Flag manifold sigma models: spin chains and integrable theories
FFlag manifold sigma models
Spin chains and integrable theories
Ian A๏ฌeck and Kyle Wamer
Department of Physics and Astronomy and Stewart Blusson Quantum Matter Institute,University of British Columbia, Vancouver, B.C., Canada, V6T1Z1 ia๏ฌ[email protected], [email protected]
Dmitri Bykov
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia [email protected], [email protected]
Abstract
This review is dedicated to two-dimensional sigma models with ๏ฌag manifold targetspaces, which are generalizations of the familiar CP ๐ โ and Grassmannian models.They naturally arise in the description of continuum limits of spin chains, and theirphase structure is sensitive to the values of the topological angles, which are determinedby the representations of spins in the chain. Gapless phases can in certain cases beexplained by the presence of discrete โt Hooft anomalies in the continuum theory. Wealso discuss integrable ๏ฌag manifold sigma models, which provide a generalization ofthe theory of integrable models with symmetric target spaces. These models, as wellas their deformations, have an alternative equivalent formulation as bosonic Gross-Neveu models, which proves useful for demonstrating that the deformed geometries aresolutions of the renormalization group (Ricci ๏ฌow) equations, as well as for the analysisof anomalies and for describing potential couplings to fermions. Prepared for
Physics Reports a r X i v : . [ h e p - t h ] J a n ontents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 1. Flag manifolds: geometry and ๏ฌrst applications . . . . . . . . . . . . 71. The geometry of SU( ๐ ) ๏ฌag manifolds . . . . . . . . . . . . . . . . . . . . . . . 71.1. The ๏ฌag manifold as a homogeneous space . . . . . . . . . . . . . . . . . 81.2. Symplectic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3. Kรคhler structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4. Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5. General (non-Kรคhler) metrics and ๐ต -๏ฌelds on the ๏ฌag manifold . . . . . . 162. Flag manifolds and elements of representation theory . . . . . . . . . . . . . . . 172.1. Mechanical particle in a non-Abelian gauge ๏ฌeld . . . . . . . . . . . . . . 182.2. โQuantizationโ of the symplectic form on ๏ฌag manifolds . . . . . . . . . . 202.3. Schwinger-Wigner quantization . . . . . . . . . . . . . . . . . . . . . . . 232.4. Holstein-Primako๏ฌ and Dyson-Maleev representations . . . . . . . . . . . 30 Chapter 2. From spin chains to sigma models . . . . . . . . . . . . . . . . . . . 333. Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1. Classical Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364. Exact Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1. Lieb-Schultz-Mattis-A๏ฌeck Theorem (LSMA) Theorem . . . . . . . . . . 374.2. A๏ฌeck-Kennedy-Lieb-Tasaki (AKLT) Constructions . . . . . . . . . . . . 395. Flavour Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406. Derivation of the continuum theory . . . . . . . . . . . . . . . . . . . . . . . . 426.1. The quantum sphere ๐ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.2. Path integral for the spin chain . . . . . . . . . . . . . . . . . . . . . . . . 446.3. Ferromagnetic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467. The antiferromagnetic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.1. Alternating representations . . . . . . . . . . . . . . . . . . . . . . . . . . 477.2. The large- ๐ limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.3. Symmetric representations and the ๏ฌag manifold as the space of Nรฉel vacua:SU ( ) case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518. The continuum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.1. The expansion around the โvacuumโ con๏ฌguration . . . . . . . . . . . . . 539. Symmetric representations: the general case . . . . . . . . . . . . . . . . . . . . 579.1. The ๏ฌag manifold sigma model from an SU ( ๐ ) spin chain . . . . . . . . . 579.2. Z ๐ symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589.3. Velocity Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . 5910. Generalized Haldane conjectures and โt Hooft anomaly matching . . . . . . . . . 6110.1. SU ( ๐ ) Haldane conjectures . . . . . . . . . . . . . . . . . . . . . . . . . 6210.2. Derivation of the mixed โt Hooft Anomaly . . . . . . . . . . . . . . . . . . 6410.3. The Z ๐ anomaly in a PSU ( ๐ ) background . . . . . . . . . . . . . . . . . . 6710.4. Examples of PSU ( ๐ ) -bundles . . . . . . . . . . . . . . . . . . . . . . . . 6910.5. Relation to the WZNW model . . . . . . . . . . . . . . . . . . . . . . . . 7211. A gas of fractional instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7311.1. Squashing to the ๐๐ -model . . . . . . . . . . . . . . . . . . . . . . . . . 7411.2. Fractional Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7511.3. Mass generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7611.4. Destructive Interference in the Presence of Topological Angles . . . . . . . 7712. More general representations: linear and quadratic dispersion . . . . . . . . . . 7812.1. Spin chain ground states . . . . . . . . . . . . . . . . . . . . . . . . . . . 7912.2. Conditions for linear dispersion and topological angles . . . . . . . . . . . 81 Chapter 3. Integrable ๏ฌag manifold sigma models and beyond . . . . . . . . . . 8413. The models and the zero-curvature representation . . . . . . . . . . . . . . . . . 8513.1. The zero-curvature representation . . . . . . . . . . . . . . . . . . . . . . 8713.2. Complex structures on ๏ฌag manifolds . . . . . . . . . . . . . . . . . . . . 8913.3. Symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9213.4. Dependence on the complex structure: Z ๐ -symmetry of the models . . . . 9414. Relation to 4D Chern-Simons theory . . . . . . . . . . . . . . . . . . . . . . . . 9514.1. The โsemi-holomorphicโ 4D Chern-Simons theory . . . . . . . . . . . . . 9714.2. The gauged linear sigma model and the ๐ฝ๐พ -systems . . . . . . . . . . . . . 9914.3. Relation to the quiver formulation . . . . . . . . . . . . . . . . . . . . . . 10215. Relation to the principal chiral model . . . . . . . . . . . . . . . . . . . . . . . 10415.1. Nilpotent orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10416. Sigma models as generalized Gross-Neveu models . . . . . . . . . . . . . . . . 10716.1. The bosonic chiral Gross-Neveu model . . . . . . . . . . . . . . . . . . . 10716.2. The deformed Gross-Neveu models . . . . . . . . . . . . . . . . . . . . . 10816.3. The ๐ฝ -function and the Ricci ๏ฌow . . . . . . . . . . . . . . . . . . . . . . 11016.4. Sigma models with polynomial interactions . . . . . . . . . . . . . . . . . 11416.5. Integrable models related to quiver varieties . . . . . . . . . . . . . . . . . 116 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119A. Kรคhler potential from the quiver quotient formulation . . . . . . . . . . . . . . . 136B. Symplectic forms on coadjoint orbits . . . . . . . . . . . . . . . . . . . . . . . 138C. Coherent states as polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 138D. Integrability of the complex structure . . . . . . . . . . . . . . . . . . . . . . . 140E. Proving the Z ๐ -โsymmetryโ of integrable models . . . . . . . . . . . . . . . . . 142F. Models with Z ๐ -graded target spaces . . . . . . . . . . . . . . . . . . . . . . . 1432 ntroduction Haldaneโs conjecture is the prediction that antiferomagnetic spin chains with integer spinhave a gap above the ground state, while those with half-odd integer spin are gapless [130].The distinction between these two cases can be seen by taking a large spin limit, in which casethe quantum ๏ฌuctuations of the antiferromagnet are governed by the O(3) nonlinear sigmamodel, with topological angle ๐ = ๐๐ . It was surprising to condensed matter physiciststhat spin chains were gapped for integer spin and surprising to high energy theorists that theO(3) non-linear sigma model was massless for ๐ = ๐ . Recently, this paradigm of mappingspin chains to relativistic quantum ๏ฌeld theories has been generalized to SU( ๐ ) chains invarious representations [69, 73, 161, 236, 237, 234]. For chains that have a rank- ๐ symmetricrepresentation at each site, the corresponding ๏ฌeld theory is a sigma model with target spaceSU ( ๐ )/[ U ( )] ๐ โ . This space is an example of a ๏ฌag manifold, which generalizes the familiarnotions of complex projective space and Grassmannian manifolds, and in this case may beparametrized in terms of ๐ mutually orthonormal ๏ฌelds ๐ง ๐ด โ C ๐ . To each of these ๏ฌeldsthere is an associated topological angle ๐ ๐ด = ๐ ๐ ๐ด / ๐ , which extends Haldaneโs originalresult, since ๐ = ๐ in SU(2). Based on this sigma model formulation, a generalization ofHaldaneโs conjecture was discovered for these SU( ๐ ) chains: When ๐ is coprime with ๐ ,gapless excitations will be present above the ground state; for all other values of ๐ , a ๏ฌniteenergy gap will occur, with a ground state degeneracy equal to ๐ / gcd ( ๐, ๐ ) .The arguments leading to this SU( ๐ ) version of Haldaneโs conjecture draw from manyareas of mathematical physics. This re๏ฌects the fact that the underlying ๏ฌag manifoldSU ( ๐ )/[ U ( )] ๐ โ has a rich geometric structure. Indeed, ๏ฌag manifolds in their own right area fascinating subject, and for this reason we commence this review in Chapter 1 by discussinggeneric ๏ฌag manifolds at great length. In particular, we will explain their symplectic, Kรคhler,and Riemannian geometries, as well as their cohomology, the latter being the key object forthe description of topological terms. In addition to providing the reader with an overview ofthe general theory of ๏ฌag manifolds, this chapter will allow us to introduce the necessarilytechnology to properly explain the mathematical underpinnings of deriving a ๏ฌag manifoldsigma model from an SU( ๐ ) spin chain. Along these lines, we also review various quantizationschemes of ๏ฌag manifolds, and how a coherent state path integral is constructed in this context.In Chapter 2, we turn to SU( ๐ ) spin chains. In the interest of being self-contained, webegin by introducing the SU( ๐ ) Heisenberg Hamiltonian, and listing various exact results thatare known for these models. Then, armed with the mathematical formalism of Chapter 1,we review in great detail how the SU ( ๐ )/[ U ( )] ๐ โ ๏ฌag manifold arises as a low-energy3igma model description of the SU( ๐ ) chain. In particular, we show how the topologicalangle ๐ ๐ด arises as the coe๏ฌcient of a Fubini-Study two-form, pulled back from CP ๐ โ tothe ๏ฌag manifold. We then proceed to discuss a technical issue that is related to the absenceof Lorentz invariance when starting with a generic SU( ๐ ) chain Hamiltonian. Having donethis, we may then ๏ฌnally review the constituent arguments that make up the SU( ๐ ) Haldaneconjecture. In particular, we discuss the notion ofโt Hooft anomaly matching, which is relatedto the inability of gauging the physical PSU( ๐ ) symmetry of the chain while maintaining adiscrete Z ๐ translation symmetry [221, 190]. We also discuss topological excitations in thesigma model, which have fractional charge and give rise to a mass generating mechanismexcept for the special values of ๐ with gcd ( ๐, ๐ ) = ๐ ) that may also be mapped to the same ๏ฌag manifold,SU ( ๐ )/[ U ( )] ๐ โ .One might expect that this would be a natural point to conclude this review: We havecovered the general properties of ๏ฌag manifolds, and explained in great detail the relationshipbetween said manifolds and SU( ๐ ) spin chains, allowing for a generalization of Haldaneโsfamed conjecture. However, this work on SU( ๐ ) chains has very recently initiated an entirelynew research program, related to integrable ๏ฌag manifold sigma models. This is the subjectof Chapter 3.The history of integrable models with an โin๏ฌnite number of degrees of freedomโ israther long. It has spanned most of the second part of the 20th century, starting with thestudy of the Korteweg-de-Vries equation [118], and continues to evolve up to the presentday. Already by the end of the 1970s the classical theory saw remarkable developmentsbased on algebro-geometric methods and the tools of ๏ฌnite-gap integration, as summarizedin the book [188]. On the other hand, the study of integrable structures of relativistic sigmamodels only started around the same time [199, 262], and the mathematical results on theclassi๏ฌcation of classical solutions were obtained substantially later [228, 132]. See [112,128] for a review of these ๏ฌndings.Whereas the classical integrability theory quickly came to be part of mathematics, thequantum theory was developed by rather di๏ฌerent methods by physicists, starting with thefamous conjecture for the ๐ -matrix in the ๐ ๐ โ sigma model [264]. The development ofthis theory then went in two directions: towards the calculation of the spectrum in ๏ฌnitevolume, using the so-called thermodynamic Bethe ansatz [259, 165, 263, 97, 39], andtowards investigating the full range of theories, to which such methods would be applicable.Within the latter research program remarkable results were achieved for models with SU ( ๐ ) symmetry, most importantly for the CP ๐ โ -model [88, 90, 91, 246, 44]. First of all, it wasfound that quantum-mechanically integrability in this model is destroyed by anomalies ofa very peculiar kind. Technically these are anomalies in a certain non-local charge ๏ฌrstconstructed by Lรผscher [168], which, when unobstructed, may be shown to generate theYangian that underpins the integrability of these models [45, 46] (see [167] for a review).This would as well lead to anomalies in the โhigherโ local charges, as anticipated earlierin [201, 121] based on simple dimensional analysis. It was also found that, by addingfermions to the pure bosonic models in various ways, one can cancel the anomalies, although4t the conceptual level the mechanism behind these cancellations remained unclear.Another major stumbling block was that the theory of integrable sigma models โ bothclassical and quantum โ seemed to require that the target space is a symmetric space, whichsubstantially narrows the space of admissible models, even within the class of homogeneousspaces. In recent years the latter issue has been resolved, at least in the classical theory,since it was shown [70, 63, 65, 75, 76] that there exist canonical models with ๏ฌag manifoldtarget spaces (which in general are not symmetric) that admit a Lax representation and sharethe virtues of the models with symmetric target spaces. This also allows one to make aconnection to the models that emerge from the spin chains discussed in Chapter 2. Althoughthe integrable models are not exactly identical to the ones that arise from the spin chains,they nevertheless share many common features with the latter. Even more recently thepaper [87] appeared, which provides a broad and uni๏ฌed framework for constructing classicalintegrable models starting from a rather exotic โfour-dimensional semi-holomorphic Chern-Simons theoryโ. In particular, the ๏ฌag manifold models may as well be obtained from thatconstruction.Quite unexpectedly, it turned out that the approach of [87], combined with the gaugedlinear sigma model approach developed earlier in [75, 76], allows one to prove the equivalenceof a wide class of sigma models with complex homogeneous target spaces (as well as theirdeformations) to bosonic and mixed bosonic/fermionic Gross-Neveu models [71]. This novelformulation provides insights into many facets of sigma model theory. For example, onecan obtain a new way of constructing supersymmetric sigma models [72], and the obscureintegrability anomalies are now conjectured to be related to the familiar chiral anomalies,which are otherwise not visible in the old approach. The Gross-Neveu formulation providesanother window into the quantum domain, related to the analysis of the ๐ฝ -function of thetheory. This is especially vivid in the deformed case. Since the deformation preserves onlya small fraction of the original symmetries of the model, the explicit calculations in thegeometric framework would be extremely cumbersome, if at all doable. In contrast, theGross-Neveu formulation results in spectacular simpli๏ฌcations, which ultimately allow oneto solve the generalized Ricci ๏ฌow equations for the deformed geometries in a very wideclass of sigma models. This is particularly important, since in the study of models with targetspaces ๐ and ๐ , the one-loop renormalizability of the deformed models was linked to theirintegrability [109, 108] (see also the more recent discussion in [230, 134] and referencestherein). We mention in passing that the subject of integrable deformations is in itself veryvast, and for more on this we refer the reader to the well-known papers [155, 154, 93, 214,137].It is unlikely that all of these exciting inter-relations are purely a coincidence. Instead,one can be optimistic that from this point the construction of the proper quantum theory ofsuch models is within reach. Additionally, the inclusion of the non-trivial ๐ -angles wouldallow one to study the phase diagram and draw parallels to the massless/massive phases ofspin chains, which would then close the logical circle that we are aiming to re๏ฌect in thisreview article. 5 otation Before we begin, we comment on the various notational choices that we have made in thisreview. โฆ A generic ๏ฌag manifold can be embedded into a copy of ๐ Grassmanians (this willbe explained in detail in Chapter 1). We use upper case Roman letter to index thesecopies. In the case ๐ = ๐ , each Grassmanian is CP ๐ โ , which we parametrize with ๐ง ๐ด โ C ๐ . When ๐ < ๐ , and multiple ๐ -component ๏ฌelds are required to parametrizethe Grassmanians, we use lower-case roman letters, i.e. ๐ง ( ๐ ) ๐ด . โฆ The ๐ components of ๐ง ๐ด are indexed using lower case greek letters: ๐ง ๐ผ๐ด โฆ Often, we will normalize the ๐ -component ๏ฌelds to satisfy | ๐ง ๐ด | =
1. In this case, wewrite ๐ข ๐ด instead of ๐ง ๐ด . โฆ We use the labels ๐, ๐, ๐... for discrete time coordinates, and the labels ๐, ๐ , ๐, ... fordiscrete spatial coordinates. For a ๏ฌeld ๐ง that is a function of ๐ and ๐ , we write ๐ง = ๐ง ( ๐, ๐ ) . When the continuum limit is taken, we write ๐ง = ๐ง ( ๐, ๐ฅ ) . โฆ The vector complex conjugate to ๐ง is written ๐ง . We write inner products in C ๐ accordingto ๐ค โฆ ๐ง = ๐ โ๏ธ ๐ผ = ๐ค ๐ผ ๐ง ๐ผ . (0.1)The norm of a vector is denoted by | ๐ง | , so that | ๐ง | = ๐ง โฆ ๐ง . โฆ From time to time we will be using the notation Hom ( C ๐ , C ๐ ) (linear maps from C ๐ to C ๐ ) for the space of ๐ ร ๐ -matrices. This notation makes it clear that ๐ is the numberof columns, and ๐ the number of rows in a matrix. Accordingly End ( C ๐ ) is the spaceof square matrices of size ๐ . 6 hapter 1. Flag manifolds: geometry and ๏ฌrst applications In the ๏ฌrst chapter of this review we recall the main facts about the rather rich geometricstructures on ๏ฌag manifolds (mostly symplectic structures and metrics), and we explainhow ๏ฌag manifolds naturally arise in representation theory. Due to this tight relation, ๏ฌagmanifolds inevitably appear in the theory of spin chains, to which the next chapter is dedicated.As a bridge between abstract mathematical structures and applications to representations ofspin operators, we describe the example of a spin carried by a mechanical particle chargedw.r.t. a non-Abelian gauge group: in this case the motion of the spin is again described by a๏ฌag manifold. ๐ ) ๏ฌag manifolds Flag manifolds are natural generalizations of both projective spaces and Grassmanians, sowe start by recalling these more familiar entities ๏ฌrst.The complex projective space CP ๐ โ is de๏ฌned as the space of ๐ -tuples of complexnumbers, which are not all zero, de๏ฌned up to multiplication by an overall factor, i.e. ( ๐ง , ยท ยท ยท , ๐ง ๐ ) โผ ฮป ( ๐ง , ยท ยท ยท , ๐ง ๐ ) . Another interpretation, which allows for generalizations moreeasily, is that CP ๐ โ is the space of lines in C ๐ , passing through the origin. Clearly, the lineis de๏ฌned by a non-zero vector ( ๐ง , ยท ยท ยท , ๐ง ๐ ) , and two vectors that di๏ฌer by an overall constantmultiple de๏ฌne the same line.This construction can be generalized by considering ๐ -dimensional planes in C ๐ , passingthrough the origin. This leads to the notion of a Grassmannian ๐บ๐ ๐,๐ , which may be de๏ฌnedas the space of ๐ ร ๐ -matrices ๐ of rank ๐ , taken up to multiplication by matrices fromGL ( ๐, C ) . The meaning of ๐ is that it comprises ๐ vectors spanning a given ๐ -dimensionalplane in C ๐ , and multiplication by GL ( ๐, C ) amounts to a change of basis and does not a๏ฌectthe plane itself. Setting ๐ = CP ๐ โ .We should point out that the equivalence relations just mentioned โ the quotients w.r.t. C โ or GL ( ๐, C ) โ are of course well-known in physics as โgauge redundanciesโ. In fact,more than once in our narrative we will encounter the formulation of the corresponding ๏ฌeldtheories as systems with gauge ๏ฌelds (the so-called โgauged linear sigma modelsโ). From themathematical perspective, choosing a gauge amounts to picking coordinates on the respectivemanifolds. The unrestricted coordinates ( ๐ง , ยท ยท ยท , ๐ง ๐ ) mentioned above, which are subject tothe equivalence relation, are known as the homogeneous coordinates on the projective space.If, say, ๐ง โ
0, then by a C โ scaling โ a gauge transformation โ we may set ๐ง =
1. This ๏ฌxes7he gauge freedom completely at the expense of e๏ฌectively excluding from considerationthe part of the space where ๐ง =
0. The corresponding coordinates ( , ๐ง , ยท ยท ยท , ๐ง ๐ ) are thenknown as the inhomogeneous coordinates . For example, on a sphere ๐ (cid:39) CP there is asingle complex inhomogeneous coordinate, which is the complex coordinate on a plane ofstereographic projection (the excluded region in this case being the point from which theprojection is performed). Finally, another option is to ๏ฌx the gauge redundancy partiallyby normalizing the coordinates ๐ (cid:205) ๐ผ = | ๐ง ๐ผ | =
1, so that the coordinates are restricted to asphere ๐ ๐ โ . This leaves the freedom of multiplying all ๐ง ๐ผ โs by the same phase, so theremaining gauge group is U ( ) . This formulation is nothing but the celebrated Hopf ๏ฌbration ๐ ๐ โ โ CP ๐ โ , with ๏ฌber U ( ) . Its advantage is that the global symmetry group SU ( ๐ ) isexplicitly maintained. Similar choices of homogeneous, inhomogeneous and other types ofcoordinates may be performed for Grassmannians and ๏ฌag manifolds as well. Both of the above examples may be concisely formulated as โspaces of subspacesโ 0 โ ๐ฟ โ C ๐ ,where ๐ฟ is a linear subspace of a given dimension. This naturally leads to the notion of a๏ฌag. A ๏ฌag in C ๐ is the sequence of nested subspaces0 โ ๐ฟ โ . . . โ ๐ฟ ๐ โ โ ๐ฟ ๐ = C ๐ (Flag) (1.1)of given dimensions dim ๐ฟ ๐ด = ๐ ๐ด . Accordingly, the ๏ฌag manifold in C ๐ may be de๏ฌned asthe manifold of such nested linear complex subspaces : F ( ๐ , . . . , ๐ ๐ ) = { โ ๐ฟ โ . . . โ ๐ฟ ๐ โ โ ๐ฟ ๐ = C ๐ } . (1.2) - Figure 1: The parabolic subgroup, stabilizing a ๏ฌag.The next important fact is that the projective space, Grassmannians and ๏ฌag manifolds areall examples of homogeneous spaces. Moreover, there are two ways to express these manifoldsas homogeneous spaces: either w.r.t. the complex symmetry group GL ( ๐, C ) , or w.r.t. itsunitary subgroup U( ๐ ) โ GL ( ๐, C ) . Let us ๏ฌrst start with the complex parametrization.The group GL( ๐, C ) acts transitively on the space of ๏ฌags of a given type: given two ๏ฌags, Reviews of the mathematical properties of ๏ฌag manifolds include [19, 28]. ๐ ๐ โ into each other, then the next-to-largest subspaces, etc. The stabilizer of any given ๏ฌag is a so-called parabolic subgroup(โa staircaseโ), consisting of matrices, depicted in Fig. 1. The reason for the non-diagonalstructure is that, given a basis of ๐ฟ ๐ด and a (larger) basis of ๐ฟ ๐ด + ๐ต , adding vectors of the ๏ฌrstbasis to vectors of the second one produces a new basis of the same sequence of spaces ๐ฟ ๐ด โ ๐ฟ ๐ด + ๐ต . Therefore we can view the ๏ฌag manifold as a homogeneous space F ( ๐ , . . . , ๐ ๐ ) = GL ( ๐, C )/H (1.3)of complex dimension dim C F ( ๐ , . . . , ๐ ๐ ) = ๐ โ ๐ (cid:205) ๐ด = ๐ ๐ด ( ๐ ๐ด โ ๐ ๐ด โ ) .As mentioned earlier, there is a second โ unitary โ parametrization of the ๏ฌag manifold.To obtain it, one picks a metric in the ambient space C ๐ and orthogonalizes the subspaces ofthe ๏ฌag. For example, we split ๐ฟ = ๐ฟ โ ( ๐ฟ ) โฅ , ๐ฟ = ๐ฟ โ ( ๐ฟ ) โฅ and so on. Altogether thissplits C ๐ into ๐ mutually orthogonal subspaces of dimensions ๐ ๐ด = ๐ ๐ด โ ๐ ๐ด โ , ๐ด = , . . . , ๐ (where we set ๐ = F ๐ ,...,๐ ๐ = U ( ๐ ) U ( ๐ ) ร . . . ร U ( ๐ ๐ ) , ๐ โ๏ธ ๐ด = ๐ ๐ด = ๐ . (1.4)Using the dimensions of the groups in the numerator and denominator, one easily computes thereal dimension of this space, which, as expected, turns out to be twice the complex dimensionof (1.3), computed earlier. Note that sometimes we will denote by F ๐ the complete ๏ฌagmanifold, i.e. the manifold (1.4), where all ๐ ๐ด = J on the ๏ฌag manifold. Since it does play a role for the integrable models introduced inChapter 3, this is explained in detail in section 13.2. For most of the exposition in the ๏ฌrsttwo chapters, in order not to dwell on this subtle issue, we will simply assume that we haveboth de๏ฌnitions at hand, and we may use any of them at our will.Throughout this paper we will mostly be interested in relativistic sigma models with ๏ฌagmanifold target spaces. Such models feature two main ingredients: the metric G and theskew-symmetric two-form ฮฉ (which is also called the ๐ต -๏ฌeld, Kalb-Ramond form, etc.) onthe target space. Particularly important are the so-called topological terms , which correspondto closed two-forms, i.e. to the case ๐ ฮฉ =
0. These do not a๏ฌect the classical equationsof motion, but might substantially alter the quantum theory. As we shall see in Chapter 2,it is precisely such topological terms that are responsible for the presence or absence of amass gap in the spectrum of the models, and of the related spin chains as well. It is thereforevery important to understand in detail, how such terms may be written in the case of ๏ฌag9anifolds. To this end, note that the condition ๐ ฮฉ =
0, together with an additional non-degeneracy assumption ฮฉ dim C F โ ฮฉ โ symplectic form.If, in addition, ฮฉ is positive in a certain sense, then ฮฉ is called a Kรคhler form. โPositivityโmeans that the corresponding symmetric tensor G = โ ฮฉ โฆ J , obtained by contracting ฮฉ with a complex structure J , is positive-de๏ฌnite and therefore a Riemannian metric on the๏ฌag manifold . To summarize we have the following embeddings:Kรคhler forms โ Symplectic forms โ Closed two-forms (topological terms)(1.5)Although these three sets do not coincide, they may all be described in a uniform manner. Inparticular, for a ๏ฌag manifold of type (1.4) they all have real dimension ๐ โ
1, and restrictingto non-degenerate forms, or positive forms, amounts to simple relations among the ๐ โ We start by describing symplectic forms on F ๐ ,...,๐ ๐ , i.e. non-degenerate, closed 2-forms ๐ ,with ๐๐ =
0. Since we will mainly be interested in SU( ๐ )-invariant models in what follows,we will accordingly restrict ourselves to SU( ๐ )-invariant symplectic forms. The main toolthat we will use is the theorem of Kirillov-Kostant that coadjoint orbits of a Lie group ๐บ admit natural symplectic forms (for a review see [152]). In our applications the Lie algebra of ๐บ =SU( ๐ ) admits a Killing metric, which may be used to relate coadjoint orbits with adjointorbits, and so we will always be talking of the latter. As the name suggests, these adjoint orbitsare de๏ฌned as follows: one picks a diagonal element ๐ = Diag ( ๐ ๐ , . . . , ๐ ๐ ๐ ๐ ) โ ๐ฐ๐ฒ ( ๐ ) ,where the ๐ ๐ด โs are distinct and Tr ( ๐ ) =
0. In this case the ๏ฌag manifold is the orbit F ๐ ,...,๐ ๐ = { ๐ ๐ ๐ โ , ๐ โ SU ( ๐ )} , (1.6)since there is an obvious gauge invariance ๐ โ ๐ ยท โ , where โ โ ๐ป = S ( U ( ๐ ) ร ยท ยท ยท ร U ( ๐ ๐ )) ,so that the orbit is really the quotient ๐บ๐ป . Introducing the Maurer-Cartan current ๐ = โ ๐ โ ๐๐, ๐ โ U ( ๐ ) , (1.7)one may write the Kirillov-Kostant symplectic form on the orbit (1.6) as ฮฉ = Tr ( ๐ ๐ โง ๐ ) . (1.8)One can check that its non-degeneracy is equivalent to the condition that all ๐ ๐ด โs are distinct.Due to the condition Tr ( ๐ง ) = ๐ โ Technically for the tensor G so de๏ฌned to be symmetric one also needs that ฮฉ is of type ( , ) , i.e. aHermitian form. This always holds in our applications, and we will not elaborate on this aspect further. ๐ฐ๐ฒ ( ๐ ) . Moreover, this embedding may be identi๏ฌedwith the image of the moment map ๐ = ๐ ๐ ๐ โ . (1.9)Let us recall what a moment map is, since it will be ubiquitous in the foregoing exposition.Whenever one has a symplectic manifold ๐ฝ with an action of a Lie group ๐บ on it thatpreserves the symplectic form, one can construct Hamiltonian functions for the action of thisgroup. The action of the group on ๐ฝ is generated by the vector ๏ฌelds ๐ฃ ๐ , ๐ = ยท ยท ยท dim ๐บ ,whose commutators satisfy the Lie algebra relations of ๐ค : [ ๐ฃ ๐ , ๐ฃ ๐ ] = ๐ ๐๐๐ ๐ฃ ๐ . To each vector๏ฌeld ๐ฃ ๐ one can put in correspondence a Hamiltonian function โ ๐ . It turns out that all ofthese Hamiltonian functions may be collected in a single matrix-valued object ๐ โ ๐ค , calledthe moment map , in such a way that โ ๐ = Tr ( ๐๐ ๐ ) . Here ๐ ๐ is the ๐ th generator of ๐ค . Onecan check from the de๏ฌnitions that the moment map de๏ฌned in (1.9) leads to the vector ๏ฌeldsgenerating the action of SU ( ๐ ) and preserving the symplectic form (1.8), cf. [73].As a simple exercise, let us write out explicitly the moment map for the Grassmannian ๐บ๐ ๐ ,๐ . To this end we set ๐ = Diag ( , ยท ยท ยท , ๐ , , ยท ยท ยท , ๐ โ ๐ ) โ ๐ ๐ ๐ , which gives ๐ ๐ = ๐ โ๏ธ ๐ = ๐ข ( ๐ ) โ ๐ข ( ๐ ) โ ๐ ๐ ๐ , (1.10)where by ๐ข ( ๐ ) we have denoted the (orthonormal) column vectors of the group element ๐ .The moment map (1.9) is the classical analogue of the SU ( ๐ ) -spin and therefore will playan important role in our treatment of spin chains in Chapter 2. Following the diagram in (1.5), we now turn to the discussion of Kรคhler forms. As explainedearlier, these involve the complex structure J in their de๏ฌnition, so we will shift to thecomplex de๏ฌnition of ๏ฌag manifolds (1.3). The Kรคhler structures can be characterizedgeometrically in at least two equivalent ways: โฆ As parameters of a linear combination of the so-called quasipotentials [31, 32] thatappear in the physics literature in [36]. โฆ As Fayet-Iliopoulos parameters related to the gauged linear sigma model representationsfor ๏ฌag manifolds [186, 96].These approaches are discussed below in sections 1.3.1, 1.3.2 respectively.11 .3.1 Explicit Kรคhler metrics on ๏ฌag manifolds
Recall that Kรคhler metrics and Kรคhler forms are in one-to-one correspondence, and are relatedby contraction with a complex structure J . It is easiest to de๏ฌne a Kรคhler metric through theso-called Kรคhler potential K , which in plain terms is a function of the complex coordinates { ๐ค ๐ , ๐ค ๐ } , such that the line element takes the form ๐๐ = (cid:205) ๐ K ๐๐ค ๐ ๐๐ค ๐ ๐๐ค ๐ ๐๐ค ๐ . A very directway of constructing a Kรคhler potential of the most general SU ( ๐ ) -invariant Kรคhler metric onthe ๏ฌag manifold (1.4) is as follows: consider the matrix ๐ = ( ๐ค , .., ๐ค ๐ ) โ GL ( ๐ ; C ) , (1.11)where each ๐ค ๐ is a column vector. We also de๏ฌne an ๐ ร ๐ ๐ด -matrix ๐ ๐ด of rank ๐ ๐ด bytruncating the matrix ๐ to the ๏ฌrst ๐ ๐ด columns: ๐ ๐ด = ( ๐ค , ..., ๐ค ๐ ๐ด ) , where ๐ ๐ด = ๐ด โ๏ธ ๐ = ๐ ๐ . (1.12)The columns of ๐ ๐ด span the vector space ๐ฟ ๐ด in the ๏ฌag (1.1). Next we introduce the function ๐ก ๐ด = det (cid:16) ๐ โ ๐ด ๐ ๐ด (cid:17) . (1.13)One can check that log ( ๐ก ๐ด ) , called the quasipotential, is the Kรคhler potential for the ๐ -normalized canonical metric on the Grassmannian ๐บ๐ ๐ ๐ด ,๐ . The potential of an arbitrarySU ( ๐ ) invariant Kรคhler metric on the ๏ฌag manifold [31, 32] may then be written as K F = ๐ โ โ๏ธ ๐ด = ๐พ ๐ด log ( ๐ก ๐ด ) , ๐พ ๐ด > . (1.14)For a detailed discussion of the geometric properties of these metrics (including the specialcase of Kรคhler-Einstein metrics) cf. [18, 4].As a simplest application of formula (1.14) let us consider the case when the ๏ฌag manifoldis the complex projective space CP ๐ โ . In this case ๐ is a column vector, and we label itscomponents ๐ง , . . . , ๐ง ๐ . The Kรคhler potential is therefore (we set ๐พ = K CP ๐ โ = ( ๐ง โฆ ๐ง ) . (1.15)The resulting Kรคhler form is the familiar Fubini-Study form: ฮฉ ๐น๐ = i ๐ง โฆ ๐ง (cid:18) ๐๐ง ๐ผ โง ๐๐ง ๐ผ โ ๐ง ๐ผ ๐๐ง ๐ผ โง ๐ง ๐ฝ ๐๐ง ๐ฝ ๐ง โฆ ๐ง (cid:19) , โซ CP ฮฉ ๐น๐ = ๐ . (1.16)In the second formula the integral is taken over a CP โ CP ๐ โ de๏ฌned by the equations ๐ง ๐ผ = , ๐ผ >
2. We will frequently use the 2 ๐ -normalized Fubini-Study form later on in ournarrative. This is the same normalization as that of the Fubini-Study metric on CP ๐ โ , i.e. the volume of a holomorphic2-sphere generating ๐ป ( ๐บ๐ ๐ ๐ด ,๐ , Z ) is ๐ . ๐ ๐ โ ๐ ๐ โ C ๐ ๐ฟ ๐ฟ ๐ฟ ๐ โ ๐ฟ ๐ โ ยท ยท ยท (1.17)Figure 2: The quiver describing the ๏ฌag manifold as a Kรคhler quotient. An attentive reader might have noticed that at the beginning of this chapter we introducedthe projective space as the quotient by the group of non-zero complex numbers C โ , and theGrassmannians as a quotient by GL ( ๐, C ) , but no similar presentation was provided for thecase of ๏ฌag manifolds. Indeed, the quotient by a subgroup H of the form found in Figure 1 isnot the same thing, as can be readily seen in the example of CP ๐ โ , where the correspondinggroup is certainly di๏ฌerent from C โ . A suitable formulation for ๏ฌag manifolds, however, doesexist, and may be formulated in terms of a so-called โquiverโ. The quiver in question has thefollowing form:Here ๐ฟ ๐ด are the vector spaces de๏ฌning the ๏ฌag (1.2), so that dim ๐ฟ ๐ด = ๐ ๐ด , and eacharrow corresponds to the space of matrices Hom ( ๐ฟ ๐ด , ๐ฟ ๐ด + ) , with ๐ ๐ด being the (linear)complex coordinates in this space. At each circular node there is an action of the gauge groupGL ( ๐ ๐ด , C ) . The main idea is that the ๏ฌag manifold may be identi๏ฌed with the quotient ofthe space of such matrices (with the requirement that each is of maximal rank) by the gaugegroup acting at the node. The projective space and the Grassmannians correspond in thislanguage to a quiver with just two nodes, corresponding to the ๏ฌag ๐ฟ โ C ๐ . To understandwhy this can be true, consider the case of complete ๏ฌags in C , i.e. the manifold U ( ) U ( ) . Oneway to parametrize this manifold is as follows. Let ๐, ๐ โ C be two linearly independentvectors.These vectors de๏ฌne a plane ๐ฟ = Span ( ๐, ๐ ) (cid:39) C โ C . (1.18)A line ๐ฟ โ ๐ฟ may be de๏ฌned as ๐ฟ = Span ( ๐ข ๐ + ๐ข ๐ ) โ ๐ฟ โ C (1.19)with ( ๐ข , ๐ข ) a ๏ฌxed non-zero two-vector.Clearly, ( ๐ข , ๐ข ) โ C , ๐ โ C , ๐ โ C uniquely de๏ฌne a given ๏ฌag ๐ฟ โ ๐ฟ โ C ,however the map is not one-to-one. Indeed, the rotated set (cid:18) (cid:101) ๐ข (cid:101) ๐ข (cid:19) = ฮป ๐ โ โฆ (cid:18) ๐ข ๐ข (cid:19) , (cid:16) (cid:101) ๐ (cid:101) ๐ (cid:17) = (cid:0) ๐ ๐ (cid:1) โฆ ๐ ngle Figure 3: Parametrization of the ๏ฌag manifold introduced in (1.18)-(1.19).with ๐ โ GL ( , C ) and ฮป โ C โ de๏ฌnes the same ๏ฌag. Therefore one has the gauge group G = C โ ร GL ( , C ) acting on the โmatter ๏ฌeldsโ constituting the linear space ๐ = ( C ) ๐ข โ( C โ C ) ๐,๐ . To make a connection to the quiver (1.17), we identify ๐ = (cid:18) ๐ข ๐ข (cid:19) and ๐ = (cid:0) ๐ ๐ (cid:1) . This is the desired generalization of the well-known presentation for theprojective space and Grassmannians that we used as our starting point at the beginning of thechapter.The quiver formulation may as well be used to describe Kรคhler metrics on the ๏ฌagmanifold by performing a symplectic reduction. This entails associating to each gauge nodeof the quiver a real constant (in the supersymmetric setup [96] these constants are calledFayet-Iliopoulos parameters), so the resulting metric depends on ๐ โ ๐พ ๐ด used in (1.14). The readerwill ๏ฌnd the details in Appendix A. It was already emphasized in the diagram (1.5) that Kรคhler and symplectic structures provideexamples of closed two-forms. Such forms are elements of the second cohomology group ๐ป (F ๐ , ยทยทยท ,๐ ๐ , R ) , which is the cohomology group most relevant for sigma model applications,since its elements are the topological terms in the action. In this section we describe anotherway of expressing the elements of this cohomology group, which is a very convenient modelto be used in the applications discussed in subsequent chapters. Let us start by writing outthe answer for the second cohomology group with integer coe๏ฌcients: ๐ป (F ๐ , ยทยทยท ,๐ ๐ , Z ) = Z ๐ โ . (1.20)One can obtain a convenient model for this cohomology group if one notes the existence ofan embedding F ๐ ,...,๐ ๐ โฉ โ ๐บ๐ ๐ ,๐ ร ยท ยท ยท ร ๐บ๐ ๐ ๐ ,๐ (1.21)of the ๏ฌag manifold into a product of Grassmannians. Indeed, a point in a ๏ฌag manifold is acollection of pairwise orthogonal planes of dimensions ๐ , ยท ยท ยท , ๐ ๐ (see section 1.1), each ofwhich is a point in the corresponding Grassmannian.To proceed, we will need the de๏ฌnition of a Lagrangian submanifold M โ N in asymplectic manifold (N , ๐ ) , which we now recall. M is Lagrangian if ๐ (cid:12)(cid:12) M = M = dim N . Let us now consider N = ๐บ๐ ๐ ,๐ ร ยท ยท ยท ร ๐บ๐ ๐ ๐ ,๐ as a symplectic manifoldwith a product symplectic form ๐ = ๐ (cid:205) ๐ด = ๐ ๐ด , where all ๐ ๐ด are normalized in the same way.In this case, as we shall now prove, M = F ๐ ,...,๐ ๐ in (1.21) is a Lagrangian submanifold, i.e. ๐ (cid:12)(cid:12) F ๐ ,...,๐๐ = . (1.22)Identifying ฮฉ ๐ด = ๐ ๐ด (cid:12)(cid:12) F ๐ ,...,๐๐ and taking into account (1.22), we obtain the relation ๐ โ๏ธ ๐ด = ฮฉ ๐ด = . (1.23)The cohomology group ๐ป (F ๐ , ยทยทยท ,๐ ๐ , Z ) is then described as the quotient ๐ป (F ๐ , ยทยทยท ,๐ ๐ , Z ) = Z [ ฮฉ , ยท ยท ยท , ฮฉ ๐ ] (cid:44) (cid:32) ๐ โ๏ธ ๐ด = ฮฉ ๐ด (cid:33) (1.24)To prove that the ๏ฌag manifold is a Lagrangian submanifold in the product of Grassman-nians, ๏ฌrst let us perform a dimensionality check. Using dim R (F ๐ , ยทยทยท ,๐ ๐ ) = ๐ โ ๐ (cid:205) ๐ด = ๐ ๐ด anddim R ( ๐บ๐ ๐ ,๐ ) = ( ๐ ยท ๐ โ ๐ ) , we obtaindim R (cid:32) ๐ (cid:214) ๐ด = ๐บ๐ ๐ ๐ด ,๐ (cid:33) = ๐ โ๏ธ ๐ด = ( ๐ ๐ด ยท ๐ โ ๐ ๐ด ) = (cid:32) ๐ โ ๐ โ๏ธ ๐ด = ๐ ๐ด (cid:33) = ยท dim R F ๐ , ยทยทยท ,๐ ๐ (1.25)We see that the dimensions match correctly. For the rest we use the following fact (whichis easy to prove starting from the de๏ฌnition): if ๐ is the moment map for the action of agroup ๐บ , the restriction of a symplectic form to a ๐บ -orbit in ๐ โ ( ) vanishes. We will nowconstruct a moment map for the diagonal action of SU ( ๐ ) on the product of Grassmanniansand prove that ๐ โ ( ) is the ๏ฌag manifold under consideration. The moment map for a singleGrassmannian was written out in (1.10), so now we sum over all Grassmannians to obtain ๐ = ๐ โ๏ธ ๐ด = ๐ ๐ ๐ด = ๐ โ๏ธ ๐ด = (cid:32) ๐ ๐ด โ๏ธ ๐ = ๐ข ( ๐ ) ๐ด โ ๐ข ( ๐ ) ๐ด (cid:33) โ ๐ . (1.26)In this formula the vectors ๐ข ( ๐ ) ๐ด inside the same group ๐ด are orthonormal: ๐ข ( ๐ ) ๐ด โฆ ๐ข ( ๐ (cid:48) ) ๐ด = ๐ฟ ๐ ๐ (cid:48) .On the other hand, it is easy to convince oneself that the set ๐ โ ( ) is composed of ๐ -tuplesof orthogonal ๐ข -vectors. It follows that the ๐ข -vectors representing di๏ฌerent ๐ ๐ด -dimensionalplanes in C ๐ ( ๐ด = ยท ยท ยท ๐ ) are mutually orthogonal as well. The set of such orthogonalsubspaces is precisely the ๏ฌag manifold F ๐ , ยทยทยท ,๐ ๐ .15efore concluding this section, let us specialize these results to the case that we willencounter most frequently below, namely the case of the complete ๏ฌag manifold, when all ๐ ๐ด =
1. The second cohomology group of the complete ๏ฌag manifold is ๐ป (F ๐ , Z ) = Z ๐ โ , (1.27)hence there exist ๐ โ ๐ป (F ๐ , Z ) .In order not to repeat ourselves, let us consider here a slightly di๏ฌerent model for ๐ป (F ๐ , Z ) .On F ๐ there are ๐ standard line bundles L , ยท ยท ยท , L ๐ , and their sum is a trivial bundle: ๐ โ ๐ด = L ๐ด = F ๐ ร C ๐ . (1.28)The ๏ฌrst Chern classes of these bundles are represented by ๐ closed 2-forms: [ ฮฉ ๐ด ] = ๐ (L ๐ด ) , ๐ด = ยท ยท ยท ๐ . Due to the condition (1.28) and the additivity of the ๏ฌrst Chern classes ๐ ( ๐ธ โ ๐น ) = ๐ ( ๐ธ ) + ๐ ( ๐น ) it is clear that the forms ฮฉ ๐ด are not independent but rather satisfythe relation ๐ โ๏ธ ๐ด = [ ฮฉ ๐ด ] = ฮฉ ๐ด ( ๐ด = ยท ยท ยท ๐ ) satisfyingthe relation (1.29), generate ๐ป (F ๐ , Z ) . Higher cohomology groups of general ๏ฌag manifoldscould as well be obtained from the relations that follow from the triviality of a sum of certainvector bundles, i.e. from a generalization of (1.28).In the present review we will only make use of cohomology, with almost no referenceto the homotopy of ๏ฌag manifolds. One reason for this is that ๏ฌag manifolds are simplyconnected, ๐ (F ๐ , ยทยทยท ,๐ ๐ ) =
0, which implies ๐ (F ๐ , ยทยทยท ,๐ ๐ ) (cid:39) ๐ป (F ๐ , ยทยทยท ,๐ ๐ , Z ) (cid:39) Z ๐ โ byHurewicz theorem, so that the two notions coincide in dimension two. In higher dimensionsthis is no longer the case. For example, ๐ป (F ๐ , ยทยทยท ,๐ ๐ , Z ) (cid:39)
0, whereas for complete ๏ฌagmanifolds ๐ (F ๐ ) (cid:39) Z . The latter is a higher-dimensional generalization of the Hopf invariant ๐ ( ๐ ) (cid:39) Z and leads to the existence of topologically non-trivial Hop๏ฌon solutions [22, 23]relevant for the Faddeev-Niemi model [105] (see also [84]). ๐ต -๏ฌelds on the ๏ฌag manifold So far we have discussed SU ( ๐ ) -invariant closed forms on ๏ฌag manifolds, as well as therelated question of invariant Kรคhler metrics. This is not the end of the story, however, as on ageneral ๏ฌag manifold (1.4) there will be large families of invariant metrics, and typically onlya small subfamily corresponds to Kรคhler metrics. Moreover, the metrics that will actuallyenter the sigma models that we discuss in Chapters 2 and 3, are in general not Kรคhler. In asimilar way, the ๐ต -๏ฌelds also come in large families and are not required to be topological ingeneral, as on a general ๏ฌag manifold there exist invariant two-forms that are not closed.To construct the general metric and ๐ต -๏ฌeld, we denote the ๏ฌag manifold SU ( ๐ ) S ( U ( ๐ )รยทยทยทร U ( ๐ ๐ )) as ๐บ๐ป and introduce the corresponding Lie algebra decomposition ๐ค = ๐ฅ โ ๐ช . Since [ ๐ฅ , ๐ช ] โ , the subgroup ๐ป is represented in the space ๐ช , and this representation may be decomposedinto irreducibles: ๐ช C = โ ๐ด โ ๐ต ๐ ๐ด๐ต , where ๐ ๐ด๐ต = C ๐ ๐ด ๐ ๐ต . (1.30)The space ๐ ๐ด๐ต of ๐ ๐ด ร ๐ ๐ต -matrices is the vector space of the bi-fundamental representationof the group U ( ๐ ๐ด ) ร U ( ๐ ๐ต ) , and moreover ๐ ๐ต๐ด = ๐ ๐ด๐ต . We decompose the Maurer-Cartancurrent ๐ = โ ๐ โ ๐๐, ๐ โ U ( ๐ ) , entering (1.8) accordingly: ๐ = [ ๐ ] ๐ฅ + [ ๐ ] ๐ช = [ ๐ ] ๐ฅ + โ๏ธ ๐ด โ ๐ต ๐ ๐ด๐ต , ๐ ๐ด๐ต โ ๐ ๐ด๐ต . (1.31)The most general invariant two-form may then be written as ฮฉ = โ๏ธ ๐ด<๐ต ๐ ๐ด๐ต Tr ( ๐ ๐ด๐ต โง ๐ ๐ต๐ด ) . (1.32)Using the zero-curvature equation for ๐ , one can check that ฮฉ is closed if and only if ๐ ๐ด๐ต = ๐ ๐ด โ ๐ ๐ต , in which case it is exactly the symplectic form (1.8) (see Appendix B). Quiteanalogously, the line element of the most general metric is ๐๐ = โ โ๏ธ ๐ด<๐ต ๐ ๐ด๐ต Tr ( ๐ ๐ด๐ต ยท ๐ ๐ต๐ด ) , (1.33)where for positivity we have to require ๐ ๐ด๐ต >
0. We conclude that there are ๐ ( ๐ โ ) realparameters de๏ฌning the most general metric, as well as ๐ ( ๐ โ ) additional parameters de๏ฌningthe most general ๐ต -๏ฌeld.As discussed earlier, the space of Kรคhler metrics is an ( ๐ โ ) -dimensional subspace inthe full space of metrics. In order to formulate the corresponding condition on the coe๏ฌcients ๐ ๐ ๐ more explicitly, one would have to specify the complex structure J (these are discussedin Chapter 3, section 13.2). In any case, the metric that will be most important for us inChapter 3 (and features in some of the most prominent examples in chapter 2) is in generalnot Kรคhler. It is the so-called normal, or reductive, metric (cf. [29]), with line element ๐๐ = โ Tr ( [ ๐ ] ๐ช ) , which corresponds to ๐ ๐ด๐ต = ๐ด, ๐ต . This metric is not a Kรคhlermetric, unless the ๏ฌag manifold is a Grassmannian (i.e. unless ๐ = Now that we are done with some formal aspects, we wish to present the ๏ฌrst example of awell-known physical situation where ๏ฌag manifolds naturally arise. Incidentally this makesa neat connection to the applications of ๏ฌag manifolds in representation theory, discussedbelow in Section 2.2. We will need the latter for our discussion of spin chains in Chapter 2.17 .1 Mechanical particle in a non-Abelian gauge ๏ฌeld
It is well-known how one can describe the motion of a classical particle on a Riemannianmanifold with metric G , interacting with an external electromagnetic ๏ฌeld ๐ด ๐ . The actionhas the form S = โซ ๐๐ก G ๐๐ (cid:164) ๐ฅ ๐ (cid:164) ๐ฅ ๐ โ โซ ๐ด = โซ ๐๐ก (cid:18) G ๐๐ (cid:164) ๐ฅ ๐ (cid:164) ๐ฅ ๐ โ ๐ด ๐ (cid:164) ๐ฅ ๐ (cid:19) . (2.1)The question is, how do we write an analogous action for the case when the gauge ๏ฌeld isnon-Abelian, or, simply speaking, when it has additional gauge indices ๐ด ๐ผ๐ฝ๐ . The answer isthat the particle should possess additional degrees of freedom. For example, in the case ofSU ( ) , the additional variables correspond to a unit vector (cid:174) ๐ โ ๐ = CP that couples tothe SU ( ) gauge ๏ฌeld (cid:174) ๐ด . More generally, the degrees of freedom associated to the โinternalspinโ take values in a certain ๏ฌag manifold, corresponding to the representation in which theparticle transforms. In other words, one should enlarge the phase space of the mechanicalsystem [215]: ๐ โ M โ ๐ โ M ร F . (2.2)Here M is the con๏ฌguration space, ๐ โ M is the cotangent bundle (i.e. phase space), and F is the ๏ฌag manifold. In SU(2), F = CP , but for larger non-Abelian groups, it is not clear apriori what the appropriate choice of F should be, since there is now choice in the parameters ๐ ๐ appearing in (1.4). We will see below that this choice is related to the di๏ฌerent families ofrepresentations under which the particle transforms.We start by rewriting the standard action of a particle in ๏ฌrst-order form: S = โซ ๐๐ก (cid:18) ๐ ๐ (cid:164) ๐ฅ ๐ โ G ๐๐ ๐ ๐ ๐ ๐ โ ๐ด ๐ (cid:164) ๐ฅ ๐ (cid:19) . (2.3)Upon enlarging the phase space we can analogously write down the non-Abelian action asfollows ( ๐ด is assumed Hermitian, and H is the Hamiltonian): S = โซ ๐ ๐ ๐๐ฅ ๐ โ โซ ๐๐ก H ( ๐ฅ, ๐ ) + โซ ( ๐ โ Tr ( ๐ด ๐ )) . (2.4)Here ๐ is the canonical (Poincarรฉ-Liouville) one-form, de๏ฌned by the condition ๐๐ = ฮฉ ( = the symplectic form on F ) , (2.5)and ๐ is the moment map for the action of the group ๐บ on F . The integral โซ ๐ is sometimescalled the Berry phase and will be an essential ingredient of the spin chain path integrals inthe next chapter. We note that the form ๐ is de๏ฌned up to the addition of a total derivative, ๐ โ ๐ + ๐โ , but the di๏ฌerence only a๏ฌects the boundary terms in the action. In the case ofperiodic boundary conditions one may even write โซ ฮ ๐ = โซ ๐ท ฮฉ , (2.6)18here ๐ท is a disc, whose boundary is the curve ฮ : ๐ ๐ท = ฮ . In fact this term is nothing butthe one-dimensional version of the Wess-Zumino-Novikov-Witten term [239, 189, 245].One needs to show that the expression in (2.4) is gauge-invariant. For simplicity let ustake as F the projective space, CP ๐ โ . Later, we will see that this corresponds to the particletransforming in the de๏ฌning representation of SU ( ๐ ) . Let us normalize the homogeneouscoordinates on CP ๐ โ : ๐ โ๏ธ ๐ผ = | ๐ข ๐ผ | = . (2.7)One still has the remaining gauge group U ( ) , which acts by multiplication of all coordinates ๐ข ๐ผ by a common phase. The Fubini-Study form (1.16) on CP ๐ โ may be simpli๏ฌed if oneuses the above normalization: ฮฉ ๐น๐ = ๐ ๐๐ข ๐ผ โง ๐๐ข ๐ผ . (2.8)Then we have the following expressions for ๐ and ๐ : ๐ = ๐ ๐ข ๐ผ ๐๐ข ๐ผ , ๐ = ๐ข โ ๐ข โ ๐ ๐ . This expression for the moment map is a special case of (1.10). The part of the actioncorresponding to the motion in the โinternalโ space (in this case the projective space) has theform (cid:101) S = โ โซ ๐๐ก ๐ข ๐ผ ( ๐ (cid:164) ๐ข ๐ผ + ( ๐ด ๐ ) ๐ผ๐ฝ (cid:164) ๐ฅ ๐ผ ๐ข ๐ฝ ) , (2.9)and one should take into account that the normalization condition (2.7) is also implied. It isevident that it is gauge-invariant w.r.t. the transformations ๐ข โ ๐ ( ๐ฅ ( ๐ก )) โฆ ๐ข ๐ด ๐ โ ๐ ๐ด ๐ ๐ โ โ ๐ ๐ ๐ ๐ ๐ โ . (2.10)To make it even more obvious, we note that the exterior derivative of the one-form ๐ โ Tr ( ๐ด ๐ ) (viewed as a form on the enlarged phase space (2.2)) produces a two-form, which is explicitlygauge-invariant: ๐ ( ๐ โ Tr ( ๐ด ๐ )) = ๐ D ๐ข ๐ผ โง D ๐ข ๐ผ โ Tr ( ๐น ๐ ) , (2.11) D ๐ข = ๐๐ข โ ๐ ๐ด ๐ข, D ๐ข = ๐๐ข + ๐ ๐ข ๐ด, ๐น = ๐๐ด โ ๐ ๐ด โง ๐ด .
Each of the two terms in (2.11) is separately gauge-invariant, however (2.11) is the only linearcombination of them, which is closed (and therefore locally is an exterior derivative of aone-form).
Now that weโve written down a gauge-invariant action for a particle coupled to a non-Abeliangauge ๏ฌeld, let us next write out the equations of motion on the ๏ฌag manifold, F . Throughout the review we will be mostly using the variable ๐ง to denote unconstrained complex coordinates,such as the homogeneous or inhomogeneous coordinates on CP ๐ โ , and the variable ๐ข to denote unit-normalizedvectors.
19o simplify the discussion, let us begin by carrying out these steps for the case of SU(2),which corresponds to F = CP = ๐ . Instead of using the spinor ( ๐ข , ๐ข ) โ C , we canparametrize F in a more standard way with the help of a unit vector (cid:174) ๐ โ R . The equationsof motion then take the form (cid:164)(cid:174) ๐ = (cid:174) ๐ด ร (cid:174) ๐, where (cid:174) ๐ด = { ๐ด ๐๐ (cid:164) ๐ฅ ๐ } ๐ = , , (2.12)is a vector of components of the gauge ๏ฌeld in the basis of Pauli matrices. We see that theequations are linear in (cid:174) ๐ , and the condition (cid:174) ๐ = const . (2.13)is a consequence of the equations, i.e. the motion takes place on a sphere in R . This is ageneral fact. Indeed, in the case of a general compact simple Lie algebra ๐ค with basis { ๐ ๐ } we can introduce a variable ๐ = (cid:205) ๐ ๐ ๐ ๐ โ ๐ค , and the equations will then take the form (cid:164) ๐ = [ ๐ด ๐ (cid:164) ๐ฅ ๐ , ๐ ] , (2.14)or, in terms of the variables ๐ ๐ , (cid:164) ๐ ๐ = ๐ ๐๐๐ ( ๐ด ๐ (cid:164) ๐ฅ ๐ ) ๐ ๐ ๐ , (2.15)where ๐ ๐๐๐ are the structure constants of ๐ค . It is in this form that this system of equationswas discovered in [257]. The motion de๏ฌned by these equations in reality takes place on ๏ฌagmanifolds embedded in ๐ค , since the โCasimirsโ ๐ถ ๐ฝ = Tr ( ๐ ๐ฝ ) , ๐ฝ = , , . . . (2.16)are integrals of motion of the system (2.14), and specifying the Casimirs is e๏ฌectively thesame as specifying the parameter ๐ of the orbit (1.6). We have thus established a connectionwith the formulation through ๏ฌag manifolds used earlier. The next question that we address is how to quantize an action of the type (2.4). Quantizationof the particle phase space coordinates ( ๐, ๐ฅ ) is standard, so the non-trivial question is howto quantize the spin phase space F โ the ๏ฌag manifold. In the case of SU(2), this willlead to the notion of spin quantization, i.e., that the particle transforms under some de๏ฌniterepresentation of SU(2), labeled by a single integer.One of the approaches to quantization is related to considering path integrals of the form โซ (cid:214) ๐, ๐ก ๐๐ ๐ ( ๐ก ) ๐ ๐ S (2.17) Another approach to the quantization of coadjoint orbits, which is also based on the path integral, wasdeveloped in [17]. ๐ ๐ parametrize F . The subtlety comes fromthe fact that the connection ๐ is not a globally-de๏ฌned one-form on the ๏ฌag manifold. Indeed,let us consider the simplest case of F = CP = ๐ . The most general invariant symplecticform is as follows : ฮฉ = ๐ ๐ ๐๐ โง ๐๐. (2.18)Here ๐ is an arbitrary constant, and ๐, ๐ are the standard angles on the sphere.Since the action S entering the exponent in (2.17) involves a term โซ ๐ , where ๐ is aconnection satisfying ๐๐ = ฮฉ , a standard argument familiar from Wess-Zumino-Novikov-Witten theory [239, 189, 245] leads to the requirement that the coe๏ฌcient ๐ is quantizedaccording to โซ ฮฉ โ ๐ Z . Let us recall the argument. To start with, we write a one-form ๐ ,well-de๏ฌned on the northern hemisphere, such that ๐๐ = ฮฉ : ๐ ๐๐ป = ๐ sin (cid:18) ๐ (cid:19) ๐๐ . (2.19)It is well-de๏ฌned at the north pole, ๐ =
0, since at that point the prefactor of ๐๐ vanishes. OnFigure 4: To apply Stokesโ theorem to the integral โซ ฮ ๐ , we choose a disc with boundary ฮ .There are two ways to do this, which lead to the domains ๐ท , ๐ท โ ๐ . One has โซ ฮ ๐ ๐๐ป = โซ ๐ท ฮฉ , โซ ฮ ๐ ๐๐ป = โ โซ ๐ท ฮฉ = โ โซ ๐ ฮฉ + โซ ฮ ๐ ๐๐ป (since in order to use Stokesโ theorem, one has topick a one-form that is well-de๏ฌned in the interior of the domain). Since โซ ๐ ฮฉ = ๐ , and thechoice of north/south poles was arbitrary, the integral โซ ฮ ๐ is only well-de๏ฌned modulo 2 ๐ .the other hand, at ๐ = ๐ it remains constant. Another way to see this is to introduce the usualround metric on the sphere ๐๐ = ๐๐ + sin ๐ ๐๐ and to calculate the norm of the di๏ฌerential ๐ : (cid:107) ๐ (cid:107) = ๐ sin (cid:16) ๐ (cid:17) ๐ . One sees that it is bounded at ๐ = ๐ = ๐ . If one It can be also written in the form
ฮฉ = โ ๐ ๐๐ง โง ๐๐ , where ๐ง = cos ๐ is the ๐ง -coordinate of a given point onthe sphere. Since the latter form is nothing but the area element of a cylinder, it implies that the projection of asphere to the cylinder preserves the area. ๐ as a connection on a line bundle, it is nevertheless well-de๏ฌned, as on the southernhemisphere we may de๏ฌne a gauge-transformed ๐ ๐๐ป = ๐ ๐๐ป โ ๐ ๐๐ = โ ๐ cos (cid:16) ๐ (cid:17) ๐๐ , whichis well-behaved at ๐ = ๐ . Therefore the integral โซ ฮ ๐ depends on which formula for theconnection we take, ๐ ๐๐ป or ๐ ๐๐ป , the di๏ฌerence being equal to 2 ๐ ๐ : โซ ฮ ๐ ๐๐ป = โซ ฮ ๐ ๐๐ป โ ๐ ๐ (see Fig. 4). If ๐ โ Z , however, the quantity ๐ ๐ โซ ๐ is de๏ฌned unambiguously. We say that | ๐ | labels the representation of SU(2) under which the particle transforms, and ๐ = | ๐ | is calledthe โspinโ of the particle.Let us turn to the ๏ฌag manifolds of SU( ๐ ), with ๐ >
2. Now there are multiple contours ฮ ๐ that must be considered, corresponding to the hemispheres of homologically distinct spheres ๐ถ ๐ โ ๐ป (F , Z ) in F . We require that each of the terms ๐ ๐ โซ ฮ ๐ ๐ is well de๏ฌned, i.e. โซ ๐ถ ๐ ฮฉ โ ๐ Z for every 2-cycle ๐ถ ๐ โ ๐ป (F , Z ) . (2.20)These quantization conditions correspond to particular representations of SU( ๐ ). Let usconstruct these 2-cycles explicitly for the case when F is a complete ๏ฌag manifold F ๐ . Lateron, we can analyze the remaining (smaller) ๏ฌag manifolds by use of a forgetful projection.The manifold F ๐ can be parametrized using ๐ orthonormal vectors ๐ข ๐ด , ๐ด = . . . ๐ , ๐ข ๐ด โฆ ๐ข ๐ต = ๐ฟ ๐ด๐ต , de๏ฌned modulo phase transformations: ๐ข ๐ด โผ ๐ ๐๐ผ ๐ด ๐ข ๐ด . As we showed insections 1.2 and 1.5, the most general symplectic form on F ๐ may be written as follows: ฮฉ = ๐ โ๏ธ ๐ด<๐ต ( ๐ ๐ด โ ๐ ๐ต ) ๐ ๐ด๐ต โง ๐ ๐ต๐ด , where ๐ ๐ด๐ต = ๐ข ๐ด โฆ ๐๐ข ๐ต (2.21)To construct the cycles ๐ถ ๐ด , note that if one ๏ฌxes ๐ โ ๐ lines de๏ฌned by the vectors ๐ข , . . . , ๐ข ๐ , the remaining free parameters de๏ฌne the con๏ฌguration space of ordered pairs ofmutually orthogonal lines, passing through the origin and laying in a plane, orthogonal to the ๐ โ CP : { ๐ข ๐ด , . . . , ๐ข ๐ด ๐ โ are ๏ฌxed , ๐ข ๐ด ๐ โ , ๐ข ๐ด ๐ โ ( ๐ข ๐ด , . . . , ๐ข ๐ด ๐ โ ) โฅ are mutually orthogonal and otherwise generic } (cid:39) ( CP ) ๐ด ๐ โ ,๐ด ๐ . Let us now ๏ฌx a permutation ( ๐ด , . . . , ๐ด ๐ ) in such a way that the ๐ ๐ด ๐ form a non-increasing sequence, i.e. ๐ ๐ด ๐ โฅ ๐ ๐ด ๐ for ๐ < ๐ . The fact that ฮฉ is non-degenerate requires thatthis sequence is actually strictly decreasing. In this case the rearrangement of ๐ โs amounts tochoosing a complex structure on F ๐ , but we will not dwell on this fact here (see Section 13.2for details). After such a permutation we may choose ( CP ) ๐ด ๐ ,๐ด ๐ + as a basis in the homologygroup ๐ป (F ๐ , Z ) . Then the integrals of the symplectic form over these cycles will be positive: The orientation of the spheres is induced by the complex structure on F ๐ . ( CP ) ๐ด๐,๐ด๐ + ฮฉ = ๐ ๐ด ๐ โ ๐ ๐ด ๐ + โ ๐ Z + , ๐ = . . . ๐ โ . (2.22)In order for the value of the integral to be an integer, one should choose ๐ ๐ด in the form ( ๐ , . . . , ๐ ๐ ) = ๐ ( , . . . , ) + ( (cid:101) ๐ , . . . , (cid:101) ๐ ๐ ) , ๐ โ R , (cid:101) ๐ ๐ โ Z . (2.23)This freedom in adding a vector ๐ ( , . . . , ) allows us to work with values { (cid:101) ๐ ๐ด } that sum tozero. According to the general theory of adjoint orbits (see Section 1.2), the ๏ฌag manifoldunder consideration is then the orbit of the element ๐ = (cid:169)(cid:173)(cid:173)(cid:171) (cid:101) ๐ ... . . . ... (cid:101) ๐ ๐ (cid:170)(cid:174)(cid:174)(cid:172) โ ๐ฐ๐ฒ ( ๐ ) . (2.24)Let us observe what happens when some of these variables (cid:101) ๐ ๐ด coincide. On the one hand, the2-form ฮฉ now becomes degenerate. On the other, we see that the corresponding adjoint orbitis no longer the complete ๏ฌag manifold, F ๐ . For example, if there are only two distinct valuesof (cid:101) ๐ ๐ , i.e. we have ๐ ๐ = Diag ( , . . . , ๐ , , . . . , ๐ โ ๐ ) โ ๐ ๐ ๐ , so that the corresponding adjoint orbitis the Grassmannian ๐บ๐ ๐ ,๐ . This demonstrates the point that we alluded to earlier, namelythat the ๏ฌag manifold F encoding the degrees of freedom of a particle coupled to an SU( ๐ )gauge ๏ฌeld is not uniquely determined by ๐ for ๐ >
2. Indeed, choosing di๏ฌerent values of ๐ ๐ด leads to di๏ฌerent ๏ฌag manifolds. In such cases when F is strictly smaller than F ๐ , onemay view the (degenerate) 2-form (2.21) on the complete ๏ฌag manifold as a non-degenerateform on the smaller F . This amounts to a forgetful projection. The general theory that wehave described is nothing but โgeometric quantizationโ for the case of ๏ฌag manifolds.The canonical quantization of the system given by the action ๐ = โซ ๐ will be treated indetail in the next section and, as we shall see, the non-negative integers ๐ ๐ด are equal to thelengths of the rows of the Young diagram characterizing a given representation of ๐ฐ๐ฒ ( ๐ ) . Forthis to make sense, one should choose ๐ in (2.23) in such a way that ๐ ๐ = Having discussed the quantization of the symplectic form ฮฉ on F , we are now ready tocanonically quantize the ๏ฌag manifold (i.e. the action corresponding to the โinternal spaceโ).To see how this works, let us ๏ฌrst canonically quantize CP ๐ โ , with action given in (2.9).Instead of working with normalized ๐ข ๐ , we ๏ฌrst write the kinetic term of the Lagrangian as L = ๐ ๐ โ๏ธ ๐ผ = ๐ง ๐ผ โฆ (cid:164) ๐ง ๐ผ , (2.25)23nd impose the normalization constraint in the form ๐ โ๏ธ ๐ผ = | ๐ง ๐ผ | = ๐ . (2.26)Therefore the canonical momentum is ๐ ๐ผ = ๐ L ๐ (cid:164) ๐ง ๐ผ = ๐๐ง ๐ผ , which leads to the algebra { ๐ง ๐ผ , ๐ง ๐ฝ } = ๐ ๐ฟ ๐ผ๐ฝ . This ultimately leads to the theory of Schwinger-Wigner quantization, which is a wayof representing spin operators using creation-annihilation operators (for a review see, forexample, [227]). In the present example it may be summarized as follows.Suppose ๐ ๐ are a set of SU ( ๐ ) generators in the fundamental representation. Introduce ๐ operators ๐ ๐ผ and their conjugates ๐ โ ,๐ผ with the canonical commutation relations [ ๐ ๐ผ , ๐ โ ,๐ฝ ] = ๐ฟ ๐ผ๐ฝ . (2.27)One can easily check that the operators ๐ ๐ = ๐ โ ,๐ผ ๐ ๐๐ผ๐ฝ ๐ ๐ฝ , (2.28)satisfy the commutation relations of ๐ฐ๐ฒ ( ๐ ) , and ๐ ๐ act irreducibly on the subspace of the fullFock space speci๏ฌed by the condition ๐ โ๏ธ ๐ผ = ๐ โ ,๐ผ ๐ ๐ผ = ๐, (2.29)where ๐ is a positive integer representing the โnumber of particlesโ. For a given ๐ therepresentation one obtains is the ๐ -th symmetric power of the fundamental representation.Now let us turn to a general ๏ฌag manifold, with the kinetic term โซ ๐ , where ๐๐ = ฮฉ isthe symplectic form (2.21). Let us rewrite it as follows: ฮฉ = โ ๐ โ๏ธ ๐ด,๐ต ( ๐ ๐ด โ ๐ ๐ต ) ๐ข ๐ด โฆ ๐๐ข ๐ต โง ๐ข ๐ต โฆ ๐๐ข ๐ด = โ ๐ โ๏ธ ๐ด,๐ต ๐ ๐ด ๐ข ๐ด โฆ ๐๐ข ๐ต โง ๐ข ๐ต โฆ ๐๐ข ๐ด = (2.30) = ๐ โ๏ธ ๐ด,๐ต ๐ ๐ด ๐๐ข ๐ด โฆ ๐ข ๐ต โง ๐ข ๐ต โฆ ๐๐ข ๐ด = using completeness = ๐ (cid:32) ๐ โ๏ธ ๐ด ๐ ๐ด ๐ข ๐ด โฆ ๐๐ข ๐ด (cid:33) De๏ฌning ๐ง ๐ด = โ ๐ ๐ด ๐ข ๐ด , we may therefore set ๐ = ๐ โ๏ธ ๐ด ๐ง ๐ด โฆ ๐๐ง ๐ด , ๐ง ๐ด โฆ ๐ง ๐ต = ๐ ๐ด ๐ฟ ๐ด๐ต . (2.31)Each ๐ ๐ด corresponds to the โnumber of particlesโ of a particular species. The canonicalquantization procedure then gives [ ๐ ๐ผ๐ด , ๐ ๐ฝ๐ต ] = ๐ฟ ๐ผ๐ฝ ๐ฟ ๐ด๐ต . (2.32)24n other words, we introduce ๐ creation-annihilation operators ๐ ๐ผ๐ด for each of the ๐ ๐ด . Then ๐ ๐ด โ Z โฅ is the occupation number of the ๐ด -th line of the Young tableau. The shift ๐ ๐ด โ ๐ ๐ด +
1, which is inessential according to the above discussion, corresponds to addinga column to a Young diagram of full length. The di๏ฌerences ๐ ๐ด โ ๐ ๐ด + are the Dynkinlabels of the representation (which are the coe๏ฌcients in the expansion of the highest weightin the highest weights of the fundamental representations). Whenever the lengths of twoconsecutive rows of the Young diagram coincide, the corresponding Dynkin label is zero,and the corresponding โsymplectic formโ degenerates, which signals that one should pass toa smaller ๏ฌag manifold. This is consistent with our discussion in the previous subsection.The next point is that the mutual orthogonality of the ๐ง ๐ด s should be re๏ฌected in theoperators ๐ ๐ผ๐ด in some way. To illustrate this, let us consider the SU ( ) adjoint representation.Let us label the six creation-anniliation operators as ๐ ๐ผ , ๐ ๐ผ (three for each non-zero row), sothat the ๐ฐ๐ฒ ( ๐ ) generators look as follows ๐ ๐ = ๐ โ ,๐ผ ๐ ๐๐ผ๐ฝ ๐ ๐ฝ + ๐ โ ,๐ผ ๐ ๐๐ผ๐ฝ ๐ ๐ฝ , (2.33)where ๐ ๐ are the ๐ ร ๐ -generators in the de๏ฌning representation. To model this representationon a subspace of the Fock space ๐น , we build the operators ๐ = ๐ โ , ๐ + ๐ โ , ๐ + ๐ โ , ๐ , ๐ = ๐ โ , ๐ + ๐ โ , ๐ + ๐ โ , ๐ (2.34) ๐ = ๐ โ , ๐ + ๐ โ , ๐ + ๐ โ , ๐ (2.35)and require the vectors | ๐ (cid:105) โ ๐น , on which the representation is built to satisfy ๐ | ๐ (cid:105) = | ๐ (cid:105) , ๐ | ๐ (cid:105) = | ๐ (cid:105) , ๐ | ๐ (cid:105) = . (2.36)The values of ๐ and ๐ correspond to the number of boxes in the ๏ฌrst and second rows ofthe Young diagram (i.e. they are โnumber operatorsโ that count the number of particles ofspecies ๐ and ๐ , respectively). Notice that the classical condition ๐ โฆ ๐ = ๐ | ๐ (cid:105) = ๐ โ | ๐ (cid:105) =
0. Indeed, the two equations would be incompatible,since [ ๐ , ๐ โ ] = ๐ โ ๐ and ( ๐ โ ๐ ) | ๐ (cid:105) = | ๐ (cid:105) โ
0. This asymmetry is the same onethat is already present in the Young diagram.Let us now explain how this generalizes to SU( ๐ ). We introduce ๐ creation operators ๐ โ ,๐ผ๐ด for each row ๐ด of the Young diagram ( ๐ด = O ๐ด๐ต | ๐ (cid:105) โก ๐ โ ๐ด โฆ ๐ ๐ต | ๐ (cid:105) = ๐ด < ๐ต. (2.37)This is a compatible set of equations, since the operators O ๐ด๐ต satisfy the algebra [O ๐ด๐ต , O ๐ถ๐ท ] = ๐ฟ ๐ต๐ถ O ๐ด๐ท โ ๐ฟ ๐ด๐ท O ๐ถ๐ต where ๐ด < ๐ต, ๐ถ < ๐ท. (2.38)The operators O ๐ด๐ต may be thus thought of as the positive roots of the Lie algebra ๐ฐ๐ฒ ( ๐ ) . InChapter 2, this algebra will reappear in the context of SU( ๐ ) spin operators.25he constraint (2.37) may be solved rather explicitly. More exactly, we are looking for thejoint kernel of the operators O ๐ด๐ต , ๐ด < ๐ต acting on states in the ( ๐ , ๐ , . . . ) -particle Fockspace: | ฮจ (cid:105) = ๐ด ๐ผ ...๐ผ ๐ | ๐ฝ ...๐ฝ ๐ | ... ๐ โ ,๐ผ . . . ๐ โ ,๐ผ ๐ ๐ โ ,๐ฝ . . . ๐ โ ,๐ฝ ๐ . . . | (cid:105) . (2.39)The kernel is a linear space, and the basis in this space may be constructed as follows.1. Assign to each row of the Young diagram a letter ๐, ๐, ๐, ยท ยท ยท . For example: ๐ ๐ ๐๐ ๐๐
2. For each column build antisymmetric combinations of the form โ๏ธ ๐ (โ) ๐ ๐ โ ,๐ ( ๐ ) ๐ โ ,๐ ( ๐ ) ๐ โ ,๐ ( ๐ ) ๐ โ ,๐ ( ๐ ) , where the number of letters participating is equal to the height of the column.3. Multiply these antisymmetric combinations (the number of โparticlesโ of type ๐ด willbe precisely equal to the length of the ๐ด -th row in the Young diagram). To see thatthese are annihilated by operators O ๐ด๐ต , ๐ด < ๐ต , note that the action of this operatorremoves the ๐ต -th letter and replaces it by the ๐ด -th letter, and since the ๐ต -th letter for ๐ต > ๐ด always enters in skew-symmetric combinations with the ๐ด -th letter, the resultwill be zero. The mathematical counterpart of the procedure that we just described is called geometricquantization. One of the main statements of the subject โ the Borel-Weil-Bott theorem โasserts that, given a representation ๐ of a group ๐บ , one can construct a holomorphic linebundle L ๐ over a suitable ๏ฌag manifold of ๐บ (in full generality one can take the manifoldof complete ๏ฌags), such that ๐ may be reconstructed as the space of holomorphic sections ๐ป (L ๐ ) of L ๐ . These holomorphic sections are polynomials, and indeed it is elementary to๏ฌnd a map from the space of states (2.39) to the space of polynomials โ this is essentiallythe Bargmann representation, as we review in Appendix C. Given the background materialaccumulated to this point, we can somewhat specify what the line bundle in the Borel-Weil-Bott theorem is: it is characterized by its ๏ฌrst Chern class that is represented by the symplecticform ฮฉ (2.30), through which the kinetic term in the action standing in the path integral isde๏ฌned. In other words, [ ฮฉ ] = ๐ (L ๐ ) โ ๐ป (F , Z ) .The ๏ฌag manifold itself that features in this construction is the manifold of โcoherentstatesโ, which by de๏ฌnition are the states in the orbit of ๐บ acting on the highest weightvector. This connection becomes perhaps more transparent if one recalls the discussion in26ection 2.2, where the integration of the symplectic form ฮฉ over various two-cycles in the๏ฌag manifold was described. We may view the cycles ( CP ) ๐ด ๐ ,๐ด ๐ + as the positive simpleroots of ๐ฐ๐ฒ ( ๐ ) , and ฮฉ as a highest weight. It is a general theorem that highest weight orbitsare Kรคhler manifolds [157]. Coherent states are important for the construction of spin chainpath integrals in Chapter 2, so we discuss them in more detail below in section 2.3.4. For ageneral discussion of geometric quantization we refer the reader to [152] (see also [73]). Let us present three example representations in SU( ๐ ). We will return to these examples lateron when we discuss coherent states. In all cases the states are built as polynomials in thecreation operators ๐ โ ,๐ผ , ๐ โ ,๐ผ , etc., acting on the vacuum state | (cid:105) . a) ๐ ๐ ๐ ๐ Symmetric powers of the fundamental representation โ Polynomials in ๐ โ , , . . . , ๐ โ ,๐ of degree 4. b) ๐ ๐๐ In this case we have linear combinations of polynomials in ๐ โ ,๐ผ and ๐ โ ,๐ผ ofthe form ๐ โ ,๐ผ ( ๐ โ ,๐ฝ ๐ โ ,๐พ โ ๐ โ ,๐พ ๐ โ ,๐ฝ ) c) ๐ ๐ ๐๐ ๐๐ Here we have linear combinations of polynomials in ๐ โ ,๐ผ , ๐ โ ,๐ผ and ๐ โ ,๐ผ ofthe form ๐ โ ,๐ผ ( ๐ โ ,๐ฝ ๐ โ ,๐พ โ ๐ โ ,๐พ ๐ โ ,๐ฝ ) ( ๐ โ ,๐พ ๐ โ ,๐ฟ ๐ โ ,๐ โ ๐ โ ,๐พ ๐ โ ,๐ ๐ โ ,๐ฟ โ ๐ โ ,๐ฟ ๐ โ ,๐พ ๐ โ ,๐ โ ๐ โ ,๐ ๐ โ ,๐ฟ ๐ โ ,๐พ + ๐ โ ,๐ฟ ๐ โ ,๐ ๐ โ ,๐พ + ๐ โ ,๐ ๐ โ ,๐พ ๐ โ ,๐ฟ ) . Apart from its aesthetic appeal, this construction o๏ฌers certain calculational bene๏ฌts. Forinstance, the calculation of values of the Casimir operators on various representations becomesa matter of simple harmonic oscillator algebra. As an example we calculate the value of thequadratic Casimir of ๐ฐ๐ฒ ( ๐ ) in the representation described schematically by the followingdiagram: ๐ (cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32) (cid:125) ๐ where we assume there are ๐ boxes in the ๏ฌrst row and ๐ boxes in thesecond one ( ๐ (cid:62) ๐ ). We assign ๐ pairs of creation/annihilation operators ๐, ๐ โ , ๐, ๐ โ toeach row. The rotation generators are ๐ ๐ = ๐ โ โฆ ๐ ๐ โฆ ๐ + ๐ โ โฆ ๐ ๐ โฆ ๐. (2.40)27he generators ๐ ๐ are unit-normalized: Tr ( ๐ ๐ ๐ ๐ ) = ๐ฟ ๐๐ . Then (cid:205) ๐ ๐ ๐ โ ๐ ๐ = ๐ โ ๐ ๐ผ , where ๐ is the permutation and ๐ผ the identity operator. Thus, for the Casimir one obtains (here forbrevity we omit the state | ๐ (cid:105) on which these operators act, but its presence is implied) ๐ถ โก โ๏ธ ๐ ๐ ๐ ๐ ๐ = ๐ โ ,๐ผ ๐ ๐ฝ ๐ โ ,๐ฝ ๐ ๐ผ = ๐ +( ๐ โ ) ๐ + ๐ โ ,๐ผ ๐ ๐ฝ ๐ โ ,๐ฝ ๐ ๐ผ = โ ๐ + ๐ โ ,๐ผ ๐ ๐ฝ ๐ โ ,๐ฝ ๐ ๐ผ = โ ๐ + ๐ โ ,๐ผ ๐ ๐ฝ ๐ โ ,๐ฝ ๐ ๐ผ = ๐ +( ๐ โ ) ๐ โโ ๐ ( ๐ โ ,๐ผ ๐ ๐ผ + ๐ โ ,๐ผ ๐ ๐ผ ) = ( ๐ + ๐ ) = ๐ + ( ๐ โ ) ๐ + ๐ + ( ๐ โ ) ๐ โ ๐ ( ๐ + ๐ ) โ ๐ This can be easily generalized to arbitrary representations of ๐ฐ๐ฒ ( ๐ ) . Indeed, consider a Youngdiagram with ๐ rows (the maximal number for ๐ฐ๐ฒ ( ๐ ) ), the row lengths being ๐ โฅ . . . โฅ ๐ ๐ โ โฅ ๐ ๐ =
0. Introducing the variable ๐ ๐ด = ๐ ๐ด โ ๐ ๐ (cid:205) ๐ต = ๐ ๐ต (so that ๐ (cid:205) ๐ด = ๐ ๐ด = ๐ถ = ๐ โ๏ธ ๐ด = ๐ ๐ด ( ๐ ๐ด โ ๐ด ) , (2.41)in accordance with the result obtained long ago [197]. Coherent states are a type of basis in a vector space on which a Lie group ๐บ is represented.One takes a highest weight vector | ๐ (cid:105) and forms its ๐บ -orbit. That is, one considers all vectorsof the form ๐ | ๐ (cid:105) , where ๐ โ ๐บ . This is a continuous basis, which is therefore overcomplete.In what follows we will be dealing solely with the case of compact ๐บ = U ( ๐ ) , however we ๏ฌndit useful to remind the reader of how the de๏ฌnition just introduced ๏ฌts into the familiar setupof quantum mechanics (cf. [153]). In this case one has a Heisenberg algebra [ ๐, ๐ โ ] = witha highest weight vector | (cid:105) , which is annihilated by ๐ (and clearly ๏ฌxed by the unit operator).The normalized coherent states are therefore given by the familiar formula | ๐ฃ (cid:105) โก ๐ โ | ๐ฃ | ๐ ๐ฃ ๐ โ | (cid:105) . (2.42)In this case coherent states are parametrized by complex numbers: ๐ฃ โ C . As wementioned earlier, it is a general fact [157] that the highest weight orbit in the projectivization ๐ ( ๐ ) of an irreducible representation ๐ of a compact Lie group is Kรคhler. For the coherentstates we ๏ฌnd below, ๐ will live in some ๏ฌag manifold.In the case of SU ( ๐ ) the coherent states can be expressed in terms of the creation-annihilation operators introduced via Schwinger-Wigner quantization above . Having the A classic reference on coherent states for compact Lie groups, suitable for a mathematically inclined reader,is [198]. A rather clear exposition of coherent states and geometric quantization can be also found in [114] and[179]. Some very explicit formulas for the coherent states of ๐ฐ๐ฒ may be found in [173]. Another approach tothe quantization of coadjoint orbits is developed in [17]. ( ๐ ) . For each of the three Young diagrams appearing inSection 2.3.2, we build them explicitly; the general case should be clear from these examples. a) The highest weight vector is ( ๐ โ , ) | (cid:105) . Since ( ๐๐ โ , ๐ โ ) = ( ๐ง โฆ ๐ โ ) for ๐ง the ๏ฌrstcolumn of ๐ โ SU ( ๐ ) , we may parameterize the coherent states in this case as | ๐ฃ (cid:105) = ( ๐ฃ โฆ ๐ โ ) | (cid:105) , ๐ฃ โ CP ๐ โ = F ,๐ โ (2.43) b) The highest weight vector is ๐ โ , ยท ( ๐ โ , ๐ โ , โ ๐ โ , ๐ โ , ) | (cid:105) , and leads to | ๐ฃ๐ค (cid:105) = ( ๐ฃ โฆ ๐ โ ) ยท [( ๐ฃ โฆ ๐ โ ) ( ๐ค โฆ ๐ โ ) โ ( ๐ค โฆ ๐ โ ) ( ๐ฃ โฆ ๐ โ )] | (cid:105) , ๐ค โฆ ๐ฃ =
0. Here ๐ฃ and ๐ค parametrize thepartial ๏ฌag manifold F , ,๐ โ . c) The highest weight vector ๐ โ , ยท ( ๐ โ , ๐ โ , โ ๐ โ , ๐ โ , ) ยท ( ๐ โ , ๐ โ , ๐ โ , โ ๐ โ , ๐ โ , ๐ โ , โ ๐ โ , ๐ โ , ๐ โ , โ ๐ โ , ๐ โ , ๐ โ , + ๐ โ , ๐ โ , ๐ โ , + ๐ โ , ๐ โ , ๐ โ , ) | (cid:105) leads to the coherent states | ๐ข๐ฃ๐ค (cid:105) = ( ๐ฃ โฆ ๐ โ ) ยท [( ๐ฃ โฆ ๐ โ ) ( ๐ค โฆ ๐ โ ) โ ( ๐ค โฆ ๐ โ ) ( ๐ฃ โฆ ๐ โ )] ยท (2.44) ยท[( ๐ฃ โฆ ๐ โ ) ( ๐ค โฆ ๐ โ ) ( ๐ข โฆ ๐ โ ) โ ( ๐ฃ โฆ ๐ โ ) ( ๐ข โฆ ๐ โ ) ( ๐ค โฆ ๐ โ ) โ ( ๐ค โฆ ๐ โ ) ( ๐ฃ โฆ ๐ โ ) ( ๐ข โฆ ๐ โ ) โโ( ๐ข โฆ ๐ โ ) ( ๐ค โฆ ๐ โ ) ( ๐ฃ โฆ ๐ โ ) + ( ๐ค โฆ ๐ โ ) ( ๐ข โฆ ๐ โ ) ( ๐ฃ โฆ ๐ โ ) + ( ๐ข โฆ ๐ โ ) ( ๐ฃ โฆ ๐ โ ) ( ๐ค โฆ ๐ โ )] | (cid:105) with ๐ค โฆ ๐ฃ = ๐ข โฆ ๐ค = ๐ข โฆ ๐ฃ =
0. These three variables parametrize F , , ,๐ โ .It is easy to see that the above vectors are highest weight vectors. It follows from therepresentation (2.40) (taking into account the obvious generalization to the case of threeoscillators ๐, ๐, ๐ ) that those generators ๐ ๐ , which are upper-triangular, correspond to thefollowing transformations of the operators ๐ โ , ๐ โ , ๐ โ : ๐ฟ๐ โ ,๐ผ = โ๏ธ ๐ฝ<๐ผ ๐ ๐ผ๐ฝ ๐ โ ,๐ฝ , ๐ฟ๐ โ ,๐ผ = โ๏ธ ๐ฝ<๐ผ ๐ ๐ผ๐ฝ ๐ โ ,๐ฝ , ๐ฟ๐ โ ,๐ผ = โ๏ธ ๐ฝ<๐ผ ๐ ๐ผ๐ฝ ๐ โ ,๐ฝ , (2.45)i.e. in the matrix ( ๐, ๐, ๐ ) the upper rows are added to the lower ones. Since the constructedstates are de๏ฌned through the upper minors of this matrix, they are invariant under suchtransformations, i.e. they are annihilated by all positive roots.One of the central properties of coherent states is that they form an overcomplete basis.This is re๏ฌected in a fundamental identity โ the so-called โpartition of unityโ. For the casewhen the manifold of coherent states is CP ๐ โ (as in (2.43)), which is the only case we willreally be using, the identity takes the form โซ ๐๐ ( ๐ฃ, ๐ฃ ) | ๐ฃ (cid:105)(cid:104) ๐ฃ |(cid:104) ๐ฃ | ๐ฃ (cid:105) = , (2.46)where ๐๐ is the suitably normalized volume form on CP ๐ โ . It is proportional to the toppower of the Fubini-Study form, ๐๐ โผ ๐ ๐ โ , and looks as follows when expressed in theinhomogeneous coordinates: ( ๐๐ ) CP ๐ โ โผ (cid:32) + ๐ โ โ๏ธ ๐ผ = ๐ฃ ๐ผ ๐ฃ ๐ผ (cid:33) โ ๐ ๐ โ (cid:214) ๐ผ = ( ๐ ๐๐ฃ ๐ผ โง ๐๐ฃ ๐ผ ) . (2.47) We use the same symbol for the matrix realization and the Fock space operator realization of a transformation ๐ โ SU ( ๐ ) . CP ๐ โ , one would have to replace ๐๐ with the corresponding volume form. In Section 2.3, we demonstrated how Schwinger-Wigner oscillators arise from the canonicalquantization of the ๏ฌag manifold phase space in homogeneous coordinates. We will nowproceed to show that the famous Holstein-Primako๏ฌ representation corresponds to the quan-tization of the sphere โ the most elementary ๏ฌag manifold โ in certain coordinates, relatedto the action-angle and to the inhomogeneous coordinates. A corresponding SU ( ๐ ) ๏ฌagmanifold version can also be developed along the same lines. We start from the ๏ฌrst-orderLagrangian L CP = ๐ ๐ ๐ง (cid:164) ๐ง โ ๐ง (cid:164) ๐ง + ๐ง๐ง = ๐ ฯ ๐๐ = ๐ ๐ ( ๐ค (cid:164) ๐ค โ ๐ค (cid:164) ๐ค ) , (2.48)where ๐ง = | ๐ง | ๐ โ ๐๐ ๐ค = ฯ ๐ โ ๐๐ , ฯ = โ + ๐ง๐ง . As explained before, upon quantization ๐ โ Z + is a positive integer encoding the representa-tion. We also need the expressions for the SU ( ) charges. If we denote the vector ๐ : = (cid:18) ๐ง (cid:19) ,the spin variables are the moment maps ๐ ๐ = ๐ ๐ โ ๐ ๐ ๐๐ โฆ ๐ , so that ๐ + = ๐ ๐ง + | ๐ง | , ๐ โ = ๐ ๐ง + | ๐ง | , ๐ ๐ง = ๐ โ | ๐ง | + | ๐ง | . (2.49)Using ๐ง = | ๐ง | ฯ ๐ค = ๐ค ( โ| ๐ค | ) / , we ๏ฌnd ๐ + = ๐ ๐ค ( โ | ๐ค | ) / , ๐ โ = ๐ ๐ค ( โ | ๐ค | ) / , ๐ ๐ง = ๐ ( โ | ๐ค | ) . (2.50)To canonically quantize the system (2.48), we denote ๐ด : = โ ๐ ๐ค , ๐ด โ : = โ ๐ ๐ค and postulatethe canonical commutation relations [ ๐ด, ๐ด โ ] =
1. Choosing the ordering compatible withthe unitary relation ๐ + = ( ๐ โ ) โ , we ๏ฌnd ๐ + = ( ๐ โ ๐ด โ ๐ด ) / ๐ด, ๐ โ = ๐ด โ ( ๐ โ ๐ด โ ๐ด ) / , ๐ ๐ง = ( ๐ โ ๐ด โ ๐ด ) , (2.51)which is the Holstein-Primako๏ฌ representation for the spin operators.We have demonstrated that the Holstein-Primako๏ฌ realization arises from the quantizationof the sphere CP which is the simplest example of a coadjoint orbit of a compact group. Thereis yet another well-known realization of the spin operators โ the so-called Dyson-Maleevrealization โ whose advantage is that the resulting expressions for the spin operators are polynomial . The reason why we wish to discuss this representation is that the correspondingsetup is very similar to the one in which the integrable models will be formulated in Chapter 3.30s we shall see there, these Dyson-Maleev variables may be used to demonstrate that theinteractions in the sigma models are polynomial.The Dyson-Maleev representation may as well be obtained in the framework of canonicalquantization, however the primary objects in this case are the orbits of the complexi๏ฌed groupSL ( ๐, C ) . In the mathematics literature this subject was initiated in [145]. The questionasked in that work was about constructing a representation of a given complex Lie algebra interms of a minimal number of Weyl pairs (i.e. ๐ ๐ , ๐ ๐ -operators, such that [ ๐ ๐ , ๐ ๐ ] = ๐๐ฟ ๐ ๐ ).As explained in [146], the solution to this problem is in considering coadjoint orbits O ofa minimal dimension of a corresponding Lie group. These are symplectic varieties, whichmay be naturally quantized in terms of ๐ Weyl pairs, where ๐ = dim C O . The classical limitof the Weyl pairs produces the Darboux coordinates on O . It was also shown in [146] that,unless the Lie algebra in question is ๐ฐ๐ฉ ( ๐ ) , the minimal orbit is nilpotent, so typically thissetup leads to the theory of nilpotent orbits. For ๐ฐ๐ฉ ( ๐ ) , which is our main case of interest,there is a continuum of semi-simple orbits, whose limiting point is a nilpotent orbit of thesame (minimal) complex dimension ๐ โ ๐ฐ๐ฉ ( ) . The semi-simple orbits may be labeled by theCartan elements (cid:18) ๐ โ ๐ (cid:19) , where ๐ โ C \ { } (the limit ๐ = ๐ โ ๐ฐ๐ฉ ( ) ) ๐ = ๐ . (2.52)Consider the following ๏ฌrst-order Lagrangian (which should be viewed as the relevant coun-terpart of (2.48)): L = โ๏ธ ๐ = (cid:16) ๐ ๐ ยท D ๐ ๐ + ๐ ๐ ยท D ๐ ๐ (cid:17) + ( ๐ A + ๐ A) , D ๐ ๐ = (cid:164) ๐ ๐ โ A ๐ ๐ . (2.53)Here ๐ ๐ , ๐ ๐ are the complex canonical variables, and the gauge ๏ฌeld A is meant to generatethe quotient by C โ . Just as before, the ๏ฌrst term in the Lagrangian is a Poincarรฉ-Liouville one-form corresponding to a certain (this time complex) symplectic form, and the introduction ofa gauge ๏ฌeld allows one to obtain the symplectic form on the orbit by means of a symplecticreduction. Here we will just take this fact for granted, but such representations are discussedin more detail in Chapter 3, in the context of integrable sigma models with ๏ฌag manifoldtarget spaces. The second term in the Lagrangian is a โFayet-Iliopoulos termโ: under gaugetransformations it shifts by a total derivative, but the action S = โซ ๐๐ก L is invariant.The group SL ( , C ) acts as ๐ โ ๐ โฆ ๐, ๐ โ ๐ โฆ ๐ โ , and from (2.53) one can derive theconserved charges corresponding to this action: ๐ = ๐ โ ๐ โ ( ๐ โฆ ๐ ) (2.54) We wish to thank K. Mkrtchyan for drawing our attention to this work and important discussions on thesubject. Some applications of the theory of โminimalโ realizations of Lie algebras, as well as a list of relatedliterature, may be found in [147]. ๐ = (cid:205) ๐ = ๐๐ ๐ โง ๐๐ ๐ , which is why wehave denoted it by ๐ . Varying the Lagrangian w.r.t. the gauge ๏ฌeld, we obtain the constraint ๐ โฆ ๐ = ๐ . As a result, ๐ satis๏ฌes the equation ๐ = ๐ , so that ๐ belongs to theorbit (2.52).Let us now choose โinhomogeneous coordinatesโ, i.e. we assume that at least one of ๐ , ๐ is non-zero, say ๐ โ
0, in which case by a C โ -transformation we may set ๐ = ๐ : = ๐ and ๐ : = ๐ . The constraint (cid:205) ๐ = ๐ ๐ ยท ๐ ๐ = ๐ may now be solved as ๐ = ๐ โ ๐ ยท ๐ . The spin matrix ๐ has the following form in these variables: ๐ = (cid:18) ๐ โ ๐ ยท ๐ ๐๐ ยท ( ๐ โ ๐ ยท ๐ ) ๐ ยท ๐ โ ๐ (cid:19) (2.55)Quantization of (2.53) in the inhomogeneous coordinates ๐, ๐ amounts to imposing thecanonical commutation relations [ ๐, ๐ ] = ๐ . In this case one has to deal with the orderingambiguity (which is still much milder than the one in (2.50) and may easily be resolved byimposing the ๐ฐ๐ฉ ( ) commutation relations), and as a result one arrives at the Dyson-Maleevrepresentation ๐ + = ๐ , ๐ โ = ๐ ( ๐ โ ๐ ๐ ) , ๐ ๐ง = ๐ โ ๐ ๐ , where [ ๐, ๐ ] = ๐ . (2.56)By identifying ๐ = โ ๐ ๐๐๐ , we also obtain the well-known di๏ฌerential operator realization ๐ + = โ ๐ ๐๐๐ , ๐ โ = ๐ (cid:18) ๐ + ๐ ๐ ๐๐๐ (cid:19) , ๐ ๐ง = ๐ + ๐ ๐ ๐๐๐ , (2.57)which for ๐ = ๐ฐ๐ฉ ( ) -operators acting on the sphere CP withinhomogeneous coordinate ๐ . 32 hapter 2. From spin chains to sigma models In this chapter, we consider quantum spin systems in one spatial dimension. In their sim-plest form, these systems are described by the Heisenberg model, and are either ferromagneticor antiferromagnetic, depending on the sign of the interaction term between neighboring spinson the chain. While the ferromagnetโs ground state is the same for both classical and quan-tum chains (it is the state with all spins aligned along a common direction), this is not truefor the antiferromagnet. Classically, the ground state is the so-called Nรฉel state, with spinsalternating between being aligned and antialigned along a common direction, but quantummechanically the Nรฉel state is no longer an eigenstate of the Heisenberg Hamiltonian. Thisfact can be understood from Colemanโs theorem, which forbids the spontaneous ordering ofa continuous symmetry in one spatial dimension [85]. The absence of an ordered ground state in the antiferromagnet has long been of interestto the physics community. Indeed, shortly after Heisenberg introduced his model of aferromagnet in 1921, Bethe discovered an exact solution of the antiferromagnetic chain withspin ๐ = at each site [47]. However, despite this initial progress, spin chains with ๐ > were not amenable to such techniques, and ๏ฌfty years would pass before their low energyproperties could be characterized. In 1981, Duncan Haldane proposed a radical classi๏ฌcationof antiferromagnetic chains: Those with integral spin ๐ have a ๏ฌnite energy gap abovetheir quantum ground states, and exponentially decaying correlation functions. Meanwhile,those chains with half-odd integral spin have gapless excitations with algebraically decayingcorrelation functions [130].Despite being consistent with Betheโs 1931 solution, Haldaneโs โconjectureโ as it came tobe known, was met with widespread skepticism [129]. This was likely due to the fact that spin-wave theory, a method that allows one to calculate the energy spectrum of antiferromagnetsin higher dimensions, largely agreed with Betheโs one dimensional results. We now knowthis to be a coincidence, but at the time, this suggested to the community that spin waveresults might be reliable in one dimension for all values of ๐ . This would imply that allantiferromagnets would exhibit gapless excitations at low energies. Of course, this wasin direct contradiction with Colemanโs theorem, that invalidated spin wave theory in onedimension, but nonetheless, by the 1980s it was widely accepted that gapless excitations wereuniversal among antiferromagnets. Colemanโs theorem is often confounded with the Mermin-Wagner-Hohenberg theorem, which forbidsan ordered grounds state in two spatial dimensions at ๏ฌnite temperature, and applies equally well to bothferromagnets and antiferromagnets [175, 136].
33n fact, Haldaneโs conjecture was met with surprise in other areas of physics as well. As wewill demonstrate below, his argument hinges on a correspondence between antiferromagnetsand the CP sigma model, a quantum ๏ฌeld theory that was being used as a toy model forquantum chromodynamics at the time [103]. The role of the spin, ๐ , manifests as a topologicalangle ๐ in the sigma model, so that integral ๐ translates to ๐ = ๐ translates to ๐ = ๐ . Thus, Haldaneโs claim about antiferromagnets was also a claim aboutmass gaps in the CP sigma model. At that time, it was widely believed that a ๏ฌnite gap wouldexist for all values of ๐ , and this was known exactly for ๐ =
0, and suggested numericallyfor small values of ๐ [264, 48, 49]. It was shown in [6] that the CP model is gapless at ๐ = ๐ but this is not true for CP ๐ โ with ๐ >
2. In that case there is a ๏ฌrst order transitionat ๐ = ๐ with the model remaining massive. This can be understood from the presence ofrelevant operators allowed by symmetry for ๐ >
2. The generalization of this behaviourto four-dimensional SU ( ๐ ) Quantum Chromodynamics is a fascinating subject [117]. Forlarge ๐ it has been established that the transition is ๏ฌrst order with a ๏ฌnite mass [243, 254].Whether or not this is true for ๐๐ ( ) is an open question.Over the next few years, Haldaneโs conjecture would defy these skeptics, thanks toveri๏ฌcation from multiple areas of research. Experimentally, neutron scattering on theorganic nickel compound NENP, which is a quasi-one dimensional ๐ = ๐ = , ๐ = CP sigma model, Monte Carlo methodswere used to numerically verify the absence of a mass gap when ๐ = ๐ [51, 33, 21, 34,111, 20], and a related integrable model was eventually discovered by the Zamolodchikovbrothers [265].In many cases, the studies carried out in order to verify Haldaneโs claims were scien-ti๏ฌc breakthroughs in their own right. Indeed, the ๏ฌelds of density matrix renormalizationgroup [241, 240, 191], and more generally tensor networks [233, 144, 232], as well as symme-try protected topological matter [185] all originated, in part, due to Haldaneโs conjecture. It isthus not a leap to claim that any generalization of Haldaneโs conjecture would be an impactfulresult to the physics community. And indeed, this is what led physicists, including A๏ฌeck,Read, Sachdev and others to extend Haldaneโs work to SU( ๐ ) generalizations of spin chainsin the late 1980s [7, 10, 6, 203]. At the time, these were purely hypothetical models with noexperimental realization, but thanks to the correspondence between spin chains and sigmamodels, they were still interesting in their own right. Another motivation was a proposedrelation between sigma models and the localization transition in the quantum Hall e๏ฌect [7,164, 104]. And while this unsolved problem remains a motivator to study such models, recentadvances from the cold atom community have revealed that SU( ๐ ) chains (with ๐ โค
10) arenow experimentally realizable, o๏ฌering a much more physical motivation [258, 138, 80, 122,50, 209, 218, 193, 267, 81, 187, 135, 192]. This fact has led to a renewed theoretical interestin the ๏ฌeld of SU( ๐ ) spin chains. As a consequence, an SU( ๐ ) version of Haldaneโs conjecturehas recently been formulated [161, 236, 237, 234].34n this chapter, we review this recent e๏ฌort of extending Haldaneโs conjecture fromSU(2) to SU( ๐ ). We begin in Section 3 by introducing the SU( ๐ ) Heisenberg chain, in therank- ๐ symmetric representation. Unlike the familiar spin chains with SU(2) symmetry, for ๐ >
2, these symmetric representations form only a small subset of all possible irreduciblerepresentations. Near the end of this chapter, we return to this issue and analyze SU( ๐ ) chainsin other representations.Next, in Section 4, we recall various exact results that exist for these SU( ๐ ) Hamiltonians.Speci๏ฌcally, we discuss the Lieb-Shultz-Mattis A๏ฌeck theorem [166, 13], and the A๏ฌeck-Kennedy-Lieb-Tasaki construction [12].In Section 5, we extend the familiar spin-wave theory to these SU( ๐ ) chains, and obtainpredictions for the velocities of low lying excitations. We observe that for ๐ >
3, there aremultiple distinct velocities, which inhibit the automatic emergence of Lorentz invariance.Sections 6 through 9 then provide a step-by-step derivation of a low-energy ๏ฌeld theorydescription of the SU( ๐ ) chain. This extends Haldaneโs original mapping of the spin chainto the CP model; now, the corresponding target space is the complete ๏ฌag manifold,SU ( ๐ )/[ U ( )] ๐ โ . Thus, via these sections we establish a direct link from SU( ๐ ) chains tothe subject matter of Chapter 1. In Section 9, we also explain how the distinct ๏ฌavour wavevelocities ๏ฌow to a common value upon renormalization.The generalized Haldane conjecture is presented in Section 10.2. This combines the exactresults of Section 4 with an analysis of mixed โt Hooft anomalies between the global symme-tries of the chain. After quoting the results, we o๏ฌer a detailed discussion of the mathematicalstructure behind these anomalies, which involves the concept of PSU( ๐ ) bundles.In Section 11.3, we reinterpret the SU( ๐ ) Haldane conjecture in terms of fractionaltopological excitations, which generalize the notion of merons in SU(2) [8]. Finally, inSection 12, we explain how SU( ๐ ) representations other than the rank- ๐ symmetric ones mayadmit a mapping to the same ๏ฌag manifold target space, SU ( ๐ )/[ U ( )] ๐ โ . This leads usto non-Lagrangian embeddings of the ๏ฌag manifold, resulting in the phenomenon that somelow energy excitations have linear dispersion, while others have quadratic dispersion. The familiar Heisenberg spin chain is characterized by a single integer, 2 ๐ , which speci๏ฌesthe irreducible representation of SU(2) that appears on each site. In SU( ๐ ), the most genericirrep is de๏ฌned by ๐ โ ๐ symmetric irreps, which have Young tableaux ๐ (cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123) . (3.1)35he simplest Hamiltonian one is tempted to write down is ๐ป = ๐ฝ โ๏ธ ๐ Tr ( ๐ ( ๐ ) ๐ ( ๐ + )) (3.2)where ๐ ( ๐ ) is an ๐ ร ๐ Hermitian matrix with Tr ( ๐ ) = ๐ , whose entries correspond to the ๐ โ ๐ ) and satisfy [ ๐ ๐ผ๐ฝ , ๐ ๐พ๐ฟ ] = ๐ฟ ๐ผ๐ฟ ๐ ๐พ๐ฟ โ ๐ฟ ๐พ๐ฝ ๐ ๐ผ๐ฟ . (3.3)Indeed, in SU(2), ๐ ๐ผ๐ฝ = (cid:174) ๐ ยท (cid:174) ๐ ๐ผ๐ฝ + ๐ I , and the Hamiltonian appearing in (3.2) equals theHeisenberg model with spin ๐ = ๐ (up to a constant). However, for ๐ >
2, this Hamiltonianpossesses local zero mode excitations that destabilize the classical ground state and inhibita low energy ๏ฌeld theory description. To remedy this, we introduce an additional ๐ โ ๐ป = โ๏ธ ๐ ๐ โ โ๏ธ ๐ = ๐ฝ ๐ Tr ( ๐ ( ๐ ) ๐ ( ๐ + ๐ )) , (3.4)where ๐ฝ couples nearest-neighbours, ๐ฝ couples next-nearest neighbours, and so on. SeeFigure 5 for a pictorial representation of these interactions. This is the Hamiltonian that wewill be studying throughout this chapter.Figure 5: Pictorial representation of the nearest (blue), next-nearest (red), and next-next-nearest (green) neighbour interactions occurring in (3.4), for the case ๐ = In the large- ๐ limit, the commutator (3.3) is subleading in ๐ , allowing us to replace ๐ by amatrix of classical numbers. To this order in ๐ , the Casimir constraints of SU( ๐ ) completelydetermine the eigenvalues of ๐ . We have ๐ ๐ผ๐ฝ = ๐๐ข ๐ผ ๐ข ๐ฝ (3.5) S(j) should be traceless; we have shifted it by a constant to simplify our calculations. ๐ข โ C ๐ with ๐ข โฆ ๐ข =
1. Note that ๐ ๐ผ๐ฝ are the components of the moment map ๐ from (1.10),up to an additive constant term. The interaction terms appearing in (3.2) reduce toTr ( ๐ ( ๐ ) ๐ ( ๐ + ๐ )) = ๐ | ๐ข ( ๐ ) โฆ ๐ข ( ๐ + ๐ ) | . (3.6)Since ๐ข lives in C ๐ , a classical ground state will posses local zero modes unless the Hamil-tonian gives rise to ๐ โ ๐ -site ordered classical ground state, which givesrise to a Z ๐ symmetry in their low energy ๏ฌeld theory description. This Z ๐ symmetry is alsopresent in the ๐ = ๐ -site unit cell through an โorder-by-disorderโmechanism that generates e๏ฌective additional couplings of order ๐ โ that lift the local zeromodes [161, 86].Since the classical ground state minimizing (3.4) has an ๐ -site order, it is characterized by ๐ normalized vectors that mutually minimize (3.6). That is, the classical ground state givesrise to an orthonormal basis of C ๐ . As we recall from section 1.1, the space of ๐ -tuples ofmutually orthogonal vectors, de๏ฌned up to a phase, is the complete ๏ฌag manifold, which isthe the mechanism how ๏ฌag manifolds arise in the context of spin chains. Due to this ๐ -foldstructure, we rewrite the Hamiltonian (3.4) as a sum over unit cells (indexed by ๐ ): ๐ป = โ๏ธ ๐ ๐ โ๏ธ ๐ด = ๐ โ โ๏ธ ๐ = ๐ฝ ๐ Tr ( ๐ ( ๐ ๐ด ) ๐ ( ๐ ๐ด + ๐ )) ๐ ๐ด : = ๐ ๐ + ( ๐ด โ ) . (3.7)In the later sections of this chapter, we will expand about this classical ground state tocharacterize the low energy physics of (3.4). But before this, we review some exact resultsthat apply to SU( ๐ ) Hamiltonians. Haldaneโs original conjecture about SU(2) chains is supported by two rigorous results per-taining to Heisenberg Hamiltonians: the Lieb-Schultz-Mattis theorem [166], and the A๏ฌeck-Kennedy-Lieb-Tasaki construction [12]. Similar results also exist for chains with SU( ๐ )symmetry, and this is what we review in this section. The LSMA theorem is a rigorous statement about ground states in translationally invariantSU( ๐ ) Hamiltonians [166, 13]: 37onsider a translationally- and SU( ๐ )-invariant Hamiltonian of a spin chainwith symmetric rank- ๐ representations at each site. If ๐ is not a multiple of ๐ ,then either the ground state is unique with gapless excitations, or there is aground state degeneracy of at least ๐ gcd ( ๐,๐ ) .Let us show how the original proof in [13] can be extended to models with further rangeinteractions. Explicitly, we consider the following Hamiltonian on a ring of ๐ฟ sites: ๐ป = ๐ โ๏ธ ๐ = ๐ป ๐ ๐ป ๐ : = ๐ฟ โ๏ธ ๐ = ๐ฝ ๐ Tr ( ๐ ( ๐ ) ๐ ( ๐ + ๐ )) (4.1)where ๐ is de๏ฌned as above. We assume that | ๐ (cid:105) is the unique ground state of ๐ป , and istranslationally invariant: ๐ | ๐ (cid:105) = | ๐ (cid:105) . We then de๏ฌne a twist operator ๐ = ๐ ๐ด ๐ด : = ๐๐๐๐ฟ ๐ฟ โ๏ธ ๐ = ๐ ๐ ( ๐ ) (4.2)with ๐ = ๐ โ โ๏ธ ๐ด = ๐ ๐ผ๐ผ โ ( ๐ โ ) ๐ ๐๐ = Tr ( ๐ ) โ ๐๐ ๐๐ = ๐ โ ๐๐ ๐๐ . (4.3)Using the commutation relations (3.3), it is easy to verify that (cid:104) Tr ( ๐ ( ๐ ) ๐ ( ๐ + ๐ )) , ๐ ( ๐ ) + ๐ ( ๐ + ๐ ) (cid:105) = ๐ โ Tr ( ๐ ( ๐ ) ๐ ( ๐ + ๐ )) ๐ = ๐ โ ๐ ๐๐๐๐ฟ ( ๐ ( ๐ + ๐ )โ ๐ ( ๐ )) Tr ( ๐ ( ๐ ) ๐ ( ๐ + ๐ )) ๐ ๐ ๐๐๐๐ฟ ( ๐ ( ๐ + ๐ )โ ๐ ( ๐ )) . (4.5)Using this, one can show that ๐ โ ๐ป๐ = ๐ป + [ ๐ป, ๐ด ] + O ( ๐ฟ โ ) (4.6)so that ๐ | ๐ (cid:105) has energy O ( ๐ฟ โ ) . Now, using the translational invariance of | ๐ (cid:105) , we ๏ฌnd (cid:104) ๐ | ๐ | ๐ (cid:105) = (cid:104) ๐ | ๐ โ ๐๐ | ๐ (cid:105) = (cid:104) ๐ | ๐๐ ๐๐๐ ๐ ( ) ๐ โ ๐๐๐๐ฟ (cid:205) ๐ฟ๐ = ๐ ( ๐ ) | ๐ (cid:105) . (4.7)Since | ๐ (cid:105) is a ground state of ๐ป , it is a SU( ๐ ) singlet, and so must be left unchanged by theglobal SU( ๐ ) transformation ๐ โ ๐๐๐๐ฟ (cid:205) ๐ฟ๐ = ๐ ( ๐ ) . Moreover, using (4.3), we have (cid:104) ๐ | ๐ | ๐ (cid:105) = ๐ ๐๐๐๐ (cid:104) ๐ | ๐๐ ๐๐๐ ๐๐ | ๐ (cid:105) . (4.8)38s shown in section 2.3, the matrices ๐ can be represented in terms of Schwinger bosons;the diagonal elements are then number operators for these bosons. Thus, ๐ ๐๐ acting on | ๐ (cid:105) will always return an integer, and ๐ ๐๐๐ ๐๐ can be dropped. Thus, we ๏ฌnd that so long as ๐ isnot a multiple of ๐ , (cid:104) ๐ | ๐ | ๐ (cid:105) = ๐ | ๐ (cid:105) is a distinct, low-lying state above | ๐ (cid:105) . This completes the proof. Finally,we may also comment on the ground state degeneracy in the event that a gap exists above theground state. Through the repeated application of (4.8), we have (cid:104) ๐ | ๐ ๐ | ๐ (cid:105) = ๐ ๐๐๐๐๐ (cid:104) ๐ | ๐ | ๐ (cid:105) . (4.10)So long as ๐ < ๐ : = ๐ / gcd ( ๐, ๐ ) , the family { ๐ ๐ | ๐ (cid:105)} is an orthogonal set of low lying states.If an energy gap is present, this suggests that the ground state is at least ๐ -fold degenerate. SeeFigures 6 and 7 for a valence bond solid picture of these degeneracies in SU(4) and SU(6),respectively.Figure 6: A valence bond construction for the predicted two-fold degenerate ground stateof SU(4) with ๐ =
2. Each node represents a fundamental ๐ = One of the ๏ฌrst results that bolstered Haldaneโs conjecture was the discovery of the so-calledAKLT model of a spin-1 chain, which exhibits a unique, translationally invariant groundstate with a ๏ฌnite excitation gap [166, 13]. In this case, the number of boxes in the Youngtableau is 2, and so the SU(2) version of the LSMA theorem does not apply. Recently, theAKLT construction has been generalized by various groups to SU( ๐ ) chains [126, 150, 187,78, 177, 207, 125]. Relevant to us are the symmetric representation AKLT Hamiltoniansintroduced in [126]. In particular, for ๐ a multiple of ๐ , Hamiltonians are constructed that39igure 7: Valence bond constructions for SU(6). The left sub๏ฌgure corresponds to ๐ = ๐ =
2, andhas a 3-fold degenerate ground state. Singlets are constructed out of 6 nodes, each of whichrepresents a fundamental irrep in SU(6).exhibit a unique, translationally invariant ground state. See Figure 8 for the case ๐ = ๐ = ๐ not a multiple of ๐ , with ๐ : = ๐ / gcd ( ๐, ๐ ) , Hamiltonians are constructedwith ๐ -fold degenerate ground states that are invariant under translations by ๐ sites (see Figures6, 7). All of these models have short range correlations, and are expected to have gappedground states, based on arguments of spinon con๏ฌnement. The fact that the construction ofa gapped, nondegenerate ground state is only possible when ๐ is a multiple of ๐ is consistentwith the LSMA theorem presented above. projection onto | ฯ i| ฯ i| ฯ i (a) p=1 (b) p=3 Figure 8: AKLT constructions in SU(3). Left: When ๐ โ ๐ , multiple valence bond solidscan be formed. The ground state is not translationally invariant and degenerate. Right: When ๐ = ๐ , a unique, translationally invariant ground state can be constructed, by projecting onto the symmetric- ๐ representation at each site. According to Colemanโs theorem [85], we do not expect spontaneous symmetry breakingof the SU( ๐ ) symmetry in the exact ground state of our Hamiltonian. Nonetheless, we maystill expand about the classical (symmetry broken) ground state to predict the Goldstonemode velocities. If the theory is asymptotically free, then at su๏ฌciently high energies theexcitations may propagate with these velocities [123]. In the familiar antiferromagnet, thisprocedure is known as spin wave theory; in SU( ๐ ), it is called ๏ฌavour wave theory [194, 195].40o begin, we introduce ๐ bosons in each unit cell to reproduce the commutation relationsof the ๐ matrices: ๐ ๐ผ๐ฝ ( ๐ ๐ด ) = ๐ โ ,๐ผ๐ด ๐ ๐ฝ๐ด . (5.1)The counting is ๐ ๏ฌavours of bosons for each of the ๐ sites of a unit cell. The conditionTr ( ๐ ) = ๐ implies there are ๐ bosons at each site. The classical ground state involves onlyโdiagonalโ bosons of the type ๐ ๐ด๐ด and ๐ โ ,๐ด๐ด . The โo๏ฌ-diagonalโ bosons are Holstein-Primako๏ฌbosons; in SU(2) they correspond to the operators ๐ด, ๐ด โ introduced in Section 2.4. Flavourwave theory allows for a small number of Holstein-Primako๏ฌ bosons at each site, capturedby ๐ ( ๐ ๐ด ) = โ๏ธ ๐ผ โ ๐ด ๐ โ ,๐ผ๐ด ๐ ๐ผ๐ด , and writes the Hamiltonian (3.4) in terms of these ๐ ( ๐ โ ) bosons. In the large ๐ (cid:29) ๐ ( ๐ ๐ด ) limit, we expand ๐ ๐ด๐ด ( ๐ฝ ๐ด ) = ๐ โ ๐ ( ๐ ๐ด ) ,๐ ๐ผ๐ด ( ๐ ๐ด ) โ โ ๐๐ โ ,๐ผ๐ด ,๐ ๐ด๐ผ ( ๐ ๐ด ) โ โ ๐๐ ๐ผ๐ด , to ๏ฌnd Tr ( ๐ ( ๐ ๐ด ) ๐ ( ๐ ๐ต )) = ๐ (cid:104) ๐ โ ,๐ด๐ต ๐ ๐ด๐ต + ๐ โ ,๐ด๐ต ๐ ๐ต๐ด + ๐ โ ,๐ต๐ด ๐ โ ,๐ด๐ต + ๐ ๐ต๐ด ๐ ๐ด๐ต (cid:105) + O ( ๐ ) . (5.2)In terms of these degrees of freedom, the Hamiltonian (3.4) decomposes into a sum ๐ป = โ๏ธ ๐ด<๐ต ๐ป ๐ด๐ต , (5.3)where ๐ป ๐ด๐ต is a Hamiltonian involving only the two boson ๏ฌavours ๐ ๐ด๐ต and ๐ ๐ต๐ด . In momentumspace, this gives ๐ ( ๐ โ ) di๏ฌerent 2 ร ๐ป ๐ด,๐ด + ๐ก = const. + โ๏ธ ๐ ๐ ๐ก ( ๐ ) โ๏ธ ๐ = (cid:18) ๐ โ ,๐๐ก ( ๐ ) ๐ ๐๐ก ( ๐ ) + (cid:19) (5.4)where ๐ ๐ก ( ๐ ) = ๐ โ๏ธ ๐ฝ ๐ก ๐ฝ ๐ โ ๐ก (cid:12)(cid:12)(cid:12)(cid:12) sin ๐๐ ๐ (cid:12)(cid:12)(cid:12)(cid:12) . (5.5)Therefore, the corresponding ๏ฌavour wave velocities are ๐ฃ ๐ก = ๐ ๐ โ๏ธ ๐ฝ ๐ก ๐ฝ ๐ โ ๐ก ๐ก = , , ยท ยท ยท , ๐ โ ๐ is odd, there are ๐ modes with each ๏ฌavour wave velocity. When ๐ is even, this istrue except for the velocity ๐ฃ ๐ , which has only ๐ modes. In each case, the number of modesadds up to ๐ ( ๐ โ ) . We note that for ๐ >
3, there is no longer a unique velocity, and theemergence of Lorentz invariance is absent. Only for a speci๏ฌc ๏ฌne tuning of the couplingscan Lorentz invariance be restored. These tuned models were the ones considered in [69]and [73]. 41
Derivation of the continuum theory
In the present section our goal is to derive a path integral representation for spin chains withHamiltonians of the type (3.7), using coherent states introduced in section 2.3.4. As a warm-up, we will start with a simpler example of a single SU(2)-spin (which may be thought of as aspin chain with one site) in Section 6.1 (such systems were considered in [55], for example).The extension to a spin chain is rather straightforward and is discussed in Section 6.2. Thereexist two di๏ฌerent continuum limits of the spin chain: one based on the ferromagnetic groundstate, which leads to a Landau-Lifschitz model with quadratic dispersion relations for the spinwaves (Section 6.3), and one based on the anti-ferromagnetic ground state (the Haldane-typelimit), which leads to linear dispersion and will be elaborated on in Section 7.3. Althoughthese two situations are rather di๏ฌerent, as we shall see in Section 12, a general spin chainwith equivalent (but arbitrary) representations at all sites leads to a continuum theory withboth linear and quadratic dispersion modes, uniting the two cases. ๐ Let us consider in detail the case of SU(2) . We introduce the notation ๐ด ( ๐, ๐ฃ ) for thenormalized matrix element between coherent states of any operator ห ๐ด (which for historicalreasons is called the kernel of ห ๐ด ): ๐ด ( ๐, ๐ฃ ) = (cid:104) ๐ | ห ๐ด | ๐ฃ (cid:105)(cid:104) ๐ | ๐ฃ (cid:105) .Once again we will consider the rank- ๐ symmetric representations , and for such repre-sentations the scalar product of coherent states is (cid:104) ๐ | ๐ฃ (cid:105) = ( ๐ โฆ ๐ฃ ) ๐ . (6.1)This can be proven, for instance, by using the Fock space expressions of the type (2.43) forthe coherent states. Viewing the Hilber space as a subspace of ( C ) โ ๐ , for the Hamiltonianacting on the quantum sphere we shall takeห ๐ = ๐ โ๏ธ ๐ = โ ... โ ๐ โ ๐ -th position โ ... โ , (6.2)which may be interpreted as an external magnetic ๏ฌeld (cid:174) ๐ป in the ๐ง -direction (under theassumption of a (cid:174) ๐ ยท (cid:174) ๐ป coupling). The kernel of ห ๐ is ๐ ( ๐, ๐ฃ ) = ๐ ๐ ๐ฃ โ ๐ ๐ฃ ๐ ๐ฃ + ๐ ๐ฃ .We will now present the derivation of the kernel of the โevolution operatorโห ๐ = ๐ โ ๐๐ผ ห ๐ . (6.3)Of course, in this ๏ฌnite-dimensional case this is merely a pedagogical exercise, since theaction of ห ๐ on a coherent state simply gives ห ๐ | ๐ฃ (cid:105) = | ๐ โ ๐๐ผ๐ โฆ ๐ฃ (cid:105) , and the corresponding General results along a similar line of reasoning were obtained in [17]. For SU(2) all representations are of this type. ๐ ( ๐, ๐ฃ ) = (cid:32) ๐ โ i ๐ผ ๐ ๐ฃ + ๐ i ๐ผ ๐ ๐ฃ ๐ ๐ฃ + ๐ ๐ฃ (cid:33) ๐ (6.4)As is standard in path integral calculations [107], in order to write a path integral repre-sentation for a matrix element ๐ ( ๐, ๐ฃ ) , one ๏ฌrst needs to know the matrix elements of thegenerator ห ๐ . Then one splits the โtimeโ interval ๐ผ into ๐ subintervals of length ๐ผ๐ and usesthe formula ห ๐ = lim ๐ โโ (cid:16) โ i ๐ผ๐ ห ๐ (cid:17) ๐ : = lim ๐ โโ ห ๐ ๐ . Inserting the completeness relation (2.46)between every two factors of ห ๐ , we arrive at the following expression for the kernel of theevolution operator: ๐ ( ๐, ๐ฃ ) = (cid:104) ๐ | ห ๐ | ๐ฃ (cid:105)(cid:104) ๐ | ๐ฃ (cid:105) = lim ๐ โโ โซ ๐ โ (cid:214) ๐ = ๐๐ ( ๐ง ( ๐ ) , ๐ง ( ๐ )) ร (6.5) ร ๐ ( ๐, ๐ง ( ๐ โ )) ยท ๐ ( ๐ง ( ๐ โ ) , ๐ง ( ๐ โ )) ยท ยท ยท ๐ ( ๐ง ( ) , ๐ง ( )) ยท ๐ ( ๐ง ( ) , ๐ฃ ) รร (cid:104) ๐ | ๐ง ( )(cid:105)(cid:104) ๐ง ( ) | ๐ง ( )(cid:105) ... (cid:104) ๐ง ( ๐ โ ) | ๐ง ( ๐ โ )(cid:105)(cid:104) ๐ง ( ๐ โ ) | ๐ฃ (cid:105)(cid:104) ๐ | ๐ฃ (cid:105)(cid:104) ๐ง ( ) | ๐ง ( )(cid:105) ... (cid:104) ๐ง ( ๐ โ ) | ๐ง ( ๐ โ )(cid:105) Here ๐ง ( ๐ ) is the coherent state of the ๐ th โtime sliceโ. To complete the derivation we use theexplicit expression ๐ ( ๐ง ( ๐ + ) , ๐ง ( ๐ )) = โ ๐ i ๐ผ๐ ๐ง ( ๐ )โฆ ๐ โฆ ๐ง ( ๐ + ) ๐ง ( ๐ )โฆ ๐ง ( ๐ + ) for ๐ and (6.1) for the scalarproduct of coherent states.We now want to โtake the limitโ in the formula (6.5), assuming that ๐ง ( ๐ + ) โ ๐ง ( ๐ ) โผ ๐ (cid:164) ๐ง ( ๐ + ) (for a justi๏ฌcation of this procedure see [268]). In order to do it we write the factors ๐ง ( ๐ )โฆ ๐ง ( ๐ + ) ๐ง ( ๐ + )โฆ ๐ง ( ๐ + ) in the following form: ๐ง ( ๐ ) โฆ ๐ง ( ๐ + ) ๐ง ( ๐ + ) โฆ ๐ง ( ๐ + ) = (cid:18) โ ( ๐ง ( ๐ + ) โ ๐ง ( ๐ )) โฆ ๐ง ( ๐ + ) ๐ง ( ๐ + ) โฆ ๐ง ( ๐ + ) (cid:19) (cid:39) โ ๐ (cid:164) ๐ง ( ๐ + ) โฆ ๐ง ( ๐ + ) ๐ง ( ๐ + ) โฆ ๐ง ( ๐ + ) for ๐ = , , ..., ๐ โ
2. Then we obtain ๐ ( ๐, ๐ฃ ) = โซ (cid:214) ๐ก โ[ , ] ๐๐ ( ๐ง ( ๐ก ) , ๐ง ( ๐ก )) (cid:18) ๐ง ( ) โฆ ๐ฃ๐ โฆ ๐ฃ (cid:19) ๐ exp (cid:169)(cid:173)(cid:171) โ ๐ โซ ๐๐ก (cid:164) ๐ง โฆ ๐ง๐ง โฆ ๐ง โ ๐ i ๐ผ โซ ๐๐ก ๐ง โฆ ๐ โฆ ๐ง๐ง โฆ ๐ง (cid:170)(cid:174)(cid:172) , (6.6)with boundary conditions ๐ง ( ) = ๐, ๐ง ( ) = ๐ . The action in the exponent should be somewhatreminiscent of the action (2.9) that we encountered in Chapter 1. Indeed, if in that formulawe set ๐ด ๐ (cid:164) ๐ฅ ๐ = ๐ผ ๐ , we would arrive exactly at the action in (6.6), upon normalizing thecoordinates as | ๐ง | = ๐ง into ๐ข ).Before concluding this section, let us demonstrate that one can actually calculate the pathintegral (6.6). To this end note that the equations of motion following from the action in theexponent of (6.6) describe the rotation of the sphere around its ๐ง -axis (the one orthogonal tothe plane of the stereographic projection). Passing to the inhomogeneous coordinate ๐ง via43 = โ +| ๐ง | , ๐ข = ๐ง โ +| ๐ง | , we ๏ฌnd that the e.o.m. are the equations of harmonic oscillations:i (cid:164) ๐ง = ๐ผ๐ง, i (cid:164) ๐ง = โ ๐ผ๐ง. (6.7)In fact with a particular choice of coordinates the Lagrangian standing in the exponent in (6.6)may be turned exactly into the canonical Lagrangian of the harmonic oscillator, but this isnot necessary for our purposes. Solving the equations with the prescribed initial conditions ๐ง ( ) = ๐ ๐ : = ๐, ๐ง ( ) = ๐ฃ ๐ฃ : = ๐ฃ , we obtain ๐ง ( ๐ก ) = ๐๐ ๐ผ๐ก and ๐ง ( ๐ก ) = ๐ฃ๐ โ ๐ผ ( ๐ก โ ) . Pluggingthis into the exponent of the path integral (6.6), we get ๐ โ ๐ i ๐ผ . The term ๐ง ( )โฆ ๐ฃ๐ โฆ ๐ฃ in front of theexponent produces + ๐ ๐ฆ ๐ ๐ผ + ๐๐ฆ , and altogether we get ๐ ( ๐, ๐ฃ ) = (cid:18) ๐ โ i ๐ผ + ๐ ๐ฃ ๐ i ๐ผ + ๐๐ฃ (cid:19) ๐ , (6.8)which, as we know from (6.4), is the right answer. Similarly to what we did in (6.6), we now want to derive a path integral expression for theevolution operator of the spin chain ห U = ๐ ๐๐ผ ห ๐ป , ห ๐ป now being a spin chain Hamiltonian. At thesame time we pass from the simple SU(2) case to the SU(3), or even SU( ๐ ) model. We startwith the Heisenberg Hamiltonianห ๐ป Heis = ๐ ๐ฟ โ๏ธ ๐ = ( Tr ( ๐ ( ๐ ) ๐ ( ๐ + )) + const. ) , (6.9)where the spin operators are assumed to be in the symmetric powers of the fundamentalrepresentation, indexed by ๐ as before, and the constant may be chosen at our will. In orderto build the path integral we ๏ฌrst need to know the matrix elements of the Hamiltonian itself,which amounts to knowing the matrix elements of P : = ๐ ( Tr ( ๐ ( ๐ ) ยท ๐ ( ๐ + )) + const. ) .This operator acts in the tensor product Sym ( C ๐ ) โ ๐ โ Sym ( C ๐ ) โ ๐ and (for a suitable choiceof the additive constant) is a restriction of the operator acting in ( C ๐ ) โ ๐ โ ( C ๐ ) โ ๐ as asum of permutations: P = ๐ ๐ (cid:205) ๐ ,๐ก = ๐ ๐ ,๐ก , where ๐ ๐ ,๐ก is the permutation of the ๐ -th and ๐ก -th C ๐ -factors in the two copies of ( C ๐ ) โ ๐ . The tensor product of coherent states has the form ( ๐ ( ๐ ) โฆ ๐ฃ ( ๐ )) ๐ ( ๐ ( ๐ + ) โฆ ๐ฃ ( ๐ + )) ๐ , andthe matrix elements of ห P is easily found to be P ( ๐ , ๐ ; ๐ฃ , ๐ฃ ) = (cid:104) ๐ , ๐ | ห P | ๐ฃ , ๐ฃ (cid:105)(cid:104) ๐ , ๐ | ๐ฃ , ๐ฃ (cid:105) = ๐ ( ๐ โฆ ๐ฃ ) ( ๐ โฆ ๐ฃ )( ๐ โฆ ๐ฃ ) ( ๐ โฆ ๐ฃ ) . (6.10)Now we can essentially repeat the steps from the previous section. The only di๏ฌculty isnotational and it comes from the fact that in this case, as opposed to the previous example,44e essentially have two โspace-timeโ directions: one โtimeโ or ๐ผ -direction, and a secondโspatialโ direction in which the spin chain is extended. As a consequence, our variables ๐ง will now take two arguments: ๐ง ( ๐, ๐ ) , where ๐ is the time index, and ๐ enumerates the sitesof the spin chain. The integrand will again split into two terms: the ๏ฌrst being a geometricphase term, and the second being the Hamiltonian: ๐ ( ๐, ๐ฃ ) = lim ๐พ โโ โซ (cid:214) ๐, ๐ ๐๐ ( ๐ง ( ๐, ๐ ) , ๐ง ( ๐, ๐ )) ร ๐ผ geom ร ๐ผ ๐ป (6.11)The geometric term is local in the spin chain index ๐ and has a simplest (nearest-neighbor,or ๏ฌrst-order) nonlocality in time, which is a general feature, since in the continuum limit itshould lead to a one-form: ๐ผ geom = (cid:214) ๐, ๐ (cid:18) ๐ง ( ๐, ๐ ) โฆ ๐ง ( ๐ + , ๐ ) ๐ง ( ๐ + , ๐ ) โฆ ๐ง ( ๐ + , ๐ ) (cid:19) ๐ (6.12)On the other hand, the Hamiltonian term has a ๏ฌrst-order nonlocality in the spin-chaindirection, but also has a ๏ฌrst-order nonlocality in the time direction, since the matrix elementsof the Hamiltonian entering the integral are always of the form ๐ (cid:104) ๐ง ( ๐ + , ๐ ) | ห ๐ป | ๐ง ( ๐, ๐ )(cid:105) . Thelatter nonlocality will not play a role, since the contribution of such matrix element alwayscomes with a damping factor ๐ , and the nonlocality being of order ๐ as well enters onlysubleading terms. In any case, the contribution of the Hamiltonian may be written as ๐ผ H = (cid:214) ๐ (cid:32) + ๐ ๐๐ผ๐ โ๏ธ ๐ ๐ง ( ๐, ๐ ) โฆ ๐ง ( ๐ + , ๐ + ) ๐ง ( ๐, ๐ ) โฆ ๐ง ( ๐ + , ๐ ) ๐ง ( ๐, ๐ + ) โฆ ๐ง ( ๐ + , ๐ ) ๐ง ( ๐, ๐ + ) โฆ ๐ง ( ๐ + , ๐ + ) (cid:33) (6.13)We may now exponentiate these expressions and take the limit ๐ โ โ , thus obtaining acontinuous time variable ๐ก : ๐ ( ๐, ๐ฃ ) = โซ (cid:214) ๐ก โ[ , ] (cid:214) ๐ ๐๐ ( ๐ง ( ๐ก, ๐ ) , ๐ง ( ๐ก, ๐ )) (cid:18) ๐ง ( , ๐ ) โฆ ๐ง ( , ๐ ) ๐ง ( , ๐ ) โฆ ๐ง ( , ๐ ) (cid:19) ๐ exp ( i S) , (6.14)where S = ๐ โซ ๐๐ก โ๏ธ ๐ (cid:18) ๐ (cid:164) ๐ง ( ๐ ) โฆ ๐ง ( ๐ )| ๐ง ( ๐ ) | + ๐ผ | ๐ง ( ๐ ) โฆ ๐ง ( ๐ + ) | | ๐ง ( ๐ ) | | ๐ง ( ๐ + ) | (cid:19) (6.15)with boundary conditions ๐ง ( , ๐ ) = ๐ ( ๐ ) , ๐ง ( , ๐ ) = ๐ฃ ( ๐ ) . We have suppressed the timeargument in the second line above. The nontrivial question is how to take the continuumlimit in the spin chain direction, indexed by โ ๐ โ, โ there are several inequivalent ways to doit. It is well-known that the isotropic (โXXXโ) spin chain has two โvacuaโ, i.e. the states (ormultiplets) with minimal and maximal energy. They also correspond to the extremal valuesof the spin: the vacuum with spin zero (or least possible spin in case the length of the chaindoes not allow for zero spin) is called antiferromagnetic, whereas the state with maximal spin(proportional to ๐ฟ โ the length of the chain) is called ferromagnetic. Which one of thesestates is the true vacuum depends, of course, on the sign of the Hamiltonian.45 .3 Ferromagnetic limit The ferromagnetic limit is especially simple. It corresponds to the case where the ๐ง โs at theneighboring sites are very close to each other, that is ๐ง ( ๐ + ) โ ๐ง ( ๐ ) โผ ๐ฟ ( ๐ฟ is the length ofthe spin chain, i.e. the number of sites). The ๏ฌrst term in (6.15) then simply produces โซ ๐๐ก โ๏ธ ๐ i (cid:164) ๐ง ( ๐ ) โฆ ๐ง ( ๐ ) ๐ง ( ๐ ) โฆ ๐ง ( ๐ ) โ ๐ฟ โซ ๐๐ก / โซ โ / ๐๐ฅ i (cid:164) ๐ง ( ๐ก, ๐ฅ ) โฆ ๐ง ( ๐ก, ๐ฅ ) ๐ง ( ๐ก, ๐ฅ ) โฆ ๐ง ( ๐ก, ๐ฅ ) , (6.16)whereas the expression in the second term can be rewritten in the same spirit: โ๏ธ ๐ | ๐ง ( ๐ ) โฆ ๐ง ( ๐ + ) | | ๐ง ( ๐ ) | | ๐ง ( ๐ + ) | = โ โ๏ธ ๐ | ฮ ๐ง ( ๐ ) | | ๐ง ( ๐ ) | โ | ๐ง ( ๐ ) โฆ ฮ ๐ง ( ๐ ) | | ๐ง ( ๐ ) | + ยท ยท ยท Upon taking the continuum limit and rescaling ๐ฅ โ ๐ฟ ๐ฅ the full action acquires the form S = ๐ โซ ๐๐ก โซ R ๐๐ฅ (cid:20) i (cid:164) ๐ง ( ๐ก, ๐ฅ ) โฆ ๐ง ( ๐ก, ๐ฅ ) ๐ง ( ๐ก, ๐ฅ ) โฆ ๐ง ( ๐ก, ๐ฅ ) โ (cid:18) ๐ ๐ฅ ๐ง โฆ ๐ ๐ฅ ๐ง๐ง โฆ ๐ง โ ( ๐ง โฆ ๐ ๐ฅ ๐ง ) ( ๐ ๐ฅ ๐ง โฆ ๐ง )( ๐ง โฆ ๐ง ) (cid:19) (cid:21) (6.17)Non-relativistic sigma-models of the type (6.17) are known as Landau-Lifshitz models .The target space of the model we have described is, obviously, CP ๐ โ (for example, in thesecond term in (6.17) one immediately recognizes the Fubini-Study metric). The simplestexample corresponds to ๐ =
2, i.e. when the target space is a usual 2-sphere. In this casethe model is also known as the classical Heisenberg ferromagnet, and it is customary to usethe unit three-vector (cid:174) ๐ instead of the complex coordinates ๐ง, ๐ง (the two parametrizations arerelated via the stereographic projection: ๐ + i ๐ = ๐ง + ๐ง๐ง , ๐ = โ ๐ง๐ง + ๐ง๐ง ). Then the e.o.m., whichfollows from Lagrangian (6.17), is: ๐ (cid:174) ๐๐๐ก = (cid:174) ๐ ร ๐ (cid:174) ๐๐๐ฅ . (6.18)Expanding around a constant magnetization direction, (cid:174) ๐ = (cid:174) ๐ + ๐ฟ (cid:174) ๐ , one obtains the linearequation ๐๐ฟ (cid:174) ๐๐๐ก = (cid:174) ๐ ร ๐ ๐ฟ (cid:174) ๐๐๐ฅ , which describes spin waves with quadratic dispersion. The antiferromagnetic limit is much more involved. The main di๏ฌerence is that in this casethe ๐ง -variables on neighboring sites are no longer close to each other. Let us ๏ฌrst elaborateon the case of the sphere, that is ๐ =
2, which was for the ๏ฌrst time explored in [130]. The mathematical structures behind such models, in particular the connection with the geometry of loopgroups, are discussed in [27].
46n this case it is intuitively clear that the antiferromagnetic limit corresponds to the casewhere the spins on the neighboring sites have opposite directions, i.e. (cid:174) ๐ ( ๐ + ) (cid:39) โ(cid:174) ๐ ( ๐ ) . Interms of the complex coordinates used above this may be written as ๐ง ( ๐ + ) (cid:39) โ ๐ง ( ๐ ) , or,using homogeneous coordinates, as ๐ง ( ๐ + ) = ๐ง ( ๐ ) , ๐ง ( ๐ + ) = โ ๐ง ( ๐ ) . Such a simpleexplanation is due to the fact that on the sphere there exists the antipodal involution, whichin that case is also unique. However, this is no longer true for CP ๐ โ with ๐ โฅ
3. This is thereason why it is not immediately obvious how one can extend the CP analysis to a higher-dimensional projective space. The answer crucially depends on the particular Hamiltonianat hand. The ๏ฌrst model after the CP -case to be successfully analyzed in [10] was the oneof an alternating spin chain, so let us now recall how this was accomplished. First of all let us consider the case of a spin chain with alternating representations: that is,on even sites one has some representation R and on odd sites the dual one R . In particular,this means that these representations can combine into a singlet and hence form an anti-ferromagnetic con๏ฌguration. For the Hamiltonian one takes the Heisenberg Hamiltonian ๐ป = ๐ฟ โ๏ธ ๐ = Tr ( ๐ ( ๐ ) ๐ ( ๐ + )) , (7.1)where it is understood that ๐ ( ๐ ) and ๐ ( ๐ + ) are in conjugate representations. For simplicitywe assume in this section that either ๐ or ๐ is the rank- ๐ symmetric representation. Thegeneralized Haldane limit for this kind of spin chain was constructed in [10]. In orderto rephrase these results one should follow the steps of the previous section to obtain thefollowing action in the ๐ก -continuum limit: S = ๐ โซ ๐๐ก โ๏ธ ๐ (cid:18) ๐ (cid:164) ๐ง ( ๐ ) โฆ ๐ง ( ๐ )| ๐ง ( ๐ ) | + ๐ผ | ๐ง ( ๐ ) โฆ ๐ง ( ๐ + ) | | ๐ง ( ๐ ) | | ๐ง ( ๐ + ) | (cid:19) . (7.2)The di๏ฌerence between the second terms in (6.15) and (7.2) precisely re๏ฌects the di๏ฌerencebetween the representations at adjacent sites. The minimum of the Hamiltonian H = โ | ๐ง ( ๐ )โฆ ๐ง ( ๐ + )| | ๐ง ( ๐ )| | ๐ง ( ๐ + )| is clearly reached for ๐ง ( ๐ + ) = ๐ง ( ๐ ) . The important observation is thatfor such con๏ฌgurations the ๏ฌrst term in (7.2) turns into a full derivative, since on everytwo neighboring sites ๐ (cid:164) ๐ง ( ๐ )โฆ ๐ง ( ๐ )| ๐ง ( ๐ )| + ๐ (cid:164) ๐ง ( ๐ + )โฆ ๐ง ( ๐ + )| ๐ง ( ๐ + )| = ๐ ๐๐๐ก ( log | ๐ง ( ๐ ) | ) . There is a simple butfundamental explanation of this fact. Consider the space CP ๐ โ ร CP ๐ โ , with the symplecticform on it being the sum of two Fubini-Study forms (1.16): ฮฉ = ฮฉ + ฮฉ . We then have thefollowing statement : The de๏ฌnition of Lagrangian submanifold was given in section 1.4. CP ๐ โ โ CP ๐ โ ร CP ๐ โ , def. by ๐ง โ ( ๐ง, ๐ง ) , is Lagrangian. (7.3)Indeed, since ฮฉ = ๐๐ , where ๐ is the one-form entering the ๏ฌrst term in (7.2), and therestriction of the symplectic form ฮฉ (cid:12)(cid:12) CP ๐ โ = ๐ (cid:12)(cid:12) CP ๐ โ = ๐ ๐ for somefunction ๐ , implying that the kinetic term is a total derivative.Let us now expand the action (7.2) around the โvacuumโ ๐ง ( ๐ + ) = ๐ง ( ๐ ) . The variables ๐ง ( ๐ + ) and ๐ง ( ๐ + ) are expressed in terms of ๐ง ( ๐ ) in the following fashion: ๐ง ( ๐ + ) = ๐ง ( ๐ ) + ๐ฟ ๐ ( ๐ ) , ๐ง ( ๐ + ) = ๐ง ( ๐ ) + ๐ฟ ๐ง ( ๐ ) (cid:48) (7.4)For convenience we introduce the projector ฮ ( ๐ ) = โ ๐ง ( ๐ )โ ๐ง ( ๐ )| ๐ง ( ๐ )| onto the subspace of C ๐ orthogonal to the vector ๐ง ( ๐ ) . Then the terms in the Hamiltonian have the following expan-sions: | ๐ง ( ๐ ) โฆ ๐ง ( ๐ + ) | | ๐ง ( ๐ ) | | ๐ง ( ๐ + ) | (cid:39) ๐ฟ ๐ ( ๐ ) โฆ ฮ ( ๐ ) โฆ ๐ ( ๐ )| ๐ง ( ๐ ) | , | ๐ง ( ๐ + ) โฆ ๐ง ( ๐ + ) | | ๐ง ( ๐ ) | | ๐ง ( ๐ + ) | (cid:39) ๐ฟ (cid:101) ๐ ( ๐ ) โฆ ฮ ( ๐ ) โฆ (cid:101) ๐ ( ๐ )| ๐ง ( ๐ ) | , (7.5)where (cid:101) ๐ ( ๐ ) = ๐ ( ๐ ) โ ๐ง ( ๐ ) (cid:48) . The kinetic terms are expanded as follows: ๐ (cid:164) ๐ง ( ๐ ) โฆ ๐ง ( ๐ )| ๐ง ( ๐ ) | + ๐ (cid:164) ๐ง ( ๐ + ) โฆ ๐ง ( ๐ + )| ๐ง ( ๐ + ) | = ๐ ๐ฟ ๐ ( ๐ ) โฆ ฮ ( ๐ ) โฆ (cid:164) ๐ง ( ๐ ) โ ๐ ( ๐ ) โฆ ฮ ( ๐ ) โฆ (cid:164) ๐ง ( ๐ )| ๐ง ( ๐ ) | + . . . (7.6)where . . . denotes a full derivative. Thus, the action (7.2) acquires the following form: S = ๐ โซ ๐๐ก โ๏ธ ๐ | ๐ง ( ๐ ) | (cid:16) ๐ ๐ฟ (cid:2) ๐ ( ๐ ) โฆ ฮ ( ๐ ) โฆ (cid:164) ๐ง ( ๐ ) โ ๐ ( ๐ ) โฆ ฮ ( ๐ ) โฆ (cid:164) ๐ง ( ๐ ) (cid:3) + (7.7) + ๐ฟ [ ๐ ( ๐ ) โฆ ฮ ( ๐ ) โฆ ๐ ( ๐ ) + ( ๐ ( ๐ ) โ ๐ง ( ๐ ) (cid:48) ) โฆ ฮ ( ๐ ) โฆ ( ๐ ( ๐ ) โ ๐ง ( ๐ ) (cid:48) )] (cid:17) . Now we simply need to โintegrate outโ the ๏ฌelds ๐, ๐ . Upon setting ๐, ๐ equal to theirstationary values we also pass to the continuum limit with respect to the โ ๐ โ index. Thisleads to the following expression: S = ๐ โซ ๐๐ก โ โซ โโ ๐๐ฅ (cid:104) ๐ ๐ ๐ง ( ๐ฅ, ๐ก ) โฆ ฮ ( ๐ง, ๐ง )| ๐ง ( ๐ฅ, ๐ก ) | โฆ ๐ ๐ ๐ง ( ๐ฅ, ๐ก ) (7.8) โ i2 ๐ ๐๐ ๐ ๐ ๐ง ( ๐ฅ, ๐ก ) โฆ ฮ ( ๐ง, ๐ง )| ๐ง ( ๐ฅ, ๐ก ) | โฆ ๐ ๐ ๐ง ( ๐ฅ, ๐ก ) (cid:105) . CP ๐ โ sigma model, whereas the secondterm is the pull-back to the worldsheet of the Kรคhler form. The second term is topologicaland corresponds to the theta-angle ๐ = ๐ ๐ mod 2 ๐ . ๐ limit A useful method for the analysis of vector-like systems (such as the CP ๐ โ model, wherethe dynamical variable is the vector ๐ง ) is the ๐ -expansion. The ๏ฌrst step is in rewriting themodel (7.8) as a gauged linear sigma model (GLSM). This is the ๏ฌrst time we encounter suchsystems in this review, but later this point of view will be useful in the discussion of anomaliesin Section 10.2, and even essential in the analysis of the integrable models in Chapter 3. TheGLSM action reads S = โซ ๐ ๐ฅ (cid:34) ๐ โ๏ธ ๐ผ = | ๐ท ๐ ๐ง ๐ผ | โ ๐ (cid:32) ๐ โ๏ธ ๐ผ = | ๐ง ๐ผ | โ ๐๐ (cid:33) (cid:35) + ๐ ๐ โซ ๐๐ด. (7.9)Here ๐ง ๐ผ are the components of ๐ง , which is normalized as | ๐ง | = ๐๐ (we have introduced theโt Hooft coupling constant ๐ of the sigma model), ๐ท ๐ is a U ( ) -covariant derivative, i.e. ๐ท ๐ ๐ง ๐ผ = ๐ ๐ ๐ง ๐ผ โ ๐ ๐ด ๐ ๐ง ๐ผ , and ๐ is a Lagrange multiplier imposing the normalization constraint.The relation to (7.8) is as follows. The model (7.8) is invariant w.r.t. complex rescalings ofthe vector ๐ง , i.e. ๐ง โ ฮป ๐ง with ฮป โ C โ , which is in accordance with the complex de๏ฌnitionof the projective space. We have used this freedom to normalize the ๐ง vector. Even moreimportantly, we have introduced a gauge ๏ฌeld ๐ด ๐ . This gauge ๏ฌeld does not have a kineticterm and enters the Lagrangian (7.9) only algebraically. It can be eliminated via its e.o.m.,which then leads one back to the system (7.8).In the model (7.8), the value of the topological angle is ๐ = ๐ ๐ , however for the presentdiscussion we prefer to leave it as a free parameter. The point of rewriting the action in theform (7.9) is that it has become quadratic in the ๐ง ๏ฌelds, so that they can be integrated out.The resulting action of the ๐ and ๐ด ๐ ๏ฌelds is S = ๐ (cid:20) ๐ Tr Log (cid:16) โ ๐ท ๐ โ ๐ (cid:17) + โซ ๐ ๐ฅ ๐๐ (cid:21) + ๐ ๐ โซ ๐๐ด. (7.10)Since this expression appears in the exponent in the integrand of the path integral, thelarge- ๐ limit corresponds to a stationary phase approximation. The critical point equation,obtained by varying w.r.t. ๐ , โ the so-called gap equation โ has the form (due to Lorentz andtranslational invariance one sets ๐ด ๐ = ๐ = const . at the critical point)1 ๐ โ โซ ๐ ๐ ( ๐ ) ๐ + ๐ = . (7.11)As the integral is UV-divergent, one imposes a cut-o๏ฌ ฮ , and the solution is ๐ = ฮ ๐ โ ๐๐ . (7.12)49 a l d a n e Dimer G a p l e ss AKLT (Exact ground state)SU ( ) fund.-fund. integrableSU ( ) spin-1 integrable ฯ Spin-1 HeisenbergSU ( ) fund.-antifund. integrable Figure 9: Phases of the bilinear-biquadratic spin chain.To get a qualitative picture of the phenomenon one may substitute this value into the originalaction (7.9), arriving at a system of ๐ massive ๏ฌelds ๐ง ๐ผ (with mass ๐ = ๐ ) interacting witha gauge ๏ฌeld ๐ด ๐ . One can show [90, 91, 246] that the e๏ฌect of the gauge ๏ฌeld is to generatean attractive ๐ -dependent potential between the โquarksโ ๐ง and โantiquarksโ ๐ง , which con๏ฌnesthem for all values of ๐ [246]. The mass of the lowest bound state is 2 ๐ + . . . , where . . . arepower-like corrections in ๐ (which also depend on ๐ ). One concludes that, for large ๐ , themodel is massive for all values of ๐ . For a review of ๐ -dependence in sigma models and ingauge theories (as well as for the references on related lattice calculations) cf. [231].An interesting relation to spin chains may be obtained by considering the so-calledbilinear-biquadratic spin chain [7] de๏ฌned by the Hamiltonian ๐ป bi =
12 cos ฯ โ๏ธ ๐ Tr ( ๐ ( ๐ ) ๐ ( ๐ + )) โ
14 sin ฯ โ๏ธ ๐ ( Tr ( ๐ ( ๐ ) ๐ ( ๐ + ))) , (7.13)where ๐ are the matrices of spin-1 generators of ๐ฐ๐ฒ ( ) . The phases of this chain as a functionof ฯ are shown in Fig. 9. For the discussion here only the Haldane and dimer phases areimportant; see [110, 176, 26] for the description of the full phase diagram and of the variousphase transitions. Before explaining the relation to sigma models, let us comment on the ๏ฌvespecial points in the diagram. They correspond to the following chains: โฆ Spin-1 Heisenberg, with Hamiltonian ๐ป = (cid:205) ๐ Tr ( ๐ ( ๐ ) ๐ ( ๐ + )) . By the originalargument of Haldane it is gapped, and the ๏ฌeld theory mapping results in an ๐ sigmamodel with ๐ = โฆ AKLT (tan ( ฯ ) = โ ): this model lies in the Haldane phase as well. The ground statemay be calculated exactly and is translationally invariant. See Section 4.2. โฆ The critical SU ( ) spin-1 integrable point ( ฯ = ๐ ). This is a higher-spin integrableextension of the spin-1/2 Heisenberg Hamiltonian [219, 35]. The continuum limit isdescribed by the SU ( ) ๐ = WZNW model.50 At ฯ = โ ๐ the symmetry is enhanced to SU ( ) and one has the SU ( ) extension [217]of the Heisenberg Hamiltonian ๐ป = (cid:205) ๐ Tr ( ๐ ( ๐ ) ๐ ( ๐ + )) , where ๐ now contain thegenerators of ๐ฐ๐ฒ ( ) in the fundamental representation at all sites. Again the spectrumis gapless, the critical theory described by SU ( ) ๐ = WZNW model. โฆ At ฯ = ๐ one again has an enhancement to SU ( ) , this time with matrices ๐ whoseentries generate alternating fundamental/anti-fundamental representations, i.e. ๐ ๐ผ๐ฝ ( ๐ + ) = ๐ ๐ผ๐ฝ ( ๐ ) . This model is integrable and gapped [5, 82, 196, 37, 156], with a two-folddegenerate ground state and broken translational invariance (the โdimerโ).The two large dots in the diagram, which correspond to the ๏ฌrst and last points in the abovelist, are particularly important for us. As already mentioned, the ฯ = ๐ sigma model with vanishing ๐ -angle. The ฯ = ๐ point corresponds to an alternatingSU ( ) spin chain, exactly of the type considered in the previous section, so that the resulting๏ฌeld theory is a CP sigma model with ๐ = ๐ ( ๐ = CP ๐ โ -models with ๐ > ๐ = ๐ , which makes them di๏ฌerent from the CP model. Atthe same time this makes it consistent with the large- ๐ description above, which predicts amass gap for the sigma models. ( ) case We now want to move forward from the Hamiltonian (7.1) and ๏ฌnd the sigma model whicharises upon taking the continuum limit around the antiferromagnetic โvacuumโ of the SU ( ) spin chain with the Hamiltonian (3.4): H = ๐ฟ โ๏ธ ๐ = ( ๐ฝ Tr ( ๐ ( ๐ ) ๐ ( ๐ + )) + ๐ฝ Tr ( ๐ ( ๐ ) ๐ ( ๐ + ))) , (7.14)where ๐ฝ > , ๐ฝ > S = ๐ โซ ๐๐ก โ๏ธ ๐ (cid:18) ๐ (cid:164) ๐ง ( ๐ ) โฆ ๐ง ( ๐ )| ๐ง ( ๐ ) | + ๐ผ ยท H ๐ (cid:19) , where (7.15) H ๐ = ๐ฝ | ๐ง ( ๐ ) โฆ ๐ง ( ๐ + ) | | ๐ง ( ๐ ) | | ๐ง ( ๐ + ) | + ๐ฝ | ๐ง ( ๐ ) โฆ ๐ง ( ๐ + ) | | ๐ง ( ๐ ) | | ๐ง ( ๐ + ) | .
51n this formula each of the variables ๐ง ( ๐ ) has an additional (hidden) index, which takes threepossible values corresponding to the fundamental representation of SU ( ) .We claim that in the case of (7.15) the antiferromagnetic vacuum con๏ฌguration is whenthe ๐ง -vectors on any 3 neighboring sites are orthogonal to each other. First of all, this isconsistent with what we had for the SU ( ) case above, since the equation 1 + ๐ง ๐ง = ๐ง = โ ๐ง , which isthe antipodal involution discussed above. When ๐ = ๐ง -vectors ๐ง , ๐ง , ๐ง sitting at these sites : ๐ง โฆ ๐ง = ๐ง โฆ ๐ง = ๐ง โฆ ๐ง = . (7.16)The submanifold of ( CP ) ร described by (7.16) is the ๏ฌag manifold F . Weโre now goingto elaborate on this simplest nontrivial example.The ๏ฌrst question is what will arise in the continuum limit from the kinetic term in (7.15).The discussion in the previous section (see (7.6)) indicates that it is natural to ๏ฌrst focus onan arbitrary set of 3 consecutive sites. Then the kinetic term in the spin chain Lagrangian isthe pull-back ๐ ๐ก of the following one-form: ๐ = i ๐๐ง โฆ ๐ง ๐ง โฆ ๐ง + i ๐๐ง โฆ ๐ง ๐ง โฆ ๐ง + i ๐๐ง โฆ ๐ง ๐ง โฆ ๐ง (7.17)This is the Poincarรฉ-Liouville one-form for the product symplectic form ฮฉ on CP ร CP ร CP , so that ๐๐ = ฮฉ . We claim that on the submanifold F , described by (7.16), this 2-formis zero. We may even formulate a slightly more general statement:The submanifold F โ ( CP ) ร , and more generally F ๐ โ ( CP ๐ โ ) ร ๐ ,is Lagrangian.In fact, we already encountered a generalization of this statement in Section 1.4, howeverhere we emphasize it due to its particular importance for the description of the antiferromag-netic interactions in spin chains. Let us focus on the consequences of this fact. It follows that ๐ | F = ๐ ๐ , where ๐ is a function (in fact, ๐ = ๐ log ( ๐ ๐ผ๐ฝ๐พ ๐ง ๐ผ ๐ง ๐ฝ ๐ง ๐พ ) ),so the integral โซ ๐ ๐ก ๐๐ก = ๐ ( ) โ ๐ ( ) reduces to the boundary term. We ignore this termin the present discussion.Let us emphasize that the geometric setup discussed in the last two sections is general,and is key to understanding the target space of the sigma model that emerges in the continuumlimit. The main conclusion is:The target space of the sigma model is the โmoduli spaceโ of Nรฉel vacua of the spinchain. It is a Lagrangian submanifold in the phase space of an elementary cell. Here we switch back to homogeneous coordinates.
The term H in (7.15) is equal to zero if we impose the background con๏ฌguration (7.16).Moreover, since 0 โค H ๐ โค ๐ฝ + ๐ฝ , one immediately sees that the ferromagnetic and antifer-romagnetic vacua saturate respectively the maximum and minimum of its possible values. Inview of the fact that we will be building an expansion around the antiferromagnetic vacuum,from this observation we deduce an important consequence, namely that this expansion muststart with a quadratic term, i.e. there is no linear term.Let us assume that the number of sites of our spin chain is a factor of 3 (this is only neededfor simplicity, and it does not play a big role for a su๏ฌciently long spin chain). In this casewe split the spin chain into ห ๐ฟ segments of length 3 and focus for the moment on just one ofthese segments, which is the elementary cell number ๐ in the chain. On each of the three sites we have a three-dimensional complex vector ๐ง ๐ด . Let us form a3 ร ๐ .The antiferromagnetic con๏ฌguration corresponds to the case where the three vectors aremutually orthogonal.Now we need to take the ๏ฌuctuations into account, and in order to build the sought forexpansion we will employ the so-called ๐ ๐ decomposition of a matrix. The
๐ ๐ decomposi-tion theorem says that an arbitrary matrix ๐ may be decomposed into a product of a unitarymatrix ๐ and an upper triangular one ๐ต + : ๐ = ๐ โฆ ๐ต + (8.1)This statement is equivalent to the Gram-Schmidt orthogonalization theorem. Let usparametrize ๐ต + at link ๐ in the following way: ๐ต + ( ๐ ) = (cid:169)(cid:173)(cid:171) ๐ฟ x ( ๐ ) ๐ฟ y ( ๐ ) ๐ฟ z ( ๐ ) (cid:170)(cid:174)(cid:172) (cid:169)(cid:173)(cid:171) a ( ๐ ) ( ๐ )
00 0 c ( ๐ ) (cid:170)(cid:174)(cid:172) . (8.2)If we denote the columns of the matrix ๐ as ( ๐ข , ๐ข , ๐ข ) , the decomposition (8.1) saysthat ๐ง ( ๐ ) = a ( ๐ ) ๐ข ( ๐ ) , ๐ง ( ๐ ) = b ( ๐ ) ( ๐ข ( ๐ ) + ๐ฟ x ( ๐ ) ๐ข ( ๐ )) , (8.3) ๐ง ( ๐ ) = c ( ๐ ) ( ๐ข ( ๐ ) + ๐ฟ y ( ๐ ) ๐ข ( ๐ ) + ๐ฟ z ( ๐ ) ๐ข ( ๐ )) ๐ฟ . A simple calculation reveals that (suppressing the index ๐ for the moment) ๐ฝ ๐ก = (cid:18) i (cid:164) ๐ง โฆ ๐ง ๐ง โฆ ๐ง + i (cid:164) ๐ง โฆ ๐ง ๐ง โฆ ๐ง + i (cid:164) ๐ง โฆ ๐ง ๐ง โฆ ๐ง (cid:19) x = y = z = โ (8.4) โ i ๐ฟ (cid:16) x ๐ข โฆ (cid:164) ๐ข + y ๐ข โฆ (cid:164) ๐ข + z ๐ข โฆ (cid:164) ๐ข โ c.c. (cid:17) + ... The hypothesis of the existence of a continuum limit implies that ๐ข ( ๐ ) , ๐ข ( ๐ ) , ๐ข ( ๐ ) varymildly with ๐ , in other words we may approximate ๐ข ๐ด ( ๐ + ) = ๐ข ๐ด ( ๐ ) + ๐ฟ ๐ข ๐ด ( ๐ ) (cid:48) + ... (8.5) H , ( k โ , k ) H , ( k โ , k ) H , ( k โ , k ) H , ( k, k ) H , ( k, k ) H , ( k, k ) k โ k Figure 10: Explanation of the various terms calculated in (8.7).Let us introduce the quantity H ๐ด,๐ต ( ๐, ๐ (cid:48) ) = | ๐ง ๐ด ( ๐ ) โฆ ๐ง ๐ต ( ๐ (cid:48) ) | | ๐ง ๐ด ( ๐ ) | | ๐ง ๐ต ( ๐ (cid:48) ) | , (8.6)which is the density of the Hamiltonian H from (7.15), if the indices ๐ด, ๐ต, ๐, ๐ (cid:48) change ina particular range. Indeed, we need to calculate H ๐ด,๐ต ( ๐, ๐ (cid:48) ) for nearest- and next-to-nearestneighbor sites, using the formulas (8.3)-(8.5) and keeping only the leading orders of ๐ฟ (seeFig. 10 for an explanation of what these terms stand for): H , ( ๐, ๐ ) = | ๐ง ( ๐ ) โฆ ๐ง ( ๐ ) | | ๐ง ( ๐ ) | | ๐ง ( ๐ ) | (cid:39) ๐ฟ | x ( ๐ ) | , H , ( ๐, ๐ ) = | ๐ง ( ๐ ) โฆ ๐ง ( ๐ ) | | ๐ง ( ๐ ) | | ๐ง ( ๐ ) | (cid:39) ๐ฟ | z ( ๐ ) | H , ( ๐, ๐ ) = | ๐ง ( ๐ ) โฆ ๐ง ( ๐ ) | | ๐ง ( ๐ ) | | ๐ง ( ๐ ) | (cid:39) ๐ฟ | y ( ๐ ) | H , ( ๐ โ , ๐ ) = | ๐ง ( ๐ โ ) โฆ ๐ง ( ๐ ) | | ๐ง ( ๐ โ ) | | ๐ง ( ๐ ) | (cid:39) ๐ฟ | โ ๐ข ( ๐ ) (cid:48) โฆ ๐ข ( ๐ ) + y ( ๐ ) | (8.7) H , ( ๐ โ , ๐ ) = | ๐ง ( ๐ โ ) โฆ ๐ง ( ๐ ) | | ๐ง ( ๐ โ ) | | ๐ง ( ๐ ) | (cid:39) ๐ฟ | โ ๐ข (cid:48) ( ๐ ) โฆ ๐ข ( ๐ ) + z ( ๐ ) | H , ( ๐ โ , ๐ ) = | ๐ง ( ๐ โ ) โฆ ๐ง ( ๐ ) | | ๐ง ( ๐ โ ) | | ๐ง ,๐ | (cid:39) ๐ฟ | โ ๐ข (cid:48) ( ๐ ) โฆ ๐ข ( ๐ ) + x ( ๐ ) | ๐ฅ ( ๐ ) , ๐ฆ ( ๐ ) , ๐ง ( ๐ ) that enter only quadratically, oneobtains the Lagrangian (we set for simplicity ๐ฝ = , ๐ฝ = ๐ ) L = + ๐ (| ๐ข โฆ (cid:164) ๐ข | โ ๐ ๐ฟ | ๐ข (cid:48) โฆ ๐ข | ) โ i ๐ + ๐ ๐ฟ ๐ ๐๐ ( ๐ข โฆ ๐ ๐ ๐ข ) ( ๐ข โฆ ๐ ๐ ๐ข ) + (8.8) + + ๐ (| ๐ข โฆ (cid:164) ๐ข | โ ๐ ๐ฟ | ๐ข โฆ ๐ข (cid:48) | ) โ i1 + ๐ ๐ฟ ๐ ๐๐ ( ๐ข โฆ ๐ ๐ ๐ข ) ( ๐ข โฆ ๐ ๐ ๐ข ) ++ + ๐ (| ๐ข โฆ (cid:164) ๐ข | โ ๐ ๐ฟ | ๐ข โฆ ๐ข (cid:48) | ) โ i ๐ + ๐ ๐ฟ ๐ ๐๐ ( ๐ข โฆ ๐ ๐ ๐ข ) ( ๐ข โฆ ๐ ๐ ๐ข ) Of course, each of the ๏ฌrst terms in the three lines above can be brought to a canonicalrelativistic-invariant form by a rescaling of the space variable . The fact that all the ๏ฌavourwave velocities are equal in (8.8) is really a coincidence that happens for ๐ =
3. For ๐ > ๐ข โก i ( ๐ข โฆ ๐๐ข ) โง ( ๐ข โฆ ๐๐ข ) (8.9) ๐ฃ โก โ i ( ๐ข โฆ ๐๐ข ) โง ( ๐ข โฆ ๐๐ข ) (8.10) ๐ค โก i ( ๐ข โฆ ๐๐ข ) โง ( ๐ข โฆ ๐๐ข ) (8.11)Then the three epsilon-terms in (8.8) are the pull-back of the following 2-form: ๐ = + ๐ ( ๐ฃ โ ๐ ๐ข โ ๐ ๐ค ) (8.12)The crucial fact is that ๐ may be split in two parts: a topological one (the ๐ -term) andthe non-topological one (the ๐ต -๏ฌeld, or Kalb-Ramond ๏ฌeld, in sigma model terminology).The splitting may be achieved by noting that the above Lagrangian possesses a Z โquasi-symmetryโ, which acts on the vectors ( ๐ข , ๐ข , ๐ข ) of the ๏ฌag by cyclically permuting them: Z : ๐ข โ ๐ข โ ๐ข โ ๐ข . (8.13)This symmetry has a transparent meaning: it arises because of the translational invariance ofthe Hamiltonian (7.14), since the transformation (8.13) corresponds to shifting the elementarycell by one site. This has to be a symmetry at the level of the partition function, however Note the convention for the epsilon-symbol: ๐ = It was noted in [73] that there exists a canonical geometric expression for the metric arising in this way.Suppose ( ๐ฝ , ๐ ) is a symplectic manifold (in this case the phase space of an elementary cell), and H a functionon it โ the classical Hamiltonian โ that attains a minimum on a Lagrangian submanifold N โ ๐ฝ (the target spaceof the sigma model). In this case one can de๏ฌne the (inverse) metric on N by the formula ๐ ๐ ๐ = ๐ ๐๐ (cid:16) ๐ H ๐๐ฅ (cid:17) ๐๐ ๐ ๐ ๐ . Z it is shifted by an integral topological term , i.e. an element of ๐ป (F , Z ) . As a result, thepartition function, which is given by a path integral of the type Z = โซ ๐ ๐ โซ ๐๐ก ๐๐ฅ L (cid:214) ๐๐ , (8.14)is unaltered. This is the same argument that is used to prove that the path integral of Chern-Simons theory is well-de๏ฌned [94, 252]. Indeed, we will now show that ๐ may be split in anon-topological part that is invariant w.r.t. Z and a ๐ -angle part that transforms non-trivially.First of all, the two-forms transform as follows under Z : Z : ๐ข โ ๐ค โ ๐ฃ โ ๐ข . (8.15)As a result, the only Z -singlet is ๐ข + ๐ฃ + ๐ค , so that one may decompose ๐ = โ ๐ ( + ๐ ) ( ๐ข + ๐ฃ + ๐ค ) โ ( ๐ข โ ๐ฃ + ๐ค โ ๐ฃ ) , (8.16)Let us now show that the second term is topological. First of all, recall from section 1.4 thatevery element of ๐ป (F , R ) is a linear combination of three forms ฮฉ ๐ด ( ๐ด = , , ) , whichare the pull-backs to the ๏ฌag manifold of the Fubini-Study forms corresponding to ๐ข , ๐ข , ๐ข .It is easy to relate these Fubini-Study forms to ๐ข, ๐ฃ, ๐ค . Recall that ฮฉ = ๐๐ข โง ๐๐ข โ ( ๐๐ข โฆ ๐ข ) โง ( ๐๐ข โฆ ๐ข ) (8.17)and similar expressions hold for ฮฉ and ฮฉ . Now let us use the identity ๐ข โ ๐ข + ๐ข โ ๐ข + ๐ข โ ๐ข = . Using this, we may rewrite the restriction of ฮฉ to the ๏ฌag manifold F in thefollowing way: ฮฉ | F = ( ๐๐ข โฆ ๐ข ) โง ( ๐๐ข โฆ ๐ข ) + ( ๐๐ข โฆ ๐ข ) โง ( ๐๐ข โฆ ๐ข ) = ๐ข โ ๐ฃ. (8.18)External di๏ฌerentiation and restriction to a submanifold are commutative operations, thereforethis restricted form is a closed 2-form on F . Analogously ฮฉ | F = ๐ค โ ๐ข, ฮฉ | F = ๐ฃ โ ๐ค . Inparticular, we see that the sum ( ฮฉ + ฮฉ + ฮฉ ) | F = ๐ป (F , Z ) . Keeping in mind (8.19), we may write ๐ as ๐ (cid:39) โ ๐ ( + ๐ ) ( ๐ข + ๐ฃ + ๐ค ) + ฮฉ + ฮฉ + ฮฉ : = ๐ top (8.20)Since the action stands in the exponent of the integrand in (8.14), ๐ is de๏ฌned moduloelements of ๐ป (F , Z ) . This is important, because under the action of Z the topological56erm ๐ top shifts precisely by such terms. Indeed, since the action of Z cyclically permutes ฮฉ , ฮฉ , ฮฉ , under its action one has Z : ๐ top โ ๐ top โ ฮฉ + ฮฉ + ฮฉ = + ฮฉ โ ๐ป (F , Z ) (cid:39) ๐ top , (8.21)where (cid:39) means โup to an element of ๐ป (F , Z ) โ. This property will be essential for thediscussion of discrete โt Hooft anomalies in section 10.2 below. In the previous section we considered the case of SU ( ) spin chains. Next we discuss thegeneralization to the SU ( ๐ ) case, where the Hamiltonian is given by (3.4). The Nรฉel state inthis case is given by a direct generalization of (7.16), namely requiring that the ๐ consecutivevectors ๐ : = ( ๐ง , . . . ๐ง ๐ ) are all pairwise orthogonal: ๐ง ๐ด โฆ ๐ง ๐ต = ๐ด โ ๐ต . (9.1)In order to derive the continuum theory, one follows the steps described in the previoussection: one ๏ฌrst introduces deviations from the Nรฉel con๏ฌguration (9.1) and performs afactorization (8.1) ๐ = ๐ โฆ ๐ต + , where ๐ โ U ( ๐ ) is unitary and ๐ต + is strictly upper-triangular.As before, the matrix ๐ต + describes the deviations from the Nรฉel state in a single unit cell.One then expands the Lagrangian to quadratic order in the elements of ๐ต + , as well as inthe cell-to-cell variations, and integrates over the ๐ต + elements that enter algebraically. Thecalculation is rather tedious, and its details can be found either in [73] or in [237]. ( ๐ ) spin chain Here we write out the resulting action of the ๏ฌeld theory describing the SU( ๐ ) chain in therank- ๐ symmetric irrep: ๐ = โ๏ธ โค ๐ด<๐ต โค ๐ โซ ๐๐ฅ๐๐ ๐ | ๐ด โ ๐ต | (cid:18) ๐ฃ | ๐ด โ ๐ต | | ๐ข ๐ด โฆ ๐ ๐ฅ ๐ข ๐ต | + ๐ฃ | ๐ด โ ๐ต | | ๐ข ๐ด โฆ ๐ ๐ ๐ข ๐ต | (cid:19) (9.2) โ ๐ ๐๐ โ๏ธ โค ๐ด<๐ต โค ๐ ๐ | ๐ด โ ๐ต | โซ ๐๐ฅ๐๐ ( ๐ข ๐ด โฆ ๐ ๐ ๐ข ๐ต ) ( ๐ข ๐ต โฆ ๐ ๐ ๐ข ๐ด ) + ๐ top , where ๐ฃ ๐ก = ๐ ๐ โ ๐ฝ ๐ก ๐ฝ ๐ โ ๐ก is the ๏ฌavour wave velocity associated with the pair of couplings ๐ฝ ๐ก and ๐ฝ ๐ โ ๐ก of the spin chain (3.4). The ๐ -terms are the generalizations of the non-topologicalcontribution to ๐ discussed in the previous section for ๐ = ๐ top = โซ ๐ top is a higher- ๐ generalization of the topological ( ๐ )-term, discussed in detail57elow. As we already mentioned earlier, the cases ๐ = ๐ = ๐ ๐ก = ๐๐ฃ ๐ก ( ๐ฝ ๐ก + ๐ฝ ๐ โ ๐ก ) (9.3)and ๐๐ ๐ก ๐ = ( ๐ โ ๐ก ) ๐ฝ ๐ โ ๐ก โ ๐ก ๐ฝ ๐ก ๐ฝ ๐ก + ๐ฝ ๐ โ ๐ก . (9.4)Since the coupling constants and velocities satisfy ๐ ๐ก = ๐ ๐ โ ๐ก and ๐ฃ ๐ก = ๐ฃ ๐ โ ๐ก , we conclude thatthere are (cid:98) ๐ (cid:99) velocities and coupling constants, where (cid:98) ๐ (cid:99) = (cid:40) ๐ ๐ even ๐ โ ๐ odd (9.5)The topological term is ๐ top : = ๐๐ ๐๐ ๐ โ๏ธ ๐ด = ๐ด ๐ ๐ด , where ๐ ๐ด : = ๐ โซ ฮฉ ๐ด (9.6)is the integral of the Fubini-Study form over the worldsheet (as such, it is a quantizedtopological charge, see (1.16)). Since (cid:205) ๐๐ด = ฮฉ ๐ด = ๐ โ๏ธ ๐ = ๐ ๐ด = , (9.7)so that there are ๐ โ ๐ -terms appearing in (9.2) are not quantized, despite the fact that they arepure imaginary in imaginary time. We give an interpretation of these terms below. In [73],these ๐ -terms were absent as a result of the same ๏ฌne-tuning that ensured a unique velocity.Indeed, the choice ๐ฝ ๐ก = โ๏ธ ๐ โ ๐ก๐ก ensures that ๐ฃ ๐ก โก const. for all ๐ก , and moreover that ๐ ๐ก = ๐ก . Z ๐ symmetry Just as in the SU ( ) case, we may introduce a discrete symmetry Z ๐ : ๐ข ๐ด โ ๐ข ๐ด + , ๐ข ๐ + โก ๐ข . (9.8)It is easy to prove that the sum of ๐ -terms is invariant under this symmetry. Indeed, let us ๏ฌx | ๐ด โ ๐ต | = ๐ก โค (cid:98) ๐ (cid:99) , then every form ๐ต ๐ก : = (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) โ๏ธ โค ๐ด<๐ต โค ๐ | ๐ด โ ๐ต | = ๐ก โ โ๏ธ โค ๐ด<๐ต โค ๐ | ๐ด โ ๐ต | = ๐ โ ๐ก (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) ( ๐ข ๐ด โฆ ๐๐ข ๐ต โง ๐ข ๐ต โฆ ๐๐ข ๐ด ) (9.9)58s Z ๐ -invariant. Using the property ๐ ๐ โ ๐ก = โ ๐ ๐ก that follows from (9.4), we may write the formentering the ๐ -terms as โ๏ธ โค ๐ด<๐ต โค ๐ ๐ | ๐ด โ ๐ต | ( ๐ข ๐ด โฆ ๐๐ข ๐ต ) โง ( ๐ข ๐ต โฆ ๐๐ข ๐ด ) = (cid:98) ๐ (cid:99) โ๏ธ ๐ก = ๐ ๐ก ๐ต ๐ก , (9.10)proving that it is also invariant. In contrast, the topological part of the action is shifted underthe Z ๐ -transformation (9.8): ๐ top โ ๐ top + ๐๐ ๐ , just as in the SU ( ) case (8.21). Thisis the ultimate reason that allows separating the topological terms from the non-topological ๐ -part of the skew-symmetric tensor ๏ฌeld. The coupling constants { ๐ ๐ก } and { ๐ ๐ก } in (9.2) correspond to the metric and torsion on the ๏ฌagmanifold, respectively [190]. However, a unique metric cannot be de๏ฌned, since the theory(9.2) lacks the Lorentz invariance that is often assumed for sigma models. Thus, we havea non-Lorentz invariant ๏ฌag manifold sigma model (the same phenomenon was observedin [236] where SU(3) chains with self-conjugate representations were considered). We willnow use the renormalization group to show that at low enough energies, it is possible for thedistinct velocities occurring to ๏ฌow to a single value, so that Lorentz invariance emerges.The Lorentz invariant versions of the above ๏ฌag manifold sigma model were studied ingreat detail in [190]. In particular, the renormalization group ๏ฌow of both the { ๐ ๐ก } and { ๐ ๐ก } were determined for general ๐ . Moreover, ๏ฌeld theoretic versions of the LSMA theorem wereformulated, using the methods of โt Hooft anomaly matching (which we review below, inSection 10). We would like to apply these results to our SU( ๐ ) chains which lack Lorentzinvariance in general. First, it will be useful to introduce dimensionless velocities, ๐ ๐ก , de๏ฌnedaccording to ๐ ๐ก : = ๐ฃ ๐ก ๐ฃ , ๐ฃ = (cid:98) ๐ (cid:99) (cid:205) (cid:98) ๐ (cid:99) ๐ก = ๐ฃ ๐ก , and introduce new spacetime coordinates by meansof a rescaling ๐ฅ โ ๐ฅ โ ๐ฃ , ๐ โ โ ๐ฃ๐. We then consider the di๏ฌerences of velocities occurringin (9.2), namely ฮ ๐ก๐ก (cid:48) : = ๐ ๐ก โ ๐ ๐ก (cid:48) , (9.11)and ask how they behave at low energies. More precisely, we calculate the one-loop betafunctions of these ฮ ๐ก๐ก (cid:48) , to orders O ( ๐ ๐ก ) and O ( ๐ ๐ก ) . We will ๏ฌnd that each of the ฮ ๐ก๐ก (cid:48) ๏ฌowsto zero under renormalization and we will show that this implies Lorentz invariance at ourorder of approximation. This is consistent with the fundamental SU( ๐ ) models with ๐ = { ๐ ๐ก } appearing in (9.2) are dimensionless, and are all proportional to ๐ .Since weโve taken a large ๐ limit, we will expand all quantities in powers of the { ๐ ๐ก } . Aswe will see below, the coe๏ฌcients { ๐ ๐ก } in (9.2) do not enter into our one-loop calculations,and so we will neglect them throughout. Since we are interested in the low energy dynamics59f these quantum ๏ฌeld theories, we make the simplifying assumption that the matrices ๐ = ( ๐ข , . . . , ๐ข ๐ ) are close to the identity matrix, and expand them in terms of the SU ( ๐ )generators.Recalling that the matrix ๐ is subject to the gauge transformations ๐ โ ๐ ยท ๐ท ๐ , where ๐ท ๐ = Diag ( ๐ ๐๐ผ , ยท ยท ยท , ๐ ๐๐ผ ๐ ) , we may ๏ฌx this gauge invariance by the followingparametrization of ๐ : ๐ = Exp (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) ๐ โ๏ธ ๐ โ o๏ฌ-diagonalgenerators โ๏ธ (cid:101) ๐ ๐ ๐ ๐ ๐ ๐ (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) = + ๐ โ๏ธ (cid:101) ๐ ๐ ๐ ๐ ๐ ๐ โ (cid:101) ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ + O ( ๐ ) . (9.12)To explain how (cid:101) ๐ ๐ are related to ๐ | ๐ด โ ๐ต | , we will assume that the following basis of o๏ฌ-diagonal generators is chosen: { ๐ ๐ } = { ๐ธ ๐ด๐ต + ๐ธ ๐ต๐ด , ๐ ( ๐ธ ๐ด๐ต โ ๐ธ ๐ต๐ด ) , ๐ด < ๐ต } , where ๐ธ ๐ด๐ต arethe elementary matrices with 1 in ๐ด๐ต -th position and 0 elsewhere. Whenever the generator ๐ ๐ corresponds to one of these two generators, we set (cid:101) ๐ ๐ : = ๐ | ๐ด โ ๐ต | . As shown in [237], inthis notation the expansion of the Lagrangian to quartic order in the ๐ โs has the form L = (cid:20) ๐ ๐ ( ๐ ๐ ๐ ๐ ) + ๐ ๐ ( ๐ ๐ฅ ๐ ๐ ) (cid:21) + โ๏ธ (cid:101) ๐ ๐ (cid:101) ๐ ๐ (cid:101) ๐ ๐ โ โ ๐ ( ๐ ) (cid:101) ๐ ๐ ๐ ๐๐๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ (9.13) โ๏ธ (cid:101) ๐ ๐ (cid:101) ๐ ๐ (cid:101) ๐ ๐ โ ๐ ( ๐ ) (cid:101) ๐ ๐ (cid:104)โ๏ธ (cid:101) ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐๐๐ ๐ ๐๐๐ + ๐ ๐๐๐ธ ๐ ๐ธ๐๐ โ๏ธ (cid:101) ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ (cid:105) + O ( ๐ ) . Here ๐ ๐๐๐ , ๐ ๐๐๐ถ (the small-letter and capital-letter indices correspond to the o๏ฌ-diagonalgenerators and all generators respectively) are the structure constants de๏ฌned by [ ๐ ๐ , ๐ ๐ ] = ๐ ๐ ๐๐๐ถ ๐ ๐ถ . Also, โ ๐ ( ๐ ) = ๐ ๐ for ๐ = ๐ and โ ๐ ( ๐ ) = ๐ ๐ for ๐ = ๐ฅ .The calculations then follow the standard procedures of renormalization theory. Werewrite the free part of the above Lagrangian in โrenormalized variablesโ, i.e. L = (cid:104) ๐ ๐๐ ๐ ๐๐ ( ๐ ๐ ๐ ๐ ) + ๐ ๐๐ ๐ ๐ฅ๐ ( ๐ ๐ฅ ๐ ๐ ) (cid:105) (and the interaction terms accordingly). The bare and renor-malized velocities are related as ๐ ๐ = ๐ ๐๐ โ๏ธ ๐ ๐ฅ๐ ๐ ๐๐ , so that one can de๏ฌne the corresponding ๐ฝ -function ๐ฝ ๐ ๐ : = ๐๐ ๐๐ ๐ log ๐ , where ๐ is the ๏ฌxed energy scale. q - kk c ba Figure 11: The diagram contributing to velocity renormalization at one loop. Throughout, repeated indices will be summed over. ๐ ๐ฅ๐ , ๐ ๐๐ so as to cancel the one-loop divergencescoming from the bubble graphs shown in Fig. 11. The details of the computation can befound in [237], the result being ๐ฝ ๐ ๐ก = ๐ ๐ก ๐ ๐ก ๐ ๐ โ โ๏ธ ๐ด = ๐ด โ ๐ก ๐ | ๐ก โ ๐ด | ๐ ๐ด (cid:20) ๐ ๐ก ๐ ๐ด โ ๐ ๐ด ๐ ๐ก (cid:21) , ๐ก = , , ยท ยท ยท , ๐ : = (cid:98) ๐ (cid:99) . (9.14)The equations for ๐ ๐ก with ๐ก > ๐ may be obtained by using the identity ๐ ๐ก = ๐ ๐ โ ๐ก . As a resultof (9.14), the renormalization group ๏ฌow equation ๐๐ ๐ก ๐ log ๐ = ๐ฝ ๐ ๐ก (we drop the superscript ๐ to simplify the notation) is a non-linear system of ODEโs for the functions ๐ ๐ก ( log ๐ ) . Onewishes to show that the โpointโ where all velocities are the same ( ๐ ๐ = ๐ ๐ for all ๐, ๐ ) is anattractor point of the system (clearly it is an equilibrium point). In general this might be aformidable task, so one can take a simpler step by linearizing the system of equations aroundthe equilibrium and proving that the spectrum of the linearization operator is positive (i.e.that the equilibrium is stable). There are ๐ โ ฮ ๐ : = ฮ ,๐ + ( ๐ = , , ยท ยท ยท , ๐ โ ) . The linearized equation then takes the form ๐๐ log ๐ ๐ซ = ๐ ๐ซ , (9.15)where ๐ is a ( ๐ โ ) ร ( ๐ โ ) matrix. The spectrum of ๐ will reveal the low energy behaviourof the ฮ ๐ก๐ก (cid:48) : if the spectrum is strictly positive, one may conclude that all velocity di๏ฌerences๏ฌow to zero in the IR. In the highly symmetric case when all of the coupling constants { ๐ ๐ก } are equal, one easily ๏ฌnds ๐ = ๐ ๐ ( ๐ โ ) ๐ โ (9.16)showing that the spectrum of ๐ is strictly positive. In the non-symmetric case, for di๏ฌer-ent choices of ๐ and values of the couplings, this has been checked numerically in [237],suggesting that the spectrum of ๐ is always positive.Up to this conjecture, we have shown that the velocity di๏ฌerences ฮ ๐ก๐ก (cid:48) ๏ฌow to zero atlow energies. Another source of Lorentz non-invariance in the Lagrangian (9.13) are thefunctions โ ๐ ( ๐ ) in the interaction terms. These are however related to the velocities ๐ ๐ andtherefore also ๏ฌow to a common value, and thus Lorentz invariance of the entire model (9.2)is possible if the velocities are initially close to each other.
10 Generalized Haldane conjectures and โt Hooft anomalymatching
Based on the renormalization group analysis in the previous section, we now argue that atlow enough energies, the SU( ๐ ) chains in the symmetric- ๐ irreps (without ๏ฌne-tuning), may61e described by a Lorentz invariant ๏ฌag manifold sigma model L = โ๏ธ ๐ด<๐ต ๐ | ๐ด โ ๐ต | | ๐ข ๐ด โฆ ๐ ๐ ๐ข ๐ต | โ ๐ ๐๐ โ๏ธ ๐ด<๐ต ๐ | ๐ด โ ๐ต | ( ๐ข ๐ด โฆ ๐ ๐ ๐ข ๐ต ) ( ๐ข ๐ต โฆ ๐ ๐ ๐ข ๐ด ) with topological theta-term ๐ top = ๐๐ ๐ โ โ๏ธ ๐ด = ๐ด ๐ ๐ด ๐ : = ๐ ๐๐ . (10.1)(10.2)These sigma models have been studied in [69, 73, 190]. In [190], the renormalization group๏ฌows of the { ๐ ๐ก } and the { ๐ ๐ก } were determined, and given a geometric interpretation. Itwas found that for ๐ >
4, the { ๐ ๐ก } ๏ฌow to a common value in the IR, and that for ๐ > { ๐ ๐ก } ๏ฌow to zero in the IR. Even for ๐ < ( ๐ ) WZNW model occurs. This is based on the observation that the ๐ term doesnโtinduce any relevant operators in the WZNW model. Thus we may expect an ๐ ๐ (permutationgroup) symmetry to emerge at low enough energies, and for ๐ >
6. It is known that in these ๐ ๐ -symmetric models, the unique coupling constant ๐ obeys [190] ๐๐๐ log ๐ = ๐ + ๐ ๐. (10.3)and the theory is asymptotically free. The ๐ ๐ -symmetric metric (with all ๐ ๐ก equal) is knownin the math literature as the โnormalโ, or reductive, metric [29]. This same metric will featurein the integrable models that we will describe in the next chapter. Interestingly, it is notKรคhler (unless ๐ =
2) but it is Einstein, with cosmological constant proportional to ๐ + ( ๐ ) Haldane conjectures
The low-energy behavior of the sigma models (10.1)-(10.2) (and of the corresponding spinchains with symmetric rank- ๐ representations at each site) depends drastically on the valuesof the ๐ -angles. In [161, 221, 190, 237] the generalizations of Haldaneโs conjecture for thisclass of models were proposed. These are summarized in Table 1.Using the notion of โt Hooft anomaly matching (which we explain below in Sections 10.2-10.3), both [190] and [221] were able to formulate a ๏ฌeld-theoretic version of the LSMAtheorem for SU( ๐ ) chains. In short, the presence of an โt Hooft anomaly signi๏ฌes nontriviallow energy physics. It was shown that in the ๏ฌag manifold models, an โt Hooft anomalyis present so long as ๐ is not a multiple of ๐ . In these cases the gapped phase must havespontaneously broken translation or PSU( ๐ ) symmetry; the latter is ruled out by the Mermin-Wagner-Coleman theorem at any ๏ฌnite temperature. In the gapped phase, the ground statedegeneracy is predicted to be ๐ gcd ( ๐,๐ ) , which is consistent with the LSMA theorem presentedin Section 4.1 above. It is interesting to note that when the classical ground state has a di๏ฌerent62 ase Conjecture Evidence ๐ โ ๐ Gapless orgapped with degenerateground states (proof) โฆ LSMA theorem [166, 13, 237]gcd ( ๐, ๐ ) = ( ๐ ) CFT โฆ โt Hooft anomalies:[221, 190, 162, 260] โฆ Fractional instanton gas [8, 235]1 < gcd ( ๐, ๐ ) < ๐ GapDegeneracy ๐ : = ๐ gcd ( ๐,๐ ) โฆ No candidate CFT [162, 260] โฆ Fractional instanton gas ๐ = ๐ Gap โฆ Numerics ( ๐ = ๐ =
3) [124] โฆ Absence of anomalies โฆ Perturbations around theintegrable WZNW point [113] โฆ Fractional instanton gas โฆ AKLT states:[126, 150, 187, 177, 207, 125]Table 1: Generalized Haldane conjectures for SU ( ๐ ) spin chains with symmetric rank- ๐ representations.structure, as in the ground state of the two-site-ordered self-conjugate SU(3) chains [236],no anomaly occurs. This is consistent with the fact that the proof of the LSMA theorem alsofails for such representations.The authors of [190] then argued that while an anomaly is present whenever ๐ mod ๐ โ ๐ and ๐ have no nontrivialcommon divisor (in Section 10.5 below we review a simple relation between WZNW and ๏ฌagmanifold models). In this case, the ๏ฌow is to SU( ๐ ) . Otherwise, the candidate IR ๏ฌxed pointis SU( ๐ ) ๐ , where ๐ = gcd ( ๐, ๐ ) , however we donโt expect SU ( ๐ ) ๐ low energy theories, with ๐ >
1, to emerge without ๏ฌne-tuning. This is because they contain relevant operators allowedby symmetry which destabilize them [226, 24, 6]. Integrable spin models are known whichdo exhibit SU ( ๐ ) ๐ low energy theories but they require ๏ฌne-tuned Hamiltonians [25, 142,143]. The most well-known example of this is the SU(2) case where integrable models ofspin ๐ have low energy SU ( ) ๐ critical points [219, 35, 11]. However, it has been establishedthat these critical points are unstable against in๏ฌnitesimal tuning of the spin Hamiltonianand one would require ๏ฌne-tuning in order for the ๏ฌag manifold sigma model to ๏ฌow there.This can already be seen from Fig. 9 in the example of the spin-1 integrable chain, whosecontinuum limit is described by SU ( ) ๐ = WZNW model. Any deformation of this chainwould lead us to one of the two massive phases, either dimer or Haldane. There is also ageneral argument that no ๏ฌow from the unstable SU ( ๐ ) ๐ theory to SU ( ๐ ) is possible, sincethis would violate the anomaly matching conditions derived in [162, 260] for generic SU( ๐ )WZNW models. For SU ( ) there is another anomaly-based argument of [115], which asserts63hat a ๏ฌow between an SU ( ) ๐ and SU ( ) ๐ (cid:48) theories is only possible if ๐ = ๐ (cid:48) mod 2.Based on these anomaly arguments, we conclude that the rank- ๐ symmetric SU( ๐ ) chainsmay ๏ฌow to a SU( ๐ ) WZNW model if ๐ and ๐ do not have a common divisor. In thiscase, we expect gapless excitations to appear in the excitation spectrum. This prediction isa natural extension of the phase diagrams occurring in [11] and [161]. See Figure 12 for asimpli๏ฌed phase diagram of the SU( ๐ ) chain in the case when ๐ and ๐ are coprime. Similarto the O(3) sigma model, we expect an RG ๏ฌow from the ๏ฌag manifold sigma model to theSU ( ๐ ) WZNW model. This model has an SU(n)-invariant interaction term (cid:205) ๐ ๐ฝ ๐๐ฟ ๐ฝ ๐๐ whichis marginal. For one sign of this coupling, it is marginally irrelevant and ๏ฌows to zero; forthe other sign it ๏ฌows to large values [162, 237]. As in the O(3) sigma model, we expect thiscoupling to have the irrelevant sign for su๏ฌciently weak coupling in the ๏ฌag-manifold sigmamodel. If the coupling gets too large then this coupling constant changes sign and there isan RG ๏ฌow to a gapped phase with broken translational symmetry, as occurs for SU(2) spinchains. g gapless gapped ( n -fold degeneracy) SU(n) Figure 12: A simpli๏ฌed phase diagram of the SU( ๐ ) chains, as a function of coupling constant ๐ (a collective notion for the (cid:98) ๐ (cid:99) di๏ฌerent coupling constants) when ๐ and ๐ are coprime.We note that when ๐ and ๐ have a common divisor, at least one of the topological anglesoccurring in (10.2) is necessarily trivial. In the instanton gas picture of Haldaneโs conjecture(reviewed below in Section 11), each type of topological excitation must have a nontrivialtopological angle in order to ensure total destructive interference in half odd integer spinchains [8]. One of the key tools in the analysis of the phase structure of the sigma model is the notion ofโt Hooft anomaly matching. To introduce this concept, ๏ฌrst one observes that the continuousglobal symmetry of the ๏ฌag model (10.1)-(10.2) is (here Z ๐ โ SU ( ๐ ) is the subgroup of thetype ๐ ๐ ๐ , where ๐ is an ๐ -th root of unity)PSU ( ๐ ) = SU ( ๐ )/ Z ๐ = U ( ๐ )/ U ( ) . (10.4)The reason is that the ๏ฌag is described by ๐ vectors ๐ : = ( ๐ข , . . . , ๐ข ๐ ) , up to U ( ) ๐ phasetransformations acting as ๐ โ ๐ ยท ๐ท , ๐ท = Diag ( ๐ ๐๐ผ , ยท ยท ยท , ๐ ๐๐ผ ๐ ) . The global symmetry isgiven by the left action ๐ โ ๐ ยท ๐ , and for ๐ โ U ( ๐ ) the Lagrangian is invariant. Onthe other hand, the action of the center U ( ) can be compensated by a gauge transformationacting on the right. As a result, the faithfully acting symmetry is PSU ( ๐ ) . Besides, the above64odels have a discrete Z ๐ -symmetry (9.8) that acts by a cyclic permutation of the vectors ๐ข , . . . , ๐ข ๐ . The claim [221, 190] is that these two symmetries have a mixed anomaly, whichwe are about to describe. Overall our exposition in this section will be split into three mainparts: โฆ Starting with a gauged linear representation for the ๏ฌag manifold models, we derivethe mixed PSU ( ๐ ) โ Z ๐ anomaly, following [221]. โฆ We discuss how PSU ( ๐ ) -bundles are related to the LSMA theorem and to fractionalinstantons โฆ Following [190], we explain how ๏ฌag manifold models may be embedded in WZNWmodels. This serves to motivate the conjecture that ๏ฌag manifold models ๏ฌow in theIR to those conformal points for certain values of the ๐ -angles ( ๐ ) global symmetry We start with the ๏ฌrst point. On several occasions we already used the fact that the complete๏ฌag manifold U ( ๐ ) U ( ) ๐ may be thought of as the space of orthonormal vectors ๐ข ๐ด โฆ ๐ข ๐ต = ๐ฟ ๐ด๐ต ,each de๏ฌned up to a phase: ๐ข ๐ด โผ ๐ ๐๐ผ ๐ด ๐ข ๐ด . It is of course standard in ๏ฌeld theory to encodesuch equivalences by means of gauge ๏ฌelds, and in the present setup this leads to the so-called โgauged linear sigma modelโ formulation, which often simpli๏ฌes the calculations (inthe example of the CP ๐ โ model we already encountered it in section 7.2). In the simplestcase ๐ ๐ก = const. , ๐ ๐ก = const. the Lagrangian (10.1) with the topological term (10.2) may berewritten as follows: ๐ = ๐ โ๏ธ ๐ด = โซ ๐๐ฅ ๐๐ (cid:20) โ ๐ | ( d + i ๐ ๐ด ) ๐ข ๐ด | + i ๐ ๐ด ๐ d ๐ ๐ด + ๐ ๐ ( ๐ข ๐ด + ยท d ๐ข ๐ด ) โง ( ๐ข ๐ด + ยท d ๐ข ๐ด ) (cid:21) , (10.5)where ๐ ๐ด are U ( ) gauge ๏ฌelds. As before, the ๏ฌrst term is the usual kinetic term of thenonlinear sigma model, and the last term is the โnon-topologicalโ part of the skew-symmetric๏ฌeld, as we explained in Sections 9.1-9.2. It is linear both in space and time derivatives,but it is not topologically quantized to integers, and will not be important for the โt Hooftanomaly matching. The second term in the above action is the topological theta term ofthe two-dimensional U ( ) ๐ gauge theory. We may set ๐ ๐ด = ๐ โซ d ๐ ๐ด , moreover the ๐ ๐ด sode๏ฌned are equivalent to the topological charges that we encountered earlier, as one can seeby eliminating the gauge ๏ฌelds through their e.o.m. Indeed, solving the e.o.m. of ๐ ๐ด , we ๏ฌnd ๐ ๐ด = i2 ( ๐ข ๐ด ยท d ๐ข ๐ด โ d ๐ข ๐ด ยท ๐ข ๐ด ) = i ๐ข ๐ด ยท d ๐ข ๐ด . (10.6)We may interpret ๐๐ ๐ด as the Fubini-Study form on the ๐ด -th copy of CP ๐ โ (see (2.8)).Moreover, as discussed in Section 1.4 the ๏ฌag manifold F ๐ is a Lagrangian submanifold of ( CP ๐ โ ) ร ๐ , which implies ๐ (cid:205) ๐ด = ๐๐ ๐ด =
0. As a result, ๐ (cid:205) ๐ด = ๐ ๐ด =
0, which is of course the65ondition that we encountered many times before, cf. (9.7). As explained in Sections 9.1-9.2,the above system (10.5) possesses a cyclic Z ๐ โquasiโ-symmetry (i.e. a symmetry up to anintegral of an element of ๐ป (F , Z ) ), if the ๐ -angles are chosen as ๐ ๐ด = ๐๐ ๐ด๐ for ๐ โ Z .Now we turn to the discussion of the mixed โt Hooft anomaly between the PSU ( ๐ ) ๏ฌavorsymmetry and the Z ๐ permutation symmetry . The anomaly manifests itself in the fact thatthe partition function of the system in a topologically non-trivial background PSU ( ๐ ) gauge๏ฌeld is not invariant under the Z ๐ permutation, even at the point ๐ โ = ๐๐ โ๐ , where the systemdescribed by the action (10.5) is invariant.How do we introduce a background ๏ฌeld for the PSU ( ๐ ) ๏ฌavor symmetry? To answerthis question, one should ๏ฌrst clarify the di๏ฌerence between SU ( ๐ ) and PSU ( ๐ ) bundles. APSU ( ๐ ) bundle ๐ over a worldsheet ฮฃ possesses an additional Z ๐ -valued invariant, which isa member of the second cohomology ๐ป ( ฮฃ , ๐ ( PSU ( ๐ ))) โ the generalized Stiefel-Whitneyclass ๐ค ( ๐ ) (see, e.g., [253, 247]). Some examples of such bundles will be provided in thenext section 10.4, and for the moment we turn to a more formal de๏ฌnition of the invariant.First, the PSU ( ๐ ) bundle ๐ may be lifted, in a non-unique way, to a vector bundle ๐ over ฮฃ , with structure group U ( ๐ ) . The latter is characterized by its ๏ฌrst Chern class ๐ ( ๐ ) (itsintegral is the โabelian ๏ฌuxโ). The non-uniqueness in choosing ๐ has to do with the factthat we could replace it with ๐ โ L , where L is any line bundle, since it would cancel outin the projective quotient anyway. Recalling that ๐ ( ๐ โ L) = ๐ ( ๐ ) + ๐ ๐ (L) , we seethat ๐ ( ๐ ) mod ๐ is a well-de๏ฌned topological quantity. This mod ๐ -reduced class is calledthe generalized Stiefel-Whitney class ๐ค ( ๐ ) โ ๐ป ( ฮฃ , Z ๐ ) = Z ๐ , which characterizes thetopologically non-trivial PSU ( ๐ ) -bundles.We will now convert this description into a relevant gauge theory formulation at the levelof the Lagrangian. To mimic the description in terms of a vector bundle ๐ with structuregroup U ( ๐ ) , we will introduce a U ( ๐ ) gauge ๏ฌeld (cid:101) ๐ด . In order to implement the quotient, onehas to postulate the additional gauge transformations ๐ ๐ด โฆโ ๐ ๐ด โ ๐, (cid:101) ๐ด โฆโ (cid:101) ๐ด + ๐ ๐ , (10.7)where ๐ is a U ( ) gauge ๏ฌeld, which simultaneously plays a role of gauge parameter here.Now, the point is that a U ( ๐ ) gauge ๏ฌeld has an integer invariant โ the ๏ฌrst Chern numberโ that is expressed as follows: โซ ฮฃ ๐ ( ๐ ) = ๐ โซ ๐ (cid:16) Tr ( (cid:101) ๐ด ) (cid:17) โ Z . Accordingly, since ๐ is aU ( ) gauge ๏ฌeld, its curvature also has quantized periods, โซ ฮฃ ๐ (L) = ๐ โซ ๐๐ โ Z . Dueto the shift symmetry (10.7) that we have imposed, the periods of (cid:101) ๐ด are shifted by multiplesof ๐ : โซ ฮฃ ๐ ( ๐ ) โฆโ โซ ฮฃ ๐ ( ๐ ) + ๐ โซ ฮฃ ๐ (L) , and as a result we get a Z ๐ (cid:39) Z (cid:14) ๐ Z invariant inplace of an integer invariant that one would have without the additional symmetry (10.7). Asexplained in [149, 116, 16], this invariant may be encoded in a 2-form Z ๐ gauge ๏ฌeld that wecall ๐ต . In our notation it is simply ๐ต : = ๐ ๐ (cid:16) Tr ( (cid:101) ๐ด ) (cid:17) . Under the shift (10.7) it changes as The discussion in [221] contains also the case of a mixed PSU ( ๐ ) โ Z ๐ (cid:48) anomaly, where ๐ (cid:48) is a divisor of ๐ , but we restrict here to the simpler case ๐ (cid:48) = ๐ . ๐ต โฆโ ๐ต + ๐๐ . (10.8)According to the above discussion, the integral12 ๐ โซ ฮฃ ๐ต โ Z ๐ (10.9)is a multiple of ๐ . To write a gauged version of the action (10.5), ๏ฌrst of all we replacethe covariant derivatives ( d + i ๐ ๐ด ) ๐ข ๐ด by the elongated derivatives ( d + i ๐ ๐ด + i (cid:101) ๐ด ) ๐ข ๐ด โ acombination invariant under the shift (10.7). Besides, the curvatures ๐๐ ๐ด are not invariantunder the shift (10.7) but the combinations ๐๐ ๐ด + ๐ต are. As a result, the e๏ฌect of subjectingthe system (10.5) to an external PSU ( ๐ ) gauge ๏ฌeld is in the following modi๏ฌcation of theaction: ๐ gauged = โ๏ธ ๐ด = โซ ๐ (cid:20) โ ๐ (cid:12)(cid:12)(cid:12) ( d + i ๐ ๐ด + i (cid:101) ๐ด ) ๐ข ๐ด (cid:12)(cid:12)(cid:12) + i ๐ ๐ด ๐ ( d ๐ ๐ด + ๐ต ) + (10.10) + ๐ ๐ { ๐ข ๐ด + ยท ( d + i (cid:101) ๐ด ) ๐ข ๐ด } โง { ๐ข ๐ด + ยท ( d + i (cid:101) ๐ด ) ๐ข ๐ด } (cid:21) . Notice that there is no need to write the ๐ ๐ด gauge ๏ฌelds in the ๐ -term, since their contributionsvanish due to the orthogonality constraint between ๐ข ๐ด . Performing the path integral, ๐ [( ๐ด, ๐ต )] = โซ D ๐ D ๐ข D ๐ข exp ( ๐ gauged ) , (10.11)we obtain the partition function ๐ [( ๐ด, ๐ต )] in the background PSU ( ๐ ) gauge ๏ฌeld. ๐ด is meantto represent the traceless part of the gauge ๏ฌeld (cid:101) ๐ด . One curious thing to notice about thepartition function ๐ is that it is no longer 2 ๐ -periodic in the ๐ -angles. Indeed, a shift of oneof the angles, say ๐ โ ๐ + ๐ , produces a phase ๐ [( ๐ด, ๐ต )] โฆโ ๐ [( ๐ด, ๐ต )] ยท ๐ ๐ โซ ๐ต . Z ๐ anomaly in a PSU ( ๐ ) background It is instructive to notice that, in the presence of the PSU ( ๐ ) background gauge ๏ฌeld, variationof the action w.r.t. ๐ โ no longer produces (10.6), and the following modi๏ฌed formula for thecurvatures holds instead: ๐ โ๏ธ ๐ด = ( ๐๐ ๐ด + ๐ต ) = . (10.12)If we consider the modi๏ฌed topological charges (cid:101) ๐ ๐ด : = ๐ โซ ฮฃ ( ๐๐ ๐ด + ๐ต ) , which sum tozero, then according to (10.9) they will be quantized in units of ๐ . We will encounter thisphenomenon in the discussion of fractional instantons below and relate it to twisted boundary67onditions, which in the present language are encoded in a non-trivial PSU ( ๐ ) gauge ๏ฌeld.Fractional instantons arising in the presence of twisted boundary conditions have also beendiscussed in the context of the resurgence program, cf. [99, 98].Let us now show that the action (10.10) of the model in an external PSU ( ๐ ) gauge๏ฌeld is not invariant under the Z ๐ shift symmetry ๐ข ๐ด โ ๐ข ๐ด + (which should be obviouslysupplemented by ๐ ๐ด โ ๐ ๐ด + ). First of all, the metric and ๐ terms are evidently invariant, soit su๏ฌces to compute the topological term. We recall that we have chosen our ๐ -angles as ๐ ๐ด = ๐๐ ๐ด๐ , so that the topological term changes as follows under a Z ๐ -shift (we set ๐ ๐ + โก ๐ ): ๐ top = i ๐ ๐ โ๏ธ ๐ด = ๐ด๐ โซ ( d ๐ ๐ด + ๐ต ) โฆโ i ๐ ๐ โ๏ธ ๐ด = ๐ด๐ โซ ( d ๐ ๐ด + + ๐ต ) = (10.13) = ๐ top โ i ๐๐ โซ ๐ โ๏ธ ๐ด = ( d ๐ ๐ด + ๐ต ) = ( . ) + i ๐ โซ ( d ๐ + ๐ต ) . As โซ d ๐ โ ๐ Z , this term drops o๏ฌ in the path-integral. However, since โซ ๐ต โ ๐๐ Z , the ๐ต -term contributes a phase, so we have ๐ [( ๐ด, ๐ต )] โฆโ ๐ [( ๐ด, ๐ต )] exp (cid:18) i ๐ โซ ๐ต (cid:19) (10.14)under the Z ๐ permutation. This is the mixed โt Hooft anomaly between PSU ( ๐ ) and Z ๐ .There is no local counter term that can eliminate the generation of the ๐ต -term under the Z ๐ exchange symmetry. Indeed the only counter-terms allowed are i ๐ โซ ๐ต where ๐ โ Z mod ๐ ,and these are invariant under the Z ๐ symmetry.In the cases of continuous global symmetries it was argued by โt Hooft long ago [139] thatthe anomalies should match between the UV and IR limits of the theory: even if the e๏ฌectivetheory in the IR looks drastically di๏ฌerent from the original UV theory, both theories shouldexhibit the same anomalies. This is typically used to derive constraints on the IR dynamics,which might be otherwise di๏ฌcult to deduce directly from the UV theory. Originally thisidea was developed for the study of chiral symmetries in QCD, however it is believed that thesame property holds for discrete symmetries or mixed continuous-discrete symmetries as inour example here. In [220] the UV/IR matching of anomalies was related to the so-calledโadiabatic continuityโ of the theory, i.e. to the smooth dependence of the physical propertiesof a theory compacti๏ฌed on a circle on the radius of the circle (cf. [216, 15, 99] for examplesof when this does or does not hold).By the anomaly matching argument, the ground state at the Z ๐ invariant point, ๐ ๐ด = ๐๐ ๐ด๐ ,with ๐ not a multiple of ๐ , cannot be trivially gapped. According to [83], in 1 + ( ๐ ) chain (see section 4.1), and the argument reviewedhere provides its ๏ฌeld-theoretic counterpart. 68 ( ๐ ) -bundles An attentive reader might have noticed that, in the above discussion, at least two of thesteps were reminiscent of what we already encountered in other contexts. First of all, thecalculation of the shift in the topological terms (10.13) is very similar to the shift calculatedin (8.21), albeit with an important distinction that in the latter case the shift was by a 2 ๐ -quantized term, which is immaterial in the path integral. Secondly, at the level of the partitionfunction (10.14) the Z ๐ -transformation is a change of variables in the path integral (10.11),so whenever exp (cid:16) i ๐ โซ ๐ต (cid:17) โ ๐ in (4.7) in the proof ofthe LSMA theorem, which leads to the vanishing of a certain matrix element. Both of thesesimilarities are not coincidences. We start with a somewhat more intuitive explanation of the background PSU ( ๐ ) gauge ๏ฌeld.In fact, in several cases the results of the previous section may well be formulated in terms of ๏ฌat background gauge ๏ฌelds, ๐๐ด โ ๐ด โง ๐ด =
0. We will explain this on two examples: thoseof a torus T and of a sphere ๐ .The torus example is somewhat easier, as there is a general statement (reviewed in [253])that, for a simple group ๐บ , every ๐บ -bundle over T admits a ๏ฌat connection . This is alsotrue for the topologically non-trivial PSU ( ๐ ) bundles of the previous section. To calculate thecorresponding invariant ๐ค , one views the gauge ๏ฌeld as an SU ( ๐ ) gauge ๏ฌeld and computesits holonomies ๐ and ๐ along two meridians of the torus. Since ๐ ( T ) = Z , in PSU ( ๐ ) theholonomies would satisfy ๐๐ = ๐๐ , which in SU ( ๐ ) is relaxed to ๐๐๐ โ ๐ โ = ๐ โ ๐ค . Notethat these holonomies ๐ and ๐ may well be non-trivial even for a ๏ฌat gauge ๏ฌeld ๐ด , which isthe reason that it su๏ฌces to consider ๏ฌat connections.Now, the ultimate use of ๏ฌat connections is that they may be completely eliminated at theexpense of imposing twisted boundary conditions on the ๏ฌelds. Indeed, a ๏ฌat connection hasthe form ๐ด = โ ๐ โ ๐๐ , where ๐ is locally a function on the worldsheet, which, when viewedglobally, encodes the holonomies ๐ and ๐ . If ๐ฅ and ๐ฆ are the coordinates along the meridiansof the torus, one has ๐ ( ๐ฅ + ๐, ๐ฆ ) = ๐ โฆ ๐ ( ๐ฅ, ๐ฆ ) and ๐ ( ๐ฅ, ๐ฆ + ๐ ) = ๐ โฆ ๐ ( ๐ฅ, ๐ฆ ) . Accordingly,if ๐ข ๐ด are the matter ๏ฌelds of the model charged under the PSU ( ๐ ) gauge group (like the unitvector ๏ฌelds of the ๏ฌag models), we can now perform a gauge transformation ๐ข ๐ด โ ๐ โฆ ๐ข ๐ด ,which completely eliminates the gauge ๏ฌeld at the expense of imposing twisted boundaryconditions ๐ข ๐ด ( ๐ฅ + ๐, ๐ฆ ) โผ ๐ โฆ ๐ข ๐ด ( ๐ฅ, ๐ฆ ) , ๐ข ๐ด ( ๐ฅ, ๐ฆ + ๐ ) โผ ๐ โฆ ๐ข ๐ด ( ๐ฅ, ๐ฆ ) ( โผ means โup to aphaseโ, since the ๐ข ๐ด take values in a projective space).In the language used before, eliminating the gauge ๏ฌeld amounts to setting ๐ด = ๐ต = The fact that ๐บ is simple is crucial here. For example, for a line bundle L with gauge group U ( ) one wouldhave an additional invariant โ the ๏ฌrst Chern number โซ ฮฃ ๐ (L) = ๐ โซ ฮฃ ๐น , expressed through the curvature ๐น ofthe connection. Gluing with twist ๐ โFigure 13: A schematic depiction showing that the twisted partition function Tr ( ๐๐ โ ๐ฝ๐ป ) ofthe spin chain leads, in the continuum limit, to a PSU ( ๐ ) bundle over a torus. This bundle isdescribed by the twist operator ๐ entering the LSMA theorem.the shifts in the topological terms: (10.13) in the presence of the ๐ด, ๐ต gauge ๏ฌelds and(8.21) without them. The point is that, with the twisted boundary conditions, the shift โซ ฮฉ in (8.21) is no longer quantized as an integer times 2 ๐ , but rather as an integer times ๐๐ , thusreproducing the shift by the ๐ต -๏ฌeld in (10.13). One way to see this is to observe that, withthe twisted boundary conditions, the ๏ฌuxes of the gauge ๏ฌelds ๐ ๐ด are quantized in multiplesof ๐๐ , and since ฮฉ = ๐๐ , this leads to the corresponding statement for โซ ฮฉ . Indeed, thetwists have the form ๐ข ๐ด ( ๐ฅ + ๐, ๐ฆ ) โผ ๐ โฆ ๐ข ๐ด ( ๐ฅ, ๐ฆ ) , where โผ means that actually ๐ โ PSU ( ๐ ) and is only de๏ฌned up to a power of the root of unity ๐ . In the formulation with ๐ ๐ด gauge๏ฌelds, to undo this ambiguity one performs a gauge transformation ๐ข ๐ด โ ๐ ๐๐ ๐ด ( ๐ฅ,๐ฆ ) ยท ๐ข ๐ด , wherethe gauge parameter ๐ ๐ด ( ๐ฅ, ๐ฆ ) has periodicity ๐ ๐ด ( ๐ฅ + ๐, ๐ฆ ) โ ๐ ๐ด ( ๐ฅ, ๐ฆ ) = ๐๐ . Since the gaugetransformation a๏ฌects the gauge ๏ฌelds ๐ ๐ด โ ๐ ๐ด โ d ๐ ๐ด , we conclude that the ๏ฌuxes of ๐ ๐ด willbe quantized as multiples of 2 ๐ / ๐ . As we recall, ๐ ๐ด = ๐ โซ ๐๐ ๐ด are the topological charges,so we come to the conclusion that these topological charges are quantized in multiples of ๐ .This means that the shift (8.21) that was immaterial for periodic boundary conditions (or,in general, for maps from a closed Riemann surface ฮฃ ), now produces a non-vanishingcontribution, equal to the one of the ๐ต -๏ฌeld in (10.13).In fact, we have already encountered an example of a non-trivial PSU ( ๐ ) -bundle over atorus (albeit in a discretized form) in the proof of the LSMA theorem in Section 4.1. Therewe de๏ฌned the so-called twist operator acting on a spin chain of length ๐ฟ : ๐ = ๐ ๐ด ๐ด : = ๐๐๐๐ฟ ๐ฟ โ๏ธ ๐ = ๐ ๐ ๐ , ๐ = ๐ โ โ๏ธ ๐ผ = ๐ ๐ผ๐ผ โ ( ๐ โ ) ๐ ๐๐ โ ๐ฐ๐ฒ ( ๐ ) . (10.15)Clearly, in the continuum limit we get ๐ = (cid:214) ๐ฅ โ[ , ๐ ) ๐ ( ๐ฅ ) , ๐ ( ๐ฅ ) = ๐ ๐ ๐ฅ๐ ๐ ๐ฅ . (10.16)In the framework of ๏ฌag manifold sigma models that arise in the continuum limit, the setupof the LSMA theorem is as follows: we wish to compute the partition function Tr ( ๐๐ โ ๐ฝ๐ป ) ๐ ,as shown in Fig. 13. Topologically this produces a PSU ( ๐ ) -bundle over a torus, whose ๐ค -invariant is characterized by the periodicity property of ๐ ( ๐ฅ ) : ๐ ( ๐ฅ + ๐ ) = ฮพ ยท ๐ ( ๐ฅ ) , where ฮพ โ Z ๐ . Looking back at (10.16), we ๏ฌnd ฮพ = ๐ ๐๐๐ ๐ . If one deals with rank- ๐ symmetricrepresentations, as in the LSMA theorem of section 4.1, the invariant is ฮพ = ๐ ๐๐๐๐ . Thisis the โanomaly factorโ that appears in (4.8) upon inserting the translation operator in (4.7),which should be seen as parallel to making a cyclic permutation ๐ข ๐ด โ ๐ข ๐ด + in the pathintegral (10.11) and obtaining a factor (10.14) as a result. ( ๐ ) bundles As a next step, we consider the worldsheet ๐ . We wish to explain that examples of sectionsof topologically non-trivial PSU ( ๐ ) bundles over a sphere ๐ are provided by the so-calledโfractional instantonsโ that we will introduce shortly. For a recent discussion of these fractionalisntantons in the context of resurgence, see the recent paper [229].The ๏ฌbers of the relevant bundles are the target spaces of the sigma model, i.e. the ๏ฌagmanifolds. We recall that a section of a topologically trivial bundle is simply a map ๐ โ F from the worldsheet to the ๏ฌag manifold target space. One can alternatively think of it as amap R โ F with โdecay conditions at in๏ฌnityโ, meaning that the in๏ฌnity of R is mappedto a single point in F . In general, ๏ฌber bundles over ๐ with structure group PSU ( ๐ ) maybe de๏ฌned using a patching function ๐ โ PSU ( ๐ ) , where ๐ = ๐ + โฉ ๐ โ is the equatorโ the intersection of the two patches ๐ ยฑ on ๐ (the northern and southern hemispheres).Topologically the bundles are determined by the class of the patching map ๐ โ PSU ( ๐ ) in the homotopy group ๐ ( PSU ( ๐ )) (cid:39) Z ๐ . Now, suppose we want to construct a sectionof such a bundle ๐ : ๐ธ โ ๐ , with ๏ฌber F and structure group PSU ( ๐ ) . Over either ๐ + or ๐ โ one can trivialize the bundle, i.e. one identi๏ฌes ๐ โ ( ๐ ยฑ ) (cid:39) ๐ ยฑ ร F . Constructing asection of ๐ธ is the same as specifying two maps ๐ ยฑ : ๐ ยฑ โ F patched across the equator,i.e. ๐ + (cid:12)(cid:12) ๐ = ๐ โฆ ๐ โ (cid:12)(cid:12) ๐ , where ๐ : ๐ โ PSU ( ๐ ) is the patching function.Let us now explain how this construction may be used for the description of fractionalinstantons. The latter, by de๏ฌnition, are maps R โ F with the following behavior at in๏ฌnity: ๐ข ๐ผ๐ด = โ ๐ ๐ ๐ด๐ผ ๐ i ๐ค ๐ผ ๐ , ๐ = ๐ ๐ i ๐ . (10.17)Here ๐ is the angle around a โcircle at in๏ฌnityโ, and ๐ค ๐ผ โ Z are the winding numbers.Unless all ๐ค ๐ผ have the same value, such maps do not satisfy decay conditions at in๏ฌnity(if ๐ค ๐ผ = const., the winding is undone by a gauge transformation). We will view R ,together with the circle at in๏ฌnity, as the upper hemisphere ๐ + , and the fractional instantonwill serve to de๏ฌne the map ๐ + : ๐ + โ F . Now, on ๐ โ we will de๏ฌne a constant map,given by (cid:101) ๐ข ๐ผ๐ด = โ ๐ ๐ ๐ด๐ผ . Clearly, along the equator the two are related by the patchingmap ๐ = ๐ โ i ๐ ( (cid:205) ๐๐ผ = ๐ค ๐ผ ) ๐ Diag ( ๐ i ๐ค ๐ , ยท ยท ยท , ๐ i ๐ค ๐ ๐ ) โ PSU ( ๐ ) . The topology of the bundle is71haracterized by [ (cid:205) ๐๐ผ = ๐ค ๐ผ mod ๐ ] โ Z ๐ = ๐ ( PSU ( ๐ )) . If ๐ค ๐ผ = const., the correspondinginvariant vanishes, in line with the discussion above. In Section 10.3 we described the mixed PSU ( ๐ ) โ Z ๐ anomalies that arise for ๏ฌag manifoldsigma models. In the case when such anomalies are present, one possibility is that therenormalization group ๏ฌow leads to a conformal ๏ฌeld theory in the IR. Moreover, as it isrecorded in Table 1 and discussed in section 10.1, one conjectures that the resulting conformal๏ฌeld theory is the SU ( ๐ ) WZNW model. To motivate this relation, we recall what is perhapsthe most vivid way to see a connection between the ๏ฌag sigma model and the WZNW model.The idea is to embed the former into the latter [221, 190]. One starts with the WZNWLagrangian, de๏ฌned as follows (here ๐บ โ SU ( ๐ ) ): ๐ WZNW : = ๐ โซ ๐ Tr ( ๐ ๐ ๐บ ๐ ๐ ๐บ โ ) + i12 ๐ ๐ โซ ๐ Tr ( ( ๐บ โ ๐๐บ ) ) , (10.18)where ๐ = ฮฃ is the two dimensional spacetime and ๐ is a three-manifold whose boundaryis ๐ , i.e. ๐ ๐ = ๐ . The coe๏ฌcient ๐ is quantized to be a positive integer. In the UVthe radius ๐ of the target space is large and the theory comprises ๐ โ ๐ = ๐ ๐ in the IR [248].The main statement is that the action (10.18), when restricted to unitary matrices ๐บ witha ๏ฌxed spectrum , produces the action of a ๏ฌag manifold sigma model, whose ๐ -angles aredictated by the spectrum of ๐บ . In other words, we will be considering matrices ๐บ of the form ๐บ = ๐ ฮฉ ๐ โ , where ฮฉ = diag ( ๐ i ๐ , ๐ i ๐ , ยท ยท ยท , ๐ i ๐ ๐ ) , ๐ โ SU ( ๐ ) . (10.19)For simplicity we assume ๐ ๐ด โ ๐ ๐ต for ๐ด โ ๐ต . In this case the matrix ๐ is de๏ฌned upto right multiplication by a diagonal matrix ๐ท , ๐ โผ ๐ ยท ๐ท , so that ๐ de๏ฌnes a point in a๏ฌag manifold. Using (10.19), the kinetic term of the WZNW action (10.18) can be easilycomputed as: ๐ ( ๐ ๐ ๐บ ๐ ๐ ๐บ โ ) = ๐ โ๏ธ ๐ด ๐ ๐ ๐ข ๐ด โฆ ๐ ๐ ๐ข ๐ด โ ๐ โ๏ธ ๐ด,๐ต ๐ i ( ๐ ๐ด โ ๐ ๐ต ) | ๐ข ๐ด โฆ ๐ ๐ ๐ข ๐ต | . (10.20)To compute the WZ term one takes ๐ = ๐ ร I where I = [ , ] is an interval withcoordinate ๐ฆ . To make sure that ๐ has a single boundary ๐ , one e๏ฌectively compacti๏ฌesthe second boundary ๐ ร { ๐ฆ = } by requiring ๐บ (cid:12)(cid:12) ๐ฆ = = . Since the WZ action (afterexponentiation) doesnโt depend on the extension of the ๏ฌelds to the bulk of ๐ , one maychoose ๐บ ( ๐ง, ๐ง, ๐ฆ ) = ๐ ( ๐ง, ๐ง ) ฮฉ ( ๐ฆ ) ๐ ( ๐ง, ๐ง ) โ , (10.21) ฮฉ ( ๐ฆ ) = Diag ( ๐ i ๐ ( ๐ฆ ) , ๐ i ๐ ( ๐ฆ ) , ยท ยท ยท , ๐ i ๐ ๐ ( ๐ฆ ) ) , where ๐ ๐ด ( ) = ๐ ๐ด , ๐ ๐ด ( ) = .
72e may now substitute (10.21) into the second term of (10.18), and one ๏ฌnds that it splitsinto a sum of two: โซ ๐ Tr ( ( ๐บ โ ๐๐บ ) ) = โซ ๐ (cid:0) ฮฉ top + ฮฉ ๐ (cid:1) .The ๏ฌrst one produces the ๐ -terms of the ๏ฌag model: โซ ๐ ฮฉ top = ๐ ๐ โซ ๐ โ๏ธ ๐ด ๐ ๐ด ๐๐ข ๐ด โงโฆ ๐๐ข ๐ด , (10.22)whereas the second one is the non-topological part of the ๐ต -๏ฌeld (the โ ๐ -termโ, as we referredto it earlier): โซ ๐ ฮฉ ๐ = โ ๐ ๐ โ๏ธ ๐ด โ ๐ต sin ( ๐ ๐ด โ ๐ ๐ต ) โซ ๐ ( ๐ข ๐ต โฆ ๐๐ข ๐ด ) โง ( ๐ข ๐ต โฆ ๐๐ข ๐ด ) . (10.23)As was shown in section 9.2, the Z ๐ -invariant values correspond to the choice of angles (withan overall subtraction so that det ฮฉ = ๐ ๐ด = ๐ ๐ด๐ โ ๐ ( ๐ + ) ๐ . (10.24)One also needs to have a mechanism to restrict the spectrum of the matrix ๐บ in (10.18)to be of the form ๐ i ๐ ๐ด , with the values (10.24). The paper [190] proposes the followingscenario: one adds to (10.18) a potential ๐ = (cid:205) (cid:98) ๐ (cid:99) ๐ = ๐ ๐ Tr ( ๐ ๐ ) Tr ( ( ๐ โ ) ๐ ) . In the limit whenall ๐ ๐ โ โ one restricts to the locus Tr ( ๐ ๐ ) = , ๐ = . . . (cid:98) ๐ (cid:99) , which can be shown to implya spectrum of the form (10.24).
11 A gas of fractional instantons
In the previous section the generalized Haldane conjectures for an SU ( ๐ ) spin chain withsymmetric rank- ๐ representations at each site were formulated. Following [235], we will nowrecall an intuitive explanation for these conjectures based on fractional topological excitations.This is a generalization of an older work in SU(2) [8], which explains the generation of theHaldane gap in terms of merons in the ๐ nonlinear sigma model.In the case of the ๐ target space the idea was to arrive at the O(2) model in a speciallimit, when a large potential is added to restrict the ๏ฌeld (cid:174) ๐ to the XY plane. In the absence ofthe ๐ -angles, it is well known that a mass gap is generated in the O(2) model, due to vortexproliferation [158]. In the case of the ๐ model with a large potential this mass gap is stillgenerated when the potential is weakened, and (cid:174) ๐ can lift o๏ฌ the plane. The resulting nonplanarvortices are known as merons [8]. This argument was used to identify merons as the mass-gap generating mechanism in the ๐ sigma model which corresponds to the purely isotropic โMeronโ means half-instanton, and refers to the fact that these con๏ฌgurations have half-integer quantizedtopological charges. The word โinstantonโ is used, in place of vortex or soliton, because one of the twodimensions in the O(2) model corresponds to Euclidean time. ๐ =
0. We will now review a generalization of this argument that suggests that amass gap is present in the SU( ๐ )/[ U ( )] ๐ โ ๏ฌag manifold sigma model (without topologicalterms), and that it is generated by topological excitations. ๐๐ -model Following [8], the strategy is to break the symmetry of the ๏ฌag manifold down to U(1),where a phase transition is well understood in terms of vortex proliferation. One starts byadding to the Lagrangian an anisotropic potential ๐ that breaks the SU( ๐ ) symmetry downto [ U ( )] ๐ โ : ๐ = ๐ ๐ โ๏ธ ๐ด = โ๏ธ ๐ผ<๐ฝ (cid:0) | ( ๐ข ๐ผ๐ด | โ | ( ๐ข ๐ฝ๐ต | (cid:1) . (11.1)It is the SU( ๐ ) generalization of adding the term (cid:205) ๐ ๐ ๐ง ( ๐ ) ๐ ๐ง ( ๐ ) to the SU(2) Hamiltonian.In the limit ๐ โ โ , the potential ๐ restricts all of the components ๐ข ๐ผ๐ด of ๐ข ๐ด to be ofequal absolute value (equal to โ ๐ due to normalization), with arbitrary phases. Taking intoaccount the gauge group U ( ) ๐ โ , this gives ๐ ( ๐ โ ) real parameters. The number of (real)orthogonality constraints is formally also ๐ ( ๐ โ ) , however one should take into account theremaining U ( ) ๐ โ global symmetry acting as ๐ข ๐ผ๐ด โ ๐ ๐๐ ๐ผ ๐ข ๐ผ๐ด . As a result, the con๏ฌgurationminimizing the potential ๐ is (up to permutations of the vectors ๐ข , ยท ยท ยท , ๐ข ๐ ) ๐ข ๐ผ๐ด = โ ๐ ๐ ๐ด๐ผ ๐ i ๐ ๐ผ , ๐ = ๐ ๐ i ๐ , ๐ ๐ผ โ [ , ๐ ] . (11.2)Orthonormality of these states follows from the identity (cid:205) ๐๐ = ๐ ๐ ๐ = ๐ โ ๐ . The Z ๐ symmetry ๐ข ๐ด โ ๐ข ๐ด + is represented on (11.2) by a shift ๐ ๐ผ โ ๐ ๐ผ + ๐๐ผ๐ . (11.3)The formula (11.2) de๏ฌnes an embedding ( ๐ ) ๐ โ โ F , and in the present setup this isthe torus of asymptotic vortex con๏ฌgurations away from the core. Substituting (11.2) into theLagrangian (10.1), one obtains a generalized XY-model with ๐ โ ๐ -valued ๏ฌelds, coupledto each other. Although this model could perhaps be studied in full generality, we would liketo make use of the known results for the standard XY-model, and to this end we will add anadditional potential ๐ = ๐ (cid:205) ๐๐ด = (cid:205) ๐ โ ๐ผ = (cid:16) Im [( ๐ข ๐ด ๐ข ๐ผ๐ด ) ๐ ] (cid:17) that will suppress all but one ๏ฌelds.The potentials ๐ and ๐ have common minima (zero locus), as one can see by rewriting ๐ in terms of (11.2): ๐ = ๐ (cid:205) ๐๐ด = (cid:205) ๐ โ ๐ผ = sin ( ๐ ( ๐ โ ๐ ๐ผ )) . Due to the factor of ๐ the set ofminima of ๐ is invariant under the Z ๐ symmetry (11.3), which corresponds to translationalinvariance in the underlying lattice model. It is clear that the e๏ฌect of ๐ is to equate all butone of the U(1) ๏ฌelds (up to the shift (11.3)). Due to residual gauge symmetry, everythingthen depends only on one variable ๐ : = ๐ ๐ โ ๐ . This is equivalent to the O(2) model of a74ector (cid:174) ๐ โ R restricted to the XY plane. By inserting this restricted form of ๐ข ๐ด into (10.1),it is easy to show that the ๐ ๐ด,๐ต terms vanish, and the resulting O(2) coupling constant is ๐ โ = โ๏ธ ๐ด,๐ต ๐ โ ๐ด,๐ต . (11.4) For ๐ = ๐ vanishes and the perturbation ๐ is equivalent to adding a massterm ๐ ( ๐ ) to the ๐ model Lagrangian, restricting (cid:174) ๐ to lie in the XY plane in the large ๐ limit. This follows from the equivalence ๐ ๐ = ๐ข โ ๐ ๐ ๐ข (we already used it earlier in (6.6),where ๐ = ๐ง โฆ ๐ โฆ ๐ง๐ง โฆ ๐ง was written in stereographic coordinates).The vortices of the model, in order to be well-de๏ฌned at the core, must become non-planar. They are called merons and have topological charge ๐ = ยฑ (the sign depending onwhether ๐ = ยฑ ๐ = ยฑ ๐ model. More exactly, to use the notation developed for ๏ฌag manifoldmodels, we will be thinking of the topological charge as a pair of charges (cid:174) ๐ = ( ๐ , ๐ ) ,such that ๐ + ๐ = ๐ = ๐ . As explained earlier (seesection 1.4, for example), this corresponds to the embedding ๐ โ ( ๐ ) ร ( ๐ ) mapping (cid:174) ๐ โ ( (cid:174) ๐, โ(cid:174) ๐ ) , in which case ๐ ๐ = ๐ โซ ฮฃ ๐ โ ฮฉ ๐ , where ฮฉ , ฮฉ are the two Fubini-Study forms,subject to ( ฮฉ + ฮฉ ) (cid:12)(cid:12) ๐ =
0, and ๐ : ฮฃ โ ๐ โ ( ๐ ) ร ( ๐ ) is a map from a worldsheet ฮฃ .In this simplest case ๐ is the area on ๐ of the image of ๐ , multiplied by the number oftimes a typical point is covered. If ฮฃ is a closed Riemann surface, such as ๐ , ๐ and ๐ are integers. In particular, a map ๐ : ๐ โ ๐ may be thought of as a map R โ ๐ witha ๏ฌxed asymptotic value at in๏ฌnity: (cid:174) ๐ (โ) = (cid:174) ๐ . In the case of a meron, on the contrary,the asymptotic behavior at in๏ฌnity is such that ๐ | โ : ๐ โ ๐ โ ๐ , where ๐ โ ๐ is theequator ๐ =
0. As a result, ๐ maps ๐ท โ ๐ , where ฮฃ = ๐ท is a disc, with the condition thatthe boundary of the disc is glued to the equator in a prescribed way. The simplest situation iswhen the disc covers once the upper or lower hemisphere, in which case the correspondingarea ๐ = ยฑ (this corresponds also to a single winding of the boundary map ๐ | โ ).The setup can be generalized to the SU( ๐ ) case as follows. As discussed earlier, theminimum of ๐ is achieved at the con๏ฌguration (11.2), which de๏ฌnes an embedding ( ๐ ) ๐ โ โF . We wish to compute the topological charges ๐ ๐ด = ๐ โซ ๐ท ๐ โ ฮฉ ๐ด for a map ๐ : ฮฃ = ๐ท โF , such that ๐ (cid:12)(cid:12) โ : ( ๐ ) WS โ ( ๐ ) ๐ โ is a map with a ๏ฌxed set of winding numbers (cid:174) ๐ค โ ๐ ( ( ๐ ) ๐ โ ) (cid:39) Z ๐ โ . From (11.2) it is natural to think of (cid:174) ๐ค as being the set of windingsof ๐ angles ๐ ๐ modulo the winding vector ( , ยท ยท ยท , ) that can be removed by an overall U ( ) gauge transformation, i.e. (cid:174) ๐ค = ( ๐ค , ยท ยท ยท , ๐ค ๐ ) mod ( , ยท ยท ยท , ) .To compute the topological numbers, we recall the ๏ฌag manifold embedding F โ( CP ๐ โ ) ร ๐ (see Section 1.4) and denote ๐ ๐ด the projection on the ๐ด -th projective space.Given a map ๐ : ฮฃ โ F , we construct a map ๐ ๐ด = ๐ ๐ด โฆ ๐ to CP ๐ โ . If ( ๐ : ยท ยท ยท : ๐ ๐ ) are the homogeneous coordinates on the projective space, we choose the standard ๐ patches75 ๐ ๐ด } ๐ด = ,...,๐ , each one given by the condition ๐ ๐ด โ ๐ด . Let us now considerthe special maps ๐ (the โelementary fractional instantonsโ) characterized by the fact that ( ๐ ) ๐ด ( ฮฃ ) โ ๐ ๐ ( ๐ด ) , ๐ being a permutation. This means that the image of each ( ๐ ) ๐ด lies in asingle patch ๐ ๐ ( ๐ด ) . In each patch ๐ ๐ด we set ๐ ๐ด = ฮฉ (cid:12)(cid:12) ๐ ๐ด = ๐๐ ๐ด ,where ๐ ๐ด = ๐ (cid:205) ๐ต ๐ ๐ต ๐๐ ๐ต + (cid:205) ๐ต โ ๐ด | ๐ ๐ด | is a well-de๏ฌned Poincarรฉ-Liouville one-form, so that by Stokes theorem ( ๐ ) ๐ด = ๐ โซ ๐ ๐ โ ๐ ๐ ( ๐ด ) = ๐ โ๏ธ ๐ต (cid:0) ๐ค ๐ต โ ๐ค ๐ ( ๐ด ) (cid:1) , (11.5)where we have substituted the asymptotic values (11.2). Clearly (cid:205) ๐ด ( ๐ ) ๐ด =
0, as required.A general fractional instanton may be thought of as a collection of instantons โon topโ of anelementary fractional instanton, resulting in the topological charge (cid:174) ๐ + ( ๐ , ยท ยท ยท , ๐ ๐ ) , where ๐ ๐ด โ Z are integers, (cid:205) ๐ด ๐ ๐ด = ๐ , and so we would like to restrict to the con๏ฌgurationsthat asymptotically minimize this potential, i.e. to the special maps ๐ท โ F , such that ( ๐ ) WS = ๐ ๐ท โ ๐ โ ( ๐ ) ๐ โ . In this case we may set ๐ค = ยท ยท ยท = ๐ค ๐ โ =
0. Apart fromthat, let us restrict to a single winding, ๐ค ๐ =
1. The topological charges of an elementaryfractional instanton then are ( ๐ ( ๐ ) = ๐ ) (cid:174) ๐ = (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) ๐ , ยท ยท ยท , ๐ โ position ๐ , ยท ยท ยท , ๐ (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) (11.6) As we have seen, there are several topological sectors. While the number of con๏ฌgurationshas increased, the original argument from SU(2) for mass generation carries over to this moregeneral case: for each species of topological excitation in this ๐ -fold family there is a speciesof particle in the Coulomb gas formalism [8]. That is, each particle has a partition functionthat is represented (at large distances) by a sine-Gordon (sG) model, L ๐๐บ = ( ๐ ๐ ๐ ) + ๐พ cos (cid:18) ๐๐ ๐ (cid:19) , (11.7)in the limit of large ๐พ , which represents the fugacity, or density, of the fractional instantongas. This expression is derived in detail in [9] and relies on the fact that all higher-loopcorrections to the (fractional) instanton gas are IR ๏ฌnite in the sG model [158]. Formallyspeaking, expanding the partition function of the sG model in ๐พ , i.e. in the cosine interaction,produces a multi-vortex Coulomb gas partition function of the XY-model (a similar trick hasbeen used in Liouville theory, cf. [222]).In (11.7) ๐ is the O(2) coupling constant in (11.4), and plays the role of temperature inthe sG model. Since all of the ๐ species arise from the same action, each will have the same76ugacity and critical ๐ , so that the above model (11.7) is merely copied ๐ times, and the SU(2)analysis from [8] can be applied directly: for large ๐ , the fractional instantons are diluteand we are in a massless boson phase. As ๐ is lowered, the e๏ฌective critical temperatureis increased until the topological excitations condense and a mass gap is produced. Thus,we conclude that fractional instantons are responsible for generating a mass gap in the ๏ฌagmanifold sigma model (10.1), in the absence of topological angles.
We now restore the topological angles ๐ ๐ผ , and study how the mass generating mechanismchanges. For large ๐ , we are in the O(2) model and the ๐ -terms do not play a role. However,as ๐ is lowered towards zero, the fugacity ๐พ in the sine-Gordon model is modi๏ฌed to ๐พ ๐ โ๏ธ ๐ด = ๐ ๐ (cid:174) ๐ ยท (cid:174) ๐ ๐ด , (11.8)where the sum is over the ๐ species of fractional instanton, and ๐ ๐ด = ๐ด๐ , with ๐ = ๐๐๐ . Using(11.6), one easily ๏ฌnds that this sum equals ๐พ (cid:205) ๐ด ๐ ๐ ๐ด . So long as ๐ is not a multiple of ๐ ,this sum vanishes, and the Coulomb gas is in its massless phase.At ๏ฌrst glance, this appears to be inconsistent with the conjecture discussed in section 10,which also predicts a gap when ๐ is not a multiple of ๐ , but has a nontrivial shared divisorwith ๐ . This discrepancy is resolved by considering higher order topological excitations.This is summarized in the following table: Case Winding Fugacity Conclusion ๐ = ๐ ๐๐พ Mass generation ๐ and ๐ coprime(no common divisor) 1 , ยท ยท ยท , ๐ โ ๐ ๐๐พ Masslessgcd ( ๐, ๐ ) โ , ๐ โ < ๐ < ๐ , ยท ยท ยท , ๐ โ ๐ ๐๐พ Mass generationHere ๐ : = ๐ gcd ( ๐,๐ ) . Winding number ๐ค refers to the map ( ๐ ) WS โ ( ๐ ) target . Forsimplest fractional instanton con๏ฌgurations the topological charge is ๐ค (cid:174) ๐ , and thefugacity is ๐พ (cid:205) ๐ด ๐ ๐ ๐ด๐ค .While objects that have winding number greater than ยฑ ๐ =
0, and itis conjectured in [235] that this holds for general ๐ . When ๐ and ๐ have a nontrivial common It is also worth noting that the exact critical exponents for the sG model are well known, and are reviewedin [9]. ๐ , con๏ฌgurations with a smaller value of the action begin to contributeto mass generation, starting with objects that have winding number ๐ . As a result, the criticalvalue ๐ is larger than at ๐ = ๐๐ (although still lower than at ๐ =
12 More general representations: linear and quadratic dis-persion
As we already discussed earlier, depending on the sign of the coupling constant ๐ฝ the groundstate of the Heisenberg chain is either ferro- or anti-ferromagnetic. The continuum limitsaround these two states lead to rather di๏ฌerent models: in the ferromagnetic case this is theLandau-Lifschitz model (section 6.3) that describes spin waves with quadratic dispersion,whereas in the anti-ferromagnetic case one obtains a relativistic sigma model (section 7),which in the gapless case describes excitations with linear dispersion.In the present section, following [234], we will consider spin chains of the following type:at each site we will place spins in an arbitrary representation R of SU ( ๐ ) , with the conditionthat this representation is the same for all sites. The representation will be characterizedby the lengths of the rows in the Young diagram, ๐ โฅ ๐ โฅ ยท ยท ยท โฅ ๐ ๐ โ โฅ
0. Amongsuch models we will pick out those that lead to sigma models with the ๏ฌag manifold targetspace U ( ๐ ) U ( ) ๐ in the continuum limit. One restriction that is imposed by this setup is that all thenonvanishing ๐ ๐ โs will be distinct (if some of the rows of the Young diagram were of the samelength, one would obtain a partial ๏ฌag manifold U ( ๐ ) U ( ๐ )รยทยทยทร U ( ๐ ๐ ) as the target space, cf. [73]).The curious feature of the general situation is that, although we will take anti-ferromagneticcouplings between the spins, some of the modes will have quadratic dispersion relation, justas in the ferromagnetic case (although others will have linear dispersion). One can then workout conditions on the representation R that ensure that only linear modes remain. For suchrepresentations we will deduce the topological angles of the resulting models, which, as wesaw earlier, are to a large extent responsible for the phase structure of these models.To start with, we recall that in the semiclassical (large spin) limit the spin opera-tors ๐ ( ๐ ) of the chain should be replaced by the corresponding moment maps ๐ ( ๐ ) = ๐ โ diag ( ๐ , ยท ยท ยท , ๐ ๐ ) ๐ , where ๐ is a unitary matrix, and ๐ ยท ยท ยท ๐ ๐ are the lengths of the rowsin the Young diagram of the representation R ( ๐ ๐ = ๐ and ๐ becomes Tr ( ๐ ( ๐ ) ๐ ( ๐ )) โ (cid:205) ๐ด,๐ต ๐ ๐ด ๐ ๐ต | ๐ข ๐ด ( ๐ ) โฆ ๐ข ๐ต ( ๐ ) | . As a result, thesimplest SU( ๐ ) chain Hamiltonian, namely the nearest-neighbour model, becomes ๐ป = ๐ฝ โ๏ธ ๐ ๐ โ โ๏ธ ๐ด,๐ต = ๐ ๐ด ๐ ๐ต | ๐ข ๐ด ( ๐ ) โฆ ๐ข ๐ต ( ๐ + ) | , ๐ฝ > . (12.1)The sums over ๐ด and ๐ต stop at ๐ โ
1, since ๐ ๐ = ๐ ) generalization of the antiferromagnetic spinchain. However, in most cases, we will be required to consider Hamiltonians with longerrange interactions if we hope to map to the ๏ฌag manifold sigma model. This should already78e clear from the discussion of the rank- ๐ symmetric representations in sections 7.3 and 9above. Since the complete ๏ฌag manifold is the space of ๐ -tuples of mutually orthogonal๏ฌelds taking values in CP ๐ โ , one must add ( ๐ โ ) -neighbour interactions in order to imposeorthogonality on the ๐ ๏ฌelds. Instead, if one couples less than ๐ sites of the chain together,there will be leftover degrees of freedom, which manifest as local zero modes, ultimatelyprohibiting any ๏ฌeld theory mapping.We will now explain how this construction generalizes as we increase the number ๐ ofnonzero ๐ ๐ .For the purposes of presentation we will consider three main examples: ๐ = ๐ = ๐ โ ๐ (cid:39) ๐ , is a generalization of the construction described in section 7.1) and the case ๐ = ๐๐ .We refer the reader to [234] for more general situations. We will be using some graphical notation for describing classical spin con๏ฌgurations ofSU( ๐ ) chains. First, let { (cid:174) ๐ ๐ } be an orthonormal basis of C ๐ .We will use coloured circles to represent the ๏ฌrst few elements of this basis, as shownin Fig 14. ~e = ~e = ~e = ~e = ~e = ~e = ~e = ~e = Figure 14: Colour dictionary for the ๏ฌrst eight basis elements in C ๐ . These coloured circleswill be used to pictorially reprsent classical states of the chain.When drawing a classical ground state, we will arrange the same-site vectors into a singlecolumn, and use a white space to separate neighbouring chain sites. For example, the Nรฉelstate of the SU(2) antiferromagnet is . (12.2)The bene๏ฌt of these ground state pictures is that it makes it easy to read o๏ฌ the energy costof a term Tr ( ๐ ( ๐ ) ๐ ( ๐ )) = (cid:205) ๐ด,๐ต ๐ ๐ด ๐ ๐ต | ๐ข ๐ด ( ๐ ) โฆ ๐ข ๐ต ( ๐ ) | . The right hand side of this expressionvanishes unless one of the complex unit vectors (i.e. one of the colours) at site ๐ equals oneof the complex unit vectors at site ๐ . In this case, the r.h.s. equals ๐ ๐ด ๐ ๐ต , where ๐ด and ๐ต are the respective positions of the unit vector/colour in column ๐ and column ๐ .With this notation in place, we will now describe the ground state structure of SU( ๐ )chains with some su๏ฌciently simple representations at each site.79 ๐ = . We begin with recalling what occurs for the symmetric representations of SU( ๐ ), with Youngtableaux that have a single row of length ๐ (see sections 7.3-9 above). For a nearest-neighbour SU( ๐ ) Heisenberg Hamiltonian any con๏ฌguration that has no energy cost per bondwill be a classical ground state. Since ๐ =
1, and only a single node is present at each site,the Nรฉel state shown above is such an example. However, for ๐ >
2, the basis at each site islarger than 2 (i.e. there are other colours available), and this leads to an in๏ฌnite number ofother ground states, resulting in a zero-energy mode that destabilizes any candidate groundstate above which we would like to derive a quantum ๏ฌeld theory. As a consequence, thenearest-neighbour Hamiltonian must be modi๏ฌed by longer-range interactions . Since thereare ๐ possible colours, we require up to ( ๐ โ ) -neighbour interactions, all of which are takento be antiferromagnetic, in order to remove the zero modes. For example, in SU(5), withinteractions up to 4th neighbour, one such ground state is . (12.3)This is a colourful depiction of the con๏ฌguration (9.1). As we recall from sections 7-9, the๏ฌag manifold U ( ๐ ) U ( ) ๐ appears here as the space of ๐ -tuples of pairwise orthogonal complexvectors, each vector coming from a copy of CP ๐ โ at one site of the chain. ๐ = ๐ โ . The second class of representations that we consider have Young tableaux with ๐ โ R already corresponds to the complete ๏ฌag manifold, a nearest-neighbour Heisenberg interaction is su๏ฌcient to derive the associated sigma model. Letus ๏ฌrst demonstrate this for the case of SU(4). The interaction term Tr ( ๐ ( ๐ ) ๐ ( ๐ + )) = (cid:205) ๐ด,๐ต = ๐ ๐ด ๐ ๐ต | ๐ข ๐ด ( ๐ ) โฆ ๐ข ๐ต ( ๐ + ) | is never zero for two adjacent sites, which requires choosingthe colour for six nodes. Using the inequality ๐ + ๐ โฅ ๐ ๐ , after a momentโs thoughtone ๏ฌnds that the ground states have the following form: (12.4)This pattern extends to general ๐ : the ๏ฌrst row of nodes establishes a Nรฉel-like state, while theremaining ๐ โ Z translation symmetry in the sigma model. These interactions may be dynamically generated from the nearest-neighbour model [86]. ๐ = ๐๐ To construct the ground state, one partitions ๐ colors into ๐ sets, with ๐ colors in each set.We would like to place each set at one of the consecutive ๐ sites, so to this end we add up ( ๐ โ ) -neighbour interactions (always with antiferromagnetic couplings) which make surethat the ๐ -planes at the consecutive sites are orthogonal to each other. On top of that, inorder to obtain the complete ๏ฌag manifold, one still needs to orthogonalize ๐ vectors insideeach ๐ -plane, which can be achieved by adding a weaker ๐ -neighbour interaction that servesto reverse order within each set of the partition. For example, in SU(6) with ๐ =
2, theHamiltonian we should consider is ๐ป = โ๏ธ ๐ (cid:16) ๐ฝ Tr ( ๐ ( ๐ ) ๐ ( ๐ + )) + ๐ฝ Tr ( ๐ ( ๐ ) ๐ ( ๐ + )) + ๐ฝ Tr ( ๐ ( ๐ ) ๐ ( ๐ + )) (cid:17) (12.5)with ๐ฝ > ๐ฝ (cid:29) ๐ฝ >
0, which has, for example, the following ground state: (12.6)The ๐ฝ and ๐ฝ terms serve to partition the colours into three sets (the โ2-planesโ): { , } , { , } , { , } , and the ๐ฝ terms serve to reverse order within each of these three sets. Clearlythe unit-cell has size 2 ๐ for these representations. In the previous section we constructed spin chain Hamiltonians, whose classical minima lieon a complete ๏ฌag manifold. This de๏ฌnes an embedding ๐ : F โ M , (12.7)where M = (cid:206) ๐๐ด = M ๐ด is the phase space of an elementary cell of length ๐ . As we recall fromsections 6.3-7, the next step in deriving a continuum theory is in evaluating the restriction ofthe symplectic form ฮฉ M = ๐ โ๏ธ ๐ด = ฮฉ ๐ด , (12.8)which comes from the kinetic term in the Lagrangian (the โBerry phaseโ), to the space ofminima of the Hamiltonian, i.e. to the ๏ฌag manifold. In the ferromagnetic situation ofSection 6.3 this restriction is non-degenerate. On the other hand, in the anti-ferromagneticsituation of section 7 the restriction is identically zero (i.e. the ๏ฌag manifold is a Lagrangiansubmanifold). The general situation is intermediate (the restricted form is degenerate but notexactly vanishing), and the relevant characteristic is the rank of the restriction ฮฉ M (cid:12)(cid:12) F , whichde๏ฌnes the number of ๏ฌelds with quadratic dispersion. By tuning the values of the integers81epresentation Conditions ๐ = ๐ = ๐ โ ๐ ๐ + ๐ ๐ โ ๐ + = ๐ ๐ even; ๐ = , ยท ยท ยท , ๐ ๐ ๐ + ๐ ๐ โ ๐ + = ๐ ๐ ๐ + = ๐ ๐ odd; ๐ = , ยท ยท ยท , ๐ โ ๐ = ๐๐ ๐ ๐ + ๐ ๐ + โ ๐ = ๐ + ๐ ๐ ๐ even; ๐ = , ยท ยท ยท , ๐ ๐ ๐ + ๐ ๐ + โ ๐ = ๐ + ๐ ๐ ๐ ๐ + = ๐ + ๐ ๐ ๐ odd; ๐ = , ยท ยท ยท , ๐ โ Table 2: Rank reduction conditions (elimination of modes with quadratic dispersion) for thetwo-form ฮฉ M (cid:12)(cid:12) F . ๐ ๐ , i.e. by suitably choosing the representation R , one can reduce the rank down to zero, inwhich case F โ M is an isotropic submanifold.At least in the linearly dispersing case (when ฮฉ M (cid:12)(cid:12) F = ๐ top = ๐ ๐ โ๏ธ ๐ด = ๐ด ยท ฮฉ ๐ด (cid:12)(cid:12) F . (12.9)We recall (cf. (2.30)) that in the general case each form ฮฉ ๐ด may be written as ฮฉ ๐ด = ๐ ๐ (cid:205) ๐ = ๐ ๐ ๐๐ข ( ๐ ) ๐ด โง โฆ ๐๐ข ( ๐ ) ๐ด , where { ๐ข ( ๐ ) ๐ด } ๐๐ = are mutually orthogonal vectors at the ๐ด -th site ofthe unit cell. These vectors are represented by the circles in a given column of the colourdiagram, such as (12.3), (12.4) or (12.6), and the โrestriction to F โ in (12.9) means replacingthe given vector ๐ข ( ๐ ) ๐ด by the vector of the ๏ฌag corresponding to the indicated colour.Let us demonstrate how this works for ๐ = ๐ โ
1. The elementary cell consists of twosites, ๐ =
2, so that ฮฉ M = ๐ (cid:205) ๐ = ๐ (cid:205) ๐ด = ๐ ๐ด ๐๐ข ๐ด ( ๐ ) โง โฆ ๐๐ข ๐ด ( ๐ ) . According to the pattern ofground states (see (12.4)), two of the colours occur once (in the ๏ฌrst position of the column),and the remaining ๐ โ For the case of spins with rectangular Young tableau at each site, when the resulting ๏ฌag manifold is themanifold of partial ๏ฌags, the same expression was obtained in [73].
82f the symplectic form is1 ๐ ฮฉ M (cid:12)(cid:12) F = ๐ ( ๐๐ข โง โฆ ๐๐ข + ๐๐ข โง โฆ ๐๐ข ) + ๐ โ๏ธ ๐ด = ( ๐ ๐ด โ + ๐ ๐ โ ๐ด + ) ๐๐ข ๐ด โง โฆ ๐๐ข ๐ด . (12.10)Here ๐ข ๐ด without the site label in brackets are meant to represent the ๐ orthogonal vectors ofthe embedded ๏ฌag manifold F . We recall that (cid:205) ๐๐ด = ๐๐ข ๐ด โงโฆ ๐๐ข ๐ด (cid:12)(cid:12) F =
0, so that the expressioncan be simpli๏ฌed: 1 ๐ ฮฉ M (cid:12)(cid:12) F = ๐ โ๏ธ ๐ด = ( ๐ ๐ด โ + ๐ ๐ โ ๐ด + โ ๐ ) ๐๐ข ๐ด โง โฆ ๐๐ข ๐ด . (12.11)Now we have up to ( ๐ โ ) ๏ฌelds with quadratic dispersion. The exact number will dependon how many of the conditions ๐ ๐ด โ + ๐ ๐ โ ๐ด + โ ๐ = ๐ (cid:39) ๐ of SU( ๐ ). The case of SU(3) with2 ๐ = ๐ was considered in detail in [236]. Similar constraints can be derived for otherrepresentations, see Table 2 and ref. [234]. Restricting in an analogous way the two-form ๐ top (12.9), one obtains the topological term. Similarly to (12.11), it may be expanded as ๐ top = ๐ ๐ โ๏ธ ๐ด = ๐ ๐ด ๐๐ข ๐ด โง โฆ ๐๐ข ๐ด . (12.12)For the simple representations that we have discussed here the values of the ๐ -angles arerecorded in Table 3. The discrete symmetry in the general case is Z ๐ : it acts on ๐ top by shifting ๐ top โ ๐ top + ฮฉ (compare with (8.21)). One could in principle derive themixed PSU ( ๐ ) โ Z ๐ anomalies in this case as well, which would provide a generalization ofHaldane-type conjectures to this type of representations.Representation Topological Angles ๐ = ๐ ๐ด = ๐๐ ๐ ( ๐ด โ ) ๐ = , , ยท ยท ยท , ๐๐ = ๐ โ ๐ ๐ด = ๐ ๐ ๐ด ๐ = , , ยท ยท ยท , ๐๐ = ๐๐ ๐ ๐ด,๐ต = ๐ ( ๐ ๐ด + ๐ ๐ + โ ๐ต ) ๐ ( ๐ต โ ) + ๐ ๐ ๐ + โ ๐ด ๐ด = , ยท ยท ยท , ๐ ; ๐ต = , ยท ยท ยท , ๐ Table 3: Possible topological angles for some representations of SU( ๐ ) chains. In the lastrow the index is split as ๐ด โ ( ๐ด, ๐ต ) . 83 hapter 3. Integrable ๏ฌag manifold sigma models and beyond In the present chapter we pass to the subject of integrable sigma models with ๏ฌag manifoldtarget spaces, as well as some more general models. Recall that the integrability of the ๐ -model [264], which predicted massive excitations over the vacuum state, was one of themotivations for Haldaneโs proposal that SU ( ) integer-spin chains have a gap in the spectrum.It was subsequently shown [265] that the ๐ = ๐ model is soluble as well, this time with amassless spectrum, in line with Haldaneโs treatment of the half-integer-spin chains. In the caseof SU ( ๐ ) chains the resulting ๏ฌag manifold sigma models described in the previous chapterare apparently not integrable, and their integrable counterparts discussed below feature avery special metric and ๐ต -๏ฌeld. One striking parallel between the two types of models is theimportant role played by the Z ๐ symmetry, as we explain below in Section 13.4. Anotherimportant feature of the proposed integrable models is their relation to nilpotent orbits, whichalso featured in our discussion of the Dyson-Maleev representation in Section 2.4 above.Sections 13, 14 and 15 are dedicated to various aspects of the classical theory of integrablesigma models. The reason why we discuss this in great detail is that, when the targetspace of the model is not symmetric, constructing even a classical integrable theory is asigni๏ฌcant challenge. In section 16 we will argue that the integrable ๏ฌag manifold sigmamodels are in fact equivalent to (generalized) chiral Gross-Neveu models. This relationallows one to take a glimpse in the quantum realm of these models, at least in the one-loopapproximation. For example, the analysis of the one-loop ๐ฝ -function in section 16.3 givesrather important insights in the structure of these models. Another quantum aspect of theproblem is the subject of chiral anomalies that we touch upon in section 16.5. Besides,rather surprisingly, the formulation of sigma models as Gross-Neveu models implies thatthe interactions in these sigma models are polynomial . This fact is based on, or perhapspartially explained by, two seemingly unrelated observations. One is that the Dyson-Maleevvariables provide a polynomial parametrization for the spin operators. The other is that, atleast in the simplest cases [58], the integrable models of the relevant class may be obtained bydimensional reductions of 4D gravity, expressed in Ashtekar variables, which are known tomake the interactions in gravity polynomial. These fascinating inter-relations are explainedin section 16.4.Before we describe the theory in full generality, let us provide an example of the relationbetween sigma models and Gross-Neveu models. Consider the bosonic Thirring model. In84erms of a Dirac spinor
ฮจ = (cid:18) ๐๐ (cid:19) the two-dimensional Thirring Lagrangian reads: L = ฮจ / ๐ ฮจ + ( ฮจ ๐พ ๐ ฮจ ) = ๐ ๐๐ + ๐๐๐ + | ๐ | | ๐ | . (12.13)To obtain the bosonic Thirring model, we now regard the variables ๐ and ๐ as bosonic.Eliminating ๐ , ๐ , we obtain the sigma model form of the system: L = ๐๐ ๐๐๐๐ . The targetspace is a cylinder with multiplicative coordinate ๐ . At the quantum level, the eliminationof ๐ , ๐ means we have to integrate over these variables in the path integral. As a result, oneshould take into account the corresponding determinant, which is the source of an emergingdilaton. In this case the dilaton ฮฆ โผ log | ๐ | is linear along the cylinder. This is thesystem describing the asymptotic region of Wittenโs cigar [250]. As we shall see, a wideclass of sigma models may be seen to arise by a very similar procedure from chiral gauged Gross-Neveu models, which are natural extensions of (12.13).
13 The models and the zero-curvature representation
We start with a more conventional formulation of sigma models by describing their metricand ๐ต -๏ฌeld in Lie-algebraic terms. In this chapter we will always assume that the worldsheet ฮฃ is a two-dimensional Riemannian manifold. As for the target-space M , in full generality wewill not require it to be a ๏ฌag manifold but rather a manifold with the following properties : โฆ M is a homogeneous space ๐บ / ๐ป, ๐บ semi-simple and compact โฆ M has an integrable ๐บ -invariant complex structure J (13.1) โฆ The Killing metric G on M is Hermitian w.r.t. J Let us explain what we mean by โKilling metricโ on a homogeneous space. To this end, wedecompose the Lie algebra ๐ค of the Lie group ๐บ as ๐ค = ๐ฅ โ ๐ช , (13.2)where ๐ช is the orthogonal complement to ๐ฅ with respect to the Killing metric on ๐ค . Wemay accordingly decompose the Maurer-Cartan current ๐ฝ = ๐ โ ๐๐ = ๐ฝ ๐ฅ โ ๐ฝ ๐ช . The โKillingmetricโ G on ๐บ๐ป is de๏ฌned by the line element ๐๐ = โ Tr ( ๐ฝ ๐ช ) . (13.3)The corresponding metric on ๐ช will be called Killing as well. In the case of trivial ๐ฅ thiswould then reduce to the canonical Killing metric, hence the name.For a target space M with the properties (13.1), one can de๏ฌne a sigma model, whoseequations of motion may be rewritten as the ๏ฌatness condition for a one-parameter family of Generalizations to non-simple groups ๐บ are also possible. ๐ด ๐ข , ๐ข โ C โ . This ๏ฌatness condition is an extension to this broader class of targetspaces of a property that is encountered in sigma models with symmetric target spaces [199,102, 262]. In the latter case, this property is an important sign of integrability of the model:it may be used to ๏ฌnd Bรคcklund transformations [228, 95], and it is a starting point for theconstruction of classical solutions of the models [132].Complex simply-connected homogeneous manifolds ๐บ / ๐ป with ๐บ semi-simple were clas-si๏ฌed long ago [238]. They are given by the following theorem: any such manifold ๐บ / ๐ป corresponds to a subgroup ๐ป , whose semi-simple part coincides with the semi-simple part ofthe centralizer of a toric subgroup of ๐บ . For the case of ๐บ = SU ( ๐ ) , for example, invariantcomplex structures exist on those of the manifolds M ๐ ,...,๐ ๐ | ๐ = SU ( ๐ ) ๐ ( U ( ๐ ) ร . . . ร U ( ๐ ๐ )) , ๐ โฅ , ๐ ๐ > , ๐ โ๏ธ ๐ = ๐ ๐ โค ๐ , (13.4)that are even-dimensional. If (cid:205) ๐๐ = ๐ ๐ = ๐ , the manifold in (13.4) is a ๏ฌag manifold. Otherwise,it is a toric bundle over a ๏ฌag manifold. The ๏ฌber U ( ) ๐ (2 ๐ = ๐ โ (cid:205) ๐๐ = ๐ ๐ ) of the toricbundle is even-dimensional, since the ๏ฌag manifold itself is even-dimensional.The models, which will be of interest for us in the present paper, are de๏ฌned by thefollowing action: S [ G , J ] : = โซ ฮฃ ๐ ๐ง (cid:107) ๐ ๐ (cid:107) G + โซ ฮฃ ๐ โ ๐, (13.5)where ๐ is the fundamental Hermitian form corresponding to the pair ( G , J ) , de๏ฌned as ๐ = G โฆ J . (13.6)In general, the Killing metric G is not Kรคhler, i.e. the fundamental Hermitian form isnot closed: ๐๐ โ
0. Even if the manifold M admits a Kรคhler metric, it is in generaldi๏ฌerent from G . As an example of such a phenomenon one can consider the ๏ฌag manifold SU ( ) ๐ ( U ( ) ) . The sigma model (13.5) for this ๏ฌag manifold was investigated in detail in [70, 63].Other examples of models of the class (13.1) are provided by Hermitian symmetric spaces โsymmetric spaces with a complex structure. These manifolds are Kรคhler, and the invariantmetric is essentially unique (up to scale), thus leading to the closedness of ๐ : ๐๐ =
0. Wewill discuss this special case in Section 13.3. For the moment let us note the followingequivalent rewriting of the action (13.5):
S [ G , J ] : = โซ ฮฃ ๐ ๐ง G ๐ ๐ ๐๐ ๐ ๐๐ ๐ , (13.7)where we have introduced complex coordinates ๐ ๐ on M . Curiously, models with the ๐ต -๏ฌeldof the form (13.6) appeared in [255] in the context of topological sigma models, and gaugedWess-Zumino-Novikov-Witten theories with this feature were studied in [119, 52].86 Let us now formulate the requirements (13.1) on the target space M = ๐บ๐ป in Lie algebraicterms, and prove that the e.o.m. that follow from the action (13.7) admit a zero-curvaturerepresentation.We will assume that the quotient space ๐บ / ๐ป possesses a ๐บ -invariant almost complexstructure J . We are not postulating that J be integrable โ this will rather follow from therequirement of the existence of a Lax connection. The almost complex structure acts on ๐ช (the subspace featuring in the decomposition (13.2)) and may be diagonalized, its eigenvaluesbeing ยฑ ๐ (see the following section for details). We denote the ยฑ ๐ -eigenspaces by ๐ช ยฑ โ ๐ช C : ๐ค C = ๐ฅ C โ ๐ช + โ ๐ช โ , J โฆ ๐ช ยฑ = ยฑ ๐ ๐ช ยฑ . (13.8) ๐บ -invariance of the almost complex structure implies that [ ๐ฅ , ๐ช ยฑ ] โ ๐ช ยฑ . We introduce thecurrent ๐ฝ = ๐ โ ๐๐ = ๐ฝ + ๐ฝ + + ๐ฝ โ , ๐ฝ โ ๐ฅ , ๐ฝ ยฑ โ ๐ช ยฑ . (13.9)It takes values in the Lie algebra ๐ค , and we have decomposed it according to the decomposition(13.8) of the Lie algebra. In these terms the action (13.5) may be rewritten as follows(henceforth we will be using bracket notation for the scalar product of two elements ๐ผ, ๐ฝ โ ๐ค in the Killing metric): S [ G , J ] : = โซ ฮฃ ๐ ๐ง (cid:104)( ๐ฝ + ) ๐ง , ( ๐ฝ โ ) ๐ง (cid:105) . (13.10) Example.
Let us consider the ๏ฌag manifolds of the group ๐บ = SU ( ๐ ) , which are the mainsubject of this review. A typical integrable complex structure on the ๏ฌag manifold de๏ฌnes theholomorphic/anti-holomorphic subspaces ๐ช ยฑ shown in Fig. 15. It is useful do decomposeFigure 15: The decomposition (13.8) of the Lie algebra. ๐ฝ ยฑ in the irreducible representations ๐ ๐ด๐ต of the stabilizer ๐ฅ , see (1.30). For this purpose weparametrize the unitary matrix ๐ as follows: ๐ = { ๐ , ๐ , . . . , ๐ ๐ โ , ๐ ๐ } , (13.11)87here ๐ . . . ๐ ๐ are groups of ๐ . . . ๐ ๐ orthonormal vectors, each group parametrizing a planeof the corresponding dimension in C ๐ . The projection of ๐ฝ ยฑ on ๐ ๐ด๐ต is given by ๐ฝ ๐ด๐ต : = ๐ โ ๐ด ๐๐ ๐ต ,and the full action (13.10) takes the form S [ G , J ] : = โซ ฮฃ ๐ ๐ง โ๏ธ ๐ด<๐ต Tr ( ( ๐ฝ ๐ต๐ด ) ๐ง ( ๐ฝ ๐ด๐ต ) ๐ง ) . (13.12)These are the integrable models with ๏ฌag manifold target spaces that we wish to study.We return to the action (13.10). The Noether current, constructed using the above action,will be denoted by ๐พ . It is derived by taking an in๏ฌnitesimal ( ๐ง, ๐ง ) -dependent variation ๐ โ ๐ ๐ ( ๐ง,๐ง ) โฆ ๐ in the above action, which leads to ๐พ = ๐ ยท (cid:0) ( ๐ฝ + ) ๐ง ๐๐ง + ( ๐ฝ โ ) ๐ง ๐๐ง (cid:1) : = ๐ ยท ๐ โ = ๐๐๐ โ (13.13)Since the target space M = ๐บ / ๐ป is homogeneous, the equations of motion of the model areequivalent to the conservation of ๐พ : ๐ โ ๐พ = . (13.14)Here, โ denotes the Hodge star operator, whose action on one-forms is de๏ฌned by โ ๐๐ง = ๐ ๐๐ง, โ ๐๐ง = โ ๐ ๐๐ง . . In order to be able to build a family of ๏ฌat connections we require that ๐พ be ๏ฌat (This will be used in (13.19)-(13.20) below.): ๐๐พ โ ๐พ โง ๐พ = . (13.15)We have to show, of course, that it is possible to satisfy this relation. Equations (13.14)-(13.15)may be rewritten in terms of ๐ (introduced in (13.13)) as follows: ๐ โ ๐ + { ๐ฝ, โ ๐ } = (13.16) = โ ๐๐๐ง โง ๐๐ง (cid:0) D ( ๐ฝ + ) ๐ง โ [( ๐ฝ + ) ๐ง , ( ๐ฝ + ) ๐ง ] + D ( ๐ฝ โ ) ๐ง + [( ๐ฝ โ ) ๐ง , ( ๐ฝ โ ) ๐ง ] (cid:1) = ๐๐ + { ๐ฝ โ ๐, ๐ } = (13.17) = โ ๐๐ง โง ๐๐ง (cid:0) D ( ๐ฝ + ) ๐ง โ [( ๐ฝ + ) ๐ง , ( ๐ฝ + ) ๐ง ] โ D ( ๐ฝ โ ) ๐ง โ [( ๐ฝ โ ) ๐ง , ( ๐ฝ โ ) ๐ง ] (cid:1) = D is the covariant derivative for the gauge group ๐ป : D ๐ ๐ ๐ : = ๐ ๐ ๐ ๐ + [( ๐ฝ ) ๐ , ๐ ๐ ] ( ๐ , ๐ = ๐ง, ๐ง ). The conditions (13.16)-(13.17) are equivalent, if [ ๐ช + , ๐ช + ] โ ๐ช + , [ ๐ช โ , ๐ช โ ] โ ๐ช โ . (13.18)We will discuss below in Section 13.2 that if the metric G is Hermitian w.r.t. the chosenalmost complex structure J , this requirement is equivalent to the integrability of J .Consider now the following family of connections ๐ด ๐ข , indexed by a parameter ๐ข โ C โ : ๐ด ๐ข = โ ๐ข ๐พ ๐ง ๐๐ง + โ ๐ข โ ๐พ ๐ง ๐๐ง . (13.19)88onservation and ๏ฌatness of the Noether current ๐พ , eqs. (13.14)-(13.15), imply that ๐ด ๐ข is ๏ฌat for all ๐ข [199]: ๐๐ด ๐ข โ ๐ด ๐ข โง ๐ด ๐ข = ๐ข โ C โ . (13.20)This completes the derivation of the zero-curvature representation for the class of mod-els (13.10), which includes the ๏ฌag manifold models (13.12). We now turn to the general theory of complex structures on ๏ฌag manifolds, which will playan important role throughout this chapter.A very detailed treatment of complex structures on homogeneous spaces was given asearly as in the classic work [53], so here we mostly present an adaptation of some of thesestatements to our needs. To start with, on the manifold U ( ๐ ) U ( ) ๐ of complete ๏ฌags in C ๐ thereare 2 ๐ ( ๐ โ ) invariant almost complex structures, with ๐ ! โค ๐ ( ๐ โ ) of them being integrable. As we already saw in (13.8), the complex structure J induces a decomposition ๐ช C =๐ช + โ ๐ช โ , where ๐ช ยฑ play the role of holomorphic tangent spaces to ๐บ / ๐ป , i.e. J โฆ ๐ = ยฑ ๐ ๐ for ๐ โ ๐ช ยฑ . In section 1.5 (formula (1.30)) we have already decomposed ๐ช C into irreduciblecomponents. Using this decomposition, we may de๏ฌne an almost complex structure on F byde๏ฌning the action of J as follows: J โฆ ๐ ๐ด๐ต = ยฑ ๐ ๐ ๐ด๐ต for 1 โค ๐ด < ๐ต โค ๐ . (13.21)As a result, one has exactly 2 ๐ ( ๐ โ ) possibilities. There are several equivalent de๏ฌnitions ofintegrability of a complex structure: โฆ Vanishing of the Nฤณenhuis tensor: [ J โฆ ๐, J โฆ ๐ ] โ J โฆ ( [ J โฆ ๐, ๐ ] + [ ๐, J โฆ ๐ ]) โ [ ๐, ๐ ] = ๐, ๐ . โฆ Using vector ๏ฌelds: the commutator of two holomorphic vector ๏ฌelds should be holo-morphic, i.e. ( โ ๐ J ) [( + ๐ J ) ๐, ( + ๐ J ) ๐ ] = . (13.23)(The property (13.23) may also be stated as the condition that the distribution ofholomorphic vector ๏ฌelds is integrable.) This is easily seen to be equivalent to (13.22). โฆ Using forms: the holomorphic forms should constitute a di๏ฌerential ideal in the algebraof forms, i.e. the following condition should be satis๏ฌed: ๐ ( ๐ฝ โ ) ๐ โผ (cid:205) ๐ ๐ ๐๐ โง ( ๐ฝ โ ) ๐ forsome one-forms ๐ ๐๐ . We note that for large ๐ , according to Stirlingโs formula, ๐ ๐ log ( ๐ ) < ๐ log ( ) ๐ .
89f the restriction to ๐ช of the adjoint-invariant metric (cid:104)โข , โข(cid:105) on ๐ฒ ( ๐ ) is Hermitian w.r.t. thechosen almost complex structure J , the last de๏ฌnition implies [ ๐ช + , ๐ช + ] โ ๐ช + , [ ๐ช โ , ๐ช โ ] โ ๐ช โ . (13.24)This is proven in Appendix D. The latter will serve us as a working de๏ฌnition of an integrablecomplex structure.On the complete ๏ฌag manifold F ๐ , one can de๏ฌne an almost complex structure by choosing ๐ ( ๐ โ ) mutually non-conjugate forms ๐ฝ ๐ด ๐ต , . . . , ๐ฝ ๐ด ๐ ( ๐ โ ) ๐ต ๐ ( ๐ โ ) and declaring them holomor-phic. The remaining ๐ ( ๐ โ ) forms will be therefore anti-holomorphic. To determine which ofthese complex structures are integrable, it is useful to use a diagrammatic representation. Wedraw ๐ vertices, as well as arrows from the node ๐ด to the node ๐ต , from ๐ด to ๐ต and so on,so that all pairs of nodes are connected (such diagrams are called โtournamentsโ, see [59]).As we shall now prove, the integrability of the almost complex structure, de๏ฌned in this way,is equivalent to the acyclicity of the graph (the condition that it should not contain closedcycles).Let us start with F , and let ๐ ๐ด , ๐ด = , , ( ๐ ๐ด ) ๐ผ = ๐ฟ ๐ด๐ผ ( ๐ผ = , , ๐ช + of the Lie algebra ( ๐ฐ๐ฒ ( )) C = ๐ฐ๐ฉ ( ) as follows: ๐ช + = Span ( ๐ธ ๐ด ๐ต , ๐ธ ๐ด ๐ต , ๐ธ ๐ด ๐ต ) , where ๐ธ ๐ด๐ต = ๐ ๐ด โ ๐ ๐ต (13.25)Integrability of the complex structure is equivalent to the requirement that ๐ช + is a subalgebra: [ ๐ช + , ๐ช + ] โ ๐ช + . On the other hand, the matrices ๐ธ ๐๐ have the commutation relations [ ๐ธ ๐ด๐ต , ๐ธ ๐ถ๐ท ] = ๐ฟ ๐ต๐ถ ๐ธ ๐ด๐ท โ ๐ฟ ๐ด๐ท ๐ธ ๐ถ๐ต (13.26)In the tournament diagram, ๐ธ ๐ด๐ต is represented by an arrow from ๐ด to ๐ต ; thus one sees thatthe closedness of ๐ช + under commutation is equivalent to the following statement:For any two consecutive arrows ๐ด โ ๐ต and ๐ต โ ๐ถ (13.27)their โshortcutโ segment ( ๐ด, ๐ถ ) has the arrow ๐ด โ ๐ถ For the diagram with three vertices, i.e. for the ๐ฐ๐ฒ ( ) case under consideration, it is clearthat the cyclic quivers are the only ones that do not lead to integrability.In the general case of the ๏ฌag manifold F ๐ , suppose we have ๐ pairwise-connectedvertices, and the graph is acyclic. Then the requirement (13.27) is satis๏ฌed, since otherwisethere would be a cycle with three vertices. Reversely, suppose the graph has a cycle. Then,using (13.27), one can โcut cornersโ to reduce again to the cycle with three vertices, which isprohibited (see Fig. 16).One can then establish that there are exactly ๐ ! acyclic diagrams. They correspond tothe total orderings of the set of ๐ vertices. This is proven in Appendix D. By the logic90 Figure 16: The procedure showing that a cycle ( , , , , ) in a graph leads to the violationof condition (13.27). Using (13.27), we replace the pair of segments ( , ) , ( , ) by ( , ) ,i.e. cut a corner. Then we replace ( , ) , ( , ) by ( , ) , arriving at the cyclic red triangle,which violates (13.27).explained above this means that there are ๐ ! complex structures on a complete ๏ฌag manifold U ( ๐ ) U ( ) ๐ . Analogously there are ๐ ! complex structures on a partial ๏ฌag manifold U ( ๐ ) U ( ๐ )รยทยทยทร U ( ๐ ๐ ) .The number of complex structures may be interpreted as follows. Choosing a complexstructure is equivalent to choosing a complex quotient space representation (1.3). In order toconstruct such a representation, one should choose a sequence of embedded linear spaces ofthe type (1.2), and the dimensions of these spaces are given by the partial sums of the integers ๐ ๐ด . These dimensions are therefore determined by an ordering of the set { ๐ ๐ด } , and there are ๐ ! such orderings.Now that we have described all invariant complex structures on an arbitrary ๏ฌag manifold,we can take a fresh glance at the spaces of closed two-forms shown in (1.5). As discussed inChapter 1, the space of closed two-forms has real dimension ๐ โ
1. Using the descriptionof the cohomology (1.24) based on the embedding of the ๏ฌag manifold into a product ofGrassmannians (1.21), one can describe the space of closed two-forms as a hyperplane ๐ฅ + . . . + ๐ฅ ๐ = R ๐ . (13.28)Consider a typical point in this vector space, where all ๐ฅ ๐ โs are distinct. This correspondsto a non-degenerate two-form, i.e. a symplectic form ฮฉ . Given any such form, one canshow that there is a unique complex structure J , such that the corresponding symmetrictensor G : = โ ฮฉ โฆ J is positive-de๏ฌnite. This is tantamount to saying that G de๏ฌnes aKรคhler metric on the ๏ฌag manifold. In the simplest case of a Grassmannian, when ๐ = ๐ผ ยท ฮฉ FS of the generalized Fubini-Study form. There arealso two invariant complex structures: J and โ J . As a result, the real line of invariantclosed two-forms parametrized by ๐ผ = ๐ฅ โ ๐ฅ is divided into two rays ๐ผ โท
0, and on each ofthese rays one picks a suitable complex structure ยฑ J to de๏ฌne a metric. The two rays areinterchanged by the action of ๐ : ๐ฅ โ ๐ฅ .Returning to the general case, one ๏ฌnds that the hyperplane (13.28) is divided into ๐ !chambers, such that the points in the interior of each chamber may be thought of as Kรคhlerforms corresponding to the same invariant complex structure. The chambers are interchanged91y the action of the permutation group on ๐ฅ , . . . , ๐ฅ ๐ , which is free provided that ๐ฅ ๐ โs areall distinct. This action is clearly synchronized with the action of ๐ ๐ that interchangesthe complex structures. The boundaries between the chambers correspond to the case whenseveral ๐ฅ ๐ โs coincide, which leads to the degeneration of the two-forms. As already mentionedat the end of section 2.2, in this case we may ๏ฌnd a suitable smaller ๏ฌag manifold, on whichthe two-form ฮฉ is non-degenerate. The smaller ๏ฌag manifold is obtained from the originalone by a forgetful projection. By induction, more and more severe degenerations willcorrespond to forgetting more and more structure of the ๏ฌag, and the extreme case when all ๐ฅ ๐ = As a ๏ฌrst example we consider the case of Hermitian symmetric spaces M , i.e. symmetricspaces that admit a complex structure. First, we recall that in terms of the decomposition (13.2)symmetric spaces are characterized by the property [ ๐ช , ๐ช ] โ ๐ฅ . The Hermitian symmetricspaces are, in turn, characterized by the relation [ ๐ช + , ๐ช + ] =
0. Indeed, this follows fromthe symmetric space property [ ๐ช , ๐ช ] โ ๐ฅ and the integrability of the complex structure [ ๐ช + , ๐ช + ] โ ๐ช + . Conversely, if [ ๐ช + , ๐ช + ] =
0, one shows, using ad-invariance of the Killingmetric on ๐ค , that [ ๐ช + , ๐ช โ ] is orthogonal to ๐ช ยฑ , and hence [ ๐ช + , ๐ช โ ] โ ๐ฅ .The case of a symmetric target space is special in that the form ๐ is closed: ๐๐ =
0. Inother words, the Killing metric G is Kรคhler. Moreover, it is the only case when this is so:The Killing metric on F = F ๐ ,...,๐ ๐ is Kรคhler if and only if F is a symmetricspace, i.e. ๐ = F = ๐บ๐ ๐ ,๐ is a Grassmannian.To prove this, we note that the components ๐ฝ ยฑ of the current ๐ฝ (see the decomposition(13.9)) represent holomorphic/anti-holomorphic one-forms. The Killing metric on F , whichis ๐๐ = โ ( ๐ฝ + ๐ฝ โ ) , is therefore Hermitian. The Kรคhler form is, accordingly, ๐ = ๐ Tr ( ๐ฝ + โง ๐ฝ โ ) . (13.29)In calculating the exterior derivative of ๐ , we will be using the ๏ฌatness equation ๐๐ฝ โ ๐ฝ โง ๐ฝ = ๐ฅ โฅ ๐ช ยฑ , ๐ช ยฑ โฅ ๐ช ยฑ (isotropy of ๐ช ยฑ ). Simplifying the resulting expression,one gets ๐๐ = ๐ ( Tr ( ๐ฝ + โง ๐ฝ โ โง ๐ฝ โ ) โ Tr ( ๐ฝ โ โง ๐ฝ + โง ๐ฝ + )) . (13.30)The three-forms in the r.h.s. are of type ( , ) and ( , ) respectively (they are complexconjugate to each other). Therefore ๐๐ = ( ๐ฝ + โง ๐ฝ โ โง ๐ฝ โ ) = . Due to thenon-degeneracy of the Killing metric, this can only hold if [ ๐ช โ , ๐ช โ ] = . (13.31)92t is easy to see that this holds if and only if ๐ = ๐ =
2, the second term inthe action (13.5) is in fact topological and therefore does not a๏ฌect the equations of motion.In this case we return to the well-known theory of integrable sigma models with symmetrictarget spaces. However the canonical Lax connection in this case is di๏ฌerent from the onein (13.19). Indeed, the connection usually employed in the analysis of sigma models withsymmetric target spaces has the form (cid:101) ๐ด ๐ = โ ๐ (cid:101) ๐พ ๐ง ๐๐ง + โ ๐ โ (cid:101) ๐พ ๐ง ๐๐ง, where (cid:101) ๐พ = ๐ ยท (cid:2) ๐ โ ๐๐ (cid:3) ๐ช ยท ๐ โ (13.32)is the Noether current derived using the canonical action S [ G ] = โซ ฮฃ ๐ ๐ง (cid:107) ๐ ๐ (cid:107) G . (13.33)In the case of a Hermitian symmetric target space the di๏ฌerence between the two actions,(13.5) and (13.33), is a topological term: S [ G , J ] โ S [ G ] = โซ ฮฃ ๐ โ ๐, (13.34)where ๐๐ = M is symmetric.Therefore the two actions lead to the same equations of motion. Nevertheless, the Noethercurrents ๐พ and (cid:101) ๐พ are di๏ฌerent, although both are ๏ฌat. For the current ๐พ this was shownin (13.16)-(13.18), whereas the ๏ฌatness ๐ (cid:101) ๐พ โ (cid:101) ๐พ โง (cid:101) ๐พ = (cid:101) ๐พ does not, in fact, require using the equations of motion โ it is purelya consequence of the structure of the Lie algebra of the symmetric space (in particular, thefact that [ ๐ช , ๐ช ] โ ๐ฅ ). Moreover, the ๏ฌatness condition may be solved, in this case, in a localfashion : (cid:101) ๐พ = โ (cid:98) ๐ โ ๐ (cid:98) ๐, where (cid:98) ๐ = ๐ ( ๐ ) ๐ โ , (13.35) ๐ being Cartanโs involution on the Lie group ๐บ . By de๏ฌnition, ๐ is a group homomorphism, ๐ ( ๐ ๐ ) = ๐ ( ๐ ) ๐ ( ๐ ) , and ๐ ( โ ) = โ for โ โ ๐ป . The formula (cid:98) ๐ = ๐ ( ๐ ) ๐ โ , viewed as amap ๐ โ ๐บ / ๐ป โ (cid:98) ๐ โ ๐บ , describes the Cartan embedding ๐บ / ๐ป โฉ โ ๐บ . (13.36)Flatness and conservation of the current (cid:101) ๐พ lead to the ๏ฌatness of the family (cid:101) ๐ด ๐ . Aquestion naturally arises of what the relation between ๐ด ๐ข and (cid:101) ๐ด ๐ is. The answer is that the โLocalโ means that (cid:98) ๐ is a local function of the ๏ฌelds of the model. Cartanโs embedding is known to be totally geodesic. By de๏ฌnition, this means that the second fundamentalform of ห ๐ ( ๐บ๐ป ) โ ๐บ vanishes: (โ ๐ ๐ ) โฅ = ๐, ๐ โ ๐ ( ห ๐ ( ๐บ๐ป )) . It is easy to check that ifห ๐ : M โ N is a totally geodesic submanifold, and ๐ : ฮฃ โ M is a harmonic map (i.e. a solution to the sigmamodel e.o.m.), then ห ๐ โฆ ๐ : ฮฃ โ N is also harmonic. This means that the classical solutions of the symmetricspace ๐บ / ๐ป model are a subset of solutions of the principal chiral model. ๐ด ๐ข and (cid:101) ๐ด ๐ are gauge-equivalent, if one makes the following identi๏ฌcation ofspectral parameters: ๐ = ๐ข / . (13.37)The gauge transformation G relating ๐ด ๐ข and (cid:101) ๐ด ๐ , (cid:101) ๐ด ๐ = G ๐ด ๐ข G โ โ G ๐ G โ , (13.38)may be constructed explicitly. The following formula holds for the case when the target-spaceis the Grassmannian ๐บ๐ ๐ ,๐ + ๐ : = SU ( ๐ + ๐ ) ๐ ( U ( ๐ )ร U ( ๐ )) and the complex structure is chosen so thatit splits ๐ช as ๐ช = (cid:16) ๐ช + ๐ช โ (cid:17) : G = ๐ ฮ ๐ โ , where ฮ = ๐ ๐ โ ๐ ๐ + ๐ Diag ( ๐ โ / , . . . , ๐ โ / ๐ , ๐ / , . . . , ๐ / ๐ ) . (13.39)Although the ๏ฌag manifold (1.4) in general is not a symmetric space, it is a so-called Z ๐ -graded space. Perhaps most well-known are the Z -graded spaces, examples of whichare provided by twistor spaces of symmetric spaces [208] and nearly Kรคhler homogeneousspaces [61] (the latter also appear in the context of string compacti๏ฌcations, cf. [169, 79,131]). In this language the ordinary symmetric spaces are 2-symmetric spaces. Similarlyto what happens for symmetric spaces, the e.o.m. of a certain class of sigma models with Z ๐ -graded target-spaces may be rewritten as ๏ฌatness conditions for a one-parameter family ofconnections. These models were introduced in [261], and the construction of Lax connectionsfor these models was elaborated in [40]. The relation to the Lax connections of Section 13.1has been recently established in [92] (this is an extension to Z ๐ of our discussion aboveregarding symmetric spaces). The fact that the integrals of motion of the models are ininvolution was proven, for instance, in [160].To summarize, there are certain relations (that we recall in Appendix F) between themodels based on the Z ๐ -graded spaces and the models discussed so far in this chapter. Ingeneral, however, we view the approach based on complex structures as rather di๏ฌerent fromthe one based on Z ๐ -gradings. This will be emphasized in the next sections, where complexstructures will be shown to play a key role through ๐ฝ๐พ -systems, as well as in the formulationof the integrable sigma models as gauged chiral Gross-Neveu models. Z ๐ -symmetry of the models As we have emphasized, the models studied in the present Chapter depend explicitly onthe complex structure J on the target space. It turns out, however, that the action (13.5),albeit depending on the complex structure, might produce the same equations of motion evenfor di๏ฌerent choices of complex structure. This is due to the fact, that for certain complexstructures, which we denote by J and J , the di๏ฌerence in the two actions may just be atopological term: S [ J ] โ S [ J ] = โซ ฮฃ O , ๐ O = . (13.40)94et us describe precisely the situation when this happens. To this end we recall that, aswas established in Section 13.2, the complex structures are in a one-to-one correspondencewith an ordering of the mutually orthogonal spaces C ๐ , . . . C ๐ ๐ constituting a ๏ฌag, i.e. apoint in a ๏ฌag manifold U ( ๐ ) U ( ๐ )รยทยทยท U ( ๐ ๐ ) . The statement is then as follows:The actions S [ J ] and S [ J ] di๏ฌer by a topological term, as in (13.40), ifand only if the corresponding sequences of spaces { C ๐ , . . . C ๐ ๐ } di๏ฌer by a cyclic permutation.This was proven in [68], and for the sake of completeness we recall the proof in Ap-pendix E. The important point is that this Z ๐ โsymmetryโ is very parallel to the Z ๐ -symmetryof sigma models arising from spin chains, which was ultimately a re๏ฌection of the transla-tional invariance of the latter and whose importance was emphasized in Sections 9.2 and 10.We write โsymmetryโ in quotation marks, because it is really a symmetry of the theory onlyin the case ๐ = ยท ยท ยท = ๐ ๐ , when it can be realized by a cyclic permutation of the groups ofvectors { ๐ข ( ) , ยท ยท ยท , ๐ข ( ) ๐ } , . . . , { ๐ข ( ๐ ) , ยท ยท ยท , ๐ข ( ๐ ) ๐ ๐ } . In all other cases this should be seen as theequivalence of di๏ฌerent theories, de๏ฌned by the action functionals S [ J ] for di๏ฌerent com-plex structures J . The same issue arises in the case of spin chains, when 1-site translationalinvariance (leading to Z ๐ -symmetry) is only present when the representations at each site areequivalent, which again leads to the condition ๐ = ยท ยท ยท = ๐ ๐ . On the other hand, continuumlimits of spin chains with di๏ฌerent representations at di๏ฌerent sites may still be described bypartial ๏ฌag manifold sigma models [73].
14 Relation to 4D Chern-Simons theory
In the recent paper [87], a novel approach to the construction of (at least classically) integrablesigma models has been proposed. The ๏ฌag manifold models of the previous sections, as wellas their deformations, may as well be obtained within this framework. Besides, as we shall seein Section 16, when combined with the gauged linear sigma model approach, this constructionprovides a novel formulation of sigma models as gauged Gross-Neveu models. Deformedmodels appear naturally in this formalism through the introduction in the Lagrangian ofthe classical ๐ -matrix. This is a very well-known object in integrable theories, but forcompleteness we shall start by recalling its de๏ฌnition and providing the simplest examplesthat we will use later on. ๐ -matrix. The classical ๐ -matrix ๐ ( ๐ข ) takes values in ๐ค โ ๐ค , where ๐ค is a semi-simple or, more generally,reductive Lie algebra and ๐ข is a parameter taking values in a complex abelian group ( C , C โ orthe elliptic curve ๐ธ ๐ , depending on whether one deals with the rational/trigonometric/elliptic95ase respectively). The ๐ -matrix satis๏ฌes the classical Yang-Baxter equation (CYBE), whichtakes values in ๐ค โ ๐ค โ ๐ค and has the following form: [ ๐ ( ๐ข ) , ๐ ( ๐ข ยท ๐ฃ )] + [ ๐ ( ๐ข ) , ๐ ( ๐ฃ )] + [ ๐ ( ๐ข ยท ๐ฃ ) , ๐ ( ๐ฃ )] = . (14.1)Since we mostly have the trigonometric case in mind, we write the equation in multiplicativeform, that is to say ๐ข, ๐ฃ โ C โ . The notation ๐ ( ๐ข ) means ๐ ( ๐ข ) = ๐ ( ๐ข ) โ , and analogouslyfor other pairs of indices. Solutions to the above equation have been extensively studied inthe classic paper [41].For the purposes of the present paper it is more convenient to think of the ๐ -matrix as a map ๐ ( ๐ข ) : ๐ค โ ๐ค , or equivalently ๐ ( ๐ข ) โ End ( ๐ค ) (cid:39) ๐ค โ ๐ค โ . In this case we will write ๐ ๐ข ( ๐ ) โ ๐ค for the ๐ -matrix acting on a Lie algebra element ๐ โ ๐ค . One also often assumes the so-calledโunitarityโ property of the ๐ -matrix:Tr ( ๐ ๐ข ( ๐ด ) ๐ต ) = โ Tr ( ๐ด ๐ ๐ข โ ( ๐ต )) . (14.2)As we will see shortly, for our purposes it will be useful to weaken this condition slightly. Inthe new notations the CYBE looks as follows: [ ๐ ๐ข ( ๐ ) , ๐ ๐ข๐ฃ ( ๐ )] + ๐ ๐ข ( [ ๐ ๐ฃ ( ๐ ) , ๐ ]) + ๐ ๐ข๐ฃ ( [ ๐, ๐ ๐ฃ โ ( ๐ )]) = . (14.3)The solution of interest has the form (for now we assume ๐ค (cid:39) ๐ฐ๐ฒ ( ๐ ) ) ๐ ๐ข = ฮฑ ๐ข ๐ + + ฮฒ ๐ข ๐ โ + ฮณ ๐ข ๐ , (14.4) ฮฑ ๐ข = ๐ข โ ๐ข , ฮฒ ๐ข = โ ๐ข , ฮณ ๐ข =
12 1 + ๐ข โ ๐ข , (14.5)where ๐ ยฑ are projections on the upper/lower-triangular matrices, and ๐ is the projection onthe diagonal. The rational limit is achieved by setting ๐ข = ๐ โ (cid:15) and taking the limit (cid:15) โ
0, inwhich case ๐ ๐ข โ (cid:15) .The ansatz ๐ ๐ข =
12 1 + ๐ข โ ๐ข Id + ๐ R (14.6)transforms the CYBE to an equation on R , which does not depend on the spectral parameter: [R ( ๐ ) , R ( ๐ )] + R ( [R ( ๐ ) , ๐ ] + [ ๐, R ( ๐ )]) โ [ ๐, ๐ ] = . (14.7)It is known in the literature as the โclassical modi๏ฌed Yang-Baxter equationโ. The solu-tion (14.4) corresponds to R = ๐ ( ๐ + โ ๐ โ ) . Another option is taking an R -matrix induced by a complex structure on the Lie group ๐บ with Liealgebra ๐ค [64]. In this case (14.7) is the condition of vanishing of the Nฤณenhuis tensor (13.22). Having the right tools in place, we proceed to explain the construction of [87], which is basedon a certain โsemi-holomorphicโ 4D Chern-Simons theory that we will now describe. In thiscase the four-dimensional โspacetimeโ is a product ฮฃ ร C , where ฮฃ is called the โtopologicalplaneโ and is endowed with coordinates ๐ง, ๐ง โ this will eventually be the worldsheet of thesigma model, โ and C is a complex curve with coordinates ๐ค, ๐ค (this is the spectral parametercurve). The latter is required to admit a nowhere-vanishing holomorphic di๏ฌerential ๐ = ๐๐ค โ
0, which means that its canonical class is trivial: ๐พ C =
0. As a result, the curveis either the complex plane, a cylinder or an elliptic curve (a torus): C (cid:39) C , C โ , ๐ธ ๐ . TheChern-Simons action of the model is ๐ CS = โ โซ ฮฃ ร C ๐ โง Tr (cid:18) ๐ด โง ( ๐๐ด + ๐ด โง ๐ด ) (cid:19) , (14.8)where ๐ด = ๐ด ๐ง ๐๐ง + ๐ด ๐ง ๐๐ง + ๐ด ๐ค ๐๐ค is a gauge ๏ฌeld corresponding to a (semi-simple) gaugegroup ๐บ . One couples this theory to certain two-dimensional systems of a very particularsort, called ๐ฝ๐พ systems. These are de๏ฌned for complex symplectic target spaces, their actionin local Darboux coordinates ( ๐, ๐ ) being ๐ ๐ฝ๐พ = โซ ฮฃ ๐ ๐ง ๐ ๐ ๐๐ ๐ . In the context of [87] oneconsiders target spaces of the form ๐ โ M , where M is a complex manifold endowed with aholomorphic action of the group ๐บ , and writes down a sum of two ๐ฝ๐พ -system actions, oneholomorphic and the other anti-holomorphic:Figure 17: Two ๐ฝ๐พ -defects located at points ๐ค , ๐ค on the spectral parameter curve C . ๐ def = โซ ฮฃ ๐ ๐ง (cid:16) ๐ ๐ ๐ท ( ๐ค ) ๐ ๐ + ๐ ๐ ๐ท ( ๐ค ) ๐ ๐ (cid:17) , (14.9)where ๐ท ( ๐ค ) ๐ ๐ = ๐๐ ๐ โ โ๏ธ ๐ (cid:16) ๐ด ( ๐ค ) ๐ง (cid:17) ๐ ๐ฃ ๐๐ and ๐ฃ ๐ are the holomorphic vector ๏ฌelds on M generating the action of ๐บ . The full actionfunctional is the sum of two: (14.8) and (14.9). The next step is to impose the โlight-coneโgauge ๐ด ๐ค =
0, in which case two of the e.o.m. become linear in ๐ด ๐ง , ๐ด ๐ง , and the third one is97he zero-curvature constraint: ๐ ๐ด ๐ง โ ๐ ๐ด ๐ง + [ ๐ด ๐ง , ๐ด ๐ง ] = , (14.10) ๐ ๐ค ๐ด ๐ง = ๐ฟ ( ) ( ๐ค โ ๐ค ) โ๏ธ ๐ ๐ ๐ ๐ฃ ๐๐ ๐ ๐ (14.11) ๐ ๐ค ๐ด ๐ง = ๐ฟ ( ) ( ๐ค โ ๐ค ) โ๏ธ ๐ ๐ ๐ ๐ฃ ๐๐ ๐ ๐ . (14.12)The delta-functions in the r.h.s. of the latter equations mean that ๐ด ๐ง ( ๐ค ) , ๐ด ๐ง ( ๐ค ) dependmeromorphically on ๐ค , and the ๏ฌrst equation is then the zero-curvature equation (on theworldsheet ฮฃ ) for the family of connections ๐ด = ๐ด ๐ง ๐๐ง + ๐ด ๐ง ๐๐ง depending on the parameter ๐ค .In order to solve the equations (14.11)-(14.12), one needs to invert the operator ๐ ๐ค . Oneof the key observations in [87] is that, with suitable boundary conditions, the Greenโs function ๐ โ ๐ค is the classical ๐ -matrix [41], viewed as an element of End ( ๐ค ) . In the rational case theGreenโs function is simply the Cauchy kernel, enhanced with additional matrix structure, i.e. ๐ ๐ค = Id ๐ค โ End ( ๐ค ) . (14.13)As we reviewed in Section 14.0.1, the ๐ -matrix is sometimes written as an element of ๐ค โ ๐ค , and the two de๏ฌnitions are simply related by raising/lowering an index, using theKilling metric on ๐ค . Accordingly, the more conventional representation for (14.13) wouldbe ๐ ( ๐ค ) = (cid:205) ๐ ๐ โ ๐ ๐ ๐ค โ ๐ค โ ๐ค , where ๐ ๐ are the generators of ๐ค . In the trigonometric case oneneeds to impose boundary conditions at the ends of the cylinder, i.e. at the two punctures on C = C โ (cid:39) CP \ { , โ} (these are shown as black dots in Fig. 17). As explained in [87]at length, the relevant boundary conditions amount to picking a decomposition of the Liealgebra ๐ค C = ๐ค + โ ๐ค โ , where ๐ค ยฑ are two isotropic subspaces of ๐ค , and requiring that ๐ด ๐ง โ ๐ค + at ๐ค = ๐ด ๐ง โ ๐ค โ at ๐ค = โ . In that case the ๐ -matrix has the form ๐ ๐ค = ฮ + โ ๐ค โ ฮ โ โ ๐ค โ โ End ( ๐ค ) , (14.14)where ฮ ยฑ are the projectors on ๐ค ยฑ , or alternatively ๐ ( ๐ค ) = (cid:205) ๐ + ๐ โ ๐ โ ๐ โ ๐ค โ (cid:205) ๐ โ ๐ โ ๐ + ๐ โ ๐ค โ โ ๐ค โ ๐ค .Once we know the ๐ -matrix, the solution to (14.11) is ๐ด ๐ง = ๐ ๐ค (cid:0)(cid:205) ๐ ๐ ๐ ๐ฃ ๐๐ ๐ ๐ (cid:1) , and analo-gously for (14.12). Substituting the solution back into the full action ๐ = ๐ CS + ๐ def (usingthe fact that the Chern-Simons action is quadratic in the ๐ด -๏ฌelds in the gauge ๐ด ๐ค = ๐, ๐ -variables: ๐ = โซ ๐ ๐ง (cid:16) ๐ ๐ ๐๐ ๐ + ๐ ๐ ๐๐ ๐ + ๐ ๐ค โ ๐ค (cid:16) ๐ ๐ ๐ฃ ๐๐ ๐ ๐ , ๐ ๐ ๐ฃ ๐๐ ๐ ๐ ) (cid:17) (cid:17) = (14.15) = โซ ๐ ๐ง (cid:18) ๐ ๐ ๐๐ ๐ + ๐ ๐ ๐๐ ๐ + ๐ค โ ๐ค โ๏ธ | ๐ ๐ ๐ฃ ๐๐ | (cid:19) , Such decomposition is the same as picking a complex structure on ๐ค , compatible with the metric. It is alsoknown in the literature as a Manin triple. ๐ -matrix. Onesees that the action is quadratic in the ๐ -variables, which are the coordinates in the ๏ฌber ofthe cotangent bundle ๐ โ M . Integrating out these variables as well, we get the sigma modelform of the action: ๐ โผ โซ ๐ ๐ง (cid:16) ๐บ ๐ ๐ ๐๐ ๐ ๐๐ ๐ (cid:17) , where ๐บ ๐ ๐ = (cid:32)โ๏ธ ๐ ๐ฃ ๐๐ ๐ฃ ๐๐ (cid:33) โ (14.16)is the metric on the target space. Note that in order for the expression for the metric ๐บ ๐ ๐ tomake sense the matrix (cid:205) ๐ ๐ฃ ๐๐ ๐ฃ ๐๐ has to be invertible. This is equivalent to the requirement that M is a homogeneous space.The model (14.16) is clearly of the same type as the general class of models (13.7)introduced earlier. In the next section, following [67], we will prove directly that, in thecase when M is a ๏ฌag manifold, the two models are equivalent (meaning that the metric ๐บ ๐ ๐ coincides with the Killing metric discussed at the beginning of Section 13). In provingthis, it will turn out extremely useful to introduce a gauged linear sigma model approach tothe models in question, which will ultimately lead us to the formulation of sigma models asgeneralized Gross-Neveu models in section 16. We will also see that the formalism describedhere, especially when combined with the GLSM-presentation, makes it very easy to constructintegrable deformations (trigonometric, and possibly even elliptic) of the sigma models bypicking the corresponding ๐ -matrices in (14.15). ๐ฝ๐พ -systems In the present section we will prove that the ๏ฌag manifold models obtained from the couplingof two ๐ฝ๐พ -systems through a four-dimensional Chern-Simons ๏ฌeld are โ in the rational caseโ equivalent to the models that we described earlier in Section 13. To this end we thereforee๏ฌectively set the ๐ -matrix to be the identity operator: ๐ = Id. Our main tool in identifyingthe two types of models will be the gauged linear sigma model representation that wasdeveloped in [75, 76] for ๏ฌag models of type (13.12). In the case when the target space isa Grassmannian, the metric G is Kรคhler, and this representation is equivalent to the Kรคhlerquotient ๐บ๐ ๐,๐ (cid:39) Hom ( C ๐ , C ๐ ) (cid:12) U ( ๐ ) . In the general case our construction leads to aquotient w.r.t. a non-reductive group and to the โKillingโ metric G , which is not Kรคhler ingeneral.The construction is as follows. We introduce the ๏ฌeld ๐ โ Hom ( C ๐ , C ๐ ) , satisfying theorthonormality condition ๐ โ ๐ = ๐ , as well as the โgaugeโ ๏ฌeld (cid:101) A = A ๐๐ง + A ๐๐ง of thespecial form shown in Fig. 18. The Lagrangian reads L = Tr (cid:16) (cid:107) ๐ท๐ (cid:107) (cid:17) , where ๐ท๐ = ๐๐ + ๐ ๐ A . (14.18)This Lagrangian is equivalent to (13.12), as can be shown by eliminating the ๏ฌeld A . Indeed,varying w.r.t. A one obtains ( ๐ท ๐ง ๐ โ โฆ ๐ ) ๐ญ ๐ =
0, where ๐ญ ๐ is the space of upper-block-99 = โ โ โ โ โ โโ โ โ โ โ โโ โ โ โโ โ โ โโ โโ โ (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) ๐ ๐ ๐ ๐ โ , A = (A) โ . (14.17)Figure 18: The gauge ๏ฌeld entering the GLSM description (14.18) of the ๏ฌag manifold sigmamodels. Here ๐ , ยท ยท ยท , ๐ ๐ โ : = ๐ are the dimensions of the complex spaces in the ๏ฌag (1.1).triangular matrices of size ๐ ร ๐ . Let us parametrize ๐ as ๐ = { ๐ , ๐ , . . . , ๐ ๐ โ } , (14.19)where ๐ . . . ๐ ๐ โ are groups of ๐ . . . ๐ ๐ โ orthonormal vectors. Adding the last group ๐ ๐ of ๐ ๐ vectors orthogonal to all the rest, we obtain the matrix ๐ from (13.11). One then has theorthonormality condition ๐ โ ๐ด ๐ ๐ต = ๐ฟ ๐ด๐ต ( ๐ด, ๐ต = , . . . , ๐ ) and the completeness relation ๐ โ๏ธ ๐ด = ๐ ๐ด ๐ โ ๐ด = ๐ , (14.20)which is equivalent to ๐๐ โ = ๐ . In this notation components of the gauge ๏ฌeld A have theform (see Fig. 14.17) [A] ๐ด๐ต = ๐ ๐ โ ๐ด ๐๐ ๐ต for ๐ด โค ๐ต ; [A] ๐ด๐ต = ๐ด > ๐ต . ๐ด, ๐ต = . . . ๐ โ . In order to compute the Lagrangian (14.18), it is useful ๏ฌrst to calculate ๐ท๐ ๐ด = ๐๐ ๐ด โ ๐ โ๏ธ ๐ต = ๐ด ๐ ๐ต ๐ โ ๐ต ๐๐ ๐ด = using ( . ) = ๐ด โ โ๏ธ ๐ต = ๐ ๐ต ๐ โ ๐ต ๐๐ ๐ด ๐ด = , . . . , ๐ โ . (14.21)Substituting into (14.18), we obtain the ๏ฌnal expression for the Lagrangian in the โnon-linearformโ L = โ๏ธ ๐ด<๐ต Tr (cid:16) ( ๐ฝ ๐ต๐ด ) โ ๐ง ( ๐ฝ ๐ต๐ด ) ๐ง (cid:17) , where ( ๐ฝ ๐ต๐ด ) ๐ง = ๐ โ ๐ต ๐๐ ๐ด . (14.22)This is clearly the same as (13.12), up to an exchange of complex structure J โ โ J .100e return to the GLSM (14.18). Due to the orthonormality condition ๐ โ ๐ = ๐ , thegauge group of the model is U ( ๐ ) ร ยท ยท ยท ร U ( ๐ ๐ ) . A natural question is whether one caninstead use a quotient w.r.t. the complex group of upper/lower-block-triangular matrices. Toanswer this, we give up the orthonormality condition and assume that ๐ โ Hom ( C ๐ , C ๐ ) isan arbitrary complex matrix of rank ๐ . We then write down the following Lagrangian: L = Tr (cid:18) ( ๐ท๐ ) โ ๐ท๐ ๐ โ ๐ (cid:19) . (14.23)It is easy to see that it is invariant w.r.t. complex gauge transformations ๐ โ ๐๐ , where ๐ โ ๐ ๐ ,...,๐ ๐ โ (a parabolic subgroup of GL ( ๐, C ) . The Gram-Schmidt orthogonalizationprocedure brings the Lagrangian (14.23) to the form (14.18), but for a number of reasonsthe complex form is preferable. In order to get rid of the denominator in the Lagrangian, weintroduce an auxiliary ๏ฌeld ๐ โ Hom ( C ๐ , C ๐ ) and write down a new Lagrangian L = Tr (cid:16) ๐ ๐ท๐ (cid:17) + Tr (cid:16) ๐ ๐ท๐ (cid:17) โ Tr (cid:16) ๐๐ โ ๐ โ ๐ (cid:17) , (14.24)that turns into the original one upon elimination of the ๏ฌeld ๐ . This is therefore a far-reachinggeneralization of the elementary example considered at the very start of this Chapter, whereusing a similar procedure we obtained a sigma model with target space the cylinder C โ .One should also keep in mind that in the process of integration over the ๐ -variables, a non-trivial one-loop determinant typically arises, which leads to a non-zero dilaton, in exactly thesame way as it happens in the context of Buscher rules for ๐ -duality [60, 225, 211] (this isparticularly important for the deformed models discussed in section 16.2). Next we performyet another quadratic transformation, in order to eliminate the quartic interaction. To this endwe introduce the complex matrix ๏ฌeld ฮฆ โ End ( C ๐ ) and its Hermitian conjugate: ฮฆ = ( ฮฆ ) โ .We write one more Lagrangian L = Tr (cid:16) ๐ D ๐ (cid:17) + Tr (cid:16) ๐ D ๐ (cid:17) + Tr (cid:16) ฮฆฮฆ (cid:17) , (14.25)where D is the โelongatedโ covariant derivative D ๐ = ๐๐ + ๐ ๐ A + ๐ ฮฆ ๐ . (14.26)Let us clarify the geometric meaning of the Lagrangian (14.25). The ๏ฌrst two terms cor-respond to a sum of the so-called ๐ฝ๐พ -systems on the ๏ฌag manifold F , in a background๏ฌeld ฮฆ [182, 256]. In the terminology of the previous section our ๏ฌeld ฮฆ should be viewedas the component ๐ด ๐ง of the Chern-Simons gauge ๏ฌeld along the โtopological planeโ (i.e. theworldsheet ฮฃ ). The quadratic form in the interaction term Tr (cid:16) ฮฆฮฆ (cid:17) in (14.25) is in this contextthe inverse propagator of the ๏ฌeld ๐ด ๐ง , which in the present (rational) case is proportional tothe identity matrix.By de๏ฌnition, such a system may be de๏ฌned for an arbitrary complex manifold M (dim C M = ๐ท ) with the help of a complex fundamental ( , ) -form ๐ = ๐ท (cid:205) ๐ = ๐ ๐ ๐๐ ๐ (the101omplex analogue of the Poincarรฉ-Liouville one-form) on ๐ โ M . Here ๐ ๐ are the complexcoordinates on M and ๐ ๐ are the complex coordinates in the ๏ฌber of the holomorphiccotangent bundle. The action of the ๐ฝ๐พ -system is then simply ๐ = โซ ฮฃ ๐ ๐ง ๐ท (cid:205) ๐ = ๐ ๐ ๐๐ ๐ . In thecase of the ๏ฌag manifold this action can be most conveniently written, using two matrices ๐ โ Hom ( C ๐ , C ๐ ) , ๐ โ Hom ( C ๐ , C ๐ ) and the gauge ๏ฌeld A . Indeed, it will be shown inthe next section that the fundamental ( , ) -form can be written as ๐ = Tr ( ๐ ๐๐ ) (cid:12)(cid:12) ๐ C = , where ( ๐, ๐ ) satisfy the condition ๐ C = ๐๐ (cid:12)(cid:12) ๐จ โ = , and ๐จ = Lie ( ๐ ๐ ,...,๐ ๐ โ ) (14.27)is the Lie algebra of the corresponding parabolic subgroup of GL ( ๐, C ) . It is also assumedthat the space of matrices, satisfying this condition, is factorized w.r.t. the action of ๐ ๐ ,...,๐ ๐ โ ,i.e. one has a complex symplectic reduction. The condition (14.27) is precisely the conditionof vanishing of the moment map ๐ C = ๐ ๐ ,...,๐ ๐ โ onthe space of matrices ( ๐, ๐ ) endowed with the symplectic form ๐ = Tr ( ๐๐ โง ๐๐ ) . As aresult, ๐๐ = ๐ red (14.28)is the complex symplectic form, arising after the reduction w.r.t. the parabolic group. Inorder to ensure the condition (14.27) at the level of the Lagrangian of the model, one needsthe gauge ๏ฌeld A โ
Lie ( ๐ ๐ ,...,๐ ๐ โ ) . Indeed, di๏ฌerentiating the Lagrangian (14.25) w.r.t. A ,one arrives at the condition (14.27). Before passing to further topics, let us clarify the relation between the complex symplecticform, as discussed in the previous section (constructed using the symplectic quotient withrespect to a parabolic subgroup), and the symplectic form that arises as a result of a reductivequotient, de๏ฌned by the so-called quiver. We recall that ๐ โ F is a hyper-Kรคhler manifoldthat may be constructed by a hyper-Kรคhler quotient of ๏ฌat space (though we stress that thereal symplectic form โ the Kรคhler form โ will not concern us here). This quotient is basedon a linear quiver diagram [180] shown in Fig. 19. This is essentially the same quiver that ๐ ๐ ๐ โ ๐ ๐ ๐ โ ๐ ๐ โ ๐ ๐ โ C ๐ ๐ฟ ๐ฟ ๐ฟ ๐ โ ๐ฟ ๐ โ ยท ยท ยท (14.29)Figure 19: Nakajima quiver for the cotangent bundle to the ๏ฌag manifold ๐ โ F .we encountered in section 1.3.2 (Fig. 2), but this time with a doubled set of arrows, which102s related to the fact that this time we have the cotangent bundle ๐ โ F rather than the ๏ฌagmanifold itself. In each node there is a vector space ๐ฟ ๐ (cid:39) C ๐ ๐ , and to each arrow from node ๐ to node ๐ corresponds a ๏ฌeld, taking values in Hom ( ๐ฟ ๐ , ๐ฟ ๐ ) . The full space of ๏ฌelds istherefore W : = โ ๐ โ ๐ = ( Hom ( ๐ฟ ๐ , ๐ฟ ๐ + ) โ Hom ( ๐ฟ ๐ + , ๐ฟ ๐ )) . (14.30)In each node there is an action of a gauge group GL ( ๐ฟ ๐ ) . We then consider the GIT-quotient W ๐ : = W / G of the stable subset W โ W w.r.t. the group G : = ๐ โ (cid:206) ๐ = GL ( ๐ฟ ๐ ) . In W ๐ we de๏ฌnea submanifold given by the vanishing conditions for the moment maps ( ๐ = , ๐ = F : = { ๐ ๐ = ๐ ๐ โ ๐ ๐ โ โ ๐ ๐ ๐ ๐ = , ๐ = , . . . , ๐ โ } โ W ๐ . (14.31)The (well-known) statement is that the resulting space is the ๏ฌag manifold (1.3)-(1.4), whichis why we have denoted it by F . On W there is a natural complex symplectic form ฮฉ = ๐ โ โ๏ธ ๐ = Tr ( ๐๐ ๐ โง ๐๐ ๐ ) . (14.32)The construction just described may be interpreted as the symplectic quotient w.r.t. thecomplex group G, and it endows F with a certain symplectic form ฮฉ F . We prove thefollowing statement: ฮฉ F = ๐ red , where ๐ red is the symplectic form (14.28) that arises as a result ofthe reduction w.r.t. a parabolic subgroup of GL ( ๐, C ) .To prove this, consider the ๏ฌelds { ๐ ๐ } . ๐ ๐ is a matrix with ๐ ๐ columns and ๐ ๐ + rows. By theaction of GL ( ๐ ๐ + , C ) one can bring ๐ ๐ to the form where the ๏ฌrst ๐ ๐ + โ ๐ ๐ rows are zero andthe last ๐ ๐ rows represent a unit matrix. The stabilizer of this canonical form w.r.t. the joint(left-right) action of GL ( ๐ ๐ + , C ) ร GL ( ๐ ๐ , C ) is the subgroup ๐ ๐ ๐ ,๐ ๐ + โ ๐ ๐ ๐ ,๐ ๐ + ร GL ( ๐ ๐ , C ) ,embedded according to the rule ๐ โ ( ๐, ๐ ๐ ( ๐ )) , where ๐ ๐ ( ๐ ) is the projection on the block ofsize ๐ ๐ ร ๐ ๐ . Iterating this procedure, i.e. bringing all matrices ๐ ๐ ( ๐ = , . . . , ๐ โ
2) to canonicalform, we arrive at the situation, when one is left with a single non-trivial matrix ๐ ๐ โ : = ๐ ,and the resulting symmetry group is precisely ๐ ๐ ,...,๐ ๐ โ . We also denote ๐ ๐ โ : = ๐ . Now,let ๐ โ ๐จ = Lie ( ๐ ๐ ,...,๐ ๐ โ ) . By de๏ฌnition of the stabilizer ๐๐ ๐ โ = ๐ ๐ โ ๐ ๐ โ ( ๐ ) , thereforeTr ( ๐๐ ๐ โ ๐ ๐ โ ) = Tr ( ๐ ๐ โ ( ๐ ) ๐ ๐ โ ๐ ๐ โ ) = Tr ( ๐ ๐ โ ( ๐ ) ๐ ๐ โ ๐ ๐ โ ) , where in the secondequality we have used the equation (14.31). Since ๐ ๐ โ ( ๐ ) โ Stab ( ๐ ๐ โ ) , we can iteratethis procedure, and at the end we will obtain Tr ( ๐๐ ๐ โ ๐ ๐ โ ) =
0. Due to the equation ๐ ๐ โ ๐ ๐ โ โ ๐๐ = ๐๐ (cid:12)(cid:12) ๐จ โ =
0, which coincides with (14.27). Besides, since thematrices ๐ ๐ ( ๐ = , . . . , ๐ โ
2) are constant, the restriction of the symplectic form ฮฉ coincideswith Tr ( ๐๐ ๐ โ โง ๐๐ ๐ โ ) = Tr ( ๐๐ โง ๐๐ ) .Let us clarify the role of the ๏ฌeld ฮฆ . Di๏ฌerentiating the Lagrangian (14.25) w.r.t. ฮฆ , weobtain ฮฆ = โ ๐ ๐๐ . (14.33)103his coincides with the expression for the ๐ง -component of the Noether current for the actionof the group GL ( ๐, C ) on the space of matrices ( ๐, ๐ ) . For the ๐ฝ๐พ -system written above ฮฆ is nothing but the moment map for the action of this group.
15 Relation to the principal chiral model
We recall that the principal chiral model is a sigma model with target space a compact Liegroup, such as SU ( ๐ ) . In the present section, following [67], we describe the relation between๏ฌag manifold models and the principal chiral model. Our starting point will be the formulas (14.27)-(14.33):
ฮฆ = โ ๐ ๐๐ , ๐ C = ๐๐ (cid:12)(cid:12) ๐จ โ = , (15.1)where ๐จ = Lie ( ๐ ๐ ,...,๐ ๐ โ ) . As a warm-up we consider the case of a Grassmannian, i.e. ๐ =
2. Then the vanishingof the moment map is simply ๐๐ =
0. Therefore ฮฆ =
0, which means that ฮฆ belongsto a nilpotent orbit of the group GL ( ๐, C ) . From the expression for ฮฆ it also follows thatIm ( ฮฆ ) โ Im ( ๐ ) โ Ker ( ฮฆ ) . As is well-known, {( ๐, ฮฆ ) : rk ( ๐ ) = ๐, ฮฆ = , Im ( ฮฆ ) โ Im ( ๐ ) โ Ker ( ฮฆ )} (cid:39) ๐ โ Gr ๐,๐ (15.2)is the cotangent bundle to a Grassmannian (notice the rank condition!), and the forgetful map ๐ โ Gr ๐,๐ โ { ฮฆ : ฮฆ = } (15.3)provides a resolution of singularities of the nilpotent orbit in the r.h.s. (the Springer res-olution). The conditions in the l.h.s. of (15.2) imply the factorization (15.1) for ฮฆ ,and the non-uniqueness in this factorization corresponds exactly to the gauge symmetry ๐ โ ๐๐, ๐ โ ๐ โ ๐ , where ๐ โ GL ( ๐, C ) .Let us now derive the equations of motion for the ๏ฌeld ฮฆ . First of all, the La-grangian (14.25) implies the equations of motion D ๐ = , D ๐ = ๐ and ๐ . Therefore D ฮฆ =
0, i.e. ๐ ฮฆ + ๐ [ ฮฆ , ฮฆ ] = . (15.4)This equation is nothing but the equation of motion of the principal chiral ๏ฌeld. Indeed,introduce a 1-form ๐ = ๐ ( ฮฆ ๐๐ง + ฮฆ ๐๐ง ) with values in the Lie algebra ๐ฒ ( ๐ ) . In this case (15.4)together with the Hermitian conjugate equation may be written in the form of two conditions ๐ โ ๐ = , ๐ ๐ โ ๐ โง ๐ = , (15.5)104hich are the e.o.m. of the principal chiral ๏ฌeld. This is consistent with the fact, reviewed insection 13.1, that the Noether current of the model (13.5) is ๏ฌat.The condition ฮฆ = ฮฆ consists of ๐ cells of sizes2 ร ๐ cells of sizes 1 ร
1. In this case ๐ = ๐ + ๐ and dim Ker ( ฮฆ ) = ๐ + ๐ .Since Im ( ๐ ) โ Ker ( ฮฆ ) and rk ( ๐ ) = ๐ , we get the condition ๐ โค ๐ + ๐ . This easilyleads to ๐ โค ๐ โ ๐ , ๐ โฅ ๐ โ ๐ .The dynamical equation (15.4) imposes severe constraints on the way in which the Jordanstructure of the matrix ฮฆ can change as one varies the point ๐ง, ๐ง on the worldsheet. Indeed,it implies that ฮฆ = ๐๐ ( ๐ง ) ๐ โ , where ๐ ( ๐ง ) is a matrix that depends holomorphically on ๐ง .The Jordan structure of the matrix ๐ ( ๐ง ) is the same as that of ฮฆ , and the vanishing of theJordan blocks occurs holomorphically in ๐ง . In particular, the Jordan structure changes onlyat โspecial pointsโ โ isolated points on the worldsheet. As a result, โalmost everywhereโ thedimension of the kernel dim Ker ( ฮฆ ) : = (cid:101) ๐ is the same, and the map ๐ : ( ๐ง, ๐ง ) โ Ker ( ฮฆ ) isa map to the Grassmannian ๐บ๐ (cid:101) ๐,๐ . A more careful analysis of the behavior of ฮฆ at a specialpoint would show that ๐ may be extended to these points. This may be summarized as follows:let ๐ ( ๐ง, ๐ง ) be a solution of the principal chiral model, i.e. a harmonic map to the group ๐บ ,satisfying the condition ฮฆ =
0, where ฮฆ : = ๐ โ ๐๐ is a component of the Noether current,and let the dimension of the kernel of ฮฆ at a typical point of the worlsheet be (cid:101) ๐ . Then onecan construct a harmonic map to the Grassmannian ๐บ๐ (cid:101) ๐,๐ by the rule ( ๐ง, ๐ง ) โ Ker ( ฮฆ ) . Let us extend the results of the previous section to more general ๏ฌag manifolds. We return๏ฌrst to the equation for ฮฆ : ๐ ฮฆ + ๐ [ ฮฆ , ฮฆ ] = , (15.6)but this time we assume that the matrix ฮฆ satis๏ฌes, in a typical point ( ๐ง, ๐ง ) โ ฮฃ , the condition ฮฆ ๐ = ฮฆ ๐ โ โ . (15.7)The matrix ฮฆ naturally de๏ฌnes a ๏ฌag ๐ : = { โ Ker ( ฮฆ ) โ Ker ( ฮฆ ) โ ยท ยท ยท โ Ker ( ฮฆ ๐ ) (cid:39) C ๐ } (15.8)The relation between the principal chiral model equations written above and the ๏ฌagmanifold models is summarized by the following assertion:Given a matrix ฮฆ satisfying (15.6)-(15.7), the map ( ๐ง, ๐ง ) โ ๐ is a solution tothe e.o.m. of the ๏ฌag manifold sigma model (13.12). The inequalities are saturated in the case Ker ( ฮฆ ) (cid:39) Im ( ๐ ) , when the number of 2 ร ๐ โ ๐ , and the number of cells of size 1 ร ๐ โ ๐ . Note that this is only possible in the case ๐ โฅ ๐ . Reduction of the number of cells of type 2 ร ฮฆ . ๐ of the form ๐ = ( ๐ ๐ โ | ยท ยท ยท | ๐ ) , where ๐ ๐ is a matrix,whose columns are the linearly independent vectors from Ker ( ฮฆ ๐ )/ Ker ( ฮฆ ๐ โ ) . Let us relatethe dimensions of these spaces to the dimensions of the Jordan cells of the matrix ฮฆ . To thisend we bring ฮฆ to the Jordan form ฮฆ ( ) = Diag { ๐ฝ ๐ , . . . , ๐ฝ ๐ โ } , โ โ๏ธ ๐ = ๐ ๐ = ๐ , (15.9)where ๐ฝ ๐ is a Jordan cell of size ๐ ร ๐ . We have chosen the ordering ๐ โฅ . . . โฅ ๐ โ , where,according to the supposition (15.7), ๐ = ๐ . We denote by ๐ ๐ the number of Jordan cells ofsize at least ๐ ( ๐ = โ ). The following two properties are obvious: โข ๐ ๐ + โค ๐ ๐ , i.e. ๐ , . . . , ๐ ๐ is a non-increasing sequence. โข dim Ker ( ฮฆ ) = ๐ , dim Ker ( ฮฆ ) = ๐ + ๐ etc.,therefore dim Ker ( ฮฆ ๐ )/ Ker ( ฮฆ ๐ โ ) = ๐ ๐ .It follows that ๐ ๐ โ Hom ( C ๐ ๐ , C ๐ ) and ๐ โ Hom ( C ๐ , C ๐ ) , where ๐ = ๐ โ (cid:205) ๐ = ๐ ๐ .Since, by construction, Im ( ๐ ) (cid:39) Ker ( ฮฆ ๐ โ ) and Im ( ฮฆ ) โ Ker ( ฮฆ ๐ โ ) , we have Im ( ฮฆ ) โ Im ( ๐ ) , i.e. there exists a matrix ๐ โ Hom ( C ๐ , C ๐ ) , such that ฮฆ = โ ๐ ๐๐ . Let us now derivethe equations of motion for the matrices ๐ and ๐ . Since ฮฆ ๐ ๐ ๐ = ๐ท ฮฆ =
0, onehas ฮฆ ๐ ๐ท๐ ๐ =
0. The columns of the matrix ( ๐ ๐ | ยท ยท ยท | ๐ ) span the kernel of ฮฆ ๐ , hence ๐ท๐ ๐ = โ ๐ (cid:205) ๐ โค ๐ ๐ ๐ A ๐ ๐ , where A ๐ ๐ are matrices of relevant sizes. Out of the matrices A ๐ ๐ (1 โค ๐ โค ๐ โค ๐ โ
1) we form a single matrix A , which schematically looks as in (14.17).Then, clearly, the following equation is satis๏ฌed: D ๐ = ๐๐ + ๐ ๐ A + ๐ ฮฆ ๐ = . (15.10)As ฮฆ = โ ๐ ๐๐ , from the non-degeneracy of ๐ it follows that D ๐ =
0. Because ๐ , . . . , ๐ ๐ โ Ker ( ฮฆ ๐ ) , in the matrix ฮฆ ๐ ๐ โผ ๐ ( ๐๐ ) ๐ the last ๐ (cid:205) ๐ = ๐ ๐ columns vanish, therefore the matrix ( ๐๐ ) ๐ is strictly-lower-triangular and has zeros on the ๏ฌrst ๐ block diagonals (the maindiagonal is counted as the ๏ฌrst one). We denote by ๐จ the parabolic subalgebra of ๐ค๐ฉ ( ๐ ) thatstabilizes the sub๏ฌag of (15.8) with the last element omitted. We have proven that ๐๐ (cid:12)(cid:12) ๐จ โ = ฮฆ ( ๐ง, ๐ง ) of the system (15.6)-(15.7) produces a solution ( ๐, ๐ ) to theequations of motion of the sigma model with target space the ๏ฌag manifoldU ( ๐ ) U ( ๐ ) ร ยท ยท ยท ร U ( ๐ ๐ ) , where (15.11) ๐ ๐ = dim Ker ( ฮฆ ๐ )/ Ker ( ฮฆ ๐ โ ) is a non-increasing sequence.The complex structure on the ๏ฌag is uniquely determined by the structure of the complex๏ฌag (15.8). 106 In the previous section we showed that the integrable ๏ฌag manifold models that we formulatedat the start of Section 13 may equivalently be written in the form (14.25) (which was motivatedby the relation to 4D Chern-Simons theory) or in the form (14.24), if one eliminates theauxiliary ๏ฌelds ฮฆ , ฮฆ . We have also seen in Section 14.1 that, in principle, there is a way ofconstructing the (trigonometric and elliptic) deformations of these models by appropriatelyinserting the ๐ -matrix into the Lagrangian. The purpose of this section is to emphasize,following [71], that all of these systems โ either deformed or undeformed โ are really examplesof chiral gauged bosonic Gross-Neveu models. This way of formulating sigma models is notmerely a simple reformulation, but o๏ฌers substantial calculational bene๏ฌts. For example, aswe shall show, these methods allow to solve the renormalization group (generalized Ricci๏ฌow) equations and arrive at a beautiful universal one-loop solution. At the quantum levelthese models have chiral anomalies, which may be cancelled by adding fermions. We willshow that this naturally leads to the notion of super-quiver-varieties and allows, among otherthings, to arrive at a novel formulation of supersymmetric models.
We start by taking a closer look at the system (14.24). It turns out very fruitful to rewrite itin Dirac form. We introduce ๐ โDirac bosonsโ ฮจ ๐ = (cid:18) ๐ ๐ ๐ ๐ (cid:19) , ๐ = , . . . , ๐ . (16.1)The Lagrangian (14.24) is, in this notation, L = ฮจ ๐ / ๐ท ฮจ ๐ + (cid:18) ฮจ ๐ + ๐พ ฮจ ๐ (cid:19) (cid:18) ฮจ ๐ โ ๐พ ฮจ ๐ (cid:19) . (16.2)The Dirac notations are standard: ๐ , are the Pauli matrices, / ๐ : = (cid:205) ๐ = ๐ ๐ ๐ ๐ and ๐พ : = ๐ ๐ ๐ .The Lagrangian (16.2) is the bosonic incarnation of the so-called chiral Gross-Neveu model(equivalently the SU ( ๐ ) Thirring model [244]) interacting with a gauge ๏ฌeld. As in (14.24)and in the example at the very start of this Chapter, the equivalence with the sigma modelformulation (with a metric, ๐ต -๏ฌeld and possibly dilaton) is established through the eliminationof ๐ and ๐ .The model (16.2) is โchiralโ, meaning that there is a symmetry ๐ โ ฮป ๐, ๐ โ ฮป โ ๐ , where ฮป โ C ร . (16.3) In Minkowski signature this would have been the usual U ( ) chiral symmetry. This di๏ฌerence in chiraltransformations has been observed in [270, 174].
107 general SU ( ๐ ) -invariant Lagrangian would only retain a U ( ) -symmetry, | ฮป | =
1, and is notinvariant under the full C โ , which arises in (16.2) due to the chiral projectors. In other words,for a Euclidean worldsheet signature, chiral symmetry is equivalent to the complexi๏ฌcationof the original (non-chiral) symmetry. This chiral symmetry is of extreme importance at leastfor two reasons: โฆ Chiral symmetry ensures that the quartic interaction terms are quadratic in the ๐ -variables, which are the โmomentaโ conjugate to ๐ . This allows integrating them outand arriving at a metric form of the sigma model. โฆ We will also be interested in sigma models on projective spaces, Grasmannians etc.,and these may be obtained by taking the quotient w.r.t. C โ , i.e. by gauging the chiralsymmetry. In doing so, one needs to verify that it is free of anomalies. As promised earlier, we proceed to discuss the deformations of the Gross-Neveu sys-tems (16.2), or equivalently of the original sigma models. As we shall see, the Gross-Neveuform of the model is particularly useful in this case. For example, the one loop renormal-ization group ๏ฌow is described in this case by a couple of elementary Feynman diagrams,which should be contrasted with the highly non-trivial generalized Ricci ๏ฌow equations thatone obtains in the geometric formulation of the sigma model. This will also serve as our ๏ฌrststep towards a de๏ฌnition and solution of these models at the quantum level.We will ๏ฌrst show how one can construct a deformation of the Lagrangian (14.25). Thiswill serve to relate our construction to the formulation in terms of the 4D Chern-Simonstheory described in section 14.1.Using the three matrices ๐ โ Hom ( C ๐ , C ๐ ) , ๐ โ Hom ( C ๐ , C ๐ ) , ฮฆ โ End ( C ๐ ) (we willalways assume ๐ โค ๐ ) that we already encountered in Section 14.2, we write down thedeformed Lagrangian L = Tr (cid:16) ๐ D ๐ (cid:17) + Tr (cid:16) ๐ D ๐ (cid:17) + Tr (cid:16) ๐ โ ๐ ( ฮฆ ) ฮฆ (cid:17) , (16.4)where ๐ ๐ is the classical ๐ -matrix, depending on the deformation parameter ๐ , that we encoun-tered in Section 14.0.1. The covariant derivative is D ๐ = ๐๐ + ๐ ฮฆ ๐ + ๐ ๐ A , where A is agauge ๏ฌeld, whose structure depends on the actual target space under consideration, and ฮฆ , ฮฆ are auxiliary ๏ฌelds. We recall that, from the perspective of [87], ฮฆ , ฮฆ are the components ofthe four-dimensional Chern-Simons gauge ๏ฌeld along the worldsheet, and the quadratic termin ฮฆ , ฮฆ contains ๐ โ ๐ , because, as discussed in section 14.1, the Greenโs function of the gauge๏ฌeld is e๏ฌectively the classical ๐ -matrix. One can eliminate these auxiliary ๏ฌelds, since theyenter the Lagrangian quadratically, arriving at the following expression: L = Tr (cid:16) ๐ ๐ท๐ (cid:17) + Tr (cid:16) ๐ ๐ท๐ (cid:17) + Tr (cid:16) ๐ ๐ ( ๐๐ ) ( ๐๐ ) โ (cid:17) , (16.5)108here ๐ท๐ = ๐๐ + ๐ ๐ A . Clearly, in Dirac notation this leads to the deformed Gross-Neveumodel of the following form: L = ฮจ ๐ / ๐ท ฮจ ๐ + ( ๐ ๐ ) ๐๐๐๐ (cid:18) ฮจ ๐ + ๐พ ฮจ ๐ (cid:19) (cid:18) ฮจ ๐ โ ๐พ ฮจ ๐ (cid:19) . (16.6)To summarize, we have obtained a chiral gauged bosonic Gross-Neveu model, where thedeformation is encoded in the classical ๐ -matrix that de๏ฌnes the quartic vertex. The main property of the system (16.5)-(16.6) is that its e.o.m. admit a zero-curvaturerepresentation. To write it down, we observe that in the undeformed case, when ๐ ๐ isproportional to the identity operator, the above Lagrangians have an SU ( ๐ ) global symmetryand a corresponding Noether current one-form K = ๐พ ๐๐ง + ๐พ ๐๐ง = ๐ ๐ ๐๐ง + ๐ ๐ ๐๐ง .
Usingthis one-form, we de๏ฌne a family of connections, following [87]: A = ๐ ๐ ( ๐พ ) ๐๐ง โ ๐ ๐ ( ๐พ ) ๐๐ง . (16.7)Here ๐ , ๐ are complex parameters that will be related below. We wish to prove that theconnection A is ๏ฌat: ๐ A + A โง A = โ ๐๐ง โง ๐๐ง (cid:16) ๐ ๐ ( ๐๐พ ) + ๐ ๐ ( ๐๐พ ) + [ ๐ ๐ ( ๐พ ) , ๐ ๐ ( ๐พ )] (cid:17) ? = . (16.8)To this end, we will use the equations of motion of the model (16.5). To write them out, wede๏ฌne the โconjugateโ operator ห ๐ by the relation Tr ( ๐ ๐ ( ๐ด ) ๐ต ) = โ Tr ( ๐ด ห ๐ ๐ โ ( ๐ต )) . When theunitarity relation (14.2) holds, ห ๐ = ๐ . The e.o.m. for the ๐ and ๐ variables may be shown toimplythe following concise equations for the โNoether currentโ ๐พ : ๐๐พ = [ ห ๐ ๐ โ ( ๐พ ) , ๐พ ] , ๐๐พ = [ ๐พ, ๐ ๐ ( ๐พ )] . (16.9)These are the deformed versions of the equations (15.6) that we encountered earlier (since ineliminating ฮฆ from (16.4) one easily sees that in the undeformed case ฮฆ โผ ๐พ ). Substitutingin the equation (16.8), we see that it is satis๏ฌed if the matrix ๐ obeys the equation ๐ ๐ ( [ ๐พ, ๐ ๐ ( ๐พ )]) + ๐ ๐ ( [ ห ๐ ๐ โ ( ๐พ ) , ๐พ ]) + [ ๐ ๐ ( ๐พ ) , ๐ ๐ ( ๐พ )] = . (16.10)The reason why in our case ห ๐ is not necessarily equal to ๐ is that we will mostly be dealing witha non-simple Lie algebra ๐ค๐ฉ ( ๐ ) = ๐ฐ๐ฉ ( ๐ ) โ C ( C โ ๐ค๐ฉ ( ๐ ) corresponds to matrices proportionalto the unit matrix). We will assume a block-diagonal ๐ -matrix, acting as follows: ๐ ๐ = ( ๐ ๐ ) ๐ฐ๐ฉ ( ๐ ) + ( ๐ ๐ ) C , ( ๐ ๐ ) ๐ฐ๐ฉ ( ๐ ) โ End ( ๐ฐ๐ฉ ( ๐ )) , ( ๐ ๐ ) C : = ๐ ( ๐ ) Tr . (16.11)109 ๐ง ๐ง โ ๐ง ๐ ๐ง ๐ง โ ๐ง โ ๐ง ๐ ๐ ๐๐๐ โ( ๐ ๐ ) ๐๐๐ ๐ Figure 20: Feynman rules of the deformed model (16.5)-(16.6), with A = ๐๐๐ ๐ ๐๐๐ ๐ Figure 21: Diagrams contributing to the ๐ฝ -function at one loop.Here ( ๐ ๐ ) ๐ฐ๐ฉ ( ๐ ) acts on traceless matrices as in (14.6), and ( ๐ ๐ ) C acts on matrices of the type ๐ผ ยท as multiplication by ๐ ๐ ( ๐ ) . In this case ห ๐ ๐ = ๐ ๐ โ ( ๐ ( ๐ โ ) + ๐ ( ๐ )) Tr. If the unitarityrelation is satis๏ฌed, the mismatch vanishes. However, in either case ๐ ( ๐ ) completely dropsout from the equation (16.10), so we will prefer allowing an arbitrary function ๐ ( ๐ ) for themoment.Postulating the relations ๐ = ๐ข, ๐ = ๐ข๐ฃ, ๐ฃ = ๐ โ (implying ๐ = ๐ ๐ ) between theparameters, we identify (16.10) with the classical Yang-Baxter equation for ๐ค = ๐ฐ๐ฉ ( ๐ ) fromSection 14.0.1. ๐ฝ -function and the Ricci ๏ฌow. In this section we turn to the analysis of the elementary quantum properties of the theoryde๏ฌned by the Lagrangian (16.5)-(16.6), in the ungauged case A =
0. The main question wepose is whether this Lagrangian preserves its form after renormalization, at least to one looporder โ in other words, whether it is su๏ฌcient to renormalize the parameters of the ๐ -matrix.To this end we write out the Feynman rules of the system in Fig. 20. At one loop the twodiagrams contributing to the renormalization of the quartic vertex are shown in Fig. 21. Thereis a relative sign between the two diagrams, due to the di๏ฌerent directions of the lines in theloops. Otherwise, the type of the divergence is the same โ it is logarithmic , proportionalto โซ ๐ ๐ง๐ง๐ง . As a result, the one-loop ๐ฝ -function is Here we are talking about UV divergences. ๐๐๐ ๐ = ๐ โ๏ธ ๐,๐ = (cid:16) ( ๐ ๐ ) ๐๐๐๐ ( ๐ ๐ ) ๐๐๐ ๐ โ ( ๐ ๐ ) ๐๐๐๐ ( ๐ ๐ ) ๐๐๐ ๐ (cid:17) (16.12)As already discussed earlier, we will assume a block-diagonal ๐ -matrix (16.11), where ( ๐ ๐ ) ๐ฐ๐ฉ ( ๐ ) acts as in (14.6), and ( ๐ ๐ ) C acts on a unit matrix as multiplication by ๐ ยท ๐ ( ๐ ) . This is translatedinto the four-index notation as follows: ( ๐ ๐ ) ๐๐๐ ๐ = ๐ ๐๐ ( ๐ ) (cid:18) ๐ฟ ๐๐ ๐ฟ ๐๐ โ ๐ ๐ฟ ๐ ๐ ๐ฟ ๐๐ (cid:19) + ๐ ( ๐ ) ๐ฟ ๐ ๐ ๐ฟ ๐๐ . (16.13)The coe๏ฌcients ๐ ๐๐ ( ๐ ) are de๏ฌned by the action of (14.6) in the standard basis: ๐ ๐ ( ๐ ๐ โ ๐ ๐ ) = ๐ ๐๐ ( ๐ ) ๐ ๐ โ ๐ ๐ . Concretely (see (14.5)), ๐ ๐ ๐ = ๏ฃฑ๏ฃด๏ฃด๏ฃด๏ฃฒ๏ฃด๏ฃด๏ฃด๏ฃณ ๐ โ ๐ = ฮฑ , ๐ < ๐ โ ๐ = ฮฒ , ๐ > ๐
12 1 + ๐ โ ๐ = ฮณ , ๐ = ๐ . (16.14)Substituting (16.13) in (16.12) and doing the summations, we obtain ๐ฝ ๐๐๐ ๐ = (cid:20) ๐๐ ( โ ๐ ) + ( ๐ โ ๐ ) ๐ ๐ ๐ (cid:21) (cid:18) ๐ฟ ๐๐ ๐ฟ ๐๐ โ ๐ ๐ฟ ๐ ๐ ๐ฟ ๐๐ (cid:19) : =ฮ ๐๐๐ ๐ . (16.15)Since ๐ ๐ ๐ =
12 1 + ๐ โ ๐ + ๐ R ๐ ๐ , the one-loop result is not of the form (16.13) (due to the termproportional to ๐ โ ๐ ). For this reason the straightforward Ricci ๏ฌow equation ๐๐ ๐ ๐๐ = ๐ฝ for ๐ ( ๐ ) does not have a solution. However, this can be easily remedied by allowing reparametrizationsof coordinates along the ๏ฌow. Let us reparametrize ๐ โ ๐ ๐, ๐ โ ๐ ๐ โ , where ๐ = Diag { ๐ , . . . , ๐ ๐ } . (16.16)The kinetic term in (16.5) is invariant, so the only e๏ฌect is in the e๏ฌective replacement ofthe ๐ -matrix by (cid:101) ๐ , where ( (cid:101) ๐ ๐ ) ๐๐๐ ๐ = ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ( ๐ ๐ ) ๐๐๐ ๐ . One has to conjugate the ๐ฝ -function tensoranalogously, and the equation we will aim to solve is ๐ (cid:101) ๐ ๐ ๐๐ = (cid:101) ๐ฝ . It may be rewritten as anequation for the original ๐ -matrix as follows: ๐๐๐ ( ๐ ๐ ) ๐๐๐ ๐ = ๐ฝ ๐๐๐ ๐ โ ๐๐๐ (cid:18) log (cid:18) ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ (cid:19) (cid:19) ( ๐ ๐ ) ๐๐๐ ๐ : = (cid:98) ๐ฝ ๐๐๐ ๐ (16.17)where (cid:98) ๐ฝ ๐๐๐ ๐ = (cid:20) ๐๐ ( โ ๐ ) + ( ๐ โ ๐ ) ๐ ๐ ๐ โ ๐๐๐ (cid:18) log (cid:18)(cid:12)(cid:12) ๐ ๐ ๐ ๐ (cid:12)(cid:12) (cid:19) (cid:19) ๐ ๐ ๐ (cid:21) ฮ ๐๐๐ ๐ The unwanted term may now be canceled by the simple substitution ๐ ๐ = ๐ ๐ ๐ . (16.18)111ecalling again the expression for ๐ ๐ ๐ ( ๐ ) , we ๏ฌnd that the remaining equations may be writtenas (cid:164) ๐ = ๐๐๐ (cid:16)
12 1 + ๐ โ ๐ (cid:17) = ๐๐ ( โ ๐ ) , or (cid:164) ๐ = ๐๐ . Therefore ๐ = const ., ๐ = ๐ ๐ ๐ (16.19)In the geometric formulation (i.e. for the sigma model de๏ฌned by a metric, ๐ต -๏ฌeld and dilaton)the RG-๏ฌow equations are the generalized Ricci ๏ฌow equations. They look as follows [89,200]: โ (cid:164) ๐ ๐ ๐ = ๐ ๐ ๐ + ๐ป ๐๐๐ ๐ป ๐๐ (cid:48) ๐ (cid:48) ๐ ๐๐ (cid:48) ๐ ๐๐ (cid:48) + โ ๐ โ ๐ ฮฆ , (16.20) โ (cid:164) ๐ต ๐ ๐ = โ โ ๐ ๐ป ๐๐ ๐ + โ ๐ ฮฆ ๐ป ๐๐ ๐ , โ (cid:164) ฮฆ = const . โ โ ๐ โ ๐ ฮฆ + โ ๐ ฮฆ โ ๐ ฮฆ + ๐ป ๐๐๐ ๐ป ๐๐๐ , where ๐ is the metric, ๐ต is the skew-symmetric ๏ฌeld with ๐ป its โcurvatureโ, and ฮฆ isthe dilaton. We arrive at the important conclusion that the trigonometrically deformedsystem (16.5)-(16.6) with A = ๐ฝ -functions of homogeneous models The calculation that we presented above was performed for the ungauged model ( A = ๐ฝ -function.Moreover, one can easily see that, even if we replace the size- ๐ vectors ๐ and ๐ by ๐ ร ๐ -matrices, which would be necessary for considering Grassmannian ๐บ๐ ๐,๐ or ๏ฌag manifoldtarget spaces, the calculation again leads to exactly the same answer, since the additionalmatrix index is simply a spectator index for the diagrams in Fig. 21. This phenomenon israther remarkable and has implications for the undeformed (i.e. homogeneous) models aswell. In the geometric formulation this means that the metric and ๐ต -๏ฌeld of all homogeneousmodels satisfy the generalized Einstein condition with the same โcosmological constantโ,equal to ๐ : ๐ ๐ ๐ + ๐ป ๐๐๐ ๐ป ๐๐ (cid:48) ๐ (cid:48) ๐ ๐๐ (cid:48) hom . ๐ ๐๐ (cid:48) hom . = ๐ ( ๐ hom . ) ๐ ๐ . (16.21)Here ๐ hom . is the homogeneous metric, and ๐ป is the curvature of the ๐ต -๏ฌeld equal to thefundamental Hermitian form of the metric. As we recall from the discussion in Section 13.3,in the case of Grassmannians the metric ๐ hom . is Kรคhler, so that the fundamental Hermitianform is closed and ๐ป =
0. In that case the equation (16.21) translates into the usual Einsteinequation ๐ ๐ ๐ = ๐ ( ๐ hom . ) ๐ ๐ . This is the well-known fact that the one-loop ๐ฝ -function forsymmetric space models is equal to the dual Coxeter number of the symmetry group ๐บ (in112articular, it is independent of the denominator ๐ป for symmetric spaces ๐บ๐ป ), cf. [269] asa general reference and [178] for the case of Hermitian symmetric spaces, as well as thelectures [266]. In the case of Kรคhler symmetric spaces โ such as Grassmannians โ there is analternative explanation: since the Ricci form represents the ๏ฌrst Chern class of the manifold,one can attribute the cosmological constant ๐ to the fact that ๐ ( ๐บ๐ ๐,๐ ) = ๐ [ C ] , where C isa generator of ๐ป ( ๐บ๐ ๐,๐ , Z ) . For non-symmetric spaces โ such as the general ๏ฌag manifoldsโ this logic does not work, and there is no direct relation between the ๏ฌrst Chern class and thedual Coxeter number. A related fact is that for non-symmetric spaces the metric in our sigmamodels is not Kรคhler, and one instead has to take into account the non-zero ๏ฌeld ๐ป in (16.21).The e๏ฌect of including this ๏ฌeld is that the two terms in the ๐ฝ -function โ the curvature andthe ๐ป -๏ฌeld term โ sum up in such a way that the ๐ฝ -function is again proportional to ๐ . Before concluding this section, let us describe a simplest example, namely the deformationof the sphere ๐ โ the so-called โsausageโ [109]. Already on this example one can see all thesalient features of the general solution described above. Let us derive this solution, startingfrom the Lagrangian (16.5) of the deformed model and making the rescalings (16.16), (16.18): L = ๐ โฆ ๐ท๐ + ๐ โฆ ๐ท๐ + Tr (cid:0) ๐ ๐ ( ๐ ๐ โ ๐ ๐ โ ) ( ๐ ๐ โ ๐ ๐ โ ) โ (cid:1) (16.22) = ๐ โฆ ๐ท๐ โ ๐ท๐ โฆ ๐ + ฮณ (| ๐ | | ๐ | + | ๐ | | ๐ | )+ ฮฑ ๐ โ ๐ | ๐ | ยท | ๐ | + ฮฒ ๐ ๐ | ๐ | ยท | ๐ | . Next we pass to the inhomogeneous gauge ๐ = ๐ : = ๐ , ๐ : = ๐ . Variationw.r.t. the gauge ๏ฌeld gives the constraint ๐ โฆ ๐ =
0, which is solved by ๐ = โ ๐ ยท ๐ . TheLagrangian acquires the following form in these coordinates: L = ๐ ยท ๐ท๐ โ ๐ ยท ๐ท๐ + (cid:16) ฮณ | ๐ | + ฮฑ ๐ โ ๐ + ฮฒ ๐ ๐ | ๐ | (cid:17) | ๐ | (16.23)Eliminating the ๏ฌelds ๐ , ๐ and using the expressions (16.14) for ฮฑ , ฮฒ , ฮณ with ๐ = ๐ ๐ , wearrive at L = ( ๐ โ ๐ โ ๐ ๐ ) | ๐ท๐ | (cid:0) ๐ ๐ + | ๐ | (cid:1) (cid:0) ๐ โ ๐ + | ๐ | (cid:1) , (16.24)which corresponds to the โsausageโ metric. As all models of the type (13.7), the La-grangian (16.24) features a ๐ต -๏ฌeld equal to the fundamental Hermitian form of the metric,however since this is a manifold of complex dimension 1, the ๐ต -๏ฌeld is closed, so that ๐ป = ฮฆ is constant (for details see [71]),so that the Ricci ๏ฌow equation is especially simple: โ ๐๐ ๐ ๐ ๐๐ = ๐ ๐ ๐ . (16.25)113he range of the Ricci time variable is ๐ โ (โโ , ) (accordingly ๐ = ๐ ๐ โ ( , ) ), and thecorresponding solution is called โancientโ in the terminology used in Ricci ๏ฌow literature.Geometrically the โsausageโ looks exactly as its name suggests:0 < ๐ < โผ | log ๐ | (16.26)The solution has two characteristic regimes. The ๏ฌrst one is ๐ โ
0, in which case ๐๐ โ | ๐๐ | | ๐ | ,so that one obtains an in๏ฌnitely long cylinder. Since the cylinder is ๏ฌat, this should beinterpreted as the UV limit with asymptotic freedom. The opposite regime is the IR limit ๐ โ
1, where one obtains a round metric on CP , albeit with a vanishing radius, which is asign of an IR singularity. The same behavior persists qualitatively in the case of CP ๐ โ [71],where Ricci ๏ฌow interpolates between a cylinder ( C โ ) ๐ โ in the UV (asymptotic freedom)and a โroundโ projective space of vanishing radius in the IR (one has a similar behavior in thecase of the so-called ๐ -deformed CP ๐ โ , as shown in [74]). In the previous sections we have claimed that the sigma models of a wide class, includingthe familiar CP ๐ โ , Grassmannian, ๏ฌag, etc. models, are equivalent to chiral Gross-Neveumodels. One remarkable consequence of this fact is that the corresponding sigma models aretherefore models with polynomial interactions, and all non-linear constraints that are usuallypresent in conventional formulations can be bypassed.The model (16.5)-(16.6) with A = CP ๐ โ -model. This system is described by (16.5), where ๐ is a column vector and ๐ a row vector, both of length ๐ , and additionally one has a C โ gauge ๏ฌeld A . It turns out thegauge ๏ฌeld may be completely eliminated by passing to the inhomogeneous C โ -gauge ๐ ๐ = . (16.27)Variation of the Lagrangian (16.5) w.r.t. A gives ๐ โฆ ๐ =
0, so that ๐ ๐ = โ ๐ โ โ๏ธ ๐ = ๐ ๐ ๐ ๐ . (16.28)This removes the gauge ๏ฌeld at the expense of modifying the Feynman rules of the theory.Dropping fermionic ๏ฌelds for the moment, we write the Lagrangian of the model (16.5) in114his gauge: L = ๐ โ โ๏ธ ๐ = (cid:16) ๐ ๐ ๐๐ ๐ โ ๐ ๐ ๐๐ ๐ + ฮฒ | ๐ ๐ | (cid:17) + (16.29) + ๐ โ โ๏ธ ๐,๐ = ๐ ๐๐ | ๐ ๐ | | ๐ ๐ | + ฮณ (cid:12)(cid:12) ๐ โ โ๏ธ ๐ = ๐ ๐ ๐ ๐ (cid:12)(cid:12) + ฮฑ (cid:32) ๐ โ โ๏ธ ๐ = | ๐ ๐ | (cid:33) (cid:12)(cid:12) ๐ โ โ๏ธ ๐ = ๐ ๐ ๐ ๐ (cid:12)(cid:12) We see that the propagators and vertices are modi๏ฌed, and on top of that sextic verticeshave appeared. A fascinating feature of the Lagrangian (16.29) is that its interaction termsare again polynomial in the ( ๐, ๐ ) -variables. In other words, instead of a nonlinear ๐ -model we have arrived at a di๏ฌerent nonlinear theory โ the theory of several bosonic ๏ฌelds(albeit with fermionic propagators) with polynomial interactions. One might also noticethat the procedure of going to inhomogeneous coordinates that we have just described is infact reminiscent of what we did at the end of section 2.4 while introducing the so calledDyson-Maleev variables. In other words, the generalized Dyson-Maleev variables allow toeliminate the gauge ๏ฌelds and turn the sigma models into models of multiple bosonic ๏ฌeldswith polynomial interactions.Another fascinating parallel that this discussion invokes is that with Ashtekar variablesin 4D general relativity. It is well-known that dimensional reduction of general relativity,possibly with additional matter ๏ฌelds, along two commuting Killing vectors leads to inte-grable sigma models [120, 42, 172] (see [183] for a review). The target space depends onthe particular gravitational system that one started with [57, 56]. For example, in the case ofpure gravity one gets ๐๐ฟ ( , R ) ๐๐ ( ) , whereas gravity with ๐ โ SU ( ๐, ) S ( U ( ๐ )ร U ( )) โ the hyperbolic analogue of projective space. These are all complex (Hermitian) symmet-ric spaces, albeit of Minkowski signature, which makes it slightly di๏ฌerent from what weencountered in most of this article, but the general structure of the models is the same. Aninteresting consequence comes from the salient property of Ashtekar variables in generalrelativity, namely that they make the interactions polynomial, cf. [30]. This of course hintson the relation to the polynomiality of interactions in the sigma models that we have justdiscussed. A more careful analysis [58] shows that the Noether currents ๐พ of the sigmamodel are bilinear combinations of Ashtekarโs canonical variables ( ๐ด, ๐ธ ) : ๐พ โผ ๐ด โ ๐ธ . Onthe other hand, these same Noether currents, when calculated from the Lagrangian (16.2)or (16.5), have the form ๐พ โผ ๐ โ ๐ , so that the canonical variables ( ๐, ๐ ) may be naturallyinterpreted as the dimensional reductions of Ashtekar variables.Apart from the polynomiality of interactions, the latter observation has yet another im-portant consequence. Since the ( ๐ด, ๐ธ ) variables are canonical, one has the Poisson brackets { ๐ด ( ๐ฅ ) , ๐ธ ( ๐ฆ )} โผ ๐ฟ ( ๐ฅ โ ๐ฆ ) , and as a result the Noether currents ๐พ have local Poisson bracketsas well, schematically of the form { ๐พ ( ๐ฅ ) , ๐พ ( ๐ฆ )} โผ ๐พ ( ๐ฅ ) ๐ฟ ( ๐ฅ โ ๐ฆ ) . This of course also im-mediately follows from the analogous Poisson structure of the ( ๐, ๐ ) variables, which canbe seen from the ๏ฌrst-order Lagrangians (16.2), or (16.5). Since the ๏ฌat connection (16.7)115s linear in the ๐พ -variables, its components also have ultralocal Poisson brackets of the stan-dard form [92] (for background see [206, 213, 159]). As simple as it may look from thisperspective, it is a rather exceptional property for sigma models, where the Poisson bracketsof Lax operators typically produce non-ultralocal terms [106, 170, 171], proportional to ๐ฟ (cid:48) ( ๐ฅ โ ๐ฆ ) . The latter cause signi๏ฌcant di๏ฌculties in the discretization of such systems, whichmay therefore be overcome for the integrable models discussed in this Chapter. The formulation of sigma models in terms of Gross-Neveu models suggests a natural, butrather far-reaching, generalization [72]. The ๏ฌrst step towards this generalization is to realizethat at the quantum level one is forced to supplement the purely bosonic models describedabove with fermions. This is necessary because in general gauged models of the type (16.2)or (16.5)-(16.6) su๏ฌer from gauge anomalies. These are in fact a property of the kinetic termin the Lagrangian (16.2), so that to this end we may omit the interaction term. The one-loopdeterminant of the matter ๏ฌelds leads us to Schwingerโs calculation [212] of the e๏ฌectiveaction of the gauge ๏ฌelds A : S e๏ฌ . = ฮพ โซ ๐๐ง ๐๐ง ๐น ๐ง๐ง (cid:52) ๐น ๐ง๐ง , ๐น ๐ง๐ง = ๐ ( ๐ A โ ๐ A) . (16.30)The coe๏ฌcient ฮพ collects some numerical factors and is proportional to the number of matter๏ฌelds we have integrated over. The action is invariant w.r.t. the gauge transformations of theoriginal U ( ) , A โ A + ๐๐ผ , A โ A + ๐๐ผ ( ๐ผ โ R ), but not w.r.t. the complexi๏ฌed C โ gauge transformations A โ A + ๐๐ผ , A โ A + ๐๐ผ ( ๐ผ โ C ). The non-Abelian analogue ofthis calculation leads to the WZNW action, as shown in [202] and discussed in [101, 100]in the Euclidean case, but even the simple abelian e๏ฌective action (16.30) su๏ฌces for mostpurposes. For example, it is clear that the anomaly may be canceled by including fermionssymmetrically with the bosons: L โ ฮจ ๐ / ๐ท ฮจ ๐ + ฮ ๐ / ๐ท ฮ ๐ , (16.31)where ฮจ are the bosons and ฮ the fermions. In this case the respective determinants cancel.We emphasize that such a simple mechanism is possible because we have rewritten thebosonic part of the theory in fermionic form in the ๏ฌrst place.In (16.31) we have dropped the interaction terms to emphasize that the anomaly is aproperty of the kinetic term in the Lagrangian. As it turns out, the kinetic term has a cleargeometric meaning and de๏ฌnes what may be called the โsuper-phase spaceโ of the model.Moreover, the whole theory of generalized integrable Gross-Neveu models of the type (16.2)can be cast in pure di๏ฌerential-geometric terms.The relevant geometric context is as follows. Suppose we have a super phase space ๐ฝ ,which is a complex symplectic (quiver) supervariety. There is a gauge (super)-group G gauge acting in the nodes of the quiver, and matter ๏ฌelds ๐ โ W , ๐ โ W are in representations116 โ W of G gauge . We assume that the quiver is โdoubledโ, meaning that every representationarises together with its dual (Nakajima quivers have this property [180, 181]). Apart from thegauge nodes, the quiver will typically have some global nodes with an action of a complexglobal symmetry (super)-group G global . We can therefore de๏ฌne the complex moment map ๐ for the action of G global (cid:8) ๐ฝ . In this setup one can, quite naturally, de๏ฌne the followingLagrangian: L = (cid:16) ๐ ยท D ๐ + ๐ ยท D ๐ (cid:17) + ฮบ STr ( ๐ ๐ ) . (16.32)The Gross-Neveu system (16.2) is a special case. The kinetic term in (16.32) corresponds tothe ๐ฝ๐พ -systems โ it is a pull-back of the canonical Poincarรฉ-Liouville one-form correspondingto the complex symplectic form of the quiver. The second term provides a coupling betweenthe holomorphic and (anti)-holomorphic ๐ฝ๐พ -systems and comes with an arbitrary coe๏ฌcient ฮบ that should be seen as a coupling constant (in the sigma model setup this is the inversesquared radius of the target space). One can directly show that the moment map ๐ satis๏ฌesthe e.o.m. ๐ ๐ = ฮบ [ ๐, ๐ ] , (16.33)which is the e.o.m. of the principal chiral model, thus once again pointing at the relationbetween the models, which has already been emphasized in section 15. In particular, wealready encountered the above equation in (15.6) and its deformed version in (16.9) (formodels with kinetic term given by the Poincarรฉ-Liouville one-form the components of theNoether current coincide with the moment map variables).One also needs to impose the chiral anomaly cancellation conditions that in the generalsetup have the form Str W ( ๐ ๐ ๐ ๐ ) = , where ๐ ๐ , ๐ ๐ โ ๐ค gauge . (16.34)These are in fact the Euclidean analogues of the standard anomaly cancellation conditions inthe WZNW models [249]. We expect that in most cases the condition Str W ( ๐ ๐ ) = CP ๐ โ -model with fermions.All known models, whose bosonic part is the CP ๐ โ sigma model, can be attributed to oneof the two cases. In both of these cases the phase spaces are complex symplectic quotientsof the form ๐ฝ = ( ๐ โ C ๐ | ๐ ) (cid:12) G gauge , (16.35)where G gauge is a subgroup of GL ( | ) โ GL ( ๐ | ๐ ) , the latter being the symmetry group of T โ C ๐ | ๐ . In this language, the two cases are distinguished by the choice of G gauge :117 The โminimal fermionsโ phase space: G gauge = C โ = (cid:26) ๐ โ SL ( | ) : ๐ = (cid:18) ฮป ฮป (cid:19) (cid:27) โฆ The โsupersymmetricโ phase space: G gauge = (cid:26) ๐ โ SL ( | ) : ๐ = (cid:18) ฮป ฮพ ฮป (cid:19) (cid:27) . In thiscase ฮป โ C โ is a bosonic element, and ฮพ โ C is a fermionic element. As a result, herethe quotient (16.35) is a genuine super-symplectic reduction.Both situations correspond to the following elementary quiver: ๐๐ C ๐ C | (16.36)Here ๐ โ Hom ( C | , C ๐ ) and ๐ โ Hom ( C ๐ , C | ) . The di๏ฌerence comes from the action ofthe gauge group G gauge on C | . In fact, one can also identify the con๏ฌguration spaces ๐ ofthe models, since in both cases ๐ฝ = ๐ โ ๐ : ๐ min = CP ๐ โ | ๐ , ๐ SUSY = ฮ ๐ ( CP ๐ โ ) . (16.37)Here ฮ ๐ stands for the โfermionic tangent bundleโ, i.e. the tangent bundle where the ๏ฌbersare assumed fermionic. Accordingly the super-projective space CP ๐ โ | ๐ may be seen as thetotal space of the super-vector bundle ฮ (O ( ) โ ยท ยท ยท โ O ( )) over CP ๐ โ , which puts the twocon๏ฌguration spaces on par with each other.Specifying the phase or con๏ฌguration space is of course not su๏ฌcient to formulate thetheory. One additionally needs to choose the global symmetry group ๐บ global , which in turnde๏ฌnes the Hamiltonian in (16.32) via the complex moment map ๐ . For ๐ min there are twochoices that lead to well-known models: โฆ ๐บ global = ๐๐ฟ ( ๐, C ) . In this case (16.32) de๏ฌnes the CP ๐ โ model with โminimallycoupled fermionsโ [1] (this is also the reason for the name of the phase space ๐ min ). โฆ ๐บ global = ๐๐๐ฟ ( ๐ | ๐, C ) . This is the sigma model with target space CP ๐ โ | ๐ that hasbeen widely studied in the literature, cf. [204, 77]. In the case ๐ = ๐ SUSY .As proven in detail in [72], in order to obtain an interacting ( , ) -supersymmetric CP ๐ โ sigma model, in this case one should choose ๐บ global = ๐๐ฟ ( ๐, C ) . This provides a newapproach to constructing worldsheet-supersymmetric models by starting from a model withtarget space supersymmetry (in this case GL ( ๐ | ๐, C ) ) and gauging part of the supergroupin such a way that the target space supersymmetry disappears and gives way to worldsheetsupersymmetry. This approach does not rely on superspace methods, and one might say thatworldsheet supersymmetry is emergent in this case.118 onclusion Despite the ubiquity of ๏ฌag manifolds in mathematics, they might not be equally familiarto the physics audience. One of the goals of this review was to ๏ฌll this gap in the physicsliterature and to introduce these rich objects, explaining that they are useful and in certaincases inevitable in physics applications.We started in Chapter 1 by de๏ฌning what ๏ฌag manifolds are, and by describing theirdi๏ฌerential-geometric structures. As a ๏ฌrst application of these methods, we considered anโalmost textbookโ example of a mechanical particle interacting with a non-Abelian gauge๏ฌeld. As we explained, the isotopic โspinโ degrees of freedom take values in a suitable ๏ฌagmanifold. The symplectic form on the ๏ฌag manifold is the classical analogue of the conceptof representation of the gauge group, w.r.t. which the particle is charged. The โBerry phaseโ,which is formulated in terms of the chosen symplectic form, serves as a kinetic term forspin motion and is nothing but a one-dimensional version of the (originally two-dimensional)WZNW-term. Exactly the same argument as in WZNW theory leads to the quantizationof the parameters entering the symplectic form. The resulting โquantum numbersโ have atransparent interpretation as the lengths of the rows in the Young diagram corresponding tothe representation of the particle.The approach that we developed on the example of the mechanical particle is in factrather universal and is colloquially known as โgeometric quantizationโ. In the rest of Chap-ter 1 we explained how the Berry phase action can be quantized, and that this leads tovarious representations of spin operators well-known in condensed matter physics, suchas the Schwinger-Wigner, Holstein-Primako๏ฌ and Dyson-Maleev representations. We alsoemphasized that the very same ๏ฌag manifolds can also be understood as the manifolds ofcoherent states for the relevant representations. We subsequently used these coherent statesto construct path integrals for spin chains in Chapter 2.Overall, in Chapters 2 and 3 we attempted to cover two major topics related to sigmamodels with a two-dimensional worldsheet and a ๏ฌag manifold target space. The ๏ฌrst topicis how such sigma models arise in the continuum limits of spin chains, and the second one isthe description of integrable ๏ฌag manifold models.In Chapter 2, we reviewed SU( ๐ ) spin chains in various representations, and discussed atlength how these representations give rise to sigma models with di๏ฌerent ๏ฌag manifold targetspaces. This can be understood by considering how the Young tableau parameters ๐ ๐ผ , whichgeneralize the notion of spin in the antiferromagnet, determine the target space of the chainโsmatrix degree of freedom, ๐ . Mathematically speaking, ๐ is a moment map, and di๏ฌerent119 ๐ผ s de๏ฌne di๏ฌerent co-adjoint orbits. For SU(2), the target space of ๐ is always ๐ , and leadsto the familiar CP model, but for ๐ >
2, the 2-sphere is promoted to some ๏ฌag manifold ofSU( ๐ ).For most of the chapter, we focused on the totally symmetric representations of SU( ๐ ),which corresponded to ๐ = ๐ , and ๐ ๐ผ = ๐ผ >
1. This de๏ฌnes a coadjoint orbitisomorphic to CP ๐ โ at each site of the chain. However, we did not end up deriving a sigmamodel with this projecitve target space. Instead, by considering Hamiltonians with longerrange interaction terms, that have classical ground states with ๐ -site order, we obtained thecomplete U ( ๐ )/[ U ( )] ๐ ๏ฌag manifold sigma model. Loosely speaking, the longer-rangeinteraction terms served to couple the di๏ฌerent CP ๐ โ sectors together, as well as imposeorthogonality. Related to this, we found that these sigma models possessed ๐ topologicalterms, each of which corresponded to the pull-back to the complete ๏ฌag manifold of theFubini-Study form on CP ๐ โ . The corresponding topological angles were found to be ๐ ๐ด = ๐๐ ๐ด๐ .Unless the various interaction terms in the Heisenberg Hamiltonian were tuned to specialvalues, we learned that these ๏ฌag manifold sigma models lack Lorentz invariance. This is dueto the fact that multiple velocities exist in the most general case. However, in Section 9.3 wereviewed how these velocities ๏ฌow to common value under renormalization, thus establishingthat Lorentz invariance does indeed emerge at low energies. This fact led to an SU( ๐ )generalization of Haldaneโs conjecture: when ๐ and ๐ are coprime, the corresponding SU( ๐ )chain will be in a gapless phase at low energies; otherwise, a ๏ฌnite energy gap will persist,with ground-state degeneracy equal to ๐ / gcd ( ๐, ๐ ) . This conjecture was supported by variousexact results, including the LSMA theorem and AKLT constructions, as well as by โt Hooftanomaly matching conditions. In short, โt Hooft anomalies are present in the ๏ฌag manifoldsigma model for all ๐ not a multiple of ๐ , but only when gcd ( ๐, ๐ ) = ๐ ) WZNW model). Theseanomalies are mixed between the PSU( ๐ ) symmetry of the model, and a global Z ๐ symmetry,which derives from the underlying ๐ -site order of SU( ๐ ) chain.Finally, we concluded Chapter 2 by reinterpreting this generalized Haldane conjecturein terms of fractional topological excitations in the sigma model. These correspond to non-trivial sections in a PSU( ๐ ) bundle, and have topological charges that are multiple of 1 / ๐ .We explained how these excitations give rise to an energy gap via a Coulomb gas mecha-nism, similar to the Kosterlitz-Thouless phase transition in the classical XY model. Whengcd ( ๐, ๐ ) =
1, these excitations interfere, resulting in an e๏ฌective fugacity of zero, and leadto a gapless phase in the sigma model. More general representations of SU( ๐ ), and how theirchains lead to sigma models with both linear and quadratic dispersion, were also reviewed atthe end of Chapter 2.The subsequent narrative was centered around a slightly di๏ฌerent circle of questionsrelated to ๏ฌag manifold sigma models. More exactly, in Chapter 3 we described a wideclass of integrable sigma models with complex homogeneous target spaces and deformationsthereof. This class includes the ๏ฌag manifold sigma models as rather representative examples.It has long been known that the construction of integrable models with target spaces that are120ot symmetric (even if homogeneous) is a signi๏ฌcant challenge already in classical theory.For this reason we started in Chapter 3 by describing from various angles the classicallyintegrable models with ๏ฌag manifold target spaces. Technically the key new ingredient thatneeds to be included to make such models integrable is a non-topological ๐ต -๏ฌeld of a specialkind. We explained that these models can be obtained by one of the three approaches: in amore conventional way by constructing Noether currents satisfying zero-curvature equations,using a remarkable relation to the principal chiral model via nilpotent orbits, or by the noveltechniques related to four-dimensional Chern-Simons theory. Besides, we showed that thesesigma models are exactly and explicitly equivalent to chiral gauged Gross-Neveu systems,whose integrability properties have been known since the 1970โs.In the latter part of Chapter 3 we concentrated on studying the proposed bosonic Gross-Neveu models, as well as their fermionic completions. It turned out that this perspective makesthe analysis of the underlying sigma models substantially easier than in the pure geometricformulation with a metric, ๐ต -๏ฌeld and dilaton. In support of this opinion we provided acalculation of the one-loop ๐ฝ -function of a wide class of trigonometrically deformed sigmamodels. This ๐ฝ -function is common for all of these models and provides a far-reachinggeneralization of the so-called โsausageโ solution that corresponds to the ๐ target space.We also explained that the general solution explains some puzzles about the undeformed,homogeneous models: for example, it provides an explanation of why the ๐ฝ -functions ofsymmetric space models depend only on the dual Coxeter number of the symmetry groupand extends this result to the (non-symmetric) ๏ฌag manifold models. Finally, we showedthat the purely bosonic gauged models su๏ฌer from chiral anomalies, which may be canceledby adding fermions. More generally, we formulated a broad di๏ฌerential-geometric setup forsigma models whose phase spaces are quiver super-varieties satisfying anomaly cancellationconditions. We demonstrated how this setup may be applied to the CP ๐ โ model withfermions, yielding all known quantum integrable models with bosonic core CP ๐ โ , andemphasized that this approach provides a new way of constructing models with worldsheetsupersymmetry by gauging models with target space supersymmetry. Acknowledgments.
We would like to thank Yu. Amari, I. Ya. Arefโeva, G. Arutyunov,A. Bourget, R. Donagi, S. Frolov, A. Hanany, E.A.Ivanov, S. Ketov, C. Klimฤรญk, M. La-jko, G. Lopes Cardoso, D. Lรผst, A. Ya. Maltsev, T. McLoughlin, F. Mila, K. Mkrtchyan,H. Nicolai, M. Nitta, V. Pestun, N. Sawado, N. Seiberg, A.G.Sergeev, V. Schomerus,E. Sharpe, S. Shatashvili, A.A.Slavnov, T. Sulejmanpasic, J. Teschner, S. Theisen, A. Tseytlin,K. Zarembo and P. Zinn-Justin for helpful discussions and N. Seiberg, K. Zarembo for com-ments on the manuscript. DB is especially grateful to A. A. Slavnov for long-term support,and to E. A. Ivanov for proposing the idea of writing a review article on the subject of ๏ฌagmanifold sigma models. DB would also like to thank the Max-Planck-Institut fรผr Physikin Munich (Germany), where part of this work was done, for hospitality. The research of121. A๏ฌeck and K. Wamer was supported by NSERC Discovery Grant 04033-2016, as well asby scholarships from NSERC and the Stewart Blusson Quantum Matter Institute. The workof D. Bykov was performed at the Steklov International Mathematical Center and supportedby the Ministry of Science and Higher Education of the Russian Federation (agreement no.075-15-2019-1614).
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A Kรคhler potential from the quiver quotient formulation
We showed in section 1.3 that there at least two ways to derive invariant Kรคhler metrics on๏ฌag manifolds: using the so-called quasipotentials and also using the Nakajima-type quivershown in Fig. 2. In this section we prove the equivalence of the two approaches.The space of matrices { ๐ ๐ด } shown in Fig. 2 is endowed with the standard symplecticform ฮฉ ๐ด = ๐ Tr ( ๐๐ ๐ด โง ๐๐ โ ๐ด ) and, accordingly, a metric ( ๐๐ ) ๐ด = Tr ( ๐๐ ๐ด ๐๐ โ ๐ด ) . The fullsymplectic form is then ฮฉ = ๐ โ โ๏ธ ๐ด = ๐ Tr ( ๐๐ ๐ด โง ๐๐ โ ๐ด ) . (A.1)At each circular node ๐ one has the action of a gauge group U ( ๐ฟ ๐ด ) : = U ( ๐ ๐ด ) โ GL ( ๐ ๐ด , C ) that preserves the symplectic form. Accordingly one can de๏ฌne the moment maps for thisaction: ๐ ๐ด = ๐ โ ๐ด ๐ ๐ด โ ๐ ๐ด โ ๐ โ ๐ด โ . The main statement is that the ๏ฌag manifold may be de๏ฌnedas a quotient: F = { ๐ ๐ด = ๐ ๐ด ๐ ๐ด , ๐ด = , . . . , ๐ โ } (cid:14) U ( ๐ฟ ) ร ยท ยท ยท ร U ( ๐ฟ ๐ โ ) , (A.2)where ๐ ๐ด > ๐ โ ๐ โ ๐ด ๐ ๐ด is non-degenerate, which implies rk ( ๐ ๐ด ) = ๐ ๐ด .The linear spaces ๐ฟ ๐ด of the ๏ฌag may be obtained as Im ( ๐ ๐ โ ยท ยท ยท ๐ ๐ด + ๐ ๐ด ) โ C ๐ : thematrix ๐ ๐ โ ยท ยท ยท ๐ ๐ด + ๐ ๐ด has rank ๐ ๐ด , so that it de๏ฌnes ๐ ๐ด vectors in C ๐ , and the quotientw.r.t. U ( ๐ฟ ๐ด ) amounts to considering the linear space spanned by these vectors (compare withthe example (1.18)-(1.19), depicted in Fig. 3). Clearly, the ๐ฟ ๐ด so de๏ฌned are nested in eachother: ๐ฟ ๐ด โ โ ๐ฟ ๐ด . 136he apparatus of symplectic quotient provides a symplectic form on F by restricting theoriginal symplectic form ฮฉ to the level set of the moment maps ฮฉ = ฮฉ (cid:12)(cid:12) ๐ = ๐ . Since thewhole setup is Kรคhler, so that there is a complex structure and metric involved, the reductionalso provides a Kรคhler metric on the ๏ฌag manifold, which should coincide with the metricgiven by the Kรคhler potential (1.14).Let us see how this happens. The strategy, known from the general theory of Kรคhlerquotients (cf. [133]), is as follows: one considers generic matrices ๐ ๐ด , ๐ โ ๐ด , not necessarilysatisfying the moment map constraints, and one needs to ๏ฌnd the complexi๏ฌed symmetrytransformation ๐ ร ยท ยท ยท ร ๐ ๐ โ โ GL ( ๐ฟ , C ) ร ยท ยท ยท ร GL ( ๐ฟ ๐ โ , C ) , such that the transformedvariables would satisfy the constraints. Introducing ๐ ๐ด = ๐ โ ๐ด ๐ ๐ด , it is easy to see that we mayrewrite this requirement as ๐ โ ๐ด ๐ ๐ด + ๐ ๐ด โ ๐ ๐ด ๐ ๐ด โ ๐ โ ๐ด โ ๐ โ ๐ด โ ๐ ๐ด = ๐ ๐ด ๐ ๐ด , ๐ ๐ = . (A.3)Given a solution ๐ , . . . , ๐ ๐ โ , we obtain the Kรคhler potential of the quotient manifold asfollows: K = ๐ โ โ๏ธ ๐ด = ๐ ๐ด Tr ( log ๐ ๐ด ) = ๐ โ โ๏ธ ๐ด = ๐ ๐ด log ( det ๐ ๐ด ) . (A.4)We proceed to compute the determinants of the matrices ๐ ๐ด . Denoting ๐ฆ ๐ด : = ๐ โ ๐ด โ ๐ โ ๐ด โ ๐ ๐ด we may rewrite (A.3) in two equivalent forms: ๐ โ ๐ด ๐ ๐ด + ๐ ๐ด = ๐ ๐ด ( ๐ ๐ด + ๐ ๐ด โ ๐ฆ ๐ด ) . (A.5) ๐ฆ ๐ด + ๐ ๐ด โ ๐ ๐ด โ ๐ฆ ๐ด = ๐ ๐ด . (A.6)Multiplying (A.5) by ๐ โ ๐ด โ from the left and by ๐ ๐ด โ from the right and using (A.6), we ๏ฌnd ๐ โ ๐ด โ ๐ โ ๐ด ๐ ๐ด + ๐ ๐ด ๐ ๐ด โ = ๐ ๐ด โ ( ๐ ๐ด โ + ๐ ๐ด โ ๐ฆ ๐ด โ ) ( ๐ ๐ด + ๐ฆ ๐ด ๐ ๐ด โ ) = (A.7) = ๐ ๐ด โ ( ๐ ๐ด โ + ๐ ๐ด โ ๐ฆ ๐ด โ ) ( ๐ ๐ด + ๐ ๐ด โ + ๐ ๐ด โ ๐ฆ ๐ด โ ) . Next we introduce the matrix ๐ ๐ต : = ๐ ๐ โ ยท ยท ยท ๐ ๐ต . As discussed above, Im ( ๐ ๐ต ) = ๐ฟ ๐ต .Recalling that ๐ ๐ = , we may continue (A.7) by induction to demonstrate that ๐ โ ๐ต ๐ ๐ต = ๐ ๐ต ยท ๐ ๐ต , (A.8)where ๐ ๐ต is a product of matrices of the type ๐ + ๐ ๐ต โ ๐ฆ ๐ต ( ๐ are constants). It turns outthat the latter matrices are triangular in a certain basis, their diagonal blocks being constant.Indeed, it follows from (A.6) that ( ๐ + ๐ ๐ต โ ๐ฆ ๐ต ) ๐ ๐ต โ = ( ๐ + ๐ ๐ต โ ) ๐ ๐ต โ + ๐ ๐ต โ ๐ ๐ต โ ๐ฆ ๐ต โ (to be continued by induction), so that the matrix ๐ + ๐ ๐ต โ ๐ฆ ๐ต = ๐ท ๐ต + ๐ ๐ต , where ๐ท ๐ต isdiagonal with eigenvalues ๐, ๐ + ๐ ๐ต โ , . . . and ๐ ๐ต is strictly triangular, in the sense that itmaps Im ( ๐ ๐ต โ ๐ ๐ต โ ยท ยท ยท ๐ ๐ต โ ๐ถ ) to Im ( ๐ ๐ต โ ๐ ๐ต โ ยท ยท ยท ๐ ๐ต โ ๐ต ๐ ๐ต โ ๐ถ โ ) for all ๐ถ . It follows thatdet ๐ ๐ต = const . โ
0, so that (A.8) implies det ( ๐ ๐ต ) โผ det ( ๐ โ ๐ต ๐ ๐ต ) , up to a constant coe๏ฌcient.Substituting into (A.4) and identifying ๐ ๐ต = ๐พ ๐ต , we ๏ฌnd agreement with (1.12)-(1.14).137 Symplectic forms on coadjoint orbits
We saw in section 1.5 that the most general invariant two-form on a ๏ฌag manifold ๐บ๐ป = ๐๐ ( ๐ ) ๐ ( ๐ ( ๐ )รยทยทยทร ๐ ( ๐ ๐ )) is ฮฉ = โ๏ธ ๐ด<๐ต ๐ ๐ด๐ต Tr ( ๐ ๐ด๐ต โง ๐ ๐ต๐ด ) . (B.1)Here we wish to prove that the requirement of it being closed leads to the Kirillov-Kostantform (1.8). To check, in which case the above two-form is closed, we will take advantage of the๏ฌatness of the Maurer-Cartan current, ๐ ๐ โ ๐ โง ๐ = . It follows that D ๐ ๐ด๐ต = (cid:205) ๐ถ โ ( ๐ด,๐ต ) ๐ ๐ด๐ถ โง ๐ ๐ถ๐ต , where D is the ๐ป -covariant derivative, de๏ฌned as follows: D ๐ ๐ด๐ต : = ๐ ๐ ๐ด๐ต โ { ๐ ๐ฅ , ๐ ๐ด๐ต } . Fromthe condition that ฮฉ is closed it follows that ๐ ๐ด๐ต + ๐ ๐ต๐ถ + ๐ ๐ถ ๐ด = ( ๐ด, ๐ต, ๐ถ ) . (B.2)The general solution to this equation is ๐ ๐ด๐ต = ๐ ๐ด โ ๐ ๐ต . (B.3)Therefore we have a family of homogeneous symplectic forms with ๐ โ ฮฉ = Tr ( ๐ ๐ โง ๐ ) , where ๐ = Diag ( ๐ ๐ , . . . , ๐ ๐ ๐ ๐ ) . (B.4)The element ๐ may be normalized to be traceless: Tr ( ๐ ) =
0. The stabilizer ๐ป may now bethought of as the stabilizer of the matrix ๐ โ ๐ฒ ๐ , and the ๏ฌag manifold itself โ as an adjointorbit: F ๐ ,...,๐ ๐ = { ๐ ๐ ๐ โ , ๐ โ SU ( ๐ )} . (B.5) C Coherent states as polynomials
In section 2.3.4 we described the coherent states of ๐๐ ( ๐ ) using the Schwinger-Wignerrepresentation in Fock space. In place of the Fock space generated by the creation operatorsacting on a vacuum state | (cid:105) we may equivalently use the space of polynomials with aGaussian inner product โ this is the celebrated Bargmann representation [38]. The map issimple: ( ๐ โ ) ๐ ยท ยท ยท ( ๐ โ ๐ ) ๐ ๐ | (cid:105) โฆโ ๐ง ๐ ยท ยท ยท ๐ง ๐ ๐ ๐ (C.1)This map is a Hilbert space isomorphism, meaning that the scalar product is preserved, ifone picks the Gaussian scalar product on the space of polynomials (here ห ๐ and ห ๐ are twopolynomials): ( ห ๐ , ห ๐ ) Bargmann = โซ ห ๐ ( ๐ง ) ห ๐ ( ๐ง ) ๐ โ ๐ (cid:205) ๐ = | ๐ง ๐ | ๐ (cid:214) ๐ผ = ( ๐ ๐๐ง ๐ผ โง ๐๐ง ๐ผ ) (C.2)138et us also discuss the relation to the de๏ฌnition of states as in homogeneous polynomialsused in the classical work [43], where coherent states were used to describe the quantizationof a sphere ๐ โผ CP โ the simplest homogeneous Kรคhler (symplectic) manifold . Tostart with, we observe that, since in our applications to representation theory the numberof oscillators is ๏ฌxed, we may introduce a new Hilbert space, isomorphic to the one ofhomogeneous polynomials (C.1). Suppose we have a rank- ๐ symmetric representation, sothat we are dealing with homogeneous polynomials of degree ๐ . It is an elementary fact thatthe following two spaces are isomorphic:Homogeneous polynomials of degree ๐ in ๐ variables โ Polynomials of degree โค ๐ in ๐ โ ๐ง =
1. This is the counterpart of passing to inhomogeneous coordinateson a projective space (see the very beginning of Chapter 1). Going backwards amountsto homogenizing the polynomial. To ๏ฌnd the correct integration measure on the space ofpolynomials of degree โค ๐ , we ๏ฌrst start with the homogeneous polynomials ห ๐ and ห ๐ ofdegree ๐ and make the change of variables { ๐ง โ ๐, ๐ง โ ๐๐ง , ยท ยท ยท , ๐ง ๐ โ ๐๐ง ๐ } . In thiscase, clearly, ห ๐ = ๐ ๐ ๐ ( ๐ง ) and analogously for ๐ , where ๐ ( ๐ง ) , ๐ ( ๐ง ) are now inhomogeneouspolynomials of ๐ โ ( ห ๐ , ห ๐ ) Bargmann = โซ ๐ โ| ๐ | (cid:18) + ๐ โ (cid:205) ๐ผ = | ๐ง ๐ผ | (cid:19) (cid:16) ๐ | ๐ | ( ๐ + ๐ โ ) ๐๐ โง ๐๐ (cid:17) ๐ ( ๐ง ) ๐ ( ๐ง ) ๐ โ (cid:214) ๐ผ = ( ๐ ๐๐ง ๐ผ โง ๐๐ง ๐ผ ) == integrating over ๐, ๐ โผ โซ ๐ ( ๐ง ) ๐ ( ๐ง ) (cid:18) + ๐ โ (cid:205) ๐ผ = | ๐ง ๐ผ | (cid:19) ๐ + ๐ ๐ โ (cid:214) ๐ผ = ( ๐ ๐๐ง ๐ผ โง ๐๐ง ๐ผ ) In other words, if ห ๐ , ห ๐ are homogeneous polynomials of degree ๐ and ๐ , ๐ are their inho-mogeneous counterparts (obtained by setting ๐ง = ( ห ๐ , ห ๐ ) Bargmann = ( ๐ , ๐ ) providedwe de๏ฌne the scalar product in the space of inhomogeneous polynomials of degree โค ๐ as Coherent states, written in inhomogeneous coordinates, are also discussed in [198]. : ( ๐ , ๐ ) = โซ ๐ ( ๐ง ) ๐ ( ๐ง ) (cid:18) + ๐ โ (cid:205) ๐ผ = | ๐ง ๐ผ | (cid:19) ๐ ๐๐ ( ๐ง, ๐ง ) , where ( ๐๐ ) CP ๐ โ โผ (cid:32) + ๐ โ โ๏ธ ๐ผ = | ๐ง ๐ผ | (cid:33) โ ๐ ๐ โ (cid:214) ๐ผ = ( ๐ ๐๐ง ๐ผ โง ๐๐ง ๐ผ ) (C.3)The measure ๐๐ โผ ฮฉ ๐ โ is in fact the volume form on CP ๐ โ , proportional to a powerof the Fubini-Study form (1.16). In the second line of (C.3) one has its expression in theinhomogeneous coordinates.The coherent state | ๐ฃ (cid:105) , when viewed as an inhomogeneous polynomial, will be denoted ๐ ๐ฃ ( ๐ง ) (this notation is borrowed from [43]). For example, if we take the state (2.43) with ๐ =
2, which in Fock space language is ( ๐ฃ ๐ โ + ๐ฃ ๐ โ ) | (cid:105) , the corresponding polynomialwould be ๐ ๐ฃ ( ๐ง ) โผ ( + ๐ฃ๐ง ) , where we have set ๐ฃ : = ๐ฃ ๐ฃ and dropped an overall factor. D Integrability of the complex structure
Here we wish to prove two claims made in section 13.2. The ๏ฌrst one is:If the restriction to ๐ช of the adjoint-invariant metric (cid:104)โข , โข(cid:105) on ๐ฒ ( ๐ ) is Hermitianw.r.t. an almost complex structure J , integrability of J (viewed as an almostcomplex structure on the ๏ฌag manifold) is equivalent to [ ๐ช + , ๐ช + ] โ ๐ช + , [ ๐ช โ , ๐ช โ ] โ ๐ช โ . (D.1)In general the integrability of an almost complex structure means that [ ๐ช + , ๐ช + ] โ ๐ช + โ ๐ฅ .To see this, note that an almost complex structure J is de๏ฌned by the conditions J โฆ ๐ฝ ยฑ = ยฑ ๐ ๐ฝ ยฑ , where ๐ฝ ยฑ are the components of a Maurer-Cartan current: ๐ฝ = โ ๐ โ ๐๐ = ๐ฝ ๐ฅ + ๐ฝ + + ๐ฝ โ , ๐ฝ ยฑ โ ๐ช ยฑ . (D.2)Since ๐๐ฝ โ ๐ฝ โง ๐ฝ =
0, we get ๐๐ฝ โ = (cid:2) โ ๐ฝ โง ๐ฝ + ( terms with ๐ฝ โ ) โ ๐ฝ + โง ๐ฝ + (cid:3) ๐ช โ . From a mathematical standpoint (which uses the Borel-Weil-Bott theorem brie๏ฌy mentioned in sec-tion 2.3.1) ๐ and ๐ are sections of the line bundle O ( ๐ ) over CP ๐ โ . The integrand may be understoodas a scalar product in the ๏ฌber at a given point ๐ง on the base. In that case ( + (cid:205) ๐ โ ๐ผ = | ๐ง ๐ผ | ) โ ๐ plays the role ofa metric in the ๏ฌber, and ( ๐ , ๐ ) is obtained by integrating the ๏ฌber scalar product over all of CP ๐ โ with thenatural measure ๐๐ . J one should have [ ๐ฝ + โง ๐ฝ + ] ๐ช โ =
0, i.e. [ ๐ช + , ๐ช + ] โ ๐ช + โ ๐ฅ .We see that the conditions (D.1) therefore de๏ฌne an integrable complex structure. Con-versely suppose we have an integrable complex structure on ๐บ / ๐ป , and ๐ช ยฑ are its respectiveholomorphic/anti-holomorphic subspaces. Then [ ๐, ๐ ] = ๐ + ๐พ , where ๐, ๐, ๐ โ ๐ช + and ๐พ โ ๐ฅ . Since (cid:104) ๐ช + , ๐ฅ (cid:105) =
0, computing the scalar product with a generic element ๐พ (cid:48) โ ๐ฅ ,we obtain (cid:104) ๐พ (cid:48) , [ ๐, ๐ ](cid:105) = (cid:104) ๐พ (cid:48) , ๐พ (cid:105) . Using the identity (cid:104)[ ๐, ๐พ (cid:48) ] , ๐ (cid:105) + (cid:104) ๐พ (cid:48) , [ ๐, ๐ ](cid:105) =
0, we get (cid:104) ๐พ (cid:48) , ๐พ (cid:105) = โ (cid:104)[ ๐, ๐พ (cid:48) ] , ๐ (cid:105) = (cid:104) ๐ (cid:48) , ๐ (cid:105) , and ๐ (cid:48) = [ ๐พ (cid:48) , ๐ ] โ ๐ช + . The subspace ๐ช + is isotropic, if themetric (cid:104)โข , โข(cid:105) is Hermitian, therefore (cid:104) ๐พ (cid:48) , ๐พ (cid:105) = ๐พ (cid:48) โ ๐ฅ , which implies ๐พ = (cid:104)โข , โข(cid:105) . The result [ ๐ช + , ๐ช + ] โ ๐ช + follows.The second statement used in section 13.2 is:There are exactly ๐ ! acyclic tournament diagrams.The statement implies that there is only one combinatorial type of diagrams, and all acyclictournament diagrams (in such diagrams, by de๏ฌnition, all pairs of nodes are connected) maybe obtained from any one of them by the action of the permutation group ๐ ๐ . Let us describethis combinatorial type. Every acyclic diagram has a โsourceโ-vertex, in which all the linesare outgoing, and a โsinkโ-vertex, in which all lines are incoming (see Fig. 22). Indeed, if thatwere not so, every vertex would contain at least, say, one outgoing line. Then one can startat any vertex and follow outgoing lines, until a loop is formed. Let us consider the โsourceโ-vertex. The diagram formed by the remaining ๐ โ ๐ outgoing lines for all ๐ = , . . . , ๐ โ
1. Thisstatement completely describes the combinatorial structure of the diagram. Equivalently,there is a total ordering on the set of vertices. Di๏ฌerent diagrams di๏ฌer just by a relabelingof the vertices. (A) (B)
Figure 22: (A) The โsinkโ vertex, (B) The โsourceโ vertex .141
Proving the Z ๐ -โsymmetryโ of integrable models In this Appendix we prove the Z ๐ symmetry property of the integrable ๏ฌag manifold models,introduced in section 13.4. The statement is that the e.o.m. of two models, in which thecomplex structures di๏ฌer by a cyclic permutation of the subspaces C ๐ , ยท ยท ยท , C ๐ ๐ , are thesame. In this case the two actions di๏ฌer only by a topological term.Let us call J the standard complex structure, whose holomorphic subspace ๐ช + is givenby upper-block-triangular matrices. Then J = ๐ ( J ) and J = ๐ ( J ) for somepermutations ๐ , ๐ โ ๐ ๐ . We recall the notation ๐ฝ ๐ด๐ต from (1.31). The correspondingKรคhler forms are ๐ = ๐ โ๏ธ ๐ด<๐ต Tr ( ๐ฝ ๐ ( ๐ด ) ๐ ( ๐ต ) โง ๐ฝ ๐ ( ๐ต ) ๐ ( ๐ด ) ) (E.1) ๐ = ๐ โ๏ธ ๐ด<๐ต Tr ( ๐ฝ ๐ ( ๐ด ) ๐ ( ๐ต ) โง ๐ฝ ๐ ( ๐ต ) ๐ ( ๐ด ) ) (E.2)Upon introducing the notation ๐ ( ๐ฝ ๐ด๐ต ) : = ๐ฝ ๐ ( ๐ด ) ๐ ( ๐ต ) , we may write the di๏ฌerence of the twoforms as ๐ โ ๐ = ๐ ๐ (cid:32) โ๏ธ ๐ด<๐ต Tr ( ๐ฝ ๐ด๐ต โง ๐ฝ ๐ต๐ด ) โ โ๏ธ ๐ด<๐ต Tr ( ๐ฝ ๐ โ ( ๐ด ) ๐ โ ( ๐ต ) โง ๐ฝ ๐ โ ( ๐ต ) ๐ โ ( ๐ด ) ) (cid:33) , (E.3)where ๐ โ = ๐ โ ๐ . This reduces the problem to that of J = J and J = ๐ โ ( J ) .Note that the exterior derivative commutes with the permutation ๐ , due to the followingsimple fact following from the Maurer-Cartan equation: ๐ ( ๐๐ฝ ๐ด๐ต ) = ๐ ( (cid:205) ๐ถ ๐ฝ ๐ด๐ถ โง ๐ฝ ๐ถ๐ต ) = (cid:205) ๐ถ ๐ ( ๐ฝ ๐ด๐ถ ) โง ๐ ( ๐ฝ ๐ถ๐ต ) = ๐๐ ( ๐ฝ ๐ด๐ต ) , where to arrive at the last equality one has to make achange of the dummy summation index ๐ถ โ ๐ ( ๐ถ ) .We wish to show that ๐ ( ๐ โ ๐ ) = ๐ is a cyclic permutation. To this endwe rewrite the above di๏ฌerence as follows: ๐ โ ๐ = ๐ ๐ (cid:32)โ๏ธ ๐ด,๐ต ๐ผ ๐ด๐ต Tr ( ๐ฝ ๐ด๐ต โง ๐ฝ ๐ต๐ด ) (cid:33) , (E.4)where ๐ผ ๐ด๐ต = (cid:0) sgn ( ๐ต โ ๐ด ) โ sgn ( ๐ ( ๐ต ) โ ๐ ( ๐ด )) (cid:1) โ {โ , , } . (E.5)(We have made a change of dummy variables ๐ด โ ๐ ( ๐ด ) and ๐ต โ ๐ ( ๐ต ) in the second sumin (E.3)). Closedness of this form requires that (see (B.2)-(B.3)) ๐ผ ๐ด๐ต = (cid:0) sgn ( ๐ต โ ๐ด ) โ sgn ( ๐ ( ๐ต ) โ ๐ ( ๐ด )) (cid:1) = ๐ ๐ด โ ๐ ๐ต . (E.6)Let us consider the case ๐ต > ๐ด . Then ๐ ๐ด = ๐ ๐ต if ๐ ( ๐ต ) > ๐ ( ๐ด ) and ๐ ๐ด = ๐ ๐ต + ๐ ( ๐ต ) < ๐ ( ๐ด ) . This means that { ๐ ๐ด } ๐ด = ...๐ form a non-increasing sequence, and moreover142he di๏ฌerence between any two elements is either zero or 1. This is only possible if the set hasthe form ( ๐, . . . , ๐ ๐พ , ๐ โ , . . . ๐ โ ๐ โ ๐พ ) . Accordingly the original sequence of ๐ consecutivenumbers can be split into two consecutive sets:1 . . . ๐ = ( ๐ผ , ๐ผ ) . (E.7)Since ๐ ( ๐ต ) < ๐ ( ๐ด ) for ( ๐ด โค ๐พ , ๐ต > ๐พ ), the permutation acts as follows: ๐ ( ๐ผ , ๐ผ ) = ( ๐ ( ๐ผ ) , ๐ ( ๐ผ )) . (E.8)Moreover, since ๐ ( ๐ด ) < ๐ ( ๐ต ) for ๐ด < ๐ต โค ๐พ and the image ๐ ( ๐ผ ) is ( ๐ โ ๐พ + , . . . ๐ ) , amomentโs thought shows that ๐ ( ๐ด ) = ๐ โ ๐พ + ๐ด for ๐ด = . . . ๐พ . Analogously ๐ ( ๐ต ) = ๐ต โ ๐พ for ๐ต = ๐พ + . . . ๐ . Therefore ๐ is nothing but a ๐พ -fold cyclic permutation โto the leftโ (or ๐ โ ๐พ -fold to the right).Since for ๐ด < ๐ต the non-zero ๐ผ ๐ด๐ต are the ones, for which ๐ ( ๐ต ) < ๐ ( ๐ด ) , this implies ๐ต = ๐พ + . . . ๐ and ๐ด = . . . ๐พ . These ๐ผ ๐ด๐ต are equal to 1, therefore ๐ โ ๐ = ๐ ๐ (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) โ๏ธ ๐ด = ... ๐พ,๐ต = ๐พ + ... ๐ Tr ( ๐ฝ ๐ด๐ต โง ๐ฝ ๐ต๐ด ) (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) , (E.9)which is easily seen to be proportional to the (generalized) Fubini-Study form on the Grass-mannian ๐บ ๐ฟ,๐ , where ๐ฟ = ๐พ (cid:205) ๐ด = ๐ ๐ด . Conversely, one shows that for a cyclic permutation thedi๏ฌerence between ๐ and ๐ is the closed form written above. F Models with Z ๐ -graded target spaces A homogeneous space ๐บ๐ป is called Z ๐ -graded (or ๐ -symmetric), if the Lie algebra ๐ค of itsisometry group admits the following decomposition: ๐ค = โ ๐ โ ๐ = ๐ค ๐ , [ ๐ค ๐ , ๐ค ๐ ] โ ๐ค ๐ + ๐ mod ๐ , ๐ค = ๐ฅ . (F.1)In this language the ordinary symmetric spaces are 2-symmetric spaces. Similarly to whathappens for symmetric spaces, the e.o.m. of a certain class of ๐ -models with Z ๐ -gradedtarget-spaces may be written as ๏ฌatness conditions of a one-parametric family of connections.These models were introduced in [261] and subsequently studied in [40]. The action has theform (cid:101) S : = โซ ฮฃ ๐ ๐ฅ (cid:107) ๐ ๐ (cid:107) ๐บ + โซ ฮฃ ๐ โ (cid:101) ๐, (F.2)where (cid:101) ๐ is a 2-form constructed using the Z ๐ -decomposition of the Lie algebra (F.1). Notethat, if (cid:101) ๐ were the fundamental Hermitian form, one would obtain precisely the action14313.5). Now we come to the precise de๏ฌnition of (cid:101) ๐ . Decompose the current ๐ฝ = โ ๐ โ ๐๐ according to (F.1): ๐ฝ = โ ๐ โ ๐๐ = ๐ โ โ๏ธ ๐ = ๐ฝ ( ๐ ) , where ๐ฝ ( ๐ ) โ ๐ค ๐ . (F.3)The form (cid:101) ๐ is de๏ฌned as follows: (cid:101) ๐ = ๐ โ โ๏ธ ๐ = ( ๐ โ ๐ ) โ ๐๐ Tr ( ๐ฝ ( ๐ ) โง ๐ฝ ( ๐ โ ๐ ) ) (F.4)This formula raises the following question. According to (F.4), the form (cid:101) ๐ depends on the Z ๐ -grading on the Lie algebra, but generally a given Lie algebra ๐ค may have many di๏ฌerentgradings (with di๏ฌerent, or same, values of ๐ ). The question is: are the models de๏ฌned by(F.2)-(F.4), corresponding to di๏ฌerent gradings of ๐ค , di๏ฌerent?Before answering this question, we review the construction of cyclic gradings on semi-simple Lie algebras [148]. Let us consider, for simplicity, the case of ๐ค = ๐ ๐ข ( ๐ ) . A cyclicgrading may be constructed as follows : one picks a system of ๐ โ ๐ผ , . . . ๐ผ ๐ โ , as well as the maximal negative root ๐ผ ๐ = โ ๐ผ โ . . . โ ๐ผ ๐ โ . Then one assignsto these ๐ roots arbitrary (non-negative integer) gradings ๐ , . . . ๐ ๐ โ , ๐ ๐ . The gradings ofall other roots are determined by the Lie algebra structure, and the value of ๐ is calculatedas ๐ = ๐ + . . . + ๐ ๐ . (F.5)In usual matrix form, this grading looks as follows: (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ๐ ๐ . . . ๐ ๐ โ ๐ ๐ (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) (F.6) Here we restrict ourselves to the grading of type ๐ด ( ) ๐ โ . In the paper of Kac [148] the roots ๐ผ , . . . ๐ผ ๐ are seen as the positive simple roots of the correspondinga๏ฌne Lie algebra (cid:98) ๐ด ๐ โ . Consider the case ๐ =
3. The simple positive roots of the loop algebra ๐ ๐ข ( )( ๐ก, ๐ก โ ) may be chosen as follows: ๐ผ = (cid:169)(cid:173)(cid:171) (cid:170)(cid:174)(cid:172) , ๐ผ = (cid:169)(cid:173)(cid:171) (cid:170)(cid:174)(cid:172) , ๐ผ = ๐ก (cid:169)(cid:173)(cid:171) (cid:170)(cid:174)(cid:172) . In this context the latter root ๐ผ โ the analog of ๐ผ ๐ โ is customarily called โimaginaryโ. In fact, the whole theoryof cyclic Lie algebra gradings is formulated by Kac naturally in terms of a๏ฌne Lie algebras and their Dynkindiagrams. ๐ค , which determines the denominator ๐ป of the quotient space ๐บ / ๐ป , isdetermined by those ๐ ๐ โs, which are zero. For example, if all ๐ ๐ >
0, the resulting space isthe manifold of complete ๏ฌags ๐๐ ( ๐ ) ๐ ( ๐ ( ) ๐ ) .In general, for a choice of grading determined by the set ๐ , . . . , ๐ ๐ some of the subspaces ๐ค ๐ will be identically zero. Therefore a natural restriction to adopt is to require that ๐ค ๐ โ ๐ ( mod ๐ ) . We will call such a grading admissible . This still leaves a wide range ofpossibilities. For example, in the case of ๐๐ ( ) the following is a complete list of admissiblegradings (up to the action of the Weyl group ๐ ): Z : (cid:169)(cid:173)(cid:171)
10 0 (cid:170)(cid:174)(cid:172) , Z : (cid:169)(cid:173)(cid:171)
22 0 (cid:170)(cid:174)(cid:172) , (cid:169)(cid:173)(cid:171)
10 0 (cid:170)(cid:174)(cid:172) , (F.7) Z : (cid:169)(cid:173)(cid:171)
23 0 (cid:170)(cid:174)(cid:172) , Z : (cid:169)(cid:173)(cid:171)
34 0 (cid:170)(cid:174)(cid:172) , (cid:169)(cid:173)(cid:171)
24 0 (cid:170)(cid:174)(cid:172) , (F.8) Z : (cid:169)(cid:173)(cid:171)
35 0 (cid:170)(cid:174)(cid:172) , Z : (cid:169)(cid:173)(cid:171)
36 0 (cid:170)(cid:174)(cid:172) (F.9)The Z -grading and the second Z -grading correspond to the homogeneous space SU ( )/ ๐ ( U ( )ร U ( )) = CP , and all other gradings correspond to the ๏ฌag manifold F , , .We will now give an answer to the question posed above: what is the relation betweenthe ๐ -models with the action (F.2), taken for di๏ฌerent gradings on the corresponding Liealgebra? Our statement is [66]:For homogeneous spaces of the unitary group, the models (F.2)-(F.4) with di๏ฌerent ๐ด ( ) ๐ โ -type gradings on ๐ค are classically equivalent to the model (13.5) with somechoice of complex structure on the target-spaceIn fact, one has a precise statement about the relation of the ๐ต -๏ฌelds in the two models.To formulate it, we โsolveโ the constraint (F.5) as follows : ๐ ๐ = ๐ ๐ โ ๐ ๐ + , (F.10)where ๐ ๐ are integers and ๐ ๐ + โก ๐ โ ๐ . We then have (see [66] for a proof): (cid:101) ๐ = ๐ โ ๐ โ๏ธ ๐ = ๐ ๐ ๐ ๐๐ฝ ๐๐ , (F.11)where ๐ฝ ๐๐ are the diagonal components of the Maurer-Cartan current. We see that, irrespectiveof the choice of the grading (which is now encoded in the integers ๐ ๐ ), the form (cid:101) ๐ di๏ฌers Formula (F.10) implies that the cyclic automorphism (cid:98) ๐ of the Lie algebra, which de๏ฌnes the Z ๐ grading,can be represented as follows: (cid:98) ๐ ( ๐ ) = ๐๐๐ โ , where ๐ = diag ( ๐ ๐๐ ๐ ๐ , . . . , ๐ ๐๐ ๐๐๐ ) . Z ๐ -graded spaces, the two classes of target spacesโ Z ๐ -graded and complex homogeneous spaces โ do not coincide. For example, one has thespace ๐บ ๐๐ ( ) (cid:39) ๐ . The stability subgroup ๐๐ ( ) acts on the tangent space ๐ช = R via ๐ โ ๐ ,where ๐ (cid:39) C is the standard representation. Therefore it has a unique almost complexstructure, which is not integrable (see the review [14]). On the other hand, it is a nearlyKรคhler manifold and is Z -graded [61]. On the other side of the story, one has the complexmanifold ๐ ร ๐ (cid:39) ๐ ( ) (see [66] for a discussion), which may be viewed as a T -bundleover CP (the simplest ๏ฌag manifold). This manifold is not a Z ๐ -graded homogeneous spaceof the group ๐บ = ๐ ( ) .We also note that the construction of Lax connections for models with Z ๐ -graded spaceswas explored in [40]. The relation to the Lax connections of section 13.1 has been recentlyestablished in [92] (this is an extension to Z ๐๐