Twisting with a Flip (the Art of Pestunization)
aa r X i v : . [ h e p - t h ] M a r UUITP-59/18
Twisting with a Flip(the Art of Pestunization)
Guido Festuccia a , Jian Qiu a,b , Jacob Winding c , Maxim Zabzine aa Department of Physics and Astronomy, Uppsala University,Box 516, SE-75120 Uppsala, Sweden b Mathematics Institute, Uppsala University,Box 480, SE-75106 Uppsala, Sweden c School of Physics, Korea Institute for Advanced Study,Seoul 130-722, Korea
Abstract
We construct N = 2 supersymmetric Yang-Mills theory on 4D manifolds with a Killingvector field with isolated fixed points. It turns out that for every fixed point onecan allocate either instanton or anti-instanton contributions to the partition function,and that this is compatible with supersymmetry. The equivariant Donaldson-Wittentheory is a special case of our construction. We present a unified treatment of Pestun’scalculation on S and equivariant Donaldson-Witten theory by generalizing the notionof self-duality on manifolds with a vector field. We conjecture the full partition functionfor a N = 2 theory on any 4D manifold with a Killing vector. Using this new notionof self-duality to localize a supersymmetric theory is what we call “Pestunization”. ontents S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 h = cos ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Varying the metric and other supergravity fields . . . . . . . . . . . . . . . . 274.3 Other choices of projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.4 Complex v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.1.2 CP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 The super-determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 The full answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Conventions 44
A.1 Flat Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44A.2 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.3 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46A.4 Other conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
B Examples arising from specific four-manifolds 46
B.1 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46B.2 CP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 C Transverse elliptic problems in 4D and 5D 48D Full cohomological complex 52E Supergravity background 53
Starting from the work [1] the localization of supersymmetric theories on compact manifoldshas attracted considerable attention (see [2] for a review of the latest developments). Su-persymmetric theories were constructed on diverse 2D to 7D compact manifolds and theirpartition functions were calculated or conjectured. Typically the manifold admits (gener-alized) Killing spinors and there is enough torus action to proceed with the localizationcalculation. In low dimensions toric geometry is not very rich; it becomes interesting start-ing from 4D and higher. Constructing supersymmetric field theories on odd dimensionalmanifolds M is simple. This is related to the fact that Killing spinors on M are related tocovariantly constant spinors on the cone over M . For example, any toric Calabi-Yau cone in6D produces a toric 5D Sasaki-Einstein geometry (i.e. a geometry with two Killing spinors).A similar story holds in 7D. In even dimensions the situation is more complicated. In thispaper we concentrate on N = 2 supersymmetric theories on 4D compact manifolds with a T -action. Our goal is to explain their structure and also to show how the localization resultfor their partition function is organized.Let us briefly review different ways of placing N = 2 theories on 4D manifolds preservingsome supersymmetry. About 30 years ago Witten [3] constructed Donaldson-Witten the-ory, which corresponds to a topological twist of N = 2 supersymmetric gauge theory andlocalizes on instantons. Donaldson-Witten theory is related to the calculation of Donaldsoninvariants [4, 5]. If the 4D manifold admits a torus action then one can define equivariant3onaldson-Witten theory. This theory has been studied in detail on R [6–9] and the corre-sponding partition function is known as the Nekrasov partition function Z inst ǫ ,ǫ ( a, q ) [10, 11].Schematically the Nekrasov function is Z inst ǫ ,ǫ ( a, q ) = Z − loop ( a ) ∞ X n =0 q n vol n ( ǫ , ǫ , a ) , (1)where vol n ( ǫ , ǫ , a ) is the equivariant volume of the moduli space of instantons of charge n and we include the perturbative 1-loop contribution Z − loop ( a ) for later convenience. Thepartition function of equivariant Donaldson-Witten theory on non-compact toric surfaceshas been later conjectured in [12] (see also [13–15] for further studies and proofs). The fullanswer is obtained gluing copies of Nekrasov partition functions associated to each fixedpoint of the torus action, with some discrete shifts associated to fluxes (related to non-trivialsecond cohomology). A second class of supersymmetric theories originates from the workof Pestun [1] who placed a N = 2 supersymmetric gauge theory on the round S . Thisconstruction was later extended to certain squashed spheres S ǫ ,ǫ in [16, 17]. The resultingtheory is not related in any obvious way to equivariant Donaldson-Witten theory. Theirpartition function on S ǫ ,ǫ can be written schematically as follows Z S ǫ ,ǫ = Z da e S cl Z inst ǫ ,ǫ ( ia, q ) Z anti − inst ǫ , − ǫ ( ia, ¯ q ) , (2)displaying the contribution of instantons over the north pole and anti-instantons over thesouth pole of S (the way we write the expression for Z S ǫ ,ǫ is clarified in section 5.3).In [18], reducing a 5D theory, a N = 2 theory was constructed on the connected sum k ( S × S ) , which is a toric manifold with T -action, and has (2+2 k ) fixed points. Similarlyto (2) the partition function for k ( S × S ) is obtained gluing ( k + 1) copies of Nekrasovfunctions for instantons and ( k + 1) -copies for anti-instantons.From the aforementioned results a question arises naturally. Consider any simply con-nected compact 4D manifold with a T -action with isolated fixed points. Distribute Nekrasovpartition functions for instantons over some of the fixed points and partition functions foranti-instantons over the rest. Does this correspond to the partition function of some su-persymmetric theory? In this paper we answer this question positively. The case when weassociate instantons to all fixed points corresponds to equivariant Donaldson invariants, (seee.g. [19, 20] and earlier works [21–23]). We present a general class of supersymmetric fieldtheories that includes on equal footing both equivariant Donaldson-Witten theory and the In conjecturing the answer for the partition function in [18] we missed the contribution of fluxes. S ǫ ,ǫ reviewed above. In the rest of this section we briefly explain the main ideasleading to our construction.Donaldson-Witten theory is a 4D gauge theory counting instantons (anti self-dual connec-tions) in some appropriate sense. Anti self-duality is a nice problem in 4D and its linearizationis related to the following elliptic complex on 4D manifolds Ω ( M ) d −→ Ω ( M ) P + d −−→ P + Ω ( M ) = Ω ( M ) , (3)where Ω p are the p -forms, d is de Rham differential and the projectors P ± = (1 ± ⋆ ) usethe Hodge star ⋆ . Ellipticity of the complex (3) (or ellipticity of the corresponding PDEs)is closely related to the existence of a finite dimensional moduli space of instantons. Usingthe projector P + the Yang-Mills action can be written as follows || F || = || P + F || + ... , (4)where the dots stand for the topological term. Donaldson-Witten field theory localizes to P + F = 0 , and provides an infinite dimensional analogue of the Euler class of a vectorbundle [24]. Eventually the calculation is reduced to a finite dimensional problem on themoduli space of instantons. Most of the known cohomological and topological field theoriesare related to underlying finite dimensional moduli spaces, and thus to some elliptic problem.In [1] Pestun uses the theory of transversely elliptic operators (correspondingly there arenotions of transversely elliptic complex and transversely elliptic PDEs) in order to calculatethe 1-loop determinants arising from localization. It is one of the goals of this paper (andits follow up [25]) to show that supersymmetry is closely related to such transversely ellipticproblems.In order to introduce the logic underlying our later constructions, we provide below aninformal explanation of the notion of transverse ellipticity. If on a compact manifold M wehave an action of a group G then we say that an operator is transversely elliptic if it is ellipticin the directions normal to the G -orbits. In this case the kernel and co-kernel of this operatorare not finite dimensional, but they can be decomposed in finite dimensional representationsof G . As an example consider the elliptic problem in 2D corresponding to the Dolbeaultoperator ¯ ∂ acting on functions , e.g. functions φ on S satisfying the condition ¯ ∂φ = 0 . Ifwe add a circle S to this problem and consider a new function φ on S × S satisfying thecondition ¯ ∂φ = 0 , we obtain an example of a 3D transversely elliptic problem with respectto the S -action. There is no finite dimensional kernel for this 3D problem; however, we can Strictly speaking ¯ ∂ sends functions to sections of the anti-canonical line bundle. For the introductionwe ignore this distinction. Similar remark applies to ¯ ∂ H on S × S and S below. φ according to L v φ n = inφ n with v being a vector field along the to S (i.e. wedecompose φ into Fourier modes). The problem ¯ ∂φ n = 0 has a finite dimensional kernel/co-kernel. This setup can be covariantized and we can consider a function φ on S with the S -action associated to the Hopf fibration and where ¯ ∂ H is the corresponding horizontalDolbeault operator on S . Again the problem ¯ ∂ H φ = 0 is an example of a transverselyelliptic problem with respect to the S -action, and this equation behaves nicely under L v (inthe same way as we just discussed, modulo some technicalities). It is crucial that the firstorder transversely elliptic operator gives rise to the standard second order elliptic operator ∆ = −L v + ∂ H ¯ ∂ H , (5)which is just the Laplace operator in this example. There is also a converse statement: if wetake a second order elliptic operator and find a decomposition like (5) then the correspondingfirst order operator ¯ ∂ will automatically be transversely elliptic. Supersymmetry typicallycomes with some Killing vector field v and it naturally produces a decomposition of a secondorder elliptic operator as in (5). This is a salient feature of many of the supersymmetricmodels on curved manifolds that are discussed in the context of supersymmetric localization.Following this informal introduction to transversely elliptic operators, we reconsider 4Dgauge theories on a four manifold M . As we have mentioned, in this context instantonsare related to the natural elliptic problem (3). We can repeat the same trick as above andconsider the 5D manifold M = S × M with trivial U (1) -action along S . There we canconstruct a 5D transversely elliptic problem by requiring anti self-duality in the transversedirections (along M ) and that the component of the gauge field along S is zero. One canrewrite this system in 5D covariant terms, see equation (144) in appendix C. This can besummarized in terms of the following transversely elliptic complex Ω ( M ) d −→ Ω ( M ) D −→ Ω H ( M ) ⊕ Ω ( M ) , (6)where H stands for horizontal. At the level of this discussion the concrete form of theoperator D is not important, what is crucial is that the 4D instanton equation admits anatural transversely elliptic 5D lift. In general this complex is defined for M with torusaction and reducing along a free S will give us some M . This reduction will naturallyproduce a new transversely elliptic problem in 4D with respect to the remaining action of avector field v . Upon the reduction the complex (6) becomes symbolically Ω ( M ) d −→ Ω ( M ) ⊕ Ω ( M ) ˜ D −→ P + ω Ω ( M ) ⊕ Ω ( M ) , (7)where P + ω is some projector which defines a sub-bundle of rank within the two forms Ω , and the system involves the 4D gauge field and a scalar field (the Ω ( M ) -part in the6iddle term). The concrete form of the operator ˜ D can be found in appendix C and willbe discussed later. As we will see N = 2 supersymmetry on M naturally selects thecorresponding transversely elliptic problem (7). The analog of the decomposition (4) for theYang-Mills action is || F || = || hP + ω F || + || f ι v F || + ... , (8)where the dots stand for lower derivative terms and f , h are some positive functions. Wewill refer to the condition P + ω F = 0 as defining a flip instanton . In this paper we willderive explicitly the form of the projector P + ω , from geometric considerations and show itsrelation to supersymmetry. The idea is that the relevant second order elliptic operator canbe decomposed in terms of first order operators as in (5) (for a gauge theory this is a subtlestatement, but it is roughly correct). We have provided the 5D perspective as a motivation,but our construction can be defined in intrinsically 4D terms, and supersymmetry is naturallyrelated to the transversely elliptic complex (6).In relating to the original Donaldson-Witten cohomological theory, the novelty here isthat on 4D manifolds with a T action, the elliptic complex (3) can be replaced by anothercomplex, which instead is transversally elliptic with respect to the T action. Not surpris-ingly this new complex, and in particular the new bundle P + ω Ω , is naturally related tosupersymmetry and the localization calculation to be carried out in this new framework. Ifwe think about Pestun’s S construction, the bundle P + ω Ω can be thought of as that of self-dual forms over the north hemisphere and of anti-selfdual forms over the south hemisphere.These two spaces are glued together using the vector field coming from the T -action. Theflip from self-duality to anti self-duality between the two hemispheres motivates why we callsolutions to P + ω F = 0 flip instantons. We present two main results: the first is the explicit construction of a N = 2 supersymmetricgauge theory on any manifold with a Killing vector field with isolated fixed points. Thesecond result is a conjecture for the full partition function for these theories. We outlinesome technical problems related to proving this conjecture.Consider a Riemannian manifold ( M, g ) with a Killing vector field v with isolated fixedpoints. Let us choose a decomposition || v || = s ˜ s in terms of two non-negative invariantfunctions s and ˜ s such that ˜ s = 0 at some fixed points (we call them plus fixed points)and s = 0 at all remaining fixed points (we call them minus fixed points). For such datawe can construct generalized Killing spinors and the corresponding N = 2 supersymmetric7auge theory for a vector multiplet. This geometrical data does not uniquely fix all auxiliarysupergravity fields; however, this redundancy does not affect the partition function. Usingthe Killing spinors we can reformulate the N = 2 vector multiplet in terms of cohomologicalvariables. These are very similar to those that are used to formulate equivariant Donaldson-Witten theory except that we change the notion of self-duality and correspondingly thedefinition of the two-form fields (denoted χ and H below). We construct a rank 3 subbundle P + ω Ω of two forms that near the plus fixed points approach self-dual two forms, and nearthe minus fixed approach anti-self-dual two forms. Away from the fixed points we use thevector field v to glue self-dual with anti-self-dual forms. Thus we define a new version ofequivariant Donaldson-Witten theory and relate it to supersymmetry.The second result of our paper is to perform the localization calculation for the partitionfunction for the N = 2 supersymmetric gauge theory we constructed. We are not able toderive the form of the partition function in the most general setting. However, we conjecturethat two types of contributions appear in the path integral: point like instantons (at plusfixed points), point like anti-instantons (at minus fixed points), as well as flux configurationsrelated to non-trivial two-cycles on M . Here we consider only vector multiplets. Assumingthat we have a T -action on M given by a vector field v = ǫ v + ǫ v the full answer can bewritten schematically as follows Z M ǫ ,ǫ = X discrete k i Z h da e − S cl p Y i =1 Z inst ǫ i ,ǫ i (cid:16) ia + k i ( ǫ i , ǫ i ) , q (cid:17) l Y i = p +1 Z anti − inst ǫ i ,ǫ i (cid:16) ia + k i ( ǫ i , ǫ i ) , ¯ q (cid:17) , (9)where we have p copies of the Nekrasov instanton partition function Z inst ǫ ,ǫ ( ia, q ) for the p plus fixed points and ( l − p ) copies of the Nekrasov anti-instanton partition function Z anti − inst ǫ ,ǫ ( ia, ¯ q ) for the ( l − p ) minus fixed points. The equivariant parameters ( ǫ i , ǫ i ) can beread off from the local action of v in the neighbourhood of the fixed points. In (9) k i ( ǫ i , ǫ i ) are vector-valued linear functions in ( ǫ i , ǫ i ) with integer coefficients that correspond to thefluxes. At the moment we are unable to characterize these functions in the general case andwe illustrate some problems related to the presence of fluxes. Looking at some exampleswe show indications that the contribution of fluxes depends on the relative distribution ofplus and minus fixed points. Assuming that M is simply connected we can express theperturbative contribution (around the zero connection) as a formal superdeterminant thatcan be calculated using index theorems for the transversely elliptic complex (7). In generalit is not clear how to consistently glue the local contributions to produce a well-definedanalytical answer. Nevertheless, we give some examples where the perturbative answer canbe written explicitly. Finally we argue that the full answer (9) depends holomorphically from ǫ and ǫ , and that this is compatible with Pestun’s answer on S for real ǫ , ǫ .8 .2 Outline of the paper We summarize the content of each section of the paper. Many formal aspects of our con-struction are only mentioned in this work and will be discussed in the follow up [25].In section 2 we start with the definition of our cohomological field theory axiomatically.Our main goal is to define a new decomposition, into two orthogonal subbundles of rank3, of the two forms Ω on a 4D manifold equipped with a vector field v . We do this bothusing the language of transition functions and by constructing the projector P + ω explicitly.Using this new decomposition of two forms we define a cohomological field theory, which is ageneralization of equivariant Donaldson-Witten theory. We briefly discuss the cohomologicalobservables in this theory.Section 3 provides the detailed construction of a N = 2 supersymmetric gauge theory ona compact manifold equipped with a Killing vector field v with isolated fixed points. Thecorresponding Killing spinors are analysed and the Lagrangian for a N = 2 vector multipletis shown. Local aspects of this construction are not novel (see e.g. [17, 26]); however, herewe establish that the supersymmetric theory is globally well defined. We provide an explicitmap between the N = 2 supersymmetric Yang-Mills theory and the cohomological fieldtheory defined in section 2. We show how the projector P + ω arises from supersymmetry.In section 4 we consider the dependence of supersymmetric observables (e.g. the par-tition function) on deformations of the geometrical and non-geometrical data entering theconstruction of the theory. The partition function is shown to be independent of many ofthese deformations. We also argue that the partition function depends holomorphically on squashing parameters.In section 5 we outline the localization calculation. In the general case we are not able tocarry it out completely. We concentrate on concrete examples and outline the technical prob-lems that arise in the general situation. We discuss both perturbative and non-perturbativeparts of the answer. Finally we conjecture the general form of the localization result for thepartition function.In section 6 we summarize the paper, point out open problems and we also provide ashort outline of our follow up work [25]. The paper has many appendices where we collect asummary of our conventions and a number of technical results. In this section we define the 4D cohomological field theory in axiomatic fashion and in thenext section we will explain its relation to supersymmetry. This cohomological theory is a9eneralization of equivariant Donaldson-Witten theory where we modify the notion of self-duality. We consider a 4D manifold with metric g and Killing vector v with isolated fixedpoints. We construct a novel subbundle of two forms Ω ( M ) that, in the neighbourhoodof a fixed point, look like either self-dual or anti-selfdual two forms. We use the vectorfield v to glue self-dual and anti-self-dual forms in one rank 3 bundle. In the next sub-section we provide the explicit construction of this bundle and later describe the associatedcohomological field theory. Before discussing the field theory we introduce a new decomposition of two forms Ω on acompact 4D manifold M . Consider the metric g on M and the corresponding Hodge star ⋆ .On the space of two forms Ω we have the scalar product h B , B i = Z M B ∧ ⋆B , B , B ∈ Ω ( M ) . (10)Thus upon choosing the orientation, the bundle Ω has the structure group SO (6) . Using theHodge star ⋆ we can introduce projectors P ± = (1 ± ⋆ ) and decompose Ω = Ω ⊕ Ω − intotwo orthogonal sub-bundles of rank . This decomposition depends only on the conformalclass of the metric g . Assume additionally that there is a vector field v which is nowhere zero(for the moment) and define its dual one form κ = g ( v ) . We can define a map m : Ω → Ω − , m : B
7→ − B + 2 ι v κ κ ∧ ι v B , (11)where ι v κ = g ( v, v ) = || v || . To see that the map (11) sends self-dual forms Ω to anti-selfdual Ω − forms and vice versa one has to use the following identity on two forms ι v ( κ ∧ ⋆B ) = ⋆ ( κ ∧ ι v B ) . (12)The map m satisfies the properties m = 1 and m ⋆ + ⋆ m = 0 . Additionally it preserves thescalar product, h m ( B ) , m ( B ) i = h B , B i . (13)Next we relax the assumptions on the vector field and let v have isolated zeros on M . Inthis case the map (11) is not globally well defined (it is singular where v = 0 ); however,we can use it as a transition function to glue self-dual and anti self-dual forms and define anew bundle. Let us be more precise. Assume that the vector field v is defined everywhere10nd that it has a finite number of isolated zeros. Without loss of generality we can choosean open covering of the manifold, M = ∪ U i , such that every open set contains exactlyone zero of v and the double intersections U i ∩ U j do not contain any zeros. Next for eachzero of v we choose a sign + or − and assign the same sign to the open set that containsthe given zero. We denote the open sets correspondingly as U + i and U − j . We associatethe self-dual forms Ω ( U + i ) to the sets U + i and anti self-dual forms Ω − ( U − j ) to the sets U − j . Over the intersections U + i ∩ U − j = ∅ we use the transition function given by the map m ij : Ω ( U + i ) → Ω − ( U − j ) defined by (11). We glue patches with the same sign through theidentity map. The maps m ij satisfy the cocycle condition, thus this gluing defines a rank subbundle of Ω ( M ) , which we denote throughout the paper as P + ω Ω . We will need a moreconcrete description of this subbundle, hence we will provide an alternative definition usinga projector operator. We will see in the following that this projector is naturally related tosupersymmetry.The projector can be derived in different ways. We first present an abstract derivationwhich is directly related to the above formal construction. For the sake of clarity let usconsider the simple situation when manifold is covered by two patches M = U + ∪ U − andthus the vector field v has only two zeros. Assume that we have a partition of unity for thiscovering, φ + + φ − = 1 and supp( φ ± ) ⊆ U ± . Further assume that our rank 3 subbundle of Ω is defined by some projector P + ω . Then on the corresponding patches we have P + ω = W − P + W on U + , P + ω = V − P − V on U − , (14)where W , V are special orthogonal transformations on U + and U − respectively, and P ± = (1 ± ⋆ ) . Thus on the intersection U + ∩ U − we have the relation P + = ( V W − ) − P − V W − , (15)which encodes the gluing map (11). To be concrete we will use by block matricescorresponding to the splitting Ω = Ω ⊕ Ω − . For example, the projector P + has the form P + = (cid:18) (cid:19) . (16)Using these notations the relation (15) can be written as follows V W − = (cid:18) m − m (cid:19) , (17)11here m is the map (11) and the minus sign guarantees that this matrix is an element of SO (6) . We can represent W and V as elements of SO (6) , W ( ρ ) = (cid:18) cos ρ m sin ρ − m sin ρ cos ρ (cid:19) , V ( ρ ) = (cid:18) − sin ρ m cos ρ − m cos ρ − sin ρ (cid:19) = W ( ρ + π , (18)where ρ is defined through the partition of unity as follows: φ + = cos ρ and φ − = 1 − cos ρ .Since φ ± ≥ , we have ρ ∈ [ − π/ , π/ . Thus around the zero of v in U + , ρ approacheszero and around the zero of v in U − , ρ approaches − π/ (or π/ depending on conventions).Hence we obtain for P + ω P + ω = W − P + W = (cid:18) cos ρ − m sin ρm sin ρ cos ρ (cid:19)(cid:18) (cid:19)(cid:18) cos ρ m sin ρ − m sin ρ cos ρ (cid:19) , (19)and using our conventions we rewrite it as follows P + ω = cos ρ P + + sin ρ P − + cos ρ sin ρ m = 12 (1 + cos 2 ρ ⋆ + sin 2 ρ m ) , (20)where ρ ∈ [ − π, π ] . The projector is well defined even at v = 0 since at those points sin 2 ρ is zero. It is straightforward to generalize this construction to the case where v has morethan two zeros and to any allocation of signs to the fixed points.Indeed we can give the following direct construction of the projector P + ω . Away from thezeros of v the following identities hold on Ω : ⋆ = 1 , m = 1 , and ⋆ m + m ⋆ = 0 . Hence,provided that α + β = 1 the combination α ⋆ + βm satisfies ( α ⋆ + βm ) = 1 . (21)It follows that (20) is the most general projector composed from ⋆ and the vector field. Wehave to ensure that it can be extended to the zeros of v and thus be well-defined over thewhole manifold. For this we need that sin 2 ρ goes to zero (at least linearly) where v = 0 .Hence at the fixed points ρ goes to either or π implying that cos 2 ρ goes to ± respectively,and the projector P + ω approaches either (1 + ⋆ ) or (1 − ⋆ ) . It is not hard to construct afunction ρ with the required properties and in appendix B we give some explicit examples.There is some redundancy in our description: as cos 2 ρ changes from +1 to − , ρ maygo either from to π or from to − π . We fix this ambiguity assuming that ρ ∈ [0 , − π ] .This allows us to perform the following change of variables − sin 2 ρ = 21 + cos ω , (22)12sing a function ω ( x ) . The projector (20) can then be expressed as follows P + ω = 11 + cos ω (cid:18) ω ⋆ − sin ω κ ∧ ι v ι v κ (cid:19) . (23)Here the function ω is chosen in such a way that the last term is well-defined at v = 0 .Moreover ω = 0 at the + fixed points and ω = π at the − fixed points. This reparametrizationof the projector is better suited for the considerations in the next section. This projectordepends only on the conformal class of the metric. Thus in a given conformal class onecan choose a representative for which || v || = sin ω (i.e., || v || ≤ ) and the projector issomewhat simpler, P + ω = 12 − || v || (cid:16) p − || v || ⋆ − κ ∧ ι v (cid:17) , (24)(Note however that v does not define ω ( x ) uniquely because of the ambiguity in taking thesquare root). Some formulas below are simpler for a metric with this special property andthe corresponding projector (24), but everything we present holds for a generic choice ofmetric.Coming back to the general projector (23) and using the identities presented in thissection we write the following useful formula for a two form F (1 + cos ω ) P + ω F ∧ ⋆P + ω F = F ∧ ⋆F + cos ω F ∧ F − sin ω || v || ι v F ∧ ⋆ι v F = 2 cos ω F + ∧ ⋆F + + 2 sin ω F − ∧ ⋆F − − sin ω || v || ι v F ∧ ⋆ι v F , (25)where F ± = 1 / ± ⋆ ) F . These identities will play a crucial role in the following.In summary, on a 4D manifold with a globally defined vector field v with isolated fixedpoints (and additional data at the fixed points as described above) and a metric, we havean alternative decomposition of two forms Ω given by the projectors P + ω + P − ω = 1Ω = P + ω Ω ⊕ P − ω Ω . (26)This provides an alternative decomposition of Ω into two orthogonal subspaces P ± ω Ω withrespect to the standard scalar product. Throughout the paper we will use these notationsfor the decomposition of two forms (26) while keeping Ω ± for the standard decompositioninto self-dual/anti-self-dual spaces. In what follows we will assume that v is a Killing vectorfield so that this decomposition is preserved by L v (provided that cos ω is invariant along v ).The typical setting we have in mind is that v arises from some T -action on M with onlyisolated fixed points. 13 .2 Cohomological complex Donaldson-Witten theory can be defined as a cohomological field theory [3, 24] which isrelated to the Donaldon invariants of four manifolds. If a four manifold admits the actionof a group (for example T ), one can further define an equivariant extension of Donaldson-Witten theory [11]. Let us review some basic facts about equivariant Donaldson-Wittencohomological field theory. Assuming that we have a Killing vector field v , we can define thefollowing odd transformations δA = i Ψ ,δ Ψ = ι v F + id A φ ,δφ = ι v Ψ ,δϕ = iη , (27) δη = L Av ϕ − [ φ, ϕ ] ,δχ = H ,δH = iL Av χ − i [ φ, χ ] , where A is a gauge connection, Ψ is an odd one-form, φ and ϕ are even scalars, η is an oddscalar, χ is a self-dual odd two form and H is a self-dual even two form. All these fields(except A ) take values in the adjoint representation of the gauge group. Throughout thepaper we use the following conventions: the covariant derivative is defined as d A = d − i [ A, ] ,the covariant version of the Lie derivative is defined as follows L Av = d A ι v + ι v d A = L v − i [ ι v A, ] and the field strength as F = dA − iA . The square of the transformations (27) is given by δ = i L v − G φ + iι v A , (28)where G ǫ stands for a gauge transformation acting on the gauge field as G ǫ A = d A ǫ , (29)and on all other field in the adjoint as G ǫ • = i [ ǫ, • ] . (30)We can further add the ghost c , anti-ghost ¯ c and Lagrangian multiplier b and extend alltransformations such that they square to the Lie derivative only (see appendix D). The co-homological theory given by the transformations (27) is not uniquely defined since we did not14pecify any reality conditions for the bosonic fields φ and ϕ . From standard supersymmetryconsiderations we know that φ and ϕ cannot be two independent complex fields. Suitablereality conditions cannot be fixed by cohomological considerations alone. Later on we will seethat supersymmetry together with the positivity of the supersymmetric Yang-Mills actionwill fix the reality conditions for φ and ϕ .So far we have reviewed the definition of equivariant Donaldson-Witten cohomologicalfield theory. Now we will describe its modification. Consider a four manifold with a Killingvector field with isolated fixed points, and specify any distribution of pluses/minuses overthe fixed points. According to the discussion in the previous subsection we can associateto these data an orthogonal decomposition of two forms P + ω Ω ⊕ P − ω Ω (26). Thus we canconsider the transformations (27) acting on the same set of fields except that we impose newconditions on χ and H P + ω χ = χ , P + ω H = H . (31)Since L v preserves the spaces P ± ω Ω , the transformations satisfy the algebra (28) as before.In this way we defined a new cohomological field theory which is ultimately related tosupersymmetric Yang-Mills as will be explained in the next section. If we distribute onlypluses (resp. minuses) for all fixed points, pick up a non-trivial function cos ω > (resp. cos ω < ) and construct the projectors P ± ω , then one can globally rotate P + ω Ω to Ω ,bringing us back to equivariant Donaldson-Witten theory. However, this is impossible if wehave both pluses and minuses over different fixed points. Hence generically P + ω Ω is notisomorphic to Ω and the corresponding cohomological theory is not related to standardequivariant Donaldson-Witten theory. It is important to remember that the bundle P + ω Ω is defined up to isomorphism, so that in general we could use different vector fields in thetransformations (27) and to define the projector (23). To avoid confusion we will postponediscussing this point to section 4.3.The observables in the new theory are constructed in the same way as in equivariantDonaldson-Witten theory, see [10, 11]. Let us comment on a particular class of observablesthat will be relevant later on. Using the transformations (27) one can observe that δ (cid:16) φ + Ψ + F (cid:17) = ( id A + ι v ) (cid:16) φ + Ψ + F (cid:17) . (32)This implies the following δ Tr (cid:16) φ + Ψ + F (cid:17) k = ( id + ι v ) Tr (cid:16) φ + Ψ + F (cid:17) k , (33)where Tr can be replaced by any Ad-invariant polynomial over a Lie algebra. If we pick an15quivariantly closed form Ω on M we can construct the observable Z M Ω ∧ Tr (cid:16) φ + Ψ + F (cid:17) k , (34)which is annihilated by δ . This observable depends only on the class of Ω in the equivariantcohomology since the shift Ω → Ω + ( id + ι v )[ ... ] will lead to a δ -exact shift in the observable.Here the imaginary i in front of d is due to our conventions and does not play any essentialrole. Once reality conditions are set we can discuss the reality and positivity of the bosonicpart of these observables. Among all observables we will be particularly interested in O = Z M (cid:16) Ω + Ω + Ω (cid:17) ∧ Tr (cid:16) φ + Ψ + F (cid:17) = Z M (cid:16) Tr ( φ )Ω + 2Ω ∧ Tr ( φF ) + Ω Tr ( F ) + Ω ∧ Tr (Ψ ) (cid:17) , (35)where (Ω + Ω + Ω ) is closed under id + ι v . As we will show in the next section, thisobservable is closely related to the supersymmetrized Yang-Mills action. We start considering a 4D spin manifold M equipped with a Riemannian metric g . We willconsider the spin c case at end of section 3.3.3. On M we have left and right handed spinors ζ iα and ¯ χ ˙ αi transforming in the fundamental of the SU (2) R R-symmetry. Here i is the SU (2) R index while α, ˙ α are spinor indices (conventions for spinors are spelled out in appendix A).In this section we construct N = 2 supersymmetric field theories on M given the followingfurther data: • We impose that g admits a smooth real Killing vector field v with at most isolatedfixed points x i . • A smooth function s on M that is positive everywhere except at a subset of the fixedpoints of v where it vanishes. We also require that v µ ∂ µ s = 0 ( s is invariant along v )and that ˜ s = s − || v || is smooth.The theories we construct will admit one supercharge δ squaring to a translation along v .Note that either s or ˜ s vanishes at each fixed point of v because || v || = s ˜ s . Thisdetermines the ± sign associated to each fixed point as described in the previous section: if ˜ s = 0 we have a positive sign, and if s = 0 we have a negative sign. The possibility that the fixed points are not isolated is not excluded but requires a case by case analysis. .1 Construction of global spinors As a first step we will use the geometrical data above to construct smooth spinors ζ iα and ¯ χ ˙ αi satisfying the reality conditions ( ζ iα ) ∗ = ζ iα and ( ¯ χ ˙ αi ) ∗ = ¯ χ i ˙ α and such that ζ i ζ i = s , ¯ χ i ¯ χ i = ˜ s , ¯ χ i ¯ σ µ ζ i = 12 v µ . (36)We cover M with open patches U k , each equipped with a choice of local frame e ak and weassume that each fixed point of v belongs to a single distinct U k . In the overlap between U k and U l the frames e ak and e al are related by an SO (4) = SU (2) l × Z SU (2) r transformation.The spinors ζ and χ are related by SU (2) l × Z SU (2) R and SU (2) r × Z SU (2) R transformationsrespectively. By identifying SU (2) l with SU (2) R we can construct a globally well definedtopologically twisted spinor ζ t whose component expression in each patch U k is given by ( ζ t ) iα = δ iα .Away from the zeros of s we can then define spinors ζ i and ¯ χ i as follows: ζ i = √ s ζ it , ¯ χ i = 1 s v µ ¯ σ µ ζ i . (37)These definitions ensure that (36) are satisfied and the spinors are real. The spinor ¯ χ i issingular at the fixed points where s = 0 , the bilinears (36) however are everywhere smooth.In a patch U k containing a zero of s the function ˜ s is strictly positive and we can definesmooth spinors: ˆ¯ χ ˙ αi = − i √ ˜ s δ ˙ αi , ˆ ζ i = − s v µ σ µ ˆ¯ χ i , (38)whose bilinears and reality properties are the same as those of ¯ χ and ζ . It follows that thehatted spinors ˆ¯ χ, ˆ ζ are related to ¯ χ and ζ by an SU (2) R transformation ¯ χ i = U ij ˆ¯ χ j , ζ i = U ij ˆ ζ j , (39)which is explicitly given by U ij = i v µ || v || σ µij . (40)Let Σ be a small three sphere surrounding a fixed point of v where s = 0 . The map U ij from Σ to SU (2) R is non-singular and of degree . We can now consider spinors thatare equal to (37) in all patches except those containing a zero of s where the spinors aregiven by (38). In going from a patch U k containing a zero of s and a second patch U l the SU (2) R transformation is (40) followed by that corresponding to topological twisting. Thisconstruction results in spinors that are smooth everywhere on M and whose bilinears aregiven by (36). This is not the same covering as in 2.1 because some of the patches need not contain a fixed point of v . .1.1 Spinor bilinears Using the spinors ζ and ¯ χ we can form other spinor bilinears besides v µ and s, ˜ s , Θ ( ij ) µν = ζ i σ µν ζ j , e Θ ( ij ) µν = ¯ χ i ¯ σ µν ¯ χ j , v ( ij ) µ = ζ i σ µ ¯ χ j + ζ j σ µ ¯ χ i . (41)These are forms valued in the adjoint of SU (2) R .The vector field v allows us to construct the family of projectors P + ω in (23). We willimpose that the function ω ∈ [0 , π ] behaves as follows near the fixed points ω = o ( √ ˜ s ) for ˜ s ∼ , ω = π − o ( √ s ) for s ∼ . (42)This guarantees smoothness of P + ω . According to the discussion in section 2.1 we see thatat the fixed points with ˜ s = 0 the projector collapses to the self-duality projector and at thefixed points with s = 0 to the anti-self-duality projector. One specific choice of ω , which isis completely specified by the Killing spinors, will be referred below as “canonical” cos ω c = s − ˜ ss + ˜ s . (43)Using Θ and e Θ in (41) we can form the combination b Θ ijµν = 41 + cos ω (cid:18) cos ( ω/ s Θ ijµν + sin ( ω/ s e Θ ijµν (cid:19) , (44)which is everywhere smooth because of (42) and enjoys the following properties P + ω b Θ ij = Θ ij , b Θ ijµν b Θ ρλij = 11 + cos ω P + ω ρλµν , b Θ ijµν b Θ µνkl = δ ik δ jl + δ il δ jk ω ) . (45) Given a supersymmetric field theory in flat space we can couple it to a supergravity back-ground. The conditions for the background to preserve supersymmetry are then encoded ingeneralised Killing spinor equations that the supersymmetry variation parameters (in ourcase ζ and ¯ χ ) have to satisfy [27, 28]. We will consider N = 2 theories with a conserved SU (2) R current whose supercurrent multiplet was studied by Sohnius [29]. These theoriescouple to the N = 2 Poincaré supergravity described in [30–33]. A supergravity background(see for instance [34]) is specified by the metric g , a choice of SU (2) R connection V µij , ascalar N, a one form G µ , a two form W µν , a scalar SU (2) R triplet S ij , and finally an closedtwo form F µν corresponding to the graviphoton field strength.18here are two sets of Killing spinor equations. The first arises from setting the variationof the gravitino to zero ( D µ − iG µ ) ζ i − i W + µρ σ ρ ¯ χ i − i σ µ ¯ η i = 0 , ( D µ + iG µ ) ¯ χ i + i W − µρ ¯ σ ρ ζ i − i σ µ η i = 0 , (46)while the second set is obtained setting the variation of the dilatino to zero (cid:16) N − R (cid:17) ¯ χ i = 4 i∂ µ G ν ¯ σ µν ¯ χ i + i (cid:0) ∇ µ + 2 iG µ (cid:1) W − µν ¯ σ ν ζ i + i ¯ σ µ (cid:0) D µ + iG µ (cid:1) η i , (cid:16) N − R (cid:17) ζ i = − i∂ µ G ν ¯ σ µν ζ i − i (cid:0) ∇ µ − iG µ (cid:1) W + µν σ ν ¯ χ i + iσ µ (cid:0) D µ − iG µ (cid:1) ¯ η i . (47)In the above equations the covariant derivative D µ includes the SU (2) R connection V µij sothat, for instance, D µ ζ i = ∇ µ ζ i − V jµ i ζ j . The spinors η i and ¯ η i are given by η i = ( F + − W + ) ζ i − G µ σ µ ¯ χ i − S ij ζ j , ¯ η i = − ( F − − W − ) ¯ χ i + 2 G µ ¯ σ µ ζ i − S ij ¯ χ j . (48)We used the shorthand notation W + = W µν σ µν and W − = W µν ¯ σ µν (similarly for F ).Provided some integrability conditions (and smoothness requirements) are satisfied, theequations above can be solved (albeit non uniquely) in terms of the spinors ζ i and ¯ χ i con-structed above. The resulting supergravity background is smooth. For the case of topologicaltwisting this was implemented in [35].In order to solve the first set of two equations in (46) we need v to be a Killing vectorfield and that s (hence ˜ s ) is invariant along v . The solution reads W µν = is + ˜ s ( ∂ µ v ν − ∂ ν v µ ) − i ( s + ˜ s ) ǫ µνρλ v ρ ∂ λ ( s − ˜ s ) − s + ˜ s ǫ µνρλ v ρ G λ ++ s − ˜ s ( s + ˜ s ) ǫ µνρλ v ρ b λ + 1 s + ˜ s ( v µ b ν − v ν b µ ) , ( V µ ) ij = 4 s + ˜ s (cid:0) ζ ( i ∇ µ ζ j ) + ¯ χ ( i ∇ µ ¯ χ j ) (cid:1) + 4 s + ˜ s (cid:18) iG ν − ∂ ν ( s − ˜ s )( s + ˜ s ) (cid:19) (Θ ij − e Θ ij ) νµ ++ 4 i ( s + ˜ s ) b ν (˜ s Θ ij + s e Θ ij ) νµ , (49)where b µ is a one form on M satisfying v µ b µ = 0 . We make the following remarks: • s + ˜ s > everywhere, hence, because the spinors ζ i , ¯ χ i and their bilinears defined insection 3.1.1 are smooth so are W and V .19 The one forms b µ and G µ are left undetermined and parametrise different solutions of(46). We could use this freedom to set W µν to zero but generically the required G µ and b µ would not be smooth. • The expression for the SU (2) R background gauge field ( V µ ) ij in a given patch is gen-erally complex. However, in transitioning from patch to patch, as described in section3.1, ( V µ ) ij changes by a real SU (2) R gauge transformation. Hence the imaginary partof ( V µ ) ij is a globally defined one form in the adjoint of SU (2) R . • In the large volume limit the supergravity background fields approach their values inflat space. Indeed the background fields W µν and ( V µ ) ij scale as r where r is the overallsize of the manifold M (assuming that G µ and b µ are also chosen of order r − ).Next we consider the second set of equations (47). They can all be solved provided that G µ and b µ are invariant along v and just determine the scalar N . This fact was alreadynoted in [17, 26]. The solution for N is everywhere smooth and of order r − . The explicitexpression for N is rather lengthy and can be found in the appendix E, equation (159).We can now use (48) to determine the remaining supergravity background fields. Asolution for the closed two form F µν is given by F µν = i∂ µ (cid:16) s + ˜ s − Ks ˜ s v ν (cid:17) − i∂ ν (cid:16) s + ˜ s − Ks ˜ s v µ (cid:17) , (50)where K is a constant. Here F µν is not just closed but exact. In order to guarantee that F µν is smooth we need to impose the extra condition that s + ˜ s approaches the same constant K at all fixed points fast enough. This can always be arranged. In principle we could ask that s + ˜ s = K everywhere on M and not just at the fixed points. This normalization, however,imposes some restriction on the metric. Namely if s is zero at some but not all of the fixedpoints of v , then s = ˜ s on some codimension 1 locus on M and the norm of the Killingvector v would have to attain its extremum there. This can always be arranged in a givenconformal class. In the following we will not assume that s + ˜ s is constant.The solution (50) is not the most general. We could add to it any closed two form ˆ f satisfying ι v ˆ f = 0 . In specific cases it may be that this freedom allows to relax the conditionthat s + ˜ s = K at the fixed points of v .The scalar triplet S ij is determined as well. The resulting expression is presented inappendix E , equation (158). Smoothness of S ij also requires s + ˜ s to approach K at fixedpoints fast enough. We can describe the freedom parametrized by b µ via a two form U µν that satisfies P + ω can U = U wherethe canonical choice for ω is as in (43). This is somewhat more general because a b µ that is singular at thefixed points can correspond to a smooth U , but it makes explicit expressions more complicated. .3 Vector multiplet An N = 2 vector multiplet comprises, in addition to the gauge field A µ , of a complex scalar X , gauginos λ iα , ˜ λ i ˙ α transforming in the fundamental of SU (2) R and an auxiliary real scalar SU (2) R triplet D ij . All these fields (except A ) transform in the adjoint of the gauge group.This multiplet can be coupled to background N = 2 supergravity as described in [34] (seealso [30–33]). In a supersymmetric background the supersymmetry variations are given by δA µ = iζ i σ µ ¯ λ i + i ¯ χ i ¯ σ µ λ i ,δ ¯ X = ¯ χ i ¯ λ i , δX = − ζ i λ i ,δD ij = iζ i σ µ (cid:0) D µ + iG µ (cid:1) ¯ λ j − i ¯ χ i ¯ σ µ (cid:0) D µ − iG µ (cid:1) λ j + 2 i [ X, ¯ χ i ¯ λ j ] + 2 i [ ¯ X, ζ i λ j ] + ( i ↔ j ) ,δ ¯ λ i = 2 i ( D µ + 2 iG µ ) ¯ X ¯ σ µ ζ i +2 (cid:0) F − − X W − (cid:1) ¯ χ i − D ij ¯ χ j − i [ X, ¯ X ] ¯ χ i + 2 ¯ X ¯ η i ,δλ i = − i ( D µ − iG µ ) Xσ µ ¯ χ i + 2 (cid:0) F + − ¯ X W + (cid:1) ζ i + D ij ζ j + 2 i [ X, ¯ X ] ζ i − Xη i . (51)Here F µν is the field strength for the gauge field A µ and we used the shorthand notation F + = F µν σ µν and F − = F µν ¯ σ µν .The composition of two transformations above on a field Φ in the vector multiplet resultsin a translation along the Killing vector field v together with an SU (2) R transformation anda gauge transformation δ Φ = i L v Φ + iv µ V µ ◦ Φ + v µ [ A µ , Φ] + i Λ ( R ) ◦ Φ − i [ s ¯ X + ˜ sX, Φ] (52)here L v is the Lie derivative along v , and ◦ denotes that Φ is acted upon according to which SU (2) R representation it belongs. Λ ( R ) is a SU (2) R transformation parameter: Λ ( R ) ij = ¯ χ i ¯ σ µ ( D µ − iG µ ) ζ j − ζ i σ µ ( D µ + iG µ ) ¯ χ j + ( i ↔ j ) . (53) From the coupling to supergravity we can also read the form of supersymmetric Lagrangians, L = g Tr h − D µ + 2 iG µ ) ¯ X ( D µ − iG µ ) X − F µν F µν − i θg π ǫ µνρλ F µν F ρ λ + D ij D ij + − X, ¯ X ] + 2 F µν ( XW − µν + ¯ XW + µν ) − X W − µν W − µν − ¯ X W + µν W + µν + 4 (cid:16) R − N (cid:17) X ¯ X i + − g Tr h iλ i σ µ (cid:16) D µ + iG µ (cid:17) ¯ λ i + i ¯ λ i ¯ σ µ (cid:16) D µ − iG µ (cid:17) λ i + 2 iλ i [ ¯ X, λ i ] + 2 i ¯ λ i [ X, ¯ λ i ] i . (54)21t short distances this Lagrangian is a small deformation of a Lagrangian for the vectormultiplet in flat space. Indeed the coupling to the supergravity background introducesterms scaling as r − or r − with the overall size r of the manifold.In general the Lagrangian (54) may not have positive real part and using it to define thepath integral is problematic. This issue depends on the concrete values of the backgroundfields and it should be addressed case by case. For example for the theory on a round S studied by [1] the Lagrangian (54) is real and positive. For the theories on squashed S considered in [16,17] the real part of (54) continues to be positive at least for small squashing.For more general backgrounds this problem becomes more complicated to analyze and it mayforce us to reconsider the reality conditions for some fields (or just for some modes of thefields). Another option can be that of adding appropriate δ -exact terms to the Lagrangian.We leave this issue aside for now. The action of supersymmetry on the components of the vector multiplet (51) and the struc-ture of the supersymmetric Lagrangian (54) are more transparent if we rewrite them usingappropriate cohomological variables. This is also how we connect the supersymmetric the-ory to the cohomological complex described in section 2.2. The change of variables willmake use of the projector P + ω defined in (23). Here we select the canonical choice for thefunction ω defined in (43) because it results in simpler expressions. We define the followingcohomological fields: η = ζ i λ i + ¯ χ i ¯ λ i ,ϕ = − i ( X − ¯ X ) , Ψ µ = ζ i σ µ ¯ λ i + ¯ χ i ¯ σ µ λ i ,φ = ˜ sX + s ¯ X ,χ µν = 2 s + ˜ ss + ˜ s (cid:18) ¯ χ i ¯ σ µν ¯ λ i − ζ i σ µν λ i + 1 s + ˜ s ( v µ Ψ ν − v ν Ψ µ ) (cid:19) ,H µν = ( P + ω c ) ρλµν (cid:20) ˆΘ ijρλ D ij − F ρλ + i X + ¯ Xs + ˜ s ( ∂ ρ v λ − ∂ λ v ρ )+ − is + ˜ s ǫ ρλγ δ v γ (cid:18)(cid:16) D δ − iG δ − i ˜ ss + ˜ s b δ (cid:17) X − (cid:16) D δ + 2 iG δ − i ss + ˜ s b δ (cid:17) ¯ X (cid:19)(cid:21) . (55)We note the following • With the standard reality conditions on the scalar fields ¯ X = X † the field ϕ is real22hile the field φ is complex. We can write φ = 12 ( s + ˜ s )( X + ¯ X ) − i s − ˜ s ) ϕ , hence the imaginary part of φ is determined by ϕ . It follows that φ and ϕ togetherhave the same degrees of freedom as the complex scalar field X . • All the cohomological fields in (55) are differential forms with values in the adjoint ofthe gauge group. They are all singlets under the SU (2) R symmetry. • The two form χ µν satisfies ( P + ω c ) χ = χ as does the two form H µν . • H µν is the only twisted variable whose definition involves the auxiliary fields in thegravity multiplet and derivatives of the Killing spinor bilinears.The change of variables (55) can be inverted, X = 1 s + ˜ s ( φ + i s ϕ ) , ¯ X = 1 s + ˜ s ( φ − i ˜ s ϕ ) , ¯ λ i = 1 s + ˜ s (cid:16) χ j (Θ µνji + e Θ µνji ) χ µν + ¯ σ µ ζ i Ψ µ + ¯ χ i η (cid:17) ,λ i = 1 s + ˜ s (cid:16) ζ j (Θ µνji + e Θ µνji ) χ µν − σ µ ¯ χ i Ψ µ − ζ i η (cid:17) ,D ij = 4 s + ˜ s ( s + ˜ s ) ˆΘ µνij ( H µν − . . . ) , (56)where in the last formula we subtract from H µν all the terms in its definition (55) that arenot proportional to D ij .Supersymmetry (51) induces the following transformations on the cohomological fields δA = i Ψ ,δϕ = iη ,δχ = H ,δφ = ι v Ψ . δ Ψ = ι v F + id A φ ,δη = L Av ϕ − [ φ, ϕ ] ,δH = i L Av χ − i [ φ, χ ] , (57)which exactly coincide with the cohomological transformations in (27) defined in section 2.2.Here L Av is the gauge covariant Lie derivative along v . Acting with δ twice reproduces thealgebra (52).At the beginning of this section we required the manifold M to be spin. However, afterswitching to cohomological variables the transformations (57) can be defined for non-spinmanifolds and a choice of rank 3 bundle P + ω Ω (e.g., for CP with the different allocations23f pluses and minuses to 3 fixed points and correspondently different P + ω Ω ). Thus in whatfollows we will use the cohomological variables for the vector multiplet and consider alsonon-spin examples.Here we do not discuss hypermultiplets which can also be coupled to the supergravitybackground. In a way similar to the case of topological twisting, (see e.g [36] for a list ofreferences) suitable cohomological variables can be defined also for hypermultiplets. Sincethe twisted hypermultiplet will contain spinors it will be defined on spin manifolds. Couplinga U (1) global symmetry acting only on the hypermultiplets to a suitable background gaugefield allows to extend the twisted theory to the spin c case [37]. We leave the hypermultipletsto further detailed study. Here we rewrite the Lagrangian (54) using cohomological variables. For simplicity we willpresent the result for the canonical choice of projector resulting in the cohomological fields(55). Later we will comment on what would change with a different choice of projector.The starting point is the following rewriting of the Yang-Mills Lagrangian
Tr [ F ∧ ⋆F ] = Tr (cid:20) (1 + cos ω )( P + ω F ) ∧ ⋆F + sin ω || v || ι v F ∧ ⋆ι v F − cos ω F ∧ F (cid:21) . Modulo δ -exact contributions and the θ term, the Lagrangian (54) arises from the super-symmetrization O of the last term in the equation above, L Vol M = 1 g O − iθ π F ∧ F + δ { . . . } . (58)Specializing to the canonical choice for ω we have O =Tr (cid:20) s − ˜ ss + ˜ s F ∧ F − (Ψ ∧ Ψ + 2 φF ) ∧ (cid:18) i s − ˜ s ( s + ˜ s ) dκ + 4 i ( s + ˜ s ) κ ∧ d ( s − ˜ s ) (cid:19) + − φ (cid:18) s − ˜ s ( s + ˜ s ) dκ ∧ dκ + 24( s + ˜ s ) κ ∧ dκ ∧ d ( s − ˜ s ) (cid:19)(cid:21) , (59)where κ = g ( v ) .We can identify O with the equivariant observable (35). We write h = cos ω c = s − ˜ ss +˜ s anddetermine the multi-form Ω = Ω + Ω + Ω to be Ω = h , Ω = − i s − ˜ s ( s + ˜ s ) dκ − i ( s + ˜ s ) κ ∧ d ( s − ˜ s ) , Ω = − s − ˜ s ( s + ˜ s ) dκ ∧ dκ − s + ˜ s ) κ ∧ dκ ∧ d ( s − ˜ s ) . (60)24ne can check that this is equivariantly closed, ( id + ι v )(Ω + Ω + Ω ) = 0 . All the forms Ω i in (60) are everywhere non-singular. S As an example we consider here the equivariantly closed form (59) on the round S . For thecoordinates, metric and other conventions see appendix B.1. We choose v = ∂ α + ∂ β and thefunctions s, ˜ s and h to be s = 2 cos θ , ˜ s = 2 sin θ , h = s − ˜ ss + ˜ s = cos θ , (61)hence h goes to ± at the poles. We also note that s + ˜ s = 2 , which leads to varioussimplifications. For these choices κ = g ( v ) satisfies the property (133) which allows us toset all the auxiliary fields in the supergravity background to zero except for the scalar S ij .This supergravity background has eight independent Killing spinors. The correspondingsupersymmetric field theory on a round S is that studied in [1]. The equivariantly closedform (60) that appears in the observable (59) takes the form Ω = h − i hdκ + 2 κ ∧ dh ) − ( 38 hdκ ∧ dκ + 32 κ ∧ dκ ∧ dh )= cos θ − i (sin θ dθ ∧ ( xdα + ( x − dβ ) + i θ sin θ dx ∧ ( dα + dβ ))+ 32 sin θ dθ ∧ dx ∧ dα ∧ dβ . (62)In particular we note that its top component is equal to S . Thus Ω is an equivariantextension of the canonical volume form. Indeed one can argue that the lowest component Ω of the equivariant extension Ω of any volume form on S (e.g., that corresponding to thesquashed S considered in [16, 17] ) always has a different sign at the two poles.Applying our observations to Pestun’s theory on a round S [1], we can see that modulothe θ -term the supersymmetric action can be rewritten as S S = Z S L Vol S = 1 g Z S (cos θ + Ω + 3Vol S ) Tr ( φ + Ψ + F ) + δ { . . . } , (63)where the two form Ω is defined in (62). If we just consider the gauge sector and rememberthat L is positive, we see that, due to cos θ , supersymmetric configurations with positivesecond Chern number will be favoured around the north pole and those with negative secondChern number around the south pole. This is just an heuristic argument that indicates theflipping behaviour realized in [1] on purely cohomological grounds.25 Deformations
In defining the theory in sections 2 and 3 we have used various geometrical and non-geometrical data (e.g., the values of background fields). In this section we determine whatpart of these data the partition function of the theory depends on. We will use the languageof cohomological field theory.When we define the fields in cohomological field theory we use the decomposition of two-forms into orthogonal bundles P + ω Ω ⊕ P − ω Ω . The fields in the complex are acted upon by δ v in (27) which depends on the vector field v . This action preserves the decomposition above.As we have discussed in section 2.1 the bundle P + ω Ω is defined up to isomorphisms bya given distribution of pluses and minuses over the fixed points of v . Thus if P + ω Ω and ˜ P + ω Ω are isomorphic and L v preserves this isomorphism we can redefine the fields χ and H in P + ω Ω to fields ˜ χ and ˜ H in ˜ P + ω Ω (very much in analogy with the logic for standardDonaldson-Witten theory [3]). This redefinition does not affect the partition function since χ and H enter the Lagrangian only through δ v -exact terms.Various data enters the Lagrangian (58) either through the cohomological observable O or through δ v -exact terms. The observable O depends only on the equivariant cohomologyclass and not on its specific representative. Finally varying any data that enters only in δ v -exact terms without changing the action of δ v does not modify the value of the pathintegral.Below we will comment in more detail on various deformations of the theory of thedifferent kinds described above. We will also consider separately deformations of δ v . h = cos ω Changing the function ω (or equivalently changing the function h = cos ω ) has two effects.First since cos ω enters the projector P + ω we need to redefine the cohomological fields χ and H in (55). Secondly the function h = cos ω also enters explicitly in various terms in theLagrangian. We will show that the partition function depends only on the values of h at thefixed points.The subbundle of two forms P + ω Ω is determined up to isomorphism once we fix the valuesof cos ω at the fixed points. When changing cos ω away from the fixed points χ and H shouldbe modified appropriately as described above. The transformations can be found using theformalism developed in section 2.1. We will not have use for their explicit expressions, whichare not illuminating. If the variation of cos ω is invariant along v the modification of χ and H commutes with δ v . Hence these variations will not affect the value of the partition function.26n the Lagrangian (58) h = cos ω enters through the cohomological observable O whichdepends on the equivariantly closed form Ω . Varying h will determine a corresponding change ∆Ω which is also equivariantly closed. Away from the fixed points ∆Ω is equivariantly exact.Indeed we can write ∆Ω = ( id + ι v ) κ ∧ ∆Ω( id + ι v ) κ . (64)This expression is everywhere non-singular provided that the variation of h and hence ∆Ω vanishes fast enough at the fixed points. Under this condition, making use of the property Z ( id + ι v )( ... ) Tr ( φ + Ψ + F ) = δ { ... } , (65)we find that the variation of the cohomological observable O determined by ∆Ω is δ -exact.Hence the partition function does not depend on the value of h away from the fixed points.It follows that in principle we can set h to zero everywhere except at the fixed points, thishowever, is a highly singular representative of the equivariant cohomology class. We can consider changing the metric on the four manifold M keeping the vector field v fixed. The variation of the metric needs to satisfy the constraint that the vector field v staysKilling. As we change the metric we can keep fixed all the cohmological fields except for χ and H . These need to be varied so that they continue to be in the kernel of P − ω with the newmetric. Because v is Killing the variations of χ and H commute with δ hence they will notaffect the partition function. The metric also enters explicitly in (58) through δ -exact terms.Therefore we can conclude that the path integral is independent of the metric provided that χ and H are changed appropriately.A similar reasoning can be followed to analyze the freedom in choosing the supergravitybackground fields. This is parametrized by G µ and b µ introduced in (49). As we change thesupergravity background all the cohmological fields in (55) are unchanged except for H . Aslong as G µ and b µ are kept invariant along v the change in H commutes with δ . Because H only enters the action through δ -exact terms, the partition function is independent on thefreedom in the supergravity background fields. Supersymmetry squares to a translation along the Killing vector field v . The projectors P ± ω also depend on a choice of vector field, which we have implicitly assumed to be v when27efining the cohomological fields in (55). In principle the vector field used in defining theprojector need not be the same as v . However, the projectors P ± ω have to be invariant along v to ensure compatibility with supersymmetry.In many cases of interest the Killing vector field v is a real linear combination of twocommuting Killing vector fields v and v v = ǫ v + ǫ v ǫ , ǫ ∈ R . (66)The two Killing vector fields v and v generate a torus action on M with isolated fixedpoints. For generic values of ǫ and ǫ the orbits of v are not closed and fill a torus in M .We can define new projectors ˜ P ± ω using a linear combination ˜ v of v and v that is differentfrom v . We use the same function cos ω to define both sets of projectors hence the subbundlesof two forms ˜ P + ω Ω and P + ω Ω are isomorphic. It follows that there is an invertible map fromthe cohomological fields χ and H in P + ω Ω to new fields ˜ χ and ˜ H in ˜ P + ω Ω . This map canbe taken to commute with the action of supersymmetry because v and ˜ v commute. Usingthe same logic as in the previous examples this modification of the complex does not changethe partition function. v Here we consider again the case where M admits a torus action generated by two commutingKilling vector fields v and v . Following the procedure described in section 3, given v asin (66) we can construct a supersymmetric field theory on M . We want to generalize thisconstruction to v ’s that are complex linear combinations of v and v . In order to constructthe corresponding Killing spinors on M we cannot directly repeat the construction of section3.1 for a number of reasons. Firstly the reality conditions on the spinors have to be relaxed.Secondly using (40) for a complex v would lead to complexified SU (2) R transformations ingoing from patch to patch.We can instead proceed as follows. First let’s consider a real v as in (66) and choosescalar functions s, ˜ s as in section 3.1. We can then construct real spinors ( ζ iα ) ∗ = ζ iα and ( ¯ χ ˙ αi ) ∗ = ¯ χ i ˙ α such that ζ i ζ i = s , ¯ χ i ¯ χ i = ˜ s , ¯ χ i ¯ σ µ ζ i = 12 v µ . (67)Consider now a complex linear combination of v and v v ′ = ǫ ′ v + ǫ ′ v ǫ ′ , ǫ ′ ∈ C . (68)28e can define the combinations ζ ′ i = a ζ i + b v ′ µ σ µ ¯ χ i , s ′ = ζ ′ i ζ ′ i , ¯ χ ′ i = c ¯ χ i + d v ′ µ ¯ σ µ ζ i , ˜ s ′ = ¯ χ ′ i ¯ χ ′ i . (69)where a, b, c, d are complex smooth scalar functions on M . For v ′ in an open neighbourhoodof v it is possible to choose these functions in such a way that • The functions a, b, c, d are invariant along v and v and the combination s ′ + ˜ s ′ isnowhere zero on M . • The complex Killing vector field v ′ is given by v ′ µ = ¯ χ ′ i ¯ σ µ ζ ′ i . • a = 1 near a fixed point of the torus action where ¯ χ i = 0 . Similarly c = 1 at thosefixed points where ζ i = 0 . This ensures that the spinors ζ ′ and ζ vanish at the samesubset of fixed points of the torus action (and similarly for ¯ χ ′ i and ¯ χ i ). • a, b, c, d and therefore ζ ′ i and ¯ χ ′ i depend holomorphically on ǫ ′ and ǫ ′ .Because we constructed the spinors ζ ′ i and ¯ χ ′ i via linear combinations of ζ i and ¯ χ i , thetransitions in going from patch to patch are the same as described in section 3.1 for ζ i and ¯ χ i . In particular these are real SU (2) R transformations. We can then determine thesupergravity background for which ζ ′ and ¯ χ ′ are Killing spinors using formulas (49),(50),(158)and (159) whose derivation did not make use of the reality conditions on the spinors. Theresulting background will depend holomorphically on ǫ ′ i . It is worth noting that we can add δ exact terms which are not holomorphic in the ǫ ′ i to the action without changing the valueof the partition function. Such terms may be useful to ensure positivity of the real part ofthe action.We can also implement the deformation to complex v at the level of the cohomologicaltheory. As discussed in 4.3 the vector field entering the definition of P ± ω need not be thesame as v . Hence when changing v we can keep fixed the projectors P ± ω used to define thesubbundle of two forms that χ and H belong to. The dependence on v of the observable O in (59) and of the δ -variations of the cohomological variables (27) can be taken to beholomorphic in the ǫ ′ i . This leads to the same conclusions as above. In summary the partitionfunction Z M ǫ ,ǫ is holomorphic with respect to ǫ and ǫ (we may refer to these as generalizedsquashing parameters ). This terminology is somewhat misleading. The metric and ǫ , can be varied independently providedthat v is Killing. In particular one can consider a v corresponding to generic ǫ and ǫ on the round S . Localization calculation
We refer to [2] for a review of the general setup of the localization procedure. The mainingredient is that one adds to the action a positive δ -exact term t δ ( · · · ) . Since this changedoes not affect the partition function one can send t → ∞ . Then the path integral reducesto finding the field configurations where δ ( · · · ) (cid:12)(cid:12) bos = 0 and performing a Gaussian integralaround them. We call these configurations the localization locus, to be analyzed in section5.1. The Gaussian integral gives the Pfaffian of the fermion quadratic term divided bythe square root of the boson quadratic term . N = 2 supersymmetry implies that thereare further cancellations between the two factors: for Donaldson-Witten’s twisted N = 2 theory, these leave only ± . More generally, e.g. for equivariant Donaldson-Witten theory,regardless of the details of the boson/fermion quadratic terms, one gets after cancellationthe superdeterminant of (using (156)) δ (cid:12)(cid:12) Localization locus = i L v + G a , (70)taken over half of the fields in the cohomology complex. More precisely δ has the struc-ture δq = p , δp = ( i L v + G a ) q , and the superdeterminant is over q ’s only. In our casethe relevant cohomological transformations are presented in Appendix D and the q ’s are ( A, ϕ, χ, c, ¯ c, ¯ a , b ) .We will analyze the localization locus next, then explain the computation of the super-determinant in section 5.2. Finally we will comment on the structure of the full answer insection 5.3. For the rest of this section we take the gauge group to be SU ( N ) . In order to localize the theory we add to the Lagrangian supersymmetric δ -exact terms.It is important that on the integration contour these terms are positive semidefinite. Weregard all fields in the theory as complex valued and specify the integration contour as ahalf dimensional subspace of the space of complexified fields. We choose this contour tobe compatible with the natural reality conditions for the dynamical bosonic fields in theLagrangian (54). Namely F is Hermitean and ϕ ∈ R , ˜ φ = φ + i s − ˜ s ) ϕ ∈ R . (71) If one is careless with signs, one can write the Pfaffian as the square root of a determinant, and simplysay that the Gaussian integral gives the square root of a super-determinant between fermions and bosons.We will use such language in what follows. H is an auxiliary field and its integration contour is chosen so that thecorresponding gaussian integral converges.The localization terms we consider are L loc Vol M = δ Tr h − H ) ∧ ⋆χ + ( ι v F − id A ( φ + i ( s − ˜ s ) ϕ )) ∧ ⋆ Ψ + ( ι v d A ϕ + [ φ, ϕ ]) η i . Here
Ω(Φ) is a two form valued in the adjoint of the gauge group that depends on the bosonicfields in the theory Φ . For the localization terms above to be supersymmetric we need that L v Ω(Φ) = Ω( L v Φ) . Moreover we will require that Ω is real on the integration contour.The bosonic terms in L loc are L B loc Vol M = Tr (cid:20) (cid:0) ι v F + d A (( s − ˜ s ) ϕ ) (cid:1) ∧ ⋆ (cid:0) ι v F + d A (( s − ˜ s ) ϕ ) (cid:1) + d A ˜ φ ∧ ⋆d A ˜ φ ++ Vol M (cid:16)(cid:0) ι v d A ϕ (cid:1) − [ ˜ φ, ϕ ] (cid:17) + 2 P + ω Ω ∧ ⋆ Ω − P + ω (cid:0) H − Ω (cid:1) ∧ ⋆ (cid:0) H − Ω (cid:1)i . (72)The last term is set to zero by integrating over the auxiliary field H . The remaining termsare positive definite on the integration contour (71) so that the path integral localizes tofield configurations such that [ ˜ φ, ϕ ] = 0 , ι v d A ϕ = 0 , d A ˜ φ = 0 , (73) ι v F + d A (( s − ˜ s ) ϕ ) = 0 , P + ω Ω = 0 . (74)Hence on the localization locus we can choose a gauge where ˜ φ and ϕ are both diagonal, ˜ φ = diag( ˜ φ a ) , ϕ = diag( ϕ a ) , a = 1 , ..., N − . (75)Generically the gauge group G is broken to its Cartan subgroup H = U (1) N − and thepath integral is over H bundles [38, 39], effectively reducing the problem to an abelian one.Topologically distinct H bundles are distinguished by their fluxes k na ∈ Z through a basis oftwo-cycles { C n } in M , π Z C n F a = k an . (76)Equation (73) implies that the ˜ φ a are constant while the ϕ a are invariant along vd ˜ φ a = 0 , ι v dϕ a = 0 . (77)Next we consider the localization conditions (74). In principle there are many possible choicesfor Ω , here we will consider the following: Ω = F − s + ˜ s ⋆ ( κ ∧ d A ϕ ) − ˆΩ( ϕ ) , (78)31 flip + − +NP SP CP flip − ++ 1 23Figure 1: The assignments of anti self-dual ( + ) and self-dual ( − ) instantons on S and CP that we consider in our localization examples.where the middle term is related to (150) (it arises naturally from the 5D perspective) and ˆΩ( ϕ ) is not yet specified. With this Ω the H field strength F a is completely determined awayfrom the fixed points by (74). For example with the canonical choice (43) for the function ω defining the projector P + ω we get F a = s + ˜ s || v || ⋆ (cid:0) κ ∧ dϕ a (cid:1) − || v || κ ∧ d (cid:0) ( s − ˜ s ) ϕ a (cid:1) + (79) + cos ω || v || ⋆ (cid:16) κ ∧ (cid:0) ι v ˆΩ a + ϕ a d ( s − ˜ s ) (cid:1)(cid:17) + (cid:16) ˆΩ a − || v || κ ∧ ι v ˆΩ a ) (cid:17) . (80)Finally we need to impose the Bianchi identity for F a . Because s, ˜ s and ϕ a are all invariantalong v this results in a single scalar constraint: ⋆ ( κ ∧ dF a ) = 0 . (81)This is a second order differential equation for ϕ a whose detailed structure depends on thechoice of ˆΩ . As we will see below in some examples this equation may have a unique solutionin every flux sector but we could not show that this is the case in general. A large class ofcases is considered in detail in [25].The discussion above determines the localization locus away from the fixed points of v .At the fixed points where cos ω = 1 the projector P + ω is the projector on self-dual two formswhile at those where cos ω = − it reduces to the projector on anti-self-dual two forms.On top of the supersymmetric configurations described above we can then add point-likeinstantons at the fixed points with cos ω = 1 and point-like anti-instantons at the fixedpoints with cos ω = − . S As a first example we consider the manifold S and determine the localization locus. Tostart we consider the round metric on S . Explicit expressions for the metric and other32bjects in the coordinates we are using can be found in appendix B.1. As in section 3.3.5,we take v = ∂ α + ∂ β while s = 2 cos ( θ/ and ˜ s = 2 sin ( θ/ which correspond to Pestun’stheory on the round S .The localization term used in [1] is Ω( ϕ ) = (cid:0) θ (cid:1) (cid:18) F − s + ˜ s ⋆ ( κ ∧ d A ϕ ) + cos θ θ ) ϕ dκ (cid:19) . (82)This is of the form (78) apart from the (1+cos θ ) prefactor whose presence does not modifythe localization locus. Proceeding as described above we determine F a away from the poles F a = 1sin θ (cid:16) ⋆ (cid:0) κ ∧ dϕ a (cid:1) − κ ∧ d (cid:0) cos θ ϕ a (cid:1)(cid:17) . (83)Imposing the Bianchi identity for F a then results in the following equation for ϕ a ∇ ϕ a = 2 ϕ a . (84)The nonzero solutions to this equation are at least of order sin( θ ) − at one or both of thetwo poles. For this singular behavior the evaluation of the observable (59) on the solution(83) gives rise to integrals diverging at the poles. Moreover in order to prove that (59) issupersymmetric one needs to perform integrations by parts that would not be allowed. Forthese reasons the only allowed solution to (84) is ϕ a = 0 . Hence the localization locus awayfrom the poles is ˜ φ a = const , ϕ a = 0 , F a = 0 . We also have localized instantons at the north pole and anti-instantons at the south pole.It is worth commenting on the relation with Pestun’s treatment of the localization locus.On a round S and with the choices made above we can make use of the special property(133) and formula (25) to rewrite the bosonic terms in the localization Lagrangian as L B loc Vol M =Tr " (cid:16) θ (cid:17)(cid:18) F + − ϕ dκ + ( θ/ (cid:19) + 2 sin (cid:16) θ (cid:17)(cid:18) F − + ϕ dκ − ( θ/ (cid:19) + ( d A ϕ ) + ( d A ˜ φ ) − Vol M [ ˜ φ, ϕ ] + 4 F ∧ d A ( ϕκ ) i . (85)The last term above vanishes upon integrating by parts and we are left with a sum of squares.Hence away from the poles we obtain d ˜ φ a = 0 , dϕ a = 0 , F a = 2sin y ϕ a κ ∧ d cos y . (86)33nforcing Bianchi results in ϕ a = 0 . This derivation implicitly assumes that integration byparts of the last term in (85) can be carried out. This excludes configurations for ϕ a thatare singular at the poles.We can deform the Killing vector away from v = ∂ α + ∂ β . This can be accompanied bychanges of the metric and other deformations as discussed in section 4. For instance we mayconsider the theories on squashed S ǫ ,ǫ described in [16, 17]. For generic ǫ i it is no longerpossible to rearrange the localization terms as in (85), which makes the determination of thelocalization locus more subtle [16]. Nevertheless, following the logic leading to (83) we canargue that for small deformations around v = ∂ α + ∂ β the differential equation (84) for ϕ a ischanged only slightly. Hence it continues to have only ϕ a = 0 as an acceptable solution. CP As a second example we consider the manifold CP and we look in detail at the structureof the localization locus. We start by choosing the Fubini study metric. Expressions for theFubini study metric and other quantities in the coordinates we use are collected in appendixB.2. We also make a choice of Killing vector field v = 2( ∂ α + ∂ β ) , κ = g ( v ) = 2(1 − x − y )( x dα + y dβ ) . The norm of v vanishes along the line x + y = 1 and not just at isolated points. This choiceof v , however, makes the analysis below simpler and prepares the ground for us to considera generic v . With these choices we can set s + ˜ s = 2 , s − ˜ s ω = 2 x + 2 y − . There is one harmonic 2-form Ω h Ω h = dx ∧ dα + dy ∧ dβ , (87)which we will use to specify the localization term (78) Ω = F − ⋆ (cid:0) κ ∧ Dϕ (cid:1) + ϕ Ω h . (88)The localization equations (73) result in the following expression for F aµν F a = 12 || v || ⋆ (cid:0) κ ∧ dϕ a (cid:1) − cos ω || v || (cid:0) κ ∧ dϕ a (cid:1) − ϕ a Ω h . (89)The Bianchi identity then gives rise to equation (81) which reads (cid:0) ∇ + 4 x∂ x + 4 y∂ y (cid:1) ϕ a = 0 . (90)34his equation is separable in the coordinates x + y and xy . An analysis similar to that on S shows that the only acceptable solution is a constant so that F a = k a Ω h , ϕ a = k a ∈ Z . (91)The constants k a take values in the integers because the flux of F is quantized as in (76).Next we can modify our choice of v to be v = 2(1 + ǫ ) ∂ α + 2(1 − ǫ ) ∂ β , − < ǫ < . (92)In order to keep the analysis of the localization locus simple it is convenient to change themetric by a nonsingular Weyl rescaling in order to be able to satisfy s + ˜ s = 2 . Becausethe Weyl rescaling of the metric is a δ -exact deformation it will not change the value ofsupersymmetric observables. Hence we set the metric to be ds = A ( x, y ) ds F S , A ( x, y ) = 1 p ǫ (1 + ǫ ) x − ǫ (1 − ǫ ) y , (93)where ds F S is the Fubini-Study metric (138). We also take cos ω = A ( x, y )(2 x (1 + ǫ ) + 2 y (1 − ǫ ) − . (94)For − < ǫ < the value of cos ω at the fixed points is unchanged from the case consideredabove. Finally we make the following choice for ΩΩ = F − ⋆ (cid:0) κ ∧ Dϕ (cid:1) − ϕ ⋆ (cid:0) κ ∧ d log( A ) (cid:1) + A ϕ Ω h . (95)Proceeding as before (81) results in an equation for ϕ that is no longer separable. The equa-tion is satisfied if Aϕ is a constant and this should be the only solution. The corresponding F field strength is F a = k a Ω h , ϕ a = A − k a , k a ∈ Z . (96)Let us comment on certain properties of this solution that warrant further study. Oursolution can be written as F a + cos ω ϕ a = k a Ω h + (2 x (1 + ǫ ) + 2 y (1 − ǫ ) − k a (97)and it is annihilated by ( d + ι v ) . It would be natural to say that it is the curvature of anequivariant line bundle. However, the concrete solution (97) is not an element in the secondintegral equivariant cohomology on CP . At this point we do not know how to interpret thisconflict which requires further detailed study.35 .2 The super-determinant Having found the localization locus, what remains is the Gaussian integral around it. Wehave explained earlier that the Gaussian integral eventually produces (the square root of)a super-determinant. Our entire computation is organized and goes along the same lines asthe original calculation of Pestun (see subsection 4.4 in [1]). However, it is instructive toformalize this computation in different terms. We will explain in a future paper [25] how N = 2 supersymmetry plus gauge fixing organizes all fields into a double complex, with δ acting vertically as an equivariant differential and a second differential ð acting horizontallycorresponding to the transversely elliptic complex in our setup. The superdeterminant isthat of the operator δ = i L v + G a taken over the cohomology of ð. Now we restrict to thetrivial gauge background, i.e. all fields are zero except one scalar field ˜ φ having a vev a . Asthe gauge bundle is trivial, one can then write sdet( i L v + G a ) = Y α ∈ roots sdet H ð (cid:0) h a , α i + i L v (cid:1) . So one may well regard a as a constant in the following computation.To compute sdet H ð (cid:0) a + i L v (cid:1) , one needs to compute H ð equivariantly, so that one canread off the weights of all the U (1) actions and thereby write down the eigenvalues of i L v .To summarize, our remaining task is to compute the equivariant cohomology H ð (in factonly the index is needed). What is the complex ( E • , ð ) for which we want to compute H ð ? Elliptic case
Let us start from the simpler setup of Donaldson-Witten theory where the correspondingcomplex is the instanton deformation complex already given in (3), → Ω ( X ) → Ω ( X ) → Ω ( X ) → , (98)which is elliptic. In the notation of the full cohomology complex (153), the three terms comefrom the fields c, A, χ respectively. One also needs to add a two term complex Ω → Ω represented by the fields ϕ and η . But since ∆ has zero index, one often ignores this lastcomplex (one says that the two terms are ’cancelled’). In the non-equivariant case the indexis easy to get with no computationind asd = b − b + b = 12 ( χ + τ ) , where χ is the Euler number and τ is the signature. Including the gauge part, say for SU (2) ,the index is ind asd = 32 ( χ + τ ) − k , k is the instanton number. Note that the dimension of the instanton (asd) modulispace is minus this index. For anti-instantons (sd), one flips the sign of τ and k .For the equivariant Donaldson-Witten theory, one computes the cohomology of (98)equivariantly. We follow the method of [40, 41] that uses equivariant localization. We followthis route because it turns out that the index of the complex we need can be obtained bysmall tweaks of the same calculation around the torus fixed points. A systematic treatmentof such computations will be left to a future paper [25]. According to [40]ind eq = X p i ∈ fixed pt χ eq ( E • p i )det(1 − df ) , (99)where χ eq ( E • p i ) stands for the equivariant Euler character of the fibre E • p i and f : M → M is the diffeomorphism induced by the group G (in our case U (1) ) action on M .We start with S . If there is an almost complex structure on M then the complex (98)is isomorphic to the direct sum of (Ω , • , ¯ ∂ ) and its conjugate. But as (99) only involves localdata of E • , we can still exploit such isomorphism to simplify computation even though S is not almost complex. We describe S as the quaternion projective space HP S ≃ { [ q , q ] | q , ∈ H } / ∼ , where [ q , q ] ∼ [ q q, q q ] , q ∈ H ∗ . We use local inhomogeneous coordinate q = q q − to cover the northern hemisphere and q − for the southern hemisphere. The U (1) isometries act by left multiplying q → sq and q → tq , where we use the same letter s, t for a phase as well as the character. Writing q = z + jw with z, w ∈ C gives us a local complex structure at the north pole. One readsoff the action of s, t on z, w as z → st − z , w → s − t − w . Then at the north pole, one gets χ eq (Ω , • ) = (1 − t/s )(1 − st ) , det(1 − df ) = (1 − s/t )(1 − t/s )(1 − s − t − )(1 − st ) . So the local contribution to the index readsnorth pole : 1(1 − s/t )(1 − s − t − ) + 1(1 − t/s )(1 − st ) . (100)At the south pole q − = z ′ + jw ′ = (¯ z − jw ) / | q | and so χ eq (Ω , • ) = (1 − s/t )(1 − st ) , while det(1 − df ) does not change. So the local contribution to the index issouth pole : 1(1 − t/s )(1 − s − t − ) + 1(1 − s/t )(1 − st ) . (101)Putting the two together one gets ind eq = 1 , which is obvious for the complex (98) for S .Set s = t = 1 , the negative of the index times 3 (for SU (2) ) gives us the expected dimension − of the instanton moduli space at a trivial background.37or the CP case, we use the inhomogeneous coordinates [ z , z , to cover one patchand we have torus action z → sz z → tz . In other patches the actions are read offfrom the standard coordinate transformation. At the fixed points z = z = 0 we have χ eq (Ω , • ) = (1 − s − )(1 − t − ) , while det(1 − df ) = (1 − s )(1 − s − )(1 − t )(1 − t − ) . The sameexpressions can be obtained in other patchesind eq,asd = 1(1 − s )(1 − t ) + 1(1 − st − )(1 − t − ) + 1(1 − s − )(1 − ts − ) + c.c. = 2 , ind eq,sd = 1(1 − s )(1 − /t ) + 1(1 − st − )(1 − t ) + 1(1 − s − )(1 − st − ) + c.c. = 1 . The difference between the two lines comes from conjugating one of the two local holomorphiccoordinates at each fixed point, in order to turn Ω in (98) into Ω − .The two indices of course agree trivially with the dimension of the instanton modulispace: dim CP , asd = 8 k − and dim CP ,sd = 8 k − , with the factor of 3 coming from SU (2) again. Transversally elliptic case
For our flipping instanton, the relevant complex to compute the equivariant index for isagain the complex that controls the deformation (7) Ω ( X ) d −→ Ω ( X ) ⊕ Ω ( X ) ˜ D −→ P + ω Ω ( X ) ⊕ Ω ( X ) , with the expression for ˜ D given by (152) (see Appendix C for further discussion). Theflipping projector P + ω approaches the self-dual projector when cos ω = 1 and anti self-dualwhen cos ω = − . These loci coincide with the torus fixed points. In contrast to theprevious elliptic case, one cannot cancel the two Ω terms since they are crucial for transversalellipticity. However, they can be cancelled at the torus fixed points, since at those loci P + ω equals the asd or sd projector and ˜ D is the direct sum ˜ D = P ± d ⊕ ∆ on the two summands.This means that as far as the localization computation of the index is concerned, one canobtain the new index from the computation done earlier plus some suitable modification toaccount for the asd/sd flips.We start from the flipping instanton case of S where the instantons/anti-instantons areat the north/south pole. We introduce the notation [1 / (1 − s )] + to mean the expansion s + s + · · · , and [1 / (1 − s )] − for − s − − s − − · · · . These two expansions are theequivariant characters of H ∂ ( C ) and H ∂ ( C ) respectively. The minus sign reflects the degree H ∂ . The two different manners of rewriting / (1 − s ) into polynomials is related to deforming ¯ ∂ with a vector field (coming from a group action) to trivialize the symbol [42] (we only statethis as a prescription, see the appendix of [1] for a review).38e observe that the equations (100), (101) can be thought of as the equivariant indexof the cohomology of ( E • , ð ) near the north/south pole, provided one expands the fractioninto power series: we rewrite (100), (101) using the new notationnorth pole : [1 / (1 − st − )] + [1 / (1 − s − t − )] + + [1 / (1 − s − t )] − [1 / (1 − st )] − , (102)south pole : [1 / (1 − s − t )] − [1 / (1 − s − t − )] + + [1 / (1 − st − )] + [1 / (1 − st )] − . (103)The north/south pole ’know about each other’ only through the choice of a vector fieldthat trivializes the symbol of the complex. Here the shift from + to − regularisation isattributable to the fact that z ′ , z have opposite weights, but w ′ , w have the same weights .Here one can check that the choice of ± at the two poles conspire so that, after expandingeach term, everything cancels except one s t = 1 , same as what we got earlier without theexpansion. This is due to the fact that we were dealing with an elliptic complex before.For the flipping instantons, the complex at the south pole is the sd complex (with pro-jector P − ), following [1] one flips the weight and regularisation at the south pole,north pole : [1 / (1 − st − )] + [1 / (1 − s − t − )] + + [1 / (1 − s − t )] − [1 / (1 − st )] − , (104)south pole : [1 / (1 − st − )] − [1 / (1 − s − t − )] − + [1 / (1 − s − t )] + [1 / (1 − st )] + . (105)Expanding everything using the given prescription and tracking down cancellations, theindex is ind eq = ( X i,j ≥ + X i,j ≥ ) (cid:0) ( st − ) i ( s − t − ) j + ( s − t ) i ( st ) j (cid:1) . (106)It is no longer a finite polynomial, but a Laurent polynomial infinite in both directions, i.e.it is a formal expression and one cannot evaluate it at concrete values of s, t . To comparewith [1], one formally sets st − = q = s − t − ind eq → X n ∈ Z | n | q n . The contribution 2 are accounted for by the ghost zero modes. Having obtained the equiv-ariant index, the super determinant reads sdet H ð (cid:0) a + i L v (cid:1) = Y i,j ≥ (cid:0) a + iǫ + jǫ (cid:1) Y i,j ≥ (cid:0) − a + iǫ + jǫ (cid:1)Y i,j ≥ (cid:0) − a + iǫ + jǫ (cid:1) Y i,j ≥ (cid:0) a + iǫ + jǫ (cid:1) , Here and next we always make the technical assumption that one can deform the symbol with a vectorfield so that the symbol of the complex is trivialized except at torus fixed points. Otherwise one has to usethe more general formula given in lecture 8 of [42]. P asd + ++ 1 23 CP flip - ++ 1 23 CP flip ′ + -- 1 23Figure 2: Three distributions of asd/sd instantons on CP , where + corresponds to antiself-dual, − to self-dual. At corner 1, we have inhomogeneous coordinates [ z , z , , corner2 [1 , z /z , /z ] and corner 3 [ z /z , , /z ] .where we have written s = e i ( ǫ − ǫ ) , t = e i ( ǫ + ǫ ) and so for, say, for a term s t in (106),one gets the i L v eigen-value − ǫ − ǫ . Here the second line corresponds to (106).Now we will modify the regularisation scheme at each fixed point to get the index for thenew instantons. For CP one can writeind CP ,asd = (cid:2) − s (cid:3) + (cid:2) − t (cid:3) + + (cid:2) − /s (cid:3) − (cid:2) − t/s (cid:3) − + (cid:2) − s/t (cid:3) + (cid:2) − /t (cid:3) − + c.c. (107)where the three terms come from the three corners of figure 2 (and the local coordinates arealso labeled there). All terms cancel except s t = 2 as before. We now consider the newinstanton complex that approaches asd at the second and third fixed points and sd at thefirst one as in the middle panel of figure 2. The modification one does to (107) is to reverseat corner 2,3 the weights of z ind CP ,flip = (cid:2) − s (cid:3) + (cid:2) − t (cid:3) + + (cid:2) − s (cid:3) − (cid:2) − t/s (cid:3) − + (cid:2) − s/t (cid:3) + (cid:2) − t (cid:3) − + c.c. This prescription is derived from regarding the flipping instanton as the reduction of 5D con-tact instantons on S along the U (1) of weight [1 , , − , and use the 5D index calculations.It is yet unclear how this prescription compares to the one used in [1], in particular, it isdesirable to have a purely 4D treatment of the index calculation.Tracking down the cancellations one gets the ind CP ,flip as the following graph plus itscomplex conjugate ( s, t → s − , t − ). ••••• ••••• ••••• ••••• ••••••••••••••••••••• t s
40 --+ ind F ,flip = • •• ••• •••• ••••••••••••••• t s Figure 3: One distribution of ± ’s for the Hirzebruch surface F and the corresponding resultof the index computation.That this picture formally looks the same as S means nothing, since the s, t parameters willbe identified differently.To compute the index for the third configuration in figure 2, we regard the flippinginstanton as the reduction from S along a U (1) of weight [ − , − , (the miss/alignmentof [ − , − , with the standard Hopf vector field [1 , , tells us whether it is asd or sd ateach corner, see [25]). Correspondinglyind C P ,flip ′ = (cid:2) − /s (cid:3) + (cid:2) − /t (cid:3) + + (cid:2) − /s (cid:3) − (cid:2) − s/t (cid:3) − + (cid:2) − t/s (cid:3) + (cid:2) − /t (cid:3) − + c.c. After cancellations, we get the same picture as the above with s, t reversed, but as we alsoadd the conjugate graph, the result is not changed.As another example, we also give the result for the Hirzebruch surface F , where F isrealised as S fibered over S with degree 1, i.e. F ≃ { [ z , z ; u , u ] } / ∼ , [ z , z ; u , u ] ∼ [ λz , λz ; λµu , µu ] , λ, µ ∈ C ∗ . The U (1) ’s s and t rotates the phase of z , u respectively. The assignment of asd/sd instan-tons and the corresponding result of the index computation is shown in figure 3. In the threeexamples above, setting s = e − iǫ and t = e − iǫ , one can write down the superdeterminantin a uniform manner. sdet H ð (cid:0) a + i L v (cid:1) = Y ( p,q ) ∈ C ∩ Z (cid:0) a + ǫ p + ǫ q (cid:1) Y ( p,q ) ∈ C ◦ ∩ Z (cid:0) − a + ǫ p + ǫ q (cid:1)Y ( p,q ) ∈ C ∩ Z (cid:0) − a + ǫ p + ǫ q (cid:1) Y ( p,q ) ∈ C ◦ ∩ Z (cid:0) a + ǫ p + ǫ q (cid:1) . (108)Here C is a rational cone and C ◦ is its interior.41ince the structure is rather universal, it is useful to define Υ C ( x | ǫ , ǫ ) = Y ( p,q ) ∈ C ∩ Z (cid:0) x + ǫ p + ǫ q (cid:1) Y ( p,q ) ∈ C ◦ ∩ Z (cid:0) − x + ǫ p + ǫ q (cid:1) . Then the superdeterminant in all cases, complete with the Lie algebra factor reads sdet( i L v + G a ) = Y α ∈ roots Υ C ( h a , α i| ǫ , ǫ )Υ C ( −h a , α i| ǫ , ǫ ) . When one computes the index by summing up local contributions from toric fixed points,one generally gets a Laurent polynomial infinite in both directions. In the above examples,the results organize into a cone and its negative, and so by reversing the negative cone, onecan in fact evaluate the infinite product (108) provided ǫ , ǫ is within the dual cone. That isfor ~n ∈ C , ~n · ~ǫ > and increases to infinity as | ~n | → ∞ . Thus one can use the zeta functionto regulate the infinite product. The Υ -function defined in (109) is closely related to thegeneralized triple sine [43] that appears in the 5D 1-loop computation on toric manifolds.Given an arbitrary assignment of asd/sd at the toric fixed points, we currently do notknow how to combine the local contributions and so we cannot write down the 1-loop partof the partition function.Secondly the parameters ǫ , are real, but one can factorize the Υ function into as manyfactors as there are toric fixed points, provided one gives a non-zero imaginary part to ǫ , .This factorization led to the conjecture that the full partition function is glued from copiesof partition functions of C eq , there being one copy for each toric fixed point. Except fortaking into account the non-zero c of the gauge bundle, the Coulomb branch parameter a is shifted between the different copies of the local contribution. Let us summarize here what we think about the general answer. Before doing this let us com-ment on the analytical properties of instanton and anti-instanton partition functions on C .With the complex coordinates ( z , z ) ∈ C the change z → z , z → ¯ z (or z → z , z → ¯ z )will induce the map from instanton equations to anti-instanton equations. Therefore we havethe following relations Z anti − inst ǫ ,ǫ ( a, ¯ q ) = Z inst ǫ , − ǫ ( a, ¯ q ) = Z inst − ǫ ,ǫ ( a, ¯ q ) , (109)where ¯ q is an instanton counting parameter. We stress that for complex ( ǫ , ǫ ) the instantonand anti-instanton partition functions are not related by simple complex conjugation Z anti − inst ǫ ,ǫ ( a, ¯ q ) = Z inst ǫ ,ǫ ( a, q ) . (110)42he answer for S ǫ ,ǫ with complex ( ǫ , ǫ ) is given by Z S ǫ ,ǫ = Z da e S cl Z inst ǫ ,ǫ ( ia, q ) Z anti − inst ǫ , − ǫ ( ia, ¯ q ) = Z da e S cl Z inst ǫ ,ǫ ( ia, q ) Z inst ǫ ,ǫ ( ia, ¯ q ) (111)and it is holomorphic in ( ǫ , ǫ ) . Only for real ǫ ’s the original formula by Pestun holds Z S ǫ ,ǫ = Z da e S cl | Z inst ǫ ,ǫ ( ia, q ) | . (112)The general answer for Z M ǫ ,ǫ with p plus points and ( l − p ) minus points will be given by X discrete k i Z h da e − S cl p Y i =1 Z inst ǫ i ,ǫ i (cid:16) ia + k i ( ǫ i , ǫ i ) , q (cid:17) l Y i = p +1 Z anti − inst ǫ i ,ǫ i (cid:16) ia + k i ( ǫ i , ǫ i ) , ¯ q (cid:17) . (113)The parameters ( ǫ i , ǫ i ) can be defined from T -action around the fixed point x i . The classicalaction can be evaluated by the localization of the observable (59) and for this we need toknow the values of φ at the fixed points (in particular k i ( ǫ i , ǫ i ) ). The main problem is tofix the functions k i ( ǫ i , ǫ i ) which corresponds to fluxes. The original idea from [12] is todeclare ( F a + cos ω ϕ a ) (we look at this combination along Cartan sub-algebra h ) to be thecurvature of an equivariant line bundle and thus to be an element of the integral equivariantcohomology class. If we accept this approach then the shift functions can be written asfollows (more details in [25]) k i ( ǫ i , ǫ i ) = k i ǫ i + k i − ǫ i , (114)where k i is vector composed from integers (the number of components of this vector isgiven by dimension of Cartan subalgebra). The sum in the final answer (113) is taken oversubset of the integers. This logic is common and a natural belief (for recent discussions seee.g. [19, 20, 44, 45]). In section 5.1.2 we have tried to analyze the concrete localization locusfor the concrete example of CP assuming only the integrality of F . This analysis results inshifts that are not compatible with the expression (114). At the moment we are unsure howto interpret this observation and we leave the problem to further study. In this work we have constructed an N = 2 supersymmetric gauge theory on a manifoldwhich admits a Killing vector field with isolated fixed points. We gave a cohomologicaldescription of this theory and thus explained the relation between Donaldson-Witten theory43nd Pestun’s calculation on S . In the follow up work [25] we study further the formalaspects of this cohomological theory and we stress the relation between supersymmetry anda transversely elliptic complex.In this work we have conjectured the answer for the partition function for the generalcase. The main remaining challenge is to prove this answer and better understand how itarises. It will be crucial to understand which geometrical data controls the fluxes, and tofurther refine the arguments presented in section 5. In [25] we are able to be more preciseabout the general answer when the theory arises from the reduction of a supersymmetric 5Dgauge theory. The 5D language appears to be very powerful for repacking the 4D answerand it allows to fix some ambiguities in 4D. Its drawback is that it does not apply to allpossible 4D theories.It would be interesting to study what are the physical ramifications of our present con-struction. For instance our construction could be interpreted from the point of view of theAGT-paradigm [46]. It is also interesting to study how our formalism relates to the “cuttingand gluing” formalism for supersymmetric path integrals described in [47, 48]. Acknowledgements:
We thank Nikita Nekrasov and Vasily Pestun for discussions. Thework of GF is supported by the ERC STG grant 639220. The work of MZ is supported inpart by Vetenskapsrådet under grant
A Conventions
Here we collect various relevant formulas and a summary of our conventions. They are basedon those of [49], adapted to Euclidean signature.
A.1 Flat Euclidean Space
The metric is δ µν , where µ, ν = 1 , . . . , . The totally antisymmetric Levi-Civita symbolis ǫ = 1 . The rotation group is SO (4) = SU (2) + × SU (2) − . A left-handed spinor ζ α is an SU (2) + doublet and has un-dotted indices. A Right-handed spinor ¯ ζ ˙ α is a doubletunder SU (2) − . It carries a bar and dotted indices. In Euclidean signature, ζ and ¯ ζ areindependent spinors as SU (2) + and SU (2) − are not related by complex conjugation. Dottedand undotted indices are raised acting on the left with the totally antisymmetric × matrix ǫ defined by ǫ = 1 . Hence we have ζ α = ǫ αβ ζ β and ¯ ζ ˙ α = ǫ ˙ α ˙ β ¯ ζ ˙ β . We write the SU (2) + ζ and η as ζ η = ζ α η α . Similarly, the SU (2) − invariant innerproduct of ¯ ζ and ¯ η is given by ¯ ζ ¯ η = ¯ ζ ˙ α ¯ η ˙ α .The sigma matrices are written in terms of the Pauli matrices ~σ = ( σ , σ , σ ) σ µα ˙ α = ( ~σ, − i ) , ¯ σ µ ˙ αα = ( − ~σ, − i ) . (115)In Euclidean signature σ µ and ¯ σ µ are not related by complex conjugation. The sigma matrices(115) satisfy the identities σ µ ¯ σ ν + σ ν ¯ σ µ = − δ µν , ¯ σ µ σ ν + ¯ σ ν σ µ = − δ µν . (116)We define the antisymmetric matrices σ µν = 14 ( σ µ ¯ σ ν − σ ν ¯ σ µ ) , ¯ σ µν = 14 (¯ σ µ σ ν − ¯ σ ν σ µ ) . (117)They are self-dual and anti-self-dual respectively, ǫ µνρλ σ ρλ = σ µν , ǫ µνρλ ¯ σ ρλ = − ¯ σ µν . (118)Given a two form ω we can separate its (2 , and (0 , components as follows: ω + αβ = 12 ω µν σ µναβ , ω − ˙ α ˙ β = 12 ω µν ¯ σ µν ˙ α ˙ β . (119) A.2 Differential Geometry
We use Greek letters µ, ν, . . . to denote curved indices and Latin letters a, b, . . . to denoteframe indices. Given a Riemannian metric g µν , we can define an orthonormal tetrad e aµ . Wedenote the Levi-Civita connection by ∇ µ . The corresponding spin connection is given by ω µab = e bν ∇ µ e aν . (120)The covariant derivatives of the spinors ζ and ¯ ζ are given by ∇ µ ζ = ∂ µ ζ + 12 ω µab σ ab ζ , ∇ µ ¯ ζ = ∂ µ ¯ ζ + 12 ω µab ¯ σ ab ¯ ζ . (121)The Riemann tensor tis R µνab = ∂ µ ω νab − ∂ ν ω µab + ω νac ω µcb − ω µac ω νcb . (122)The Ricci tensor is R µν = R µρν ρ , and R = R µµ is the Ricci scalar. Note that, with theseconventions, a round sphere has negative Ricci scalar.45 .3 Differential forms We use the following conventions when switching to index free notation for differential forms.An n -form is given by ω ( n ) : ω ( n ) = 1 n ! ω µ µ ...µ n dx µ ∧ dx µ ∧ ... ∧ dx µ n . (123)The Hodge star operator acts as follows: ( ⋆ω ) (4 − n ) µ ...µ − n = 1 n ! ω ( n ) ν ...ν n ǫ ν ...ν n µ ...µ − n . (124)It satisfies ⋆⋆ω ( n ) = ( − n ω ( n ) . Given a vector field v = v µ ∂∂x µ the contraction of a differential n -form ω ( n ) with v is ( ι v ω ) ( n − µ ...µ n − = v ν ω ( n ) νµ ...µ n − . (125)Consider the -form κ obtained from v via the metric κ µ = g µν v ν , (126)then the following relation holds ι v ⋆ ω ( n ) = ( − n ⋆ ( κ ∧ ω ( n ) ) . (127) A.4 Other conventions
We denote symmetrization over indices using round brackets, A ( i ...i p ) = 1 p ! X σ ∈ S p A i σ (1) ...i σ ( p ) . (128)Anti-symmetrization is denoted by square brackets A [ i ...i p ] = 1 p ! X σ ∈ S p ( − σ A i σ (1) ...i σ ( p ) . (129) B Examples arising from specific four-manifolds
B.1 S Here we collect the basic conventions we follow when using S as an example. Let usintroduce the coordinates θ, x, α, β on S where θ ∈ [0 , π ] , x ∈ [0 , and α, β ∈ [0 , π ] areangles. The round metric in these coordinates takes the form ds = dθ + sin θ (cid:18) dx x (1 − x ) + xdα + (1 − x ) dβ (cid:19) . (130)46hese coordinates allow us to think of S as a degenerating T fibration (the two anglesspanning T ) over the 2d surface with boundaries spanned by θ, x , see figure 1. The cycleparametrized by α shrinks when x = 0 , and the β -cycle shrinks when x = 1 . The north polecorresponds to θ = 0 and the south pole to θ = π .To compare with the work of Pestun we choose the Killing vector field v and its dual1-form κ as v = ∂ α + ∂ β , κ = g ( v ) = sin θ ( xdα + (1 − x ) dβ ) . (131)We also select cos( ω ) = cos θ , which goes from +1 on the north pole to − on the southpole. The projector on 2-forms we introduced in section 2 is P + θ = 11 + cos θ (1 + cos θ ⋆ − sin θ ˆ κ ∧ ι v )= 11 + cos θ (1 + cos θ ⋆ − κ ∧ ι v ) , (132)which is everywhere smooth since κ and v are globally well defined. Since v and κ both goto zero at the poles, we can see that P + θ collapses to the self-dual projector over the northpole and and to the anti-self-dual projector over the south pole. For the specific choice (131)one can check that dκ satisfies the following property P + θ dκ = dκ . (133)The volume form in these coordinates is Vol S = 12 sin θ dθ ∧ dx ∧ dα ∧ dβ. (134) B.2 CP We introduce the following coordinate system on CP . Two of the coordinates x, y parametrizea simplex in R , x ≥ , y ≥ , x + y ≤ , (135)see the right part of figure 1. On any point of the simplex ( x, y ) there is a torus parametrizedby two angles α and β . The 1-cycle on the torus parametrized by α shrinks to zero for x = 0 ,the one parametrized by β shrinks to zero for y = 0 while the 1-cycle at constant α − β shrinkson the remaining boundary of the simplex x + y = 1 . The relation of these coordinates tothe inhomogeneous coordinates [ z , z , used in section 5.2 is as follows: z = (cid:18) x − x − y (cid:19) e iα , z = (cid:18) y − x − y (cid:19) e iβ . (136)47here are three fixed points of the torus action at the corners of the simplex x = 0 , y = 0) , x = 1 , y = 0) , x = 0 , y = 1) . (137)Consider the Fubini-study metric on CP , which in the coordinates we just introduced isgiven by ds F S = dx x + ( dx + dy ) − x − y ) + dy y + (1 − x ) xdα − xydαdβ + (1 − y ) ydβ . (138)The volume form in these coordinates is Vol CP = 14 dx ∧ dy ∧ dα ∧ dβ. (139)Next consider the Killing vector field v = 2( ∂ α + ∂ β ) , κ = g ( v ) = 2(1 − x − y )( x dα + y dβ ) . The norm of v is || v || = 4(1 − x − y )( x + y ) . This v does not vanish at isolated points because || v || = 0 for x + y = 1 ; however, it is auseful starting point to analyze more general choices. With these choices we can set s + ˜ s = 2 , h = s − ˜ s ω = 2 x + 2 y − , so that cos ω = 1 − || v || .The projector P + ω for these choices takes the form P + ω = 11 + (2 x + 2 y − (cid:0) x + 2 y − ⋆ +4(1 − x − y )( x dα + y dβ ) ∧ ι ∂ α + ∂ β (cid:1) , (140)which is nowhere singular and goes to P − at fixed point and to P + along the line x + y = 1 .Note that cos ω is equal to − at fixed point and is equal to +1 at fixed points and .Hence the cohomological theory built using P + ω in section 2.2 would not be equivalent to theequivariant topological twist on CP . C Transverse elliptic problems in 4D and 5D
In this appendix we collect explicit expressions for the transversally elliptic problems whichappear in the context of gauge theories in 4D and 5D. Since we are interested in symbols48f operators, we consider only the case of an abelian theory and expand around the triv-ial connection. All our geometrical considerations are local in nature since ellipticity andtransversal ellipticity are local notions.Let us start from anti self-duality in 4D, P + F = 0 . If we add the gauge fixing conditionwe obtain the following system of PDEs d † A = 0 ,F + F = 0 ,F + F = 0 , (141) F + F = 0 . The symbol of the corresponding operator is σ sd D = p p p p − p p − p p − p p p − p − p − p p p , (142)and this is an elliptic symbol. If we add a fifth direction as a trivial circle and label thisdirection by zero then we can lift this elliptic problem to a transversely elliptic one (withrespect to new U (1) circle) simply by declaring that A = 0 . However, we would like torewrite this system in 5D gauge covariant terms ∂ F + ∂ F + ∂ F + ∂ F = 0 ,d † A = 0 ,F + F = 0 , (143) F + F = 0 ,F + F = 0 , where now the Lorentz gauge condition d † A = 0 is understood as a five dimensional condition.This system can be covariantized as follows d † ι R F = 0 ,d † A = 0 , (144) F + H = 0 , where R is the vector field associated with the action of U (1) (or a more general toric action).The condition F + H = 0 is anti self-duality in the horizontal space with respect to R , which49equires a one form κ with the property ι R κ = 1 . The symbol for the PDEs (143) can bewritten as follows σ tsd D = − p · p p p p p p p p p p p p p p − p p − p p − p p p − p − p − p p p , (145)where p · p = p + p + p + p . This is the symbol of a transversely elliptic operator withrespect to zeroth direction (setting p = 0 ). Thus on 5D manifolds with a torus action thesystem of PDEs (144) gives rise to a transversely elliptic problem and these PDEs are centralfor for the treatment of 5D supersymmetric Yang-Mills theory [50].However, here our main interest is in 4D gauge theory systems. Hence we discuss thereduction of the 5D system (144) down to 4D. There is always a trivial reduction along R (zeroth direction in our notations) which will bring us back to the elliptic problem in 4D.We are interested in dimensional reduction along a different direction. Let us assume thatwe have at least a T action on the five manifold and that it is generated by ∂ and ∂ . Wecan introduce another basis, ˜ ∂ and ˜ ∂ such that ∂ = cos ω ˜ ∂ + sin ω ˜ ∂ ,∂ = − sin ω ˜ ∂ + cos ω ˜ ∂ , where ω is some function invariant under the torus actions. We stress again that our dis-cussion is local, at the level of symbols and the idea is to perform the reduction along ˜ ∂ byrequiring all fields to be independent of this direction. Almost by definition, the resultingsystem of PDEs will be transversally elliptic with respect to the remaining U (1) action. Atthe level of the symbol (145) we just have to make a substitution p = sin ω ˜ p ≡ s ω ˜ p and p = cos ω ˜ p ≡ c ω ˜ p with ˜ p = 0 σ tsd D = − c ω ˜ p − ~p · ~p c ω s ω ˜ p s ω p ˜ p s ω p ˜ p s ω p ˜ p s ω ˜ p c ω ˜ p p p p − p c ω ˜ p − p p − p p c ω ˜ p − p − p − p p c ω ˜ p , (146)where ~p · ~p = p + p + p . This matrix, however, is written in the wrong basis ( A , A , A , A , A ) .Since we perform the reduction along ˜ ∂ , the 4D boson Φ is defined as the ˜ ∂ -component of50D gauge field. Thus we have the following definition A = cos ω Φ + sin ω ˜ A , (147) A = − sin ω Φ + cos ω ˜ A , (148)and in the new basis (Φ , ˜ A , A , A , A ) the symbol (145) takes the form ˜ σ tsd D = − c ω (˜ p + ~p · ~p ) − s ω ~p · ~p s ω p ˜ p s ω p ˜ p s ω p ˜ p p p p p s ω p − c ω p c ω ˜ p − p p s ω p − c ω p p c ω ˜ p − p s ω p − c ω p − p p c ω ˜ p . (149)This is exactly the same symbol as in Pestun’s analysis on S (see equation (4.32) in [1], upto an overall minus in first row) with ω being identified with the θ -angle on S . In our moregeneral story ω is an effective angle in 5D which controls the relative direction between R and ˜ ∂ . In 4D we have a fixed point when sin ω = 0 and there the symbol (149) will correspondto the symbol of either the self-dual connection or of the anti self-dual connection problem,depending on the sign of cos ω at the fixed point.All these considerations can be reformulated in terms of a transversely elliptic complex.The 5D language appears to be a natural way to repack the 4D expressions. It is possi-ble, however, to formulate the corresponding transversally elliptic problem intrinsically in4D terms. Let us assume that there is a U (1) -action on the four manifold and that thecorresponding vector field is v . Then one can write down the following PDEs for the gaugepotential A and a scalar Φ (in the context of N = 2 the scalar Φ is identified with theappropriate combination of the scalars in supersymmetric gauge theory), d † (cid:16) ι v F − d (cos ω Φ) (cid:17) = 0 ,d † A = 0 , (150) P + ω (cid:16) F + ι v ( ⋆d Φ) (cid:17) = 0 , where the projector P + ω is defined in (23) and the function ω is invariant under v and hasa specific behavior around the fixed points (see subsection 2.1 for further details). Herefor the sake of clarity we assume that the metric is such that || v || = sin ω . Up to lowerderivative terms these PDEs can be obtained by the dimensional reduction of equations(144). However, they can also be defined intrinsically in 4D terms. In order to calculatethe symbol it is convenient to make a choice of coordinates adapted to the isometry, namely v = sin ω p (by construction sin ω = 0 at the fixed point). Then using the form of P + ω
51n these coordinates we obtain the symbol of (150) which is exactly the same as in theexpression (149). For a more general metric (that is dropping the condition || v || = sin ω )the first and third equations in (150) should be slightly modified by some additional scalingfactors in order to obtain the same symbol (149).The equations (150) can be encoded into the transversely elliptic complex Ω ( M ) d −→ Ω ( M ) ⊕ Ω ( M ) ˜ D −→ P + ω Ω ( M ) ⊕ Ω ( M ) , (151)which we denote as ( E • , ð ) . Here the explicit form of the operator ˜ D can be read off fromthe first and third equations in (150) and it is given by ˜ D = P + ω d P + ω ι v ⋆ dd † ι v d − d † d cos ω ! (152)written in the basis Ω ( M ) ⊕ Ω ( M ) . Remember that here we expand around the zeroconnection for the abelian problem. It is straightforward to generalize the equations (144),(150) and (151) to the non-abelian case and around a non-zero connection. However, thiswill not affect our discussion of the symbols. D Full cohomological complex
In subsection 2.2 we defined the cohomological supersymmetry transformation (27) on theset of fields ( A, Ψ , φ, ϕ, η, χ, H ) . In the full theory we have to include more fields related tothe gauge fixing and combine appropriately the cohomological supersymmetry (27) with theBRST transformations. Let us introduce the ghost c which is an odd zero form in the adjointrepresentation, the anti-ghost ¯ c which is an odd zero form in the adjoint representation anda Lagrangian multiplier b which is an even zero form in the adjoint representation. The fullcohomological transformations are defined as follows δA = i Ψ + d A c ,δϕ = iη + i [ c, ϕ ] ,δχ = H + i { c, χ } ,δc = φ + iι v A + i { c, c } / ,δ ¯ c = b + i { c, ¯ c } , δ Ψ = ι v F + id A φ + i { c, Ψ } ,δη = L Av ϕ − [ φ, ϕ ] + i { c, η } ,δH = i L Av χ − i [ φ, χ ] + i [ c, H ] ,δφ = ι v Ψ + i [ c, φ ] ,δb = i L Av ¯ c − i [ φ, ¯ c ] + i [ c, b ] , (153)where we assume the same conventions as in subsection 2.2 and on all fields we have δ = i L v .Actually we have to choose a background connection around which we expand and in above52ransformations we expand around zero connection. These cohomological transformationsand their linearization have an intricate structure which is discussed in detail in [25].For localization we need to treat zero modes more systematically. Following [1] we in-troduce the zero mode sector ( a , ¯ a , c , ¯ c , b ) . In the transformation (153) we modify thetransformation for the ghost c as follows δc = a + φ + iι v A + i { c, c } , (154)and in addition we define the following transformations for zero modes δ ¯ a = ¯ c ,δb = c ,δa = 0 . δ ¯ c = i [ a , ¯ a ] ,δc = i [ a , b ] , (155)The modified transformations satisfy the following algebra δ = i L v + G a , (156)on all fields. E Supergravity background
Here we collect together explicit expressions for the background supergravity fields necessaryto solve the generalized Killing spinor equations as described in section3.2. These formulaemake use of the spinor bilinears Θ ij , e Θ ij and v µij that are as defined in (41).The SU(2) R connection ( V µ ) ij and two form W µν are given by W µν = is + ˜ s ( ∂ µ v ν − ∂ ν v µ ) − i ( s + ˜ s ) ǫ µνρλ v ρ ∂ λ ( s − ˜ s ) − s + ˜ s ǫ µνρλ v ρ G λ ++ s − ˜ s ( s + ˜ s ) ǫ µνρλ v ρ b λ + 1 s + ˜ s ( v µ b ν − v ν b µ ) , ( V µ ) ij = 4 s + ˜ s (cid:0) ζ ( i ∇ µ ζ j ) + ¯ χ ( i ∇ µ ¯ χ j ) (cid:1) + 4 s + ˜ s (cid:18) iG ν − ∂ ν ( s − ˜ s )( s + ˜ s ) (cid:19) (Θ ij − e Θ ij ) νµ ++ 4 i ( s + ˜ s ) b ν (˜ s Θ ij + s e Θ ij ) νµ , (157)The graviphoton field strength and the scalars S ij take the form F µν = i∂ µ (cid:16) s + ˜ s − Ks ˜ s v ν (cid:17) − i∂ ν (cid:16) s + ˜ s − Ks ˜ s v µ (cid:17) ,S ij = 8 i ( s + ˜ s ) (Θ ij + e Θ ij ) µν ∂ µ v ν − i s + ˜ s − K ( s ˜ s ) (˜ s Θ ij + s e Θ ij ) µν ∂ µ v ν ++ 2 s + ˜ s (cid:16) G µ − s − ˜ ss + ˜ s b µ − i s − ˜ ss ˜ s ∂ µ ( s + ˜ s ) (cid:17) v µij . (158)53inally the scalar combination R/ − N is given by (cid:16) R − N (cid:17) = s − ˜ s ( s + ˜ s ) (cid:3) ( s − ˜ s ) + ∂ [ µ v ν ] ∂ [ µ v ν ] ( s + ˜ s ) − s − ˜ s ( s + ˜ s ) ǫ µνρλ ( ∂ µ v ν )( ∂ ρ v λ )+ − ǫ µνρλ ( s + ˜ s ) v µ ( ∂ ν v ρ ) ∂ λ ( s − ˜ s ) + 2 s ˜ s ( s + ˜ s ) ∂ µ ( s − ˜ s ) ∂ µ ( s − ˜ s )+ − s − ˜ s ( s + ˜ s ) ∂ µ ( s − ˜ s ) ∂ µ ( s + ˜ s ) − is ˜ s ( s + ˜ s ) ∇ µ b µ + 2( s ˜ s ) ( s + ˜ s ) b µ b µ + − i s − ˜ s ( s + ˜ s ) ǫ µνρλ ( ∂ µ v ν ) v ρ b λ + 3 i s − ˜ s ( s + ˜ s ) b µ (˜ s∂ µ s − s∂ µ ˜ s )++ 2 i s + ˜ s ( s + ˜ s ) b µ ∂ µ ( s ˜ s ) − i s − ˜ ss + ˜ s ∇ µ G µ + 4 s + ˜ s ( s + ˜ s ) G µ G µ + 8( v µ G µ ) ( s + ˜ s ) ++ 4 s ˜ s s − ˜ s ( s + ˜ s ) G µ b µ + 4 iǫ µνρλ ( s + ˜ s ) ( ∂ µ v ν ) v ρ G λ + 4 i s − ˜ s ( s + ˜ s ) G µ ∂ µ ( s ˜ s ) . (159) References [1] V. Pestun, “ Localization of gauge theory on a four-sphere and supersymmetric Wilsonloops,”
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