∗ G ( · ) is a contravariant functor between Top (or
Man ) andthe category of H ∗ ( BG )-modules. ppendix B. Mathematical background on equivariant cohomology 167 B.3 Fixed point sets and Borel localization
We now spend a few words about a procedure that we used many times without manyworries, that is to “algebraically localize” the space of equivariant differential forms Ω( M ) U (1) [ φ ]with respect to the indeterminate φ , setting it to φ = −
1. This localization was useful tosimplify the notation in many occasions, but it really has a non-trivial deeper meaning. In fact,it allows to show in a more algebraic way that the G -equivariant cohomology of the smooth G -manifold M is encoded in the fixed point set F of the G -action, at least when G is a torus.The fundamental theorem concerning this point is the so-called Borel localization theorem , thatsometimes allows to obtain the ring structure of the equivariant cohomology of the manifoldfrom that of its fixed point set. We consider the case of a circle action here.First, let us recall what localization in algebra means. If R is a commutative ring, the localization of R with respect to a closed subset S ⊆ R is a way to formally introduce amultiplicative inverse for every element of S in R , so to introduce fractions in R , analogouslyto what one does in the construction of the rational numbers Q from the integers Z . Thisprocedure makes the former commutative ring into a field (in the algebraic sense). Since we areinterested in U (1)-equivariant cohomologies, let us consider an R [ φ ]-module N , and practicallydefine the localization of N with respect to φ as N φ ∼ = (cid:26) xφ n (cid:12)(cid:12)(cid:12)(cid:12) x ∈ N, n ∈ N (cid:27) , (B.25)identifying elements in N φ as xφ n ∼ yφ m ⇔ ∃ k ∈ N : φ k ( φ m x − φ n y ) = 0 in N. (B.26)The simplest example of such a localized module is just R [ φ ] φ ∼ = R [ φ − , φ ], i.e. the Laurentpolynomials in φ . Notice that there is always an R [ φ ]-module homomorphism that makes N inject into N φ , i : N → N φ such that i ( x ) := x/φ . If f : N → M is an R [ φ ]-modulehomomorphism, then there is a well-defined induced homomorphism between the localizedmodules f φ : N φ → M φ such that f ( x/φ n ) := f ( x ) /φ n . The important algebraic property oflocalization for what concerns this discussion is that it commutes with cohomology : if ( A, d ) isa differential complex, A (0) d −→ A (1) d −→ · · · , d = 0 , (B.27)where A ( i ) are R [ φ ]-modules, then also ( A φ , d φ ) is a differential complex, and H ∗ ( A, d ) φ ∼ = H ∗ ( A φ , d φ ) . (B.28)Quite analogously, from Example 2.4.1 onward we substitute the indeterminate φ ∈ S ( u (1) ∗ )with a variable , and then set it to the value φ = − U (1)-equivariant differential forms Ω( M ) U (1) [ φ ],that has clearly an R [ φ ]-module structure, and localize it to Ω( M ) U (1) [ φ ] φ , so introducing φ also at the denominator. This puts φ on the same footing as a real variable, so that we areallowed to fix it to some value, for convenience only. Notice that operations like (3.8), wherewe “invert” an equivariant form, are allowed only in the localized module Ω( M ) U (1) [ φ ] φ , wherethe division by φ is meaningful. From the result (B.28), we understand that this localization of68 B.4. Equivariant integration and Stokes’ theoremthe Cartan model does not spoil the resulting equivariant cohomology H ∗ U (1) ( M ), because thetwo operations commute. The Borel localization theorem relates really the localized equivariant cohomologies of the U (1)-manifold M and of its fixed point locus F . To understand what this has to say aboutthe actual equivariant cohomology of M , we recall first some other algebraic facts. A torsion element in a module N over a ring R , is an element x ∈ N such that ∃ r (cid:54) = 0 ∈ R : rx = 0. If N is an R [ φ ] module, the element x is said to be φ - torsion if it exists some power of φ thatannihilates it: φ k x = 0 for some k ∈ N . The module N is φ -torsion if every one of its elementsis φ -torsion. It is easy to see that N is φ -torsion ⇔ N φ = 0 . (B.29)Applying this to the case of N = H ∗ U (1) ( M ), we can see that the equivariant cohomology inthe case of a free U (1)-action on M is φ -torsion. In fact, if the action is free, we can easilycompute H ∗ U (1) ( M ) = H ∗ ( M/U (1)), so that H kU (1) ( M ) = 0 in some degree k > dim( M/U (1)).This means that φ k · H ∗ U (1) ( M ) = 0 for some k high enough. The first argument in Section 3.1in fact is the proof that more is true: H ∗ U (1) ( M ) is φ -torsion if the U (1)-action is locally free on M , since we found essentially ( H ∗ U (1) ( M )) φ = 0 as the Poincar´e lemma, after having introduced φ at the denominator. This motivates the following theorem, that states that, up to torsion ,the U (1)-equivariant cohomology of M is concentrated on its fixed point set. A proof can befound in [18, 22]. Theorem B.3.1 (Borel localization) . Let U (1) act smoothly on the manifold M , with compactfixed point set F . The inclusion i : F (cid:44) → M induces an isomorphism of algebras over R [ φ ], i ∗ φ : H ∗ U (1) ( M ) φ → H ∗ U (1) ( F ) φ . This theorem is an “abstract version” of the localization theorems described in Chapter 3,and intuitively gives another way to see that they have to be true, without having to travelthrough all the smooth algebraic models and the integration theory that we described in duetime. It shows that localization is something present at a very low level of structure, originatingjust from the topological nature of equivariant cohomology.
B.4 Equivariant integration and Stokes’ theorem
In this section we define what it means to integrate a G -equivariant differential form ω ∈ Ω G ( M ) over a smooth, oriented G -manifold M of dimension dim ( M ) = n and we report anextended version of Stokes’ theorem that applies in the equivariant setup. Let G be a connectedLie group acting (smoothly) on the left on M , being { φ a } a =1 , ··· ,dim ( g ) a basis for g ∗ := Lie ( G ) ∗ .If the equivariant form ω is of degree k , we can express it as ω = ω ( k ) + ω ( k − a φ a + ω ( k − ab φ a φ b + · · · = (cid:88) p ≥ ω ( k − p ) a ··· a p φ a · · · φ a p (B.30) Notice that H ∗ U (1) ( M ) has generically an R [ φ ]-module structure, by the discussion in Appendix B.2 and theapplication of the Weil model (see Section 2.4) H ∗ ( BG ) ∼ = S ( u (1) ∗ ) ∼ = R [ φ ]. Just consider that in the localized module x ∼ φ k φ k x , so if x is φ -torsion it is equivalent to 0 in N φ . In Section 3.1 M is the manifold without its fixed point set, there called ˜ M . ppendix B. Mathematical background on equivariant cohomology 169where the coefficients are differential forms on M , tensor products have been suppressed and werequire ω to be G -invariant. The natural way to define integration of such objects is obtainedjust making the integral (cid:82) M act on the coefficients ω ( k − p ) a ··· a p of the φ -expansion of ω . In this way,one obtains a map (cid:90) M : Ω G ( M ) → S ( g ∗ ) ≡ R [ φ a ] . (B.31)Thanks to the equivariant Stokes’ theorem (to be stated later), this descends also in equivariantcohomology, (cid:82) M : H ∗ G ( M ) → S ( g ∗ ), analogously to the standard (non-equivariant) case. Definition B.4.1.
The integral on M of the G -equivariant form ω of deg( ω ) = k is defined as (cid:90) M ω := (cid:88) p ≥ (cid:18)(cid:90) M ω ( k − p ) a ··· a p (cid:19) φ a · · · φ a p . Notice that if n and k are of different parity, the integral is automatically zero. If instead k = n + 2 m for some m ∈ Z , then (cid:90) M ω = (cid:40)(cid:16)(cid:82) M ω ( n ) a ··· a m (cid:17) φ a · · · φ a m k (cid:62) n k < n. (B.32)In particular, if we have a top form ω ∈ Ω( M ) on M and ˜ ω is any equivariant extension of ω inΩ G ( M ), then we can deform the integral (cid:90) M ω = (cid:90) M ˜ ω (B.33)without changing its value.We can then prove the equivariant version of the Stokes’ theorem. Theorem B.4.1.
Let G be a connected Lie group acting (smoothly) on the left on a smoothmanifold M with boundary ∂M . If ω ∈ Ω G ( M ) of deg( ω ) = k , then (cid:90) M d C ω = (cid:90) ∂M ω where d C = 1 ⊗ d + φ a ⊗ ι a is the Cartan differential and ι a ≡ ι T a , with { T a } a basis of g ∗ dualto { φ a } . Proof.
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