Flag manifold sigma models: spin chains and integrable theories
FFlag manifold sigma models
Spin chains and integrable theories
Ian Affleck and Kyle Wamer
Department of Physics and Astronomy and Stewart Blusson Quantum Matter Institute,University of British Columbia, Vancouver, B.C., Canada, V6T1Z1 iaffl[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 flag manifold targetspaces, which are generalizations of the familiar CP 𝑛 − 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 ’t Hooft anomalies in the continuum theory. Wealso discuss integrable flag 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 flow) 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 first applications . . . . . . . . . . . . 71. The geometry of SU( 𝑛 ) flag manifolds . . . . . . . . . . . . . . . . . . . . . . . 71.1. The flag manifold as a homogeneous space . . . . . . . . . . . . . . . . . 81.2. Symplectic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3. Kähler structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4. Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5. General (non-Kähler) metrics and 𝐵 -fields on the flag manifold . . . . . . 162. Flag manifolds and elements of representation theory . . . . . . . . . . . . . . . 172.1. Mechanical particle in a non-Abelian gauge field . . . . . . . . . . . . . . 182.2. ‘Quantization’ of the symplectic form on flag manifolds . . . . . . . . . . 202.3. Schwinger-Wigner quantization . . . . . . . . . . . . . . . . . . . . . . . 232.4. Holstein-Primakoff 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-Affleck Theorem (LSMA) Theorem . . . . . . . . . . 374.2. Affleck-Kennedy-Lieb-Tasaki (AKLT) Constructions . . . . . . . . . . . . 395. Flavour Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406. Derivation of the continuum theory . . . . . . . . . . . . . . . . . . . . . . . . 426.1. The quantum sphere 𝑆 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.2. Path integral for the spin chain . . . . . . . . . . . . . . . . . . . . . . . . 446.3. Ferromagnetic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467. The antiferromagnetic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.1. Alternating representations . . . . . . . . . . . . . . . . . . . . . . . . . . 477.2. The large- 𝑛 limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.3. Symmetric representations and the flag manifold as the space of Néel vacua:SU ( ) case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518. The continuum limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.1. The expansion around the “vacuum” configuration . . . . . . . . . . . . . 539. Symmetric representations: the general case . . . . . . . . . . . . . . . . . . . . 579.1. The flag manifold sigma model from an SU ( 𝑛 ) spin chain . . . . . . . . . 579.2. Z 𝑛 symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589.3. Velocity Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . 5910. Generalized Haldane conjectures and ’t Hooft anomaly matching . . . . . . . . . 6110.1. SU ( 𝑛 ) Haldane conjectures . . . . . . . . . . . . . . . . . . . . . . . . . 6210.2. Derivation of the mixed ’t Hooft Anomaly . . . . . . . . . . . . . . . . . . 6410.3. The Z 𝑛 anomaly in a PSU ( 𝑛 ) background . . . . . . . . . . . . . . . . . . 6710.4. Examples of PSU ( 𝑛 ) -bundles . . . . . . . . . . . . . . . . . . . . . . . . 6910.5. Relation to the WZNW model . . . . . . . . . . . . . . . . . . . . . . . . 7211. A gas of fractional instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7311.1. Squashing to the 𝑋𝑌 -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 flag manifold sigma models and beyond . . . . . . . . . . 8413. The models and the zero-curvature representation . . . . . . . . . . . . . . . . . 8513.1. The zero-curvature representation . . . . . . . . . . . . . . . . . . . . . . 8713.2. Complex structures on flag manifolds . . . . . . . . . . . . . . . . . . . . 8913.3. Symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9213.4. Dependence on the complex structure: Z 𝑚 -symmetry of the models . . . . 9414. Relation to 4D Chern-Simons theory . . . . . . . . . . . . . . . . . . . . . . . . 9514.1. The ‘semi-holomorphic’ 4D Chern-Simons theory . . . . . . . . . . . . . 9714.2. The gauged linear sigma model and the 𝛽𝛾 -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 𝛽 -function and the Ricci flow . . . . . . . . . . . . . . . . . . . . . . 11016.4. Sigma models with polynomial interactions . . . . . . . . . . . . . . . . . 11416.5. Integrable models related to quiver varieties . . . . . . . . . . . . . . . . . 116 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119A. Kähler 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 𝑚 -‘symmetry’ of integrable models . . . . . . . . . . . . . . . . . 142F. Models with Z 𝑚 -graded target spaces . . . . . . . . . . . . . . . . . . . . . . . 1432 ntroduction Haldane’s 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 fluctuations of the antiferromagnet are governed by the O(3) nonlinear sigmamodel, with topological angle 𝜃 = 𝜋𝑆 . 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 𝜃 = 𝜋 . Recently, this paradigm of mappingspin chains to relativistic quantum field theories has been generalized to SU( 𝑛 ) chains invarious representations [69, 73, 161, 236, 237, 234]. For chains that have a rank- 𝑝 symmetricrepresentation at each site, the corresponding field theory is a sigma model with target spaceSU ( 𝑛 )/[ U ( )] 𝑛 − . This space is an example of a flag manifold, which generalizes the familiarnotions of complex projective space and Grassmannian manifolds, and in this case may beparametrized in terms of 𝑛 mutually orthonormal fields 𝑧 𝐴 ∈ C 𝑛 . To each of these fieldsthere is an associated topological angle 𝜃 𝐴 = 𝜋 𝑝 𝐴 / 𝑛 , which extends Haldane’s originalresult, since 𝑝 = 𝑆 in SU(2). Based on this sigma model formulation, a generalization ofHaldane’s conjecture was discovered for these SU( 𝑛 ) chains: When 𝑝 is coprime with 𝑛 ,gapless excitations will be present above the ground state; for all other values of 𝑝 , a finiteenergy gap will occur, with a ground state degeneracy equal to 𝑛 / gcd ( 𝑛, 𝑝 ) .The arguments leading to this SU( 𝑛 ) version of Haldane’s conjecture draw from manyareas of mathematical physics. This reflects the fact that the underlying flag manifoldSU ( 𝑛 )/[ U ( )] 𝑛 − has a rich geometric structure. Indeed, flag manifolds in their own right area fascinating subject, and for this reason we commence this review in Chapter 1 by discussinggeneric flag manifolds at great length. In particular, we will explain their symplectic, Kähler,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 flag manifolds, this chapter will allow us to introduce the necessarilytechnology to properly explain the mathematical underpinnings of deriving a flag manifoldsigma model from an SU( 𝑛 ) spin chain. Along these lines, we also review various quantizationschemes of flag manifolds, and how a coherent state path integral is constructed in this context.In Chapter 2, we turn to SU( 𝑛 ) spin chains. In the interest of being self-contained, webegin by introducing the SU( 𝑛 ) 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 ( 𝑛 )/[ U ( )] 𝑛 − flag manifold arises as a low-energy3igma model description of the SU( 𝑛 ) chain. In particular, we show how the topologicalangle 𝜃 𝐴 arises as the coefficient of a Fubini-Study two-form, pulled back from CP 𝑛 − tothe flag manifold. We then proceed to discuss a technical issue that is related to the absenceof Lorentz invariance when starting with a generic SU( 𝑛 ) chain Hamiltonian. Having donethis, we may then finally review the constituent arguments that make up the SU( 𝑛 ) Haldaneconjecture. In particular, we discuss the notion of‘t Hooft anomaly matching, which is relatedto the inability of gauging the physical PSU( 𝑛 ) symmetry of the chain while maintaining adiscrete Z 𝑛 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 𝑝 with gcd ( 𝑛, 𝑝 ) = 𝑛 ) that may also be mapped to the same flag manifold,SU ( 𝑛 )/[ U ( )] 𝑛 − .One might expect that this would be a natural point to conclude this review: We havecovered the general properties of flag manifolds, and explained in great detail the relationshipbetween said manifolds and SU( 𝑛 ) spin chains, allowing for a generalization of Haldane’sfamed conjecture. However, this work on SU( 𝑛 ) chains has very recently initiated an entirelynew research program, related to integrable flag manifold sigma models. This is the subjectof Chapter 3.The history of integrable models with an ‘infinite number of degrees of freedom’ 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 finite-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 theclassification of classical solutions were obtained substantially later [228, 132]. See [112,128] for a review of these findings.Whereas the classical integrability theory quickly came to be part of mathematics, thequantum theory was developed by rather different methods by physicists, starting with thefamous conjecture for the 𝑆 -matrix in the 𝑆 𝑛 − sigma model [264]. The development ofthis theory then went in two directions: towards the calculation of the spectrum in finitevolume, 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 ( 𝑛 ) symmetry, most importantly for the CP 𝑛 − -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 firstconstructed by Lüscher [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 ‘higher’ 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 – bothclassical and quantum – 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 flag 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 unified framework for constructing classicalintegrable models starting from a rather exotic ‘four-dimensional semi-holomorphic Chern-Simons theory’. In particular, the flag 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 𝛽 -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 simplifications, which ultimately allow oneto solve the generalized Ricci flow equations for the deformed geometries in a very wideclass of sigma models. This is particularly important, since in the study of models with targetspaces 𝑆 and 𝑆 , 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 𝜃 -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 reflect in thisreview article. 5 otation Before we begin, we comment on the various notational choices that we have made in thisreview. ◦ A generic flag manifold can be embedded into a copy of 𝑚 Grassmanians (this willbe explained in detail in Chapter 1). We use upper case Roman letter to index thesecopies. In the case 𝑚 = 𝑛 , each Grassmanian is CP 𝑛 − , which we parametrize with 𝑧 𝐴 ∈ C 𝑛 . When 𝑚 < 𝑛 , and multiple 𝑛 -component fields are required to parametrizethe Grassmanians, we use lower-case roman letters, i.e. 𝑧 ( 𝑘 ) 𝐴 . ◦ The 𝑛 components of 𝑧 𝐴 are indexed using lower case greek letters: 𝑧 𝛼𝐴 ◦ Often, we will normalize the 𝑛 -component fields to satisfy | 𝑧 𝐴 | =
1. In this case, wewrite 𝑢 𝐴 instead of 𝑧 𝐴 . ◦ We use the labels 𝑎, 𝑏, 𝑐... for discrete time coordinates, and the labels 𝑖, 𝑗 , 𝑘, ... fordiscrete spatial coordinates. For a field 𝑧 that is a function of 𝑎 and 𝑗 , we write 𝑧 = 𝑧 ( 𝑎, 𝑗 ) . When the continuum limit is taken, we write 𝑧 = 𝑧 ( 𝜏, 𝑥 ) . ◦ The vector complex conjugate to 𝑧 is written 𝑧 . We write inner products in C 𝑛 accordingto 𝑤 ◦ 𝑧 = 𝑛 ∑︁ 𝛼 = 𝑤 𝛼 𝑧 𝛼 . (0.1)The norm of a vector is denoted by | 𝑧 | , so that | 𝑧 | = 𝑧 ◦ 𝑧 . ◦ From time to time we will be using the notation Hom ( C 𝑝 , C 𝑞 ) (linear maps from C 𝑝 to C 𝑞 ) for the space of 𝑝 × 𝑞 -matrices. This notation makes it clear that 𝑝 is the numberof columns, and 𝑞 the number of rows in a matrix. Accordingly End ( C 𝑝 ) is the spaceof square matrices of size 𝑝 . 6 hapter 1. Flag manifolds: geometry and first applications In the first chapter of this review we recall the main facts about the rather rich geometricstructures on flag manifolds (mostly symplectic structures and metrics), and we explainhow flag manifolds naturally arise in representation theory. Due to this tight relation, flagmanifolds 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 aflag manifold. 𝑛 ) flag manifolds Flag manifolds are natural generalizations of both projective spaces and Grassmanians, sowe start by recalling these more familiar entities first.The complex projective space CP 𝑛 − is defined as the space of 𝑛 -tuples of complexnumbers, which are not all zero, defined up to multiplication by an overall factor, i.e. ( 𝑧 , · · · , 𝑧 𝑛 ) ∼ λ ( 𝑧 , · · · , 𝑧 𝑛 ) . Another interpretation, which allows for generalizations moreeasily, is that CP 𝑛 − is the space of lines in C 𝑛 , passing through the origin. Clearly, the lineis defined by a non-zero vector ( 𝑧 , · · · , 𝑧 𝑛 ) , and two vectors that differ by an overall constantmultiple define the same line.This construction can be generalized by considering 𝑘 -dimensional planes in C 𝑛 , passingthrough the origin. This leads to the notion of a Grassmannian 𝐺𝑟 𝑘,𝑛 , which may be definedas the space of 𝑘 × 𝑛 -matrices 𝑍 of rank 𝑘 , taken up to multiplication by matrices fromGL ( 𝑘, C ) . The meaning of 𝑍 is that it comprises 𝑘 vectors spanning a given 𝑘 -dimensionalplane in C 𝑛 , and multiplication by GL ( 𝑘, C ) amounts to a change of basis and does not affectthe plane itself. Setting 𝑘 = CP 𝑛 − .We should point out that the equivalence relations just mentioned – the quotients w.r.t. C ∗ or GL ( 𝑘, C ) – are of course well-known in physics as ‘gauge redundancies’. In fact,more than once in our narrative we will encounter the formulation of the corresponding fieldtheories as systems with gauge fields (the so-called ‘gauged linear sigma models’). From themathematical perspective, choosing a gauge amounts to picking coordinates on the respectivemanifolds. The unrestricted coordinates ( 𝑧 , · · · , 𝑧 𝑛 ) mentioned above, which are subject tothe equivalence relation, are known as the homogeneous coordinates on the projective space.If, say, 𝑧 ≠
0, then by a C ∗ scaling – a gauge transformation – we may set 𝑧 =
1. This fixes7he gauge freedom completely at the expense of effectively excluding from considerationthe part of the space where 𝑧 =
0. The corresponding coordinates ( , 𝑧 , · · · , 𝑧 𝑛 ) are thenknown as the inhomogeneous coordinates . For example, on a sphere 𝑆 (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 fix the gauge redundancy partiallyby normalizing the coordinates 𝑛 (cid:205) 𝛼 = | 𝑧 𝛼 | =
1, so that the coordinates are restricted to asphere 𝑆 𝑛 − . This leaves the freedom of multiplying all 𝑧 𝛼 ’s by the same phase, so theremaining gauge group is U ( ) . This formulation is nothing but the celebrated Hopf fibration 𝑆 𝑛 − → CP 𝑛 − , with fiber U ( ) . Its advantage is that the global symmetry group SU ( 𝑛 ) isexplicitly maintained. Similar choices of homogeneous, inhomogeneous and other types ofcoordinates may be performed for Grassmannians and flag manifolds as well. Both of the above examples may be concisely formulated as ‘spaces of subspaces’ 0 ⊂ 𝐿 ⊂ C 𝑛 ,where 𝐿 is a linear subspace of a given dimension. This naturally leads to the notion of aflag. A flag in C 𝑛 is the sequence of nested subspaces0 ⊂ 𝐿 ⊂ . . . ⊂ 𝐿 𝑚 − ⊂ 𝐿 𝑚 = C 𝑛 (Flag) (1.1)of given dimensions dim 𝐿 𝐴 = 𝑑 𝐴 . Accordingly, the flag manifold in C 𝑛 may be defined asthe manifold of such nested linear complex subspaces : F ( 𝑑 , . . . , 𝑑 𝑚 ) = { ⊂ 𝐿 ⊂ . . . ⊂ 𝐿 𝑚 − ⊂ 𝐿 𝑚 = C 𝑛 } . (1.2) - Figure 1: The parabolic subgroup, stabilizing a flag.The next important fact is that the projective space, Grassmannians and flag 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 ( 𝑛, C ) , or w.r.t. itsunitary subgroup U( 𝑛 ) ⊂ GL ( 𝑛, C ) . Let us first start with the complex parametrization.The group GL( 𝑛, C ) acts transitively on the space of flags of a given type: given two flags, Reviews of the mathematical properties of flag manifolds include [19, 28]. 𝑑 𝑚 − into each other, then the next-to-largest subspaces, etc. The stabilizer of any given flag is a so-called parabolic subgroup(‘a staircase’), consisting of matrices, depicted in Fig. 1. The reason for the non-diagonalstructure is that, given a basis of 𝐿 𝐴 and a (larger) basis of 𝐿 𝐴 + 𝐵 , adding vectors of the firstbasis to vectors of the second one produces a new basis of the same sequence of spaces 𝐿 𝐴 ⊂ 𝐿 𝐴 + 𝐵 . Therefore we can view the flag manifold as a homogeneous space F ( 𝑑 , . . . , 𝑑 𝑚 ) = GL ( 𝑛, C )/H (1.3)of complex dimension dim C F ( 𝑑 , . . . , 𝑑 𝑚 ) = 𝑛 − 𝑚 (cid:205) 𝐴 = 𝑑 𝐴 ( 𝑑 𝐴 − 𝑑 𝐴 − ) .As mentioned earlier, there is a second – unitary – parametrization of the flag manifold.To obtain it, one picks a metric in the ambient space C 𝑛 and orthogonalizes the subspaces ofthe flag. For example, we split 𝐿 = 𝐿 ⊕ ( 𝐿 ) ⊥ , 𝐿 = 𝐿 ⊕ ( 𝐿 ) ⊥ and so on. Altogether thissplits C 𝑛 into 𝑚 mutually orthogonal subspaces of dimensions 𝑛 𝐴 = 𝑑 𝐴 − 𝑑 𝐴 − , 𝐴 = , . . . , 𝑚 (where we set 𝑑 = F 𝑛 ,...,𝑛 𝑚 = U ( 𝑛 ) U ( 𝑛 ) × . . . × U ( 𝑛 𝑚 ) , 𝑚 ∑︁ 𝐴 = 𝑛 𝐴 = 𝑛 . (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 𝑛 the complete flagmanifold, i.e. the manifold (1.4), where all 𝑛 𝐴 = J on the flag 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 firsttwo chapters, in order not to dwell on this subtle issue, we will simply assume that we haveboth definitions 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 flagmanifold target spaces. Such models feature two main ingredients: the metric G and theskew-symmetric two-form Ω (which is also called the 𝐵 -field, 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 𝑑 Ω =
0. These do not affect 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 flag9anifolds. To this end, note that the condition 𝑑 Ω =
0, together with an additional non-degeneracy assumption Ω dim C F ≠ Ω ≠ symplectic form.If, in addition, Ω is positive in a certain sense, then Ω is called a Kähler form. ‘Positivity’means that the corresponding symmetric tensor G = − Ω ◦ J , obtained by contracting Ω with a complex structure J , is positive-definite and therefore a Riemannian metric on theflag manifold . To summarize we have the following embeddings:Kähler forms ⊂ Symplectic forms ⊂ 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 flag manifold of type (1.4) they all have real dimension 𝑚 −
1, and restrictingto non-degenerate forms, or positive forms, amounts to simple relations among the 𝑚 − We start by describing symplectic forms on F 𝑛 ,...,𝑛 𝑚 , i.e. non-degenerate, closed 2-forms 𝜔 ,with 𝑑𝜔 =
0. Since we will mainly be interested in SU( 𝑛 )-invariant models in what follows,we will accordingly restrict ourselves to SU( 𝑛 )-invariant symplectic forms. The main toolthat we will use is the theorem of Kirillov-Kostant that coadjoint orbits of a Lie group 𝐺 admit natural symplectic forms (for a review see [152]). In our applications the Lie algebra of 𝐺 =SU( 𝑛 ) 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 defined as follows: one picks a diagonal element 𝑝 = Diag ( 𝑝 𝑛 , . . . , 𝑝 𝑚 𝑛 𝑚 ) ∈ 𝔰𝔲 ( 𝑛 ) ,where the 𝑝 𝐴 ’s are distinct and Tr ( 𝑝 ) =
0. In this case the flag manifold is the orbit F 𝑛 ,...,𝑛 𝑚 = { 𝑔 𝑝 𝑔 − , 𝑔 ∈ SU ( 𝑛 )} , (1.6)since there is an obvious gauge invariance 𝑔 → 𝑔 · ℎ , where ℎ ∈ 𝐻 = S ( U ( 𝑛 ) × · · · × U ( 𝑛 𝑚 )) ,so that the orbit is really the quotient 𝐺𝐻 . Introducing the Maurer-Cartan current 𝑗 = − 𝑔 − 𝑑𝑔, 𝑔 ∈ U ( 𝑛 ) , (1.7)one may write the Kirillov-Kostant symplectic form on the orbit (1.6) as Ω = Tr ( 𝑝 𝑗 ∧ 𝑗 ) . (1.8)One can check that its non-degeneracy is equivalent to the condition that all 𝑝 𝐴 ’s are distinct.Due to the condition Tr ( 𝑧 ) = 𝑚 − Technically for the tensor G so defined to be symmetric one also needs that Ω is of type ( , ) , i.e. aHermitian form. This always holds in our applications, and we will not elaborate on this aspect further. 𝔰𝔲 ( 𝑛 ) . Moreover, this embedding may be identifiedwith the image of the moment map 𝜇 = 𝑔 𝑝 𝑔 − . (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 𝚽 with an action of a Lie group 𝐺 on it thatpreserves the symplectic form, one can construct Hamiltonian functions for the action of thisgroup. The action of the group on 𝚽 is generated by the vector fields 𝑣 𝑎 , 𝑎 = · · · dim 𝐺 ,whose commutators satisfy the Lie algebra relations of 𝔤 : [ 𝑣 𝑎 , 𝑣 𝑏 ] = 𝑓 𝑐𝑎𝑏 𝑣 𝑐 . To each vectorfield 𝑣 𝑎 one can put in correspondence a Hamiltonian function ℎ 𝑎 . It turns out that all ofthese Hamiltonian functions may be collected in a single matrix-valued object 𝜇 ∈ 𝔤 , calledthe moment map , in such a way that ℎ 𝑎 = Tr ( 𝜇𝑇 𝑎 ) . Here 𝑇 𝑎 is the 𝑎 th generator of 𝔤 . Onecan check from the definitions that the moment map defined in (1.9) leads to the vector fieldsgenerating the action of SU ( 𝑛 ) and preserving the symplectic form (1.8), cf. [73].As a simple exercise, let us write out explicitly the moment map for the Grassmannian 𝐺𝑟 𝑠,𝑛 . To this end we set 𝑝 = Diag ( , · · · , 𝑠 , , · · · , 𝑛 − 𝑠 ) − 𝑠𝑛 𝑛 , which gives 𝜇 𝑠 = 𝑠 ∑︁ 𝑘 = 𝑢 ( 𝑘 ) ⊗ 𝑢 ( 𝑘 ) − 𝑠𝑛 𝑛 , (1.10)where by 𝑢 ( 𝑘 ) we have denoted the (orthonormal) column vectors of the group element 𝑔 .The moment map (1.9) is the classical analogue of the SU ( 𝑛 ) -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 Kähler forms. As explainedearlier, these involve the complex structure J in their definition, so we will shift to thecomplex definition of flag manifolds (1.3). The Kähler structures can be characterizedgeometrically in at least two equivalent ways: ◦ As parameters of a linear combination of the so-called quasipotentials [31, 32] thatappear in the physics literature in [36]. ◦ As Fayet-Iliopoulos parameters related to the gauged linear sigma model representationsfor flag manifolds [186, 96].These approaches are discussed below in sections 1.3.1, 1.3.2 respectively.11 .3.1 Explicit Kähler metrics on flag manifolds
Recall that Kähler metrics and Kähler forms are in one-to-one correspondence, and are relatedby contraction with a complex structure J . It is easiest to define a Kähler metric through theso-called Kähler potential K , which in plain terms is a function of the complex coordinates { 𝑤 𝑎 , 𝑤 𝑎 } , such that the line element takes the form 𝑑𝑠 = (cid:205) 𝜕 K 𝜕𝑤 𝑎 𝜕𝑤 𝑏 𝑑𝑤 𝑎 𝑑𝑤 𝑏 . A very directway of constructing a Kähler potential of the most general SU ( 𝑛 ) -invariant Kähler metric onthe flag manifold (1.4) is as follows: consider the matrix 𝑊 = ( 𝑤 , .., 𝑤 𝑛 ) ∈ GL ( 𝑛 ; C ) , (1.11)where each 𝑤 𝑖 is a column vector. We also define an 𝑛 × 𝑑 𝐴 -matrix 𝑊 𝐴 of rank 𝑑 𝐴 bytruncating the matrix 𝑊 to the first 𝑑 𝐴 columns: 𝑊 𝐴 = ( 𝑤 , ..., 𝑤 𝑑 𝐴 ) , where 𝑑 𝐴 = 𝐴 ∑︁ 𝑙 = 𝑛 𝑙 . (1.12)The columns of 𝑊 𝐴 span the vector space 𝐿 𝐴 in the flag (1.1). Next we introduce the function 𝑡 𝐴 = det (cid:16) 𝑊 † 𝐴 𝑊 𝐴 (cid:17) . (1.13)One can check that log ( 𝑡 𝐴 ) , called the quasipotential, is the Kähler potential for the 𝜋 -normalized canonical metric on the Grassmannian 𝐺𝑟 𝑑 𝐴 ,𝑛 . The potential of an arbitrarySU ( 𝑛 ) invariant Kähler metric on the flag manifold [31, 32] may then be written as K F = 𝑚 − ∑︁ 𝐴 = 𝛾 𝐴 log ( 𝑡 𝐴 ) , 𝛾 𝐴 > . (1.14)For a detailed discussion of the geometric properties of these metrics (including the specialcase of Kähler-Einstein metrics) cf. [18, 4].As a simplest application of formula (1.14) let us consider the case when the flag manifoldis the complex projective space CP 𝑛 − . In this case 𝑊 is a column vector, and we label itscomponents 𝑧 , . . . , 𝑧 𝑛 . The Kähler potential is therefore (we set 𝛾 = K CP 𝑛 − = ( 𝑧 ◦ 𝑧 ) . (1.15)The resulting Kähler form is the familiar Fubini-Study form: Ω 𝐹𝑆 = i 𝑧 ◦ 𝑧 (cid:18) 𝑑𝑧 𝛼 ∧ 𝑑𝑧 𝛼 − 𝑧 𝛼 𝑑𝑧 𝛼 ∧ 𝑧 𝛽 𝑑𝑧 𝛽 𝑧 ◦ 𝑧 (cid:19) , ∫ CP Ω 𝐹𝑆 = 𝜋 . (1.16)In the second formula the integral is taken over a CP ⊂ CP 𝑛 − defined by the equations 𝑧 𝛼 = , 𝛼 >
2. We will frequently use the 2 𝜋 -normalized Fubini-Study form later on in ournarrative. This is the same normalization as that of the Fubini-Study metric on CP 𝑛 − , i.e. the volume of a holomorphic2-sphere generating 𝐻 ( 𝐺𝑟 𝑑 𝐴 ,𝑛 , Z ) is 𝜋 . 𝑈 𝑚 − 𝑈 𝑚 − C 𝑛 𝐿 𝐿 𝐿 𝑚 − 𝐿 𝑚 − · · · (1.17)Figure 2: The quiver describing the flag manifold as a Kähler 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 ∗ , and theGrassmannians as a quotient by GL ( 𝑘, C ) , but no similar presentation was provided for thecase of flag 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 𝑛 − , where the correspondinggroup is certainly different from C ∗ . A suitable formulation for flag manifolds, however, doesexist, and may be formulated in terms of a so-called ‘quiver’. The quiver in question has thefollowing form:Here 𝐿 𝐴 are the vector spaces defining the flag (1.2), so that dim 𝐿 𝐴 = 𝑑 𝐴 , and eacharrow corresponds to the space of matrices Hom ( 𝐿 𝐴 , 𝐿 𝐴 + ) , with 𝑈 𝐴 being the (linear)complex coordinates in this space. At each circular node there is an action of the gauge groupGL ( 𝑑 𝐴 , C ) . The main idea is that the flag manifold may be identified 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 flag 𝐿 ⊂ C 𝑛 . To understandwhy this can be true, consider the case of complete flags in C , i.e. the manifold U ( ) U ( ) . Oneway to parametrize this manifold is as follows. Let 𝑙, 𝑝 ∈ C be two linearly independentvectors.These vectors define a plane 𝐿 = Span ( 𝑙, 𝑝 ) (cid:39) C ⊂ C . (1.18)A line 𝐿 ⊂ 𝐿 may be defined as 𝐿 = Span ( 𝑢 𝑙 + 𝑢 𝑝 ) ⊂ 𝐿 ⊂ C (1.19)with ( 𝑢 , 𝑢 ) a fixed non-zero two-vector.Clearly, ( 𝑢 , 𝑢 ) ∈ C , 𝑙 ∈ C , 𝑝 ∈ C uniquely define a given flag 𝐿 ⊂ 𝐿 ⊂ C ,however the map is not one-to-one. Indeed, the rotated set (cid:18) (cid:101) 𝑢 (cid:101) 𝑢 (cid:19) = λ 𝑔 − ◦ (cid:18) 𝑢 𝑢 (cid:19) , (cid:16) (cid:101) 𝑙 (cid:101) 𝑝 (cid:17) = (cid:0) 𝑙 𝑝 (cid:1) ◦ 𝑔 ngle Figure 3: Parametrization of the flag manifold introduced in (1.18)-(1.19).with 𝑔 ∈ GL ( , C ) and λ ∈ C ∗ defines the same flag. Therefore one has the gauge group G = C ∗ × GL ( , C ) acting on the ‘matter fields’ constituting the linear space 𝑉 = ( C ) 𝑢 ⊕( C ⊗ C ) 𝑙,𝑝 . To make a connection to the quiver (1.17), we identify 𝑈 = (cid:18) 𝑢 𝑢 (cid:19) and 𝑈 = (cid:0) 𝑙 𝑝 (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 Kähler metrics on the flagmanifold 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 𝑚 − 𝛾 𝐴 used in (1.14). The readerwill find the details in Appendix A. It was already emphasized in the diagram (1.5) that Kähler and symplectic structures provideexamples of closed two-forms. Such forms are elements of the second cohomology group 𝐻 (F 𝑛 , ··· ,𝑛 𝑚 , 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 coefficients: 𝐻 (F 𝑛 , ··· ,𝑛 𝑚 , Z ) = Z 𝑚 − . (1.20)One can obtain a convenient model for this cohomology group if one notes the existence ofan embedding F 𝑛 ,...,𝑛 𝑚 ↩ → 𝐺𝑟 𝑛 ,𝑛 × · · · × 𝐺𝑟 𝑛 𝑚 ,𝑛 (1.21)of the flag manifold into a product of Grassmannians. Indeed, a point in a flag manifold is acollection of pairwise orthogonal planes of dimensions 𝑛 , · · · , 𝑛 𝑚 (see section 1.1), each ofwhich is a point in the corresponding Grassmannian.To proceed, we will need the definition of a Lagrangian submanifold M ⊂ N in asymplectic manifold (N , 𝜔 ) , which we now recall. M is Lagrangian if 𝜔 (cid:12)(cid:12) M = M = dim N . Let us now consider N = 𝐺𝑟 𝑛 ,𝑛 × · · · × 𝐺𝑟 𝑛 𝑚 ,𝑛 as a symplectic manifoldwith a product symplectic form 𝜔 = 𝑚 (cid:205) 𝐴 = 𝜔 𝐴 , where all 𝜔 𝐴 are normalized in the same way.In this case, as we shall now prove, M = F 𝑛 ,...,𝑛 𝑚 in (1.21) is a Lagrangian submanifold, i.e. 𝜔 (cid:12)(cid:12) F 𝑛 ,...,𝑛𝑚 = . (1.22)Identifying Ω 𝐴 = 𝜔 𝐴 (cid:12)(cid:12) F 𝑛 ,...,𝑛𝑚 and taking into account (1.22), we obtain the relation 𝑚 ∑︁ 𝐴 = Ω 𝐴 = . (1.23)The cohomology group 𝐻 (F 𝑛 , ··· ,𝑛 𝑚 , Z ) is then described as the quotient 𝐻 (F 𝑛 , ··· ,𝑛 𝑚 , Z ) = Z [ Ω , · · · , Ω 𝑚 ] (cid:44) (cid:32) 𝑚 ∑︁ 𝐴 = Ω 𝐴 (cid:33) (1.24)To prove that the flag manifold is a Lagrangian submanifold in the product of Grassman-nians, first let us perform a dimensionality check. Using dim R (F 𝑛 , ··· ,𝑛 𝑚 ) = 𝑁 − 𝑚 (cid:205) 𝐴 = 𝑛 𝐴 anddim R ( 𝐺𝑟 𝑠,𝑛 ) = ( 𝑠 · 𝑛 − 𝑠 ) , we obtaindim R (cid:32) 𝑚 (cid:214) 𝐴 = 𝐺𝑟 𝑛 𝐴 ,𝑛 (cid:33) = 𝑚 ∑︁ 𝐴 = ( 𝑛 𝐴 · 𝑁 − 𝑛 𝐴 ) = (cid:32) 𝑛 − 𝑚 ∑︁ 𝐴 = 𝑛 𝐴 (cid:33) = · dim R F 𝑛 , ··· ,𝑛 𝑚 (1.25)We see that the dimensions match correctly. For the rest we use the following fact (whichis easy to prove starting from the definition): if 𝜇 is the moment map for the action of agroup 𝐺 , the restriction of a symplectic form to a 𝐺 -orbit in 𝜇 − ( ) vanishes. We will nowconstruct a moment map for the diagonal action of SU ( 𝑛 ) on the product of Grassmanniansand prove that 𝜇 − ( ) is the flag manifold under consideration. The moment map for a singleGrassmannian was written out in (1.10), so now we sum over all Grassmannians to obtain 𝜇 = 𝑚 ∑︁ 𝐴 = 𝜇 𝑛 𝐴 = 𝑚 ∑︁ 𝐴 = (cid:32) 𝑛 𝐴 ∑︁ 𝑘 = 𝑢 ( 𝑘 ) 𝐴 ⊗ 𝑢 ( 𝑘 ) 𝐴 (cid:33) − 𝑛 . (1.26)In this formula the vectors 𝑢 ( 𝑘 ) 𝐴 inside the same group 𝐴 are orthonormal: 𝑢 ( 𝑘 ) 𝐴 ◦ 𝑢 ( 𝑘 (cid:48) ) 𝐴 = 𝛿 𝑘 𝑘 (cid:48) .On the other hand, it is easy to convince oneself that the set 𝜇 − ( ) is composed of 𝑛 -tuplesof orthogonal 𝑢 -vectors. It follows that the 𝑢 -vectors representing different 𝑛 𝐴 -dimensionalplanes in C 𝑛 ( 𝐴 = · · · 𝑚 ) are mutually orthogonal as well. The set of such orthogonalsubspaces is precisely the flag manifold F 𝑛 , ··· ,𝑛 𝑚 .15efore concluding this section, let us specialize these results to the case that we willencounter most frequently below, namely the case of the complete flag manifold, when all 𝑛 𝐴 =
1. The second cohomology group of the complete flag manifold is 𝐻 (F 𝑛 , Z ) = Z 𝑛 − , (1.27)hence there exist 𝑛 − 𝐻 (F 𝑛 , Z ) .In order not to repeat ourselves, let us consider here a slightly different model for 𝐻 (F 𝑛 , Z ) .On F 𝑛 there are 𝑛 standard line bundles L , · · · , L 𝑛 , and their sum is a trivial bundle: 𝑛 ⊕ 𝐴 = L 𝐴 = F 𝑛 × C 𝑛 . (1.28)The first Chern classes of these bundles are represented by 𝑛 closed 2-forms: [ Ω 𝐴 ] = 𝑐 (L 𝐴 ) , 𝐴 = · · · 𝑛 . Due to the condition (1.28) and the additivity of the first Chern classes 𝑐 ( 𝐸 ⊕ 𝐹 ) = 𝑐 ( 𝐸 ) + 𝑐 ( 𝐹 ) it is clear that the forms Ω 𝐴 are not independent but rather satisfythe relation 𝑛 ∑︁ 𝐴 = [ Ω 𝐴 ] = Ω 𝐴 ( 𝐴 = · · · 𝑛 ) satisfyingthe relation (1.29), generate 𝐻 (F 𝑛 , Z ) . Higher cohomology groups of general flag 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 flag manifolds. One reason for this is that flag manifolds are simplyconnected, 𝜋 (F 𝑛 , ··· ,𝑛 𝑚 ) =
0, which implies 𝜋 (F 𝑛 , ··· ,𝑛 𝑚 ) (cid:39) 𝐻 (F 𝑛 , ··· ,𝑛 𝑚 , Z ) (cid:39) Z 𝑚 − byHurewicz theorem, so that the two notions coincide in dimension two. In higher dimensionsthis is no longer the case. For example, 𝐻 (F 𝑛 , ··· ,𝑛 𝑚 , Z ) (cid:39)
0, whereas for complete flagmanifolds 𝜋 (F 𝑛 ) (cid:39) Z . The latter is a higher-dimensional generalization of the Hopf invariant 𝜋 ( 𝑆 ) (cid:39) Z and leads to the existence of topologically non-trivial Hopfion solutions [22, 23]relevant for the Faddeev-Niemi model [105] (see also [84]). 𝐵 -fields on the flag manifold So far we have discussed SU ( 𝑛 ) -invariant closed forms on flag manifolds, as well as therelated question of invariant Kähler metrics. This is not the end of the story, however, as on ageneral flag manifold (1.4) there will be large families of invariant metrics, and typically onlya small subfamily corresponds to Kähler metrics. Moreover, the metrics that will actuallyenter the sigma models that we discuss in Chapters 2 and 3, are in general not Kähler. In asimilar way, the 𝐵 -fields also come in large families and are not required to be topological ingeneral, as on a general flag manifold there exist invariant two-forms that are not closed.To construct the general metric and 𝐵 -field, we denote the flag manifold SU ( 𝑛 ) S ( U ( 𝑛 )×···× U ( 𝑛 𝑚 )) as 𝐺𝐻 and introduce the corresponding Lie algebra decomposition 𝔤 = 𝔥 ⊕ 𝔪 . Since [ 𝔥 , 𝔪 ] ⊂ , the subgroup 𝐻 is represented in the space 𝔪 , and this representation may be decomposedinto irreducibles: 𝔪 C = ⊕ 𝐴 ≠ 𝐵 𝑉 𝐴𝐵 , where 𝑉 𝐴𝐵 = C 𝑛 𝐴 𝑛 𝐵 . (1.30)The space 𝑉 𝐴𝐵 of 𝑛 𝐴 × 𝑛 𝐵 -matrices is the vector space of the bi-fundamental representationof the group U ( 𝑛 𝐴 ) × U ( 𝑛 𝐵 ) , and moreover 𝑉 𝐵𝐴 = 𝑉 𝐴𝐵 . We decompose the Maurer-Cartancurrent 𝑗 = − 𝑔 − 𝑑𝑔, 𝑔 ∈ U ( 𝑛 ) , entering (1.8) accordingly: 𝑗 = [ 𝑗 ] 𝔥 + [ 𝑗 ] 𝔪 = [ 𝑗 ] 𝔥 + ∑︁ 𝐴 ≠ 𝐵 𝑗 𝐴𝐵 , 𝑗 𝐴𝐵 ∈ 𝑉 𝐴𝐵 . (1.31)The most general invariant two-form may then be written as Ω = ∑︁ 𝐴<𝐵 𝑏 𝐴𝐵 Tr ( 𝑗 𝐴𝐵 ∧ 𝑗 𝐵𝐴 ) . (1.32)Using the zero-curvature equation for 𝑗 , one can check that Ω is closed if and only if 𝑏 𝐴𝐵 = 𝑝 𝐴 − 𝑝 𝐵 , in which case it is exactly the symplectic form (1.8) (see Appendix B). Quiteanalogously, the line element of the most general metric is 𝑑𝑠 = − ∑︁ 𝐴<𝐵 𝑎 𝐴𝐵 Tr ( 𝑗 𝐴𝐵 · 𝑗 𝐵𝐴 ) , (1.33)where for positivity we have to require 𝑎 𝐴𝐵 >
0. We conclude that there are 𝑚 ( 𝑚 − ) realparameters defining the most general metric, as well as 𝑚 ( 𝑚 − ) additional parameters definingthe most general 𝐵 -field.As discussed earlier, the space of Kähler metrics is an ( 𝑚 − ) -dimensional subspace inthe full space of metrics. In order to formulate the corresponding condition on the coefficients 𝑎 𝑖 𝑗 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 Kähler. It is the so-called normal, or reductive, metric (cf. [29]), with line element 𝑑𝑠 = − Tr ( [ 𝑗 ] 𝔪 ) , which corresponds to 𝑎 𝐴𝐵 = 𝐴, 𝐵 . This metric is not a Kählermetric, unless the flag manifold is a Grassmannian (i.e. unless 𝑚 = Now that we are done with some formal aspects, we wish to present the first example of awell-known physical situation where flag manifolds naturally arise. Incidentally this makesa neat connection to the applications of flag 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 field
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 field 𝐴 𝜇 . The actionhas the form S = ∫ 𝑑𝑡 G 𝜇𝜈 (cid:164) 𝑥 𝜇 (cid:164) 𝑥 𝜈 − ∫ 𝐴 = ∫ 𝑑𝑡 (cid:18) G 𝜇𝜈 (cid:164) 𝑥 𝜇 (cid:164) 𝑥 𝜈 − 𝐴 𝜇 (cid:164) 𝑥 𝜇 (cid:19) . (2.1)The question is, how do we write an analogous action for the case when the gauge field isnon-Abelian, or, simply speaking, when it has additional gauge indices 𝐴 𝛼𝛽𝜇 . 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) 𝑛 ∈ 𝑆 = CP that couples tothe SU ( ) gauge field (cid:174) 𝐴 . More generally, the degrees of freedom associated to the ‘internalspin’ take values in a certain flag manifold, corresponding to the representation in which theparticle transforms. In other words, one should enlarge the phase space of the mechanicalsystem [215]: 𝑇 ∗ M → 𝑇 ∗ M × F . (2.2)Here M is the configuration space, 𝑇 ∗ M is the cotangent bundle (i.e. phase space), and F is the flag 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 𝑛 𝑖 appearing in (1.4). We will see below that this choice is related to the different families ofrepresentations under which the particle transforms.We start by rewriting the standard action of a particle in first-order form: S = ∫ 𝑑𝑡 (cid:18) 𝑝 𝜇 (cid:164) 𝑥 𝜇 − G 𝜇𝜈 𝑝 𝜇 𝑝 𝜈 − 𝐴 𝜇 (cid:164) 𝑥 𝜇 (cid:19) . (2.3)Upon enlarging the phase space we can analogously write down the non-Abelian action asfollows ( 𝐴 is assumed Hermitian, and H is the Hamiltonian): S = ∫ 𝑝 𝜇 𝑑𝑥 𝜇 − ∫ 𝑑𝑡 H ( 𝑥, 𝑝 ) + ∫ ( 𝜃 − Tr ( 𝐴 𝜇 )) . (2.4)Here 𝜃 is the canonical (Poincaré-Liouville) one-form, defined by the condition 𝑑𝜃 = Ω ( = the symplectic form on F ) , (2.5)and 𝜇 is the moment map for the action of the group 𝐺 on F . The integral ∫ 𝜃 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 𝜃 is defined up to the addition of a total derivative, 𝜃 → 𝜃 + 𝑑ℎ , but the difference only affects the boundary terms in the action. In the case ofperiodic boundary conditions one may even write ∫ Γ 𝜃 = ∫ 𝐷 Ω , (2.6)18here 𝐷 is a disc, whose boundary is the curve Γ : 𝜕 𝐷 = Γ . 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 𝑛 − . Later, we will see that this corresponds to the particletransforming in the defining representation of SU ( 𝑛 ) . Let us normalize the homogeneouscoordinates on CP 𝑛 − : 𝑛 ∑︁ 𝛼 = | 𝑢 𝛼 | = . (2.7)One still has the remaining gauge group U ( ) , which acts by multiplication of all coordinates 𝑢 𝛼 by a common phase. The Fubini-Study form (1.16) on CP 𝑛 − may be simplified if oneuses the above normalization: Ω 𝐹𝑆 = 𝑖 𝑑𝑢 𝛼 ∧ 𝑑𝑢 𝛼 . (2.8)Then we have the following expressions for 𝜃 and 𝜇 : 𝜃 = 𝑖 𝑢 𝛼 𝑑𝑢 𝛼 , 𝜇 = 𝑢 ⊗ 𝑢 − 𝑛 𝑛 . This expression for the moment map is a special case of (1.10). The part of the actioncorresponding to the motion in the ‘internal’ space (in this case the projective space) has theform (cid:101) S = − ∫ 𝑑𝑡 𝑢 𝛼 ( 𝑖 (cid:164) 𝑢 𝛼 + ( 𝐴 𝜇 ) 𝛼𝛽 (cid:164) 𝑥 𝛼 𝑢 𝛽 ) , (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 𝑢 → 𝑔 ( 𝑥 ( 𝑡 )) ◦ 𝑢 𝐴 𝜇 → 𝑔 𝐴 𝜇 𝑔 − − 𝑖 𝜕 𝜇 𝑔 𝑔 − . (2.10)To make it even more obvious, we note that the exterior derivative of the one-form 𝜃 − Tr ( 𝐴 𝜇 ) (viewed as a form on the enlarged phase space (2.2)) produces a two-form, which is explicitlygauge-invariant: 𝑑 ( 𝜃 − Tr ( 𝐴 𝜇 )) = 𝑖 D 𝑢 𝛼 ∧ D 𝑢 𝛼 − Tr ( 𝐹 𝜇 ) , (2.11) D 𝑢 = 𝑑𝑢 − 𝑖 𝐴 𝑢, D 𝑢 = 𝑑𝑢 + 𝑖 𝑢 𝐴, 𝐹 = 𝑑𝐴 − 𝑖 𝐴 ∧ 𝐴 .
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 we’ve written down a gauge-invariant action for a particle coupled to a non-Abeliangauge field, let us next write out the equations of motion on the flag manifold, F . Throughout the review we will be mostly using the variable 𝑧 to denote unconstrained complex coordinates,such as the homogeneous or inhomogeneous coordinates on CP 𝑛 − , and the variable 𝑢 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 = 𝑆 . Instead of using the spinor ( 𝑢 , 𝑢 ) ∈ C , we canparametrize F in a more standard way with the help of a unit vector (cid:174) 𝑛 ∈ R . The equationsof motion then take the form (cid:164)(cid:174) 𝑛 = (cid:174) 𝐴 × (cid:174) 𝑛, where (cid:174) 𝐴 = { 𝐴 𝑎𝜇 (cid:164) 𝑥 𝜇 } 𝑎 = , , (2.12)is a vector of components of the gauge field in the basis of Pauli matrices. We see that theequations are linear in (cid:174) 𝑛 , and the condition (cid:174) 𝑛 = 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 𝔤 with basis { 𝑇 𝑎 } we can introduce a variable 𝑛 = (cid:205) 𝑛 𝑎 𝑇 𝑎 ∈ 𝔤 , and the equations will then take the form (cid:164) 𝑛 = [ 𝐴 𝜇 (cid:164) 𝑥 𝜇 , 𝑛 ] , (2.14)or, in terms of the variables 𝑛 𝑎 , (cid:164) 𝑛 𝑎 = 𝑓 𝑎𝑏𝑐 ( 𝐴 𝜇 (cid:164) 𝑥 𝜇 ) 𝑏 𝑛 𝑐 , (2.15)where 𝑓 𝑎𝑏𝑐 are the structure constants of 𝔤 . It is in this form that this system of equationswas discovered in [257]. The motion defined by these equations in reality takes place on flagmanifolds embedded in 𝔤 , since the ‘Casimirs’ 𝐶 𝐽 = Tr ( 𝑛 𝐽 ) , 𝐽 = , , . . . (2.16)are integrals of motion of the system (2.14), and specifying the Casimirs is effectively thesame as specifying the parameter 𝑝 of the orbit (1.6). We have thus established a connectionwith the formulation through flag 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 ( 𝑝, 𝑥 ) is standard, so the non-trivial question is howto quantize the spin phase space F – the flag manifold. In the case of SU(2), this willlead to the notion of spin quantization, i.e., that the particle transforms under some definiterepresentation of SU(2), labeled by a single integer.One of the approaches to quantization is related to considering path integrals of the form ∫ (cid:214) 𝑗, 𝑡 𝑑𝜑 𝑗 ( 𝑡 ) 𝑒 𝑖 S (2.17) Another approach to the quantization of coadjoint orbits, which is also based on the path integral, wasdeveloped in [17]. 𝜑 𝑖 parametrize F . The subtlety comes fromthe fact that the connection 𝜃 is not a globally-defined one-form on the flag manifold. Indeed,let us consider the simplest case of F = CP = 𝑆 . The most general invariant symplecticform is as follows : Ω = 𝑝 𝜗 𝑑𝜗 ∧ 𝑑𝜙. (2.18)Here 𝑝 is an arbitrary constant, and 𝜗, 𝜙 are the standard angles on the sphere.Since the action S entering the exponent in (2.17) involves a term ∫ 𝜃 , where 𝜃 is aconnection satisfying 𝑑𝜃 = Ω , a standard argument familiar from Wess-Zumino-Novikov-Witten theory [239, 189, 245] leads to the requirement that the coefficient 𝑝 is quantizedaccording to ∫ Ω ∈ 𝜋 Z . Let us recall the argument. To start with, we write a one-form 𝜃 ,well-defined on the northern hemisphere, such that 𝑑𝜃 = Ω : 𝜃 𝑁𝐻 = 𝑝 sin (cid:18) 𝜗 (cid:19) 𝑑𝜙 . (2.19)It is well-defined at the north pole, 𝜗 =
0, since at that point the prefactor of 𝑑𝜙 vanishes. OnFigure 4: To apply Stokes’ theorem to the integral ∫ Γ 𝜃 , we choose a disc with boundary Γ .There are two ways to do this, which lead to the domains 𝐷 , 𝐷 ⊂ 𝑆 . One has ∫ Γ 𝜃 𝑁𝐻 = ∫ 𝐷 Ω , ∫ Γ 𝜃 𝑆𝐻 = − ∫ 𝐷 Ω = − ∫ 𝑆 Ω + ∫ Γ 𝜃 𝑁𝐻 (since in order to use Stokes’ theorem, one has topick a one-form that is well-defined in the interior of the domain). Since ∫ 𝑆 Ω = 𝜋 , and thechoice of north/south poles was arbitrary, the integral ∫ Γ 𝜃 is only well-defined modulo 2 𝜋 .the other hand, at 𝜗 = 𝜋 it remains constant. Another way to see this is to introduce the usualround metric on the sphere 𝑑𝑠 = 𝑑𝜗 + sin 𝜗 𝑑𝜙 and to calculate the norm of the differential 𝜃 : (cid:107) 𝜃 (cid:107) = 𝑝 sin (cid:16) 𝜗 (cid:17) 𝜗 . One sees that it is bounded at 𝜃 = 𝜃 = 𝜋 . If one It can be also written in the form
Ω = − 𝑝 𝑑𝑧 ∧ 𝑑𝜙 , where 𝑧 = cos 𝜗 is the 𝑧 -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. 𝜃 as a connection on a line bundle, it is nevertheless well-defined, as on the southernhemisphere we may define a gauge-transformed 𝜃 𝑆𝐻 = 𝜃 𝑁𝐻 − 𝑝 𝑑𝜙 = − 𝑝 cos (cid:16) 𝜗 (cid:17) 𝑑𝜙 , whichis well-behaved at 𝜃 = 𝜋 . Therefore the integral ∫ Γ 𝜃 depends on which formula for theconnection we take, 𝜃 𝑆𝐻 or 𝜃 𝑁𝐻 , the difference being equal to 2 𝜋 𝑝 : ∫ Γ 𝜃 𝑆𝐻 = ∫ Γ 𝜃 𝑁𝐻 − 𝜋 𝑝 (see Fig. 4). If 𝑝 ∈ Z , however, the quantity 𝑒 𝑖 ∫ 𝜃 is defined unambiguously. We say that | 𝑝 | labels the representation of SU(2) under which the particle transforms, and 𝑠 = | 𝑝 | is calledthe ‘spin’ of the particle.Let us turn to the flag manifolds of SU( 𝑛 ), with 𝑛 >
2. Now there are multiple contours Γ 𝑖 that must be considered, corresponding to the hemispheres of homologically distinct spheres 𝐶 𝑖 ∈ 𝐻 (F , Z ) in F . We require that each of the terms 𝑒 𝑖 ∫ Γ 𝑖 𝜃 is well defined, i.e. ∫ 𝐶 𝑖 Ω ∈ 𝜋 Z for every 2-cycle 𝐶 𝑖 ∈ 𝐻 (F , Z ) . (2.20)These quantization conditions correspond to particular representations of SU( 𝑛 ). Let usconstruct these 2-cycles explicitly for the case when F is a complete flag manifold F 𝑛 . Lateron, we can analyze the remaining (smaller) flag manifolds by use of a forgetful projection.The manifold F 𝑛 can be parametrized using 𝑛 orthonormal vectors 𝑢 𝐴 , 𝐴 = . . . 𝑛 , 𝑢 𝐴 ◦ 𝑢 𝐵 = 𝛿 𝐴𝐵 , defined modulo phase transformations: 𝑢 𝐴 ∼ 𝑒 𝑖𝛼 𝐴 𝑢 𝐴 . As we showed insections 1.2 and 1.5, the most general symplectic form on F 𝑛 may be written as follows: Ω = 𝑖 ∑︁ 𝐴<𝐵 ( 𝑝 𝐴 − 𝑝 𝐵 ) 𝑗 𝐴𝐵 ∧ 𝑗 𝐵𝐴 , where 𝑗 𝐴𝐵 = 𝑢 𝐴 ◦ 𝑑𝑢 𝐵 (2.21)To construct the cycles 𝐶 𝐴 , note that if one fixes 𝑛 − 𝑛 lines defined by the vectors 𝑢 , . . . , 𝑢 𝑛 , the remaining free parameters define the configuration space of ordered pairs ofmutually orthogonal lines, passing through the origin and laying in a plane, orthogonal to the 𝑛 − CP : { 𝑢 𝐴 , . . . , 𝑢 𝐴 𝑛 − are fixed , 𝑢 𝐴 𝑛 − , 𝑢 𝐴 𝑛 ∈ ( 𝑢 𝐴 , . . . , 𝑢 𝐴 𝑛 − ) ⊥ are mutually orthogonal and otherwise generic } (cid:39) ( CP ) 𝐴 𝑛 − ,𝐴 𝑛 . Let us now fix a permutation ( 𝐴 , . . . , 𝐴 𝑛 ) in such a way that the 𝑝 𝐴 𝑖 form a non-increasing sequence, i.e. 𝑝 𝐴 𝑖 ≥ 𝑝 𝐴 𝑗 for 𝑖 < 𝑗 . The fact that Ω is non-degenerate requires thatthis sequence is actually strictly decreasing. In this case the rearrangement of 𝑝 ’s amounts tochoosing a complex structure on F 𝑛 , but we will not dwell on this fact here (see Section 13.2for details). After such a permutation we may choose ( CP ) 𝐴 𝑖 ,𝐴 𝑖 + as a basis in the homologygroup 𝐻 (F 𝑛 , 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 𝑛 . ( CP ) 𝐴𝑖,𝐴𝑖 + Ω = 𝑝 𝐴 𝑖 − 𝑝 𝐴 𝑖 + ∈ 𝜋 Z + , 𝑖 = . . . 𝑛 − . (2.22)In order for the value of the integral to be an integer, one should choose 𝑝 𝐴 in the form ( 𝑝 , . . . , 𝑝 𝑛 ) = 𝜆 ( , . . . , ) + ( (cid:101) 𝑝 , . . . , (cid:101) 𝑝 𝑛 ) , 𝜆 ∈ R , (cid:101) 𝑝 𝑖 ∈ Z . (2.23)This freedom in adding a vector 𝜆 ( , . . . , ) allows us to work with values { (cid:101) 𝑝 𝐴 } that sum tozero. According to the general theory of adjoint orbits (see Section 1.2), the flag manifoldunder consideration is then the orbit of the element 𝑝 = (cid:169)(cid:173)(cid:173)(cid:171) (cid:101) 𝑝 ... . . . ... (cid:101) 𝑝 𝑛 (cid:170)(cid:174)(cid:174)(cid:172) ∈ 𝔰𝔲 ( 𝑛 ) . (2.24)Let us observe what happens when some of these variables (cid:101) 𝑝 𝐴 coincide. On the one hand, the2-form Ω now becomes degenerate. On the other, we see that the corresponding adjoint orbitis no longer the complete flag manifold, F 𝑛 . For example, if there are only two distinct valuesof (cid:101) 𝑝 𝑖 , i.e. we have 𝑝 𝑠 = Diag ( , . . . , 𝑠 , , . . . , 𝑛 − 𝑠 ) − 𝑠𝑛 𝑛 , so that the corresponding adjoint orbitis the Grassmannian 𝐺𝑟 𝑠,𝑛 . This demonstrates the point that we alluded to earlier, namelythat the flag manifold F encoding the degrees of freedom of a particle coupled to an SU( 𝑛 )gauge field is not uniquely determined by 𝑛 for 𝑛 >
2. Indeed, choosing different values of 𝑝 𝐴 leads to different flag manifolds. In such cases when F is strictly smaller than F 𝑛 , onemay view the (degenerate) 2-form (2.21) on the complete flag manifold as a non-degenerateform on the smaller F . This amounts to a forgetful projection. The general theory that wehave described is nothing but ‘geometric quantization’ for the case of flag manifolds.The canonical quantization of the system given by the action 𝑆 = ∫ 𝜃 will be treated indetail in the next section and, as we shall see, the non-negative integers 𝑝 𝐴 are equal to thelengths of the rows of the Young diagram characterizing a given representation of 𝔰𝔲 ( 𝑛 ) . Forthis to make sense, one should choose 𝜆 in (2.23) in such a way that 𝑝 𝑛 = Having discussed the quantization of the symplectic form Ω on F , we are now ready tocanonically quantize the flag manifold (i.e. the action corresponding to the ‘internal space’).To see how this works, let us first canonically quantize CP 𝑛 − , with action given in (2.9).Instead of working with normalized 𝑢 𝑖 , we first write the kinetic term of the Lagrangian as L = 𝑖 𝑛 ∑︁ 𝛼 = 𝑧 𝛼 ◦ (cid:164) 𝑧 𝛼 , (2.25)23nd impose the normalization constraint in the form 𝑛 ∑︁ 𝛼 = | 𝑧 𝛼 | = 𝑝 . (2.26)Therefore the canonical momentum is 𝜋 𝛼 = 𝜕 L 𝜕 (cid:164) 𝑧 𝛼 = 𝑖𝑧 𝛼 , which leads to the algebra { 𝑧 𝛼 , 𝑧 𝛽 } = 𝑝 𝛿 𝛼𝛽 . 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 𝜏 𝑎 are a set of SU ( 𝑛 ) generators in the fundamental representation. Introduce 𝑛 operators 𝑎 𝛼 and their conjugates 𝑎 † ,𝛼 with the canonical commutation relations [ 𝑎 𝛼 , 𝑎 † ,𝛽 ] = 𝛿 𝛼𝛽 . (2.27)One can easily check that the operators 𝑆 𝑎 = 𝑎 † ,𝛼 𝜏 𝑎𝛼𝛽 𝑎 𝛽 , (2.28)satisfy the commutation relations of 𝔰𝔲 ( 𝑛 ) , and 𝑆 𝑎 act irreducibly on the subspace of the fullFock space specified by the condition 𝑛 ∑︁ 𝛼 = 𝑎 † ,𝛼 𝑎 𝛼 = 𝑝, (2.29)where 𝑝 is a positive integer representing the ‘number of particles’. For a given 𝑝 therepresentation one obtains is the 𝑝 -th symmetric power of the fundamental representation.Now let us turn to a general flag manifold, with the kinetic term ∫ 𝜃 , where 𝑑𝜃 = Ω isthe symplectic form (2.21). Let us rewrite it as follows: Ω = − 𝑖 ∑︁ 𝐴,𝐵 ( 𝑝 𝐴 − 𝑝 𝐵 ) 𝑢 𝐴 ◦ 𝑑𝑢 𝐵 ∧ 𝑢 𝐵 ◦ 𝑑𝑢 𝐴 = − 𝑖 ∑︁ 𝐴,𝐵 𝑝 𝐴 𝑢 𝐴 ◦ 𝑑𝑢 𝐵 ∧ 𝑢 𝐵 ◦ 𝑑𝑢 𝐴 = (2.30) = 𝑖 ∑︁ 𝐴,𝐵 𝑝 𝐴 𝑑𝑢 𝐴 ◦ 𝑢 𝐵 ∧ 𝑢 𝐵 ◦ 𝑑𝑢 𝐴 = using completeness = 𝑑 (cid:32) 𝑖 ∑︁ 𝐴 𝑝 𝐴 𝑢 𝐴 ◦ 𝑑𝑢 𝐴 (cid:33) Defining 𝑧 𝐴 = √ 𝑝 𝐴 𝑢 𝐴 , we may therefore set 𝜃 = 𝑖 ∑︁ 𝐴 𝑧 𝐴 ◦ 𝑑𝑧 𝐴 , 𝑧 𝐴 ◦ 𝑧 𝐵 = 𝑝 𝐴 𝛿 𝐴𝐵 . (2.31)Each 𝑝 𝐴 corresponds to the ‘number of particles’ of a particular species. The canonicalquantization procedure then gives [ 𝑎 𝛼𝐴 , 𝑎 𝛽𝐵 ] = 𝛿 𝛼𝛽 𝛿 𝐴𝐵 . (2.32)24n other words, we introduce 𝑛 creation-annihilation operators 𝑎 𝛼𝐴 for each of the 𝑝 𝐴 . Then 𝑝 𝐴 ∈ Z ≥ is the occupation number of the 𝐴 -th line of the Young tableau. The shift 𝑝 𝐴 → 𝑝 𝐴 +
1, which is inessential according to the above discussion, corresponds to addinga column to a Young diagram of full length. The differences 𝑝 𝐴 − 𝑝 𝐴 + are the Dynkinlabels of the representation (which are the coefficients 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 ‘symplectic form’ degenerates, which signals that one should pass toa smaller flag manifold. This is consistent with our discussion in the previous subsection.The next point is that the mutual orthogonality of the 𝑧 𝐴 s should be reflected in theoperators 𝑎 𝛼𝐴 in some way. To illustrate this, let us consider the SU ( ) adjoint representation.Let us label the six creation-anniliation operators as 𝑎 𝛼 , 𝑏 𝛼 (three for each non-zero row), sothat the 𝔰𝔲 ( 𝑛 ) generators look as follows 𝑆 𝑎 = 𝑎 † ,𝛼 𝜏 𝑎𝛼𝛽 𝑎 𝛽 + 𝑏 † ,𝛼 𝜏 𝑎𝛼𝛽 𝑏 𝛽 , (2.33)where 𝜏 𝑎 are the 𝑛 × 𝑛 -generators in the defining representation. To model this representationon a subspace of the Fock space 𝐹 , we build the operators 𝑁 = 𝑎 † , 𝑎 + 𝑎 † , 𝑎 + 𝑎 † , 𝑎 , 𝑁 = 𝑏 † , 𝑏 + 𝑏 † , 𝑏 + 𝑏 † , 𝑏 (2.34) 𝑂 = 𝑎 † , 𝑏 + 𝑎 † , 𝑏 + 𝑎 † , 𝑏 (2.35)and require the vectors | 𝜓 (cid:105) ∈ 𝐹 , on which the representation is built to satisfy 𝑁 | 𝜓 (cid:105) = | 𝜓 (cid:105) , 𝑁 | 𝜓 (cid:105) = | 𝜓 (cid:105) , 𝑂 | 𝜓 (cid:105) = . (2.36)The values of 𝑁 and 𝑁 correspond to the number of boxes in the first and second rows ofthe Young diagram (i.e. they are ‘number operators’ that count the number of particles ofspecies 𝑎 and 𝑏 , respectively). Notice that the classical condition 𝑎 ◦ 𝑏 = 𝑂 | 𝜓 (cid:105) = 𝑂 † | 𝜓 (cid:105) =
0. Indeed, the two equations would be incompatible,since [ 𝑂 , 𝑂 † ] = 𝑁 − 𝑁 and ( 𝑁 − 𝑁 ) | 𝜓 (cid:105) = | 𝜓 (cid:105) ≠
0. This asymmetry is the same onethat is already present in the Young diagram.Let us now explain how this generalizes to SU( 𝑛 ). We introduce 𝑛 creation operators 𝑎 † ,𝛼𝐴 for each row 𝐴 of the Young diagram ( 𝐴 = O 𝐴𝐵 | 𝜓 (cid:105) ≡ 𝑎 † 𝐴 ◦ 𝑎 𝐵 | 𝜓 (cid:105) = 𝐴 < 𝐵. (2.37)This is a compatible set of equations, since the operators O 𝐴𝐵 satisfy the algebra [O 𝐴𝐵 , O 𝐶𝐷 ] = 𝛿 𝐵𝐶 O 𝐴𝐷 − 𝛿 𝐴𝐷 O 𝐶𝐵 where 𝐴 < 𝐵, 𝐶 < 𝐷. (2.38)The operators O 𝐴𝐵 may be thus thought of as the positive roots of the Lie algebra 𝔰𝔲 ( 𝑛 ) . InChapter 2, this algebra will reappear in the context of SU( 𝑛 ) spin operators.25he constraint (2.37) may be solved rather explicitly. More exactly, we are looking for thejoint kernel of the operators O 𝐴𝐵 , 𝐴 < 𝐵 acting on states in the ( 𝑁 , 𝑁 , . . . ) -particle Fockspace: | Ψ (cid:105) = 𝐴 𝛼 ...𝛼 𝑁 | 𝛽 ...𝛽 𝑁 | ... 𝑎 † ,𝛼 . . . 𝑎 † ,𝛼 𝑁 𝑏 † ,𝛽 . . . 𝑏 † ,𝛽 𝑁 . . . | (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 𝑎, 𝑏, 𝑐, · · · . For example: 𝑎 𝑎 𝑎𝑏 𝑏𝑐
2. For each column build antisymmetric combinations of the form ∑︁ 𝜎 (−) 𝜎 𝑎 † ,𝜎 ( 𝑖 ) 𝑏 † ,𝜎 ( 𝑗 ) 𝑐 † ,𝜎 ( 𝑘 ) 𝑑 † ,𝜎 ( 𝑙 ) , where the number of letters participating is equal to the height of the column.3. Multiply these antisymmetric combinations (the number of ‘particles’ of type 𝐴 willbe precisely equal to the length of the 𝐴 -th row in the Young diagram). To see thatthese are annihilated by operators O 𝐴𝐵 , 𝐴 < 𝐵 , note that the action of this operatorremoves the 𝐵 -th letter and replaces it by the 𝐴 -th letter, and since the 𝐵 -th letter for 𝐵 > 𝐴 always enters in skew-symmetric combinations with the 𝐴 -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 – the Borel-Weil-Bott theorem –asserts that, given a representation 𝑉 of a group 𝐺 , one can construct a holomorphic linebundle L 𝑉 over a suitable flag manifold of 𝐺 (in full generality one can take the manifoldof complete flags), such that 𝑉 may be reconstructed as the space of holomorphic sections 𝐻 (L 𝑉 ) of L 𝑉 . These holomorphic sections are polynomials, and indeed it is elementary tofind a map from the space of states (2.39) to the space of polynomials – 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 first Chern class that is represented by the symplecticform Ω (2.30), through which the kinetic term in the action standing in the path integral isdefined. In other words, [ Ω ] = 𝑐 (L 𝑉 ) ∈ 𝐻 (F , Z ) .The flag manifold itself that features in this construction is the manifold of ‘coherentstates’, which by definition are the states in the orbit of 𝐺 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 Ω over various two-cycles in theflag manifold was described. We may view the cycles ( CP ) 𝐴 𝑖 ,𝐴 𝑖 + as the positive simpleroots of 𝔰𝔲 ( 𝑛 ) , and Ω as a highest weight. It is a general theorem that highest weight orbitsare Kähler 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( 𝑛 ). We will return to these examples lateron when we discuss coherent states. In all cases the states are built as polynomials in thecreation operators 𝑎 † ,𝛼 , 𝑏 † ,𝛼 , etc., acting on the vacuum state | (cid:105) . a) 𝑎 𝑎 𝑎 𝑎 Symmetric powers of the fundamental representation ⇒ Polynomials in 𝑎 † , , . . . , 𝑎 † ,𝑛 of degree 4. b) 𝑎 𝑎𝑏 In this case we have linear combinations of polynomials in 𝑎 † ,𝛼 and 𝑏 † ,𝛼 ofthe form 𝑎 † ,𝛼 ( 𝑎 † ,𝛽 𝑏 † ,𝛾 − 𝑎 † ,𝛾 𝑏 † ,𝛽 ) c) 𝑎 𝑎 𝑎𝑏 𝑏𝑐 Here we have linear combinations of polynomials in 𝑎 † ,𝛼 , 𝑏 † ,𝛼 and 𝑐 † ,𝛼 ofthe form 𝑎 † ,𝛼 ( 𝑎 † ,𝛽 𝑏 † ,𝛾 − 𝑎 † ,𝛾 𝑏 † ,𝛽 ) ( 𝑎 † ,𝛾 𝑏 † ,𝛿 𝑐 † ,𝜆 − 𝑎 † ,𝛾 𝑏 † ,𝜆 𝑐 † ,𝛿 − 𝑎 † ,𝛿 𝑏 † ,𝛾 𝑐 † ,𝜆 − 𝑎 † ,𝜆 𝑏 † ,𝛿 𝑐 † ,𝛾 + 𝑎 † ,𝛿 𝑏 † ,𝜆 𝑐 † ,𝛾 + 𝑎 † ,𝜆 𝑏 † ,𝛾 𝑐 † ,𝛿 ) . Apart from its aesthetic appeal, this construction offers certain calculational benefits. 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 𝔰𝔲 ( 𝑛 ) in the representation described schematically by the followingdiagram: 𝑝 (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) 𝑝 where we assume there are 𝑝 boxes in the first row and 𝑝 boxes in thesecond one ( 𝑝 (cid:62) 𝑝 ). We assign 𝑛 pairs of creation/annihilation operators 𝑎, 𝑎 † , 𝑏, 𝑏 † toeach row. The rotation generators are 𝑆 𝑎 = 𝑎 † ◦ 𝑇 𝑎 ◦ 𝑎 + 𝑏 † ◦ 𝑇 𝑎 ◦ 𝑏. (2.40)27he generators 𝑇 𝑎 are unit-normalized: Tr ( 𝑇 𝑎 𝑇 𝑏 ) = 𝛿 𝑎𝑏 . Then (cid:205) 𝑎 𝑇 𝑎 ⊗ 𝑇 𝑎 = 𝑃 − 𝑛 𝐼 , where 𝑃 is the permutation and 𝐼 the identity operator. Thus, for the Casimir one obtains (here forbrevity we omit the state | 𝜓 (cid:105) on which these operators act, but its presence is implied) 𝐶 ≡ ∑︁ 𝑎 𝑆 𝑎 𝑆 𝑎 = 𝑎 † ,𝛼 𝑎 𝛽 𝑎 † ,𝛽 𝑎 𝛼 = 𝑝 +( 𝑛 − ) 𝑝 + 𝑎 † ,𝛼 𝑎 𝛽 𝑏 † ,𝛽 𝑏 𝛼 = − 𝑝 + 𝑏 † ,𝛼 𝑏 𝛽 𝑎 † ,𝛽 𝑎 𝛼 = − 𝑝 + 𝑏 † ,𝛼 𝑏 𝛽 𝑏 † ,𝛽 𝑏 𝛼 = 𝑝 +( 𝑛 − ) 𝑝 −− 𝑛 ( 𝑎 † ,𝛼 𝑎 𝛼 + 𝑏 † ,𝛼 𝑏 𝛼 ) = ( 𝑝 + 𝑝 ) = 𝑝 + ( 𝑛 − ) 𝑝 + 𝑝 + ( 𝑛 − ) 𝑝 − 𝑛 ( 𝑝 + 𝑝 ) − 𝑝 This can be easily generalized to arbitrary representations of 𝔰𝔲 ( 𝑛 ) . Indeed, consider a Youngdiagram with 𝑛 rows (the maximal number for 𝔰𝔲 ( 𝑛 ) ), the row lengths being 𝑝 ≥ . . . ≥ 𝑝 𝑛 − ≥ 𝑝 𝑛 =
0. Introducing the variable 𝑠 𝐴 = 𝑝 𝐴 − 𝑛 𝑛 (cid:205) 𝐵 = 𝑝 𝐵 (so that 𝑛 (cid:205) 𝐴 = 𝑠 𝐴 = 𝐶 = 𝑛 ∑︁ 𝐴 = 𝑠 𝐴 ( 𝑠 𝐴 − 𝐴 ) , (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 𝐺 is represented.One takes a highest weight vector | 𝜓 (cid:105) and forms its 𝐺 -orbit. That is, one considers all vectorsof the form 𝑔 | 𝜓 (cid:105) , where 𝑔 ∈ 𝐺 . This is a continuous basis, which is therefore overcomplete.In what follows we will be dealing solely with the case of compact 𝐺 = U ( 𝑛 ) , however we findit useful to remind the reader of how the definition just introduced fits into the familiar setupof quantum mechanics (cf. [153]). In this case one has a Heisenberg algebra [ 𝑎, 𝑎 † ] = witha highest weight vector | (cid:105) , which is annihilated by 𝑎 (and clearly fixed by the unit operator).The normalized coherent states are therefore given by the familiar formula | 𝑣 (cid:105) ≡ 𝑒 − | 𝑣 | 𝑒 𝑣 𝑎 † | (cid:105) . (2.42)In this case coherent states are parametrized by complex numbers: 𝑣 ∈ C . As wementioned earlier, it is a general fact [157] that the highest weight orbit in the projectivization 𝑃 ( 𝑉 ) of an irreducible representation 𝑉 of a compact Lie group is Kähler. For the coherentstates we find below, 𝜈 will live in some flag manifold.In the case of SU ( 𝑛 ) 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 𝔰𝔲 may be found in [173]. Another approach tothe quantization of coadjoint orbits is developed in [17]. ( 𝑛 ) . 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 ( 𝑎 † , ) | (cid:105) . Since ( 𝑔𝑎 † , 𝑔 − ) = ( 𝑧 ◦ 𝑎 † ) for 𝑧 the firstcolumn of 𝑔 ∈ SU ( 𝑛 ) , we may parameterize the coherent states in this case as | 𝑣 (cid:105) = ( 𝑣 ◦ 𝑎 † ) | (cid:105) , 𝑣 ∈ CP 𝑛 − = F ,𝑛 − (2.43) b) The highest weight vector is 𝑎 † , · ( 𝑎 † , 𝑏 † , − 𝑎 † , 𝑏 † , ) | (cid:105) , and leads to | 𝑣𝑤 (cid:105) = ( 𝑣 ◦ 𝑎 † ) · [( 𝑣 ◦ 𝑎 † ) ( 𝑤 ◦ 𝑏 † ) − ( 𝑤 ◦ 𝑎 † ) ( 𝑣 ◦ 𝑏 † )] | (cid:105) , 𝑤 ◦ 𝑣 =
0. Here 𝑣 and 𝑤 parametrize thepartial flag manifold F , ,𝑛 − . c) The highest weight vector 𝑎 † , · ( 𝑎 † , 𝑏 † , − 𝑎 † , 𝑏 † , ) · ( 𝑎 † , 𝑏 † , 𝑐 † , − 𝑎 † , 𝑏 † , 𝑐 † , − 𝑎 † , 𝑏 † , 𝑐 † , − 𝑎 † , 𝑏 † , 𝑐 † , + 𝑎 † , 𝑏 † , 𝑐 † , + 𝑎 † , 𝑏 † , 𝑐 † , ) | (cid:105) leads to the coherent states | 𝑢𝑣𝑤 (cid:105) = ( 𝑣 ◦ 𝑎 † ) · [( 𝑣 ◦ 𝑎 † ) ( 𝑤 ◦ 𝑏 † ) − ( 𝑤 ◦ 𝑎 † ) ( 𝑣 ◦ 𝑏 † )] · (2.44) ·[( 𝑣 ◦ 𝑎 † ) ( 𝑤 ◦ 𝑏 † ) ( 𝑢 ◦ 𝑐 † ) − ( 𝑣 ◦ 𝑎 † ) ( 𝑢 ◦ 𝑏 † ) ( 𝑤 ◦ 𝑐 † ) − ( 𝑤 ◦ 𝑎 † ) ( 𝑣 ◦ 𝑏 † ) ( 𝑢 ◦ 𝑐 † ) −−( 𝑢 ◦ 𝑎 † ) ( 𝑤 ◦ 𝑏 † ) ( 𝑣 ◦ 𝑐 † ) + ( 𝑤 ◦ 𝑎 † ) ( 𝑢 ◦ 𝑏 † ) ( 𝑣 ◦ 𝑐 † ) + ( 𝑢 ◦ 𝑎 † ) ( 𝑣 ◦ 𝑏 † ) ( 𝑤 ◦ 𝑐 † )] | (cid:105) with 𝑤 ◦ 𝑣 = 𝑢 ◦ 𝑤 = 𝑢 ◦ 𝑣 =
0. These three variables parametrize F , , ,𝑛 − .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 𝑎, 𝑏, 𝑐 ) that those generators 𝑇 𝑎 , which are upper-triangular, correspond to thefollowing transformations of the operators 𝑎 † , 𝑏 † , 𝑐 † : 𝛿𝑎 † ,𝛼 = ∑︁ 𝛽<𝛼 𝜅 𝛼𝛽 𝑎 † ,𝛽 , 𝛿𝑏 † ,𝛼 = ∑︁ 𝛽<𝛼 𝜅 𝛼𝛽 𝑏 † ,𝛽 , 𝛿𝑐 † ,𝛼 = ∑︁ 𝛽<𝛼 𝜅 𝛼𝛽 𝑐 † ,𝛽 , (2.45)i.e. in the matrix ( 𝑎, 𝑏, 𝑐 ) the upper rows are added to the lower ones. Since the constructedstates are defined 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 reflected in a fundamental identity – the so-called ‘partition of unity’. For the casewhen the manifold of coherent states is CP 𝑛 − (as in (2.43)), which is the only case we willreally be using, the identity takes the form ∫ 𝑑𝜇 ( 𝑣, 𝑣 ) | 𝑣 (cid:105)(cid:104) 𝑣 |(cid:104) 𝑣 | 𝑣 (cid:105) = , (2.46)where 𝑑𝜇 is the suitably normalized volume form on CP 𝑛 − . It is proportional to the toppower of the Fubini-Study form, 𝑑𝜇 ∼ 𝜔 𝑛 − , and looks as follows when expressed in theinhomogeneous coordinates: ( 𝑑𝜇 ) CP 𝑛 − ∼ (cid:32) + 𝑛 − ∑︁ 𝛼 = 𝑣 𝛼 𝑣 𝛼 (cid:33) − 𝑛 𝑛 − (cid:214) 𝛼 = ( 𝑖 𝑑𝑣 𝛼 ∧ 𝑑𝑣 𝛼 ) . (2.47) We use the same symbol for the matrix realization and the Fock space operator realization of a transformation 𝑔 ∈ SU ( 𝑛 ) . CP 𝑛 − , one would have to replace 𝑑𝜇 with the corresponding volume form. In Section 2.3, we demonstrated how Schwinger-Wigner oscillators arise from the canonicalquantization of the flag manifold phase space in homogeneous coordinates. We will nowproceed to show that the famous Holstein-Primakoff representation corresponds to the quan-tization of the sphere – the most elementary flag manifold – in certain coordinates, relatedto the action-angle and to the inhomogeneous coordinates. A corresponding SU ( 𝑛 ) flagmanifold version can also be developed along the same lines. We start from the first-orderLagrangian L CP = 𝑝 𝑖 𝑧 (cid:164) 𝑧 − 𝑧 (cid:164) 𝑧 + 𝑧𝑧 = 𝑝 ρ 𝑑𝜑 = 𝑝 𝑖 ( 𝑤 (cid:164) 𝑤 − 𝑤 (cid:164) 𝑤 ) , (2.48)where 𝑧 = | 𝑧 | 𝑒 − 𝑖𝜑 𝑤 = ρ 𝑒 − 𝑖𝜑 , ρ = − + 𝑧𝑧 . As explained before, upon quantization 𝑝 ∈ Z + is a positive integer encoding the representa-tion. We also need the expressions for the SU ( ) charges. If we denote the vector 𝑍 : = (cid:18) 𝑧 (cid:19) ,the spin variables are the moment maps 𝑆 𝑎 = 𝑝 𝑍 † 𝜏 𝑎 𝑍𝑍 ◦ 𝑍 , so that 𝑆 + = 𝑝 𝑧 + | 𝑧 | , 𝑆 − = 𝑝 𝑧 + | 𝑧 | , 𝑆 𝑧 = 𝑝 − | 𝑧 | + | 𝑧 | . (2.49)Using 𝑧 = | 𝑧 | ρ 𝑤 = 𝑤 ( −| 𝑤 | ) / , we find 𝑆 + = 𝑝 𝑤 ( − | 𝑤 | ) / , 𝑆 − = 𝑝 𝑤 ( − | 𝑤 | ) / , 𝑆 𝑧 = 𝑝 ( − | 𝑤 | ) . (2.50)To canonically quantize the system (2.48), we denote 𝐴 : = √ 𝑝 𝑤 , 𝐴 † : = √ 𝑝 𝑤 and postulatethe canonical commutation relations [ 𝐴, 𝐴 † ] =
1. Choosing the ordering compatible withthe unitary relation 𝑆 + = ( 𝑆 − ) † , we find 𝑆 + = ( 𝑝 − 𝐴 † 𝐴 ) / 𝐴, 𝑆 − = 𝐴 † ( 𝑝 − 𝐴 † 𝐴 ) / , 𝑆 𝑧 = ( 𝑝 − 𝐴 † 𝐴 ) , (2.51)which is the Holstein-Primakoff representation for the spin operators.We have demonstrated that the Holstein-Primakoff 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 – the so-called Dyson-Maleevrealization – 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 complexified groupSL ( 𝑛, 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. 𝑞 𝑗 , 𝑝 𝑗 -operators, such that [ 𝑞 𝑗 , 𝑝 𝑘 ] = 𝑖𝛿 𝑗 𝑘 ).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 𝑠 Weyl pairs, where 𝑠 = 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 𝔰𝔩 ( 𝑛 ) , the minimal orbit is nilpotent, so typically thissetup leads to the theory of nilpotent orbits. For 𝔰𝔩 ( 𝑛 ) , 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 𝑛 − 𝔰𝔩 ( ) . The semi-simple orbits may be labeled by theCartan elements (cid:18) 𝑎 − 𝑎 (cid:19) , where 𝑎 ∈ C \ { } (the limit 𝑎 = 𝑀 ∈ 𝔰𝔩 ( ) ) 𝑀 = 𝑎 . (2.52)Consider the following first-order Lagrangian (which should be viewed as the relevant coun-terpart of (2.48)): L = ∑︁ 𝑗 = (cid:16) 𝑉 𝑖 · D 𝑈 𝑖 + 𝑉 𝑖 · D 𝑈 𝑖 (cid:17) + ( 𝑎 A + 𝑎 A) , D 𝑈 𝑖 = (cid:164) 𝑈 𝑖 − A 𝑈 𝑖 . (2.53)Here 𝑈 𝑖 , 𝑉 𝑖 are the complex canonical variables, and the gauge field A is meant to generatethe quotient by C ∗ . Just as before, the first term in the Lagrangian is a Poincaré-Liouville one-form corresponding to a certain (this time complex) symplectic form, and the introduction ofa gauge field 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 flag manifoldtarget spaces. The second term in the Lagrangian is a ‘Fayet-Iliopoulos term’: under gaugetransformations it shifts by a total derivative, but the action S = ∫ 𝑑𝑡 L is invariant.The group SL ( , C ) acts as 𝑈 → 𝑔 ◦ 𝑈, 𝑉 → 𝑉 ◦ 𝑔 − , and from (2.53) one can derive theconserved charges corresponding to this action: 𝜇 = 𝑈 ⊗ 𝑉 − ( 𝑉 ◦ 𝑈 ) (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 ‘minimal’ realizations of Lie algebras, as well as a list of relatedliterature, may be found in [147]. 𝜔 = (cid:205) 𝑖 = 𝑑𝑉 𝑖 ∧ 𝑑𝑈 𝑖 , which is why wehave denoted it by 𝜇 . Varying the Lagrangian w.r.t. the gauge field, we obtain the constraint 𝑉 ◦ 𝑈 = 𝑎 . As a result, 𝜇 satisfies the equation 𝜇 = 𝑎 , so that 𝜇 belongs to theorbit (2.52).Let us now choose ‘inhomogeneous coordinates’, i.e. we assume that at least one of 𝑈 , 𝑈 is non-zero, say 𝑈 ≠
0, in which case by a C ∗ -transformation we may set 𝑈 = 𝑈 : = 𝑈 and 𝑉 : = 𝑉 . The constraint (cid:205) 𝑖 = 𝑉 𝑖 · 𝑈 𝑖 = 𝑎 may now be solved as 𝑉 = 𝑎 − 𝑉 · 𝑈 . The spin matrix 𝜇 has the following form in these variables: 𝜇 = (cid:18) 𝑎 − 𝑉 · 𝑈 𝑉𝑈 · ( 𝑎 − 𝑉 · 𝑈 ) 𝑉 · 𝑈 − 𝑎 (cid:19) (2.55)Quantization of (2.53) in the inhomogeneous coordinates 𝑈, 𝑉 amounts to imposing thecanonical commutation relations [ 𝑈, 𝑉 ] = 𝑖 . 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 𝔰𝔩 ( ) commutation relations), and as a result one arrives at the Dyson-Maleevrepresentation 𝑆 + = 𝑉 , 𝑆 − = 𝑈 ( 𝑎 − 𝑈 𝑉 ) , 𝑆 𝑧 = 𝑎 − 𝑈 𝑉 , where [ 𝑈, 𝑉 ] = 𝑖 . (2.56)By identifying 𝑉 = − 𝑖 𝜕𝜕𝑈 , we also obtain the well-known differential operator realization 𝑆 + = − 𝑖 𝜕𝜕𝑈 , 𝑆 − = 𝑈 (cid:18) 𝑎 + 𝑖 𝑈 𝜕𝜕𝑈 (cid:19) , 𝑆 𝑧 = 𝑎 + 𝑖 𝑈 𝜕𝜕𝑈 , (2.57)which for 𝑎 = 𝔰𝔩 ( ) -operators acting on the sphere CP withinhomogeneous coordinate 𝑈 . 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 ferromagnet’s 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 Néel state, with spinsalternating between being aligned and antialigned along a common direction, but quantummechanically the Néel state is no longer an eigenstate of the Heisenberg Hamiltonian. Thisfact can be understood from Coleman’s 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 𝑆 = at each site [47]. However, despite this initial progress, spin chains with 𝑠 > were not amenable to such techniques, and fifty years would pass before their low energyproperties could be characterized. In 1981, Duncan Haldane proposed a radical classificationof antiferromagnetic chains: Those with integral spin 𝑠 have a finite 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 Bethe’s 1931 solution, Haldane’s “conjecture” 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 Bethe’s 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 𝑆 . This would imply that allantiferromagnets would exhibit gapless excitations at low energies. Of course, this wasin direct contradiction with Coleman’s theorem, that invalidated spin wave theory in onedimension, but nonetheless, by the 1980s it was widely accepted that gapless excitations wereuniversal among antiferromagnets. Coleman’s theorem is often confounded with the Mermin-Wagner-Hohenberg theorem, which forbidsan ordered grounds state in two spatial dimensions at finite temperature, and applies equally well to bothferromagnets and antiferromagnets [175, 136].
33n fact, Haldane’s 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 field theory that was being used as a toy model forquantum chromodynamics at the time [103]. The role of the spin, 𝑠 , manifests as a topologicalangle 𝜃 in the sigma model, so that integral 𝑠 translates to 𝜃 = 𝑠 translates to 𝜃 = 𝜋 . Thus, Haldane’s claim about antiferromagnets was also a claim aboutmass gaps in the CP sigma model. At that time, it was widely believed that a finite gap wouldexist for all values of 𝜃 , and this was known exactly for 𝜃 =
0, and suggested numericallyfor small values of 𝜃 [264, 48, 49]. It was shown in [6] that the CP model is gapless at 𝜃 = 𝜋 but this is not true for CP 𝑛 − with 𝑛 >
2. In that case there is a first order transitionat 𝜃 = 𝜋 with the model remaining massive. This can be understood from the presence ofrelevant operators allowed by symmetry for 𝑛 >
2. The generalization of this behaviourto four-dimensional SU ( 𝑛 ) Quantum Chromodynamics is a fascinating subject [117]. Forlarge 𝑛 it has been established that the transition is first order with a finite mass [243, 254].Whether or not this is true for 𝑆𝑈 ( ) is an open question.Over the next few years, Haldane’s conjecture would defy these skeptics, thanks toverification from multiple areas of research. Experimentally, neutron scattering on theorganic nickel compound NENP, which is a quasi-one dimensional 𝑠 = 𝑆 = , 𝑆 = CP sigma model, Monte Carlo methodswere used to numerically verify the absence of a mass gap when 𝜃 = 𝜋 [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 Haldane’s claims were scien-tific breakthroughs in their own right. Indeed, the fields 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 Haldane’s conjecture. It isthus not a leap to claim that any generalization of Haldane’s conjecture would be an impactfulresult to the physics community. And indeed, this is what led physicists, including Affleck,Read, Sachdev and others to extend Haldane’s work to SU( 𝑛 ) 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 effect [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( 𝑛 ) chains (with 𝑛 ≤
10) arenow experimentally realizable, offering 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 field of SU( 𝑛 ) spin chains. As a consequence, an SU( 𝑛 ) version of Haldane’s conjecturehas recently been formulated [161, 236, 237, 234].34n this chapter, we review this recent effort of extending Haldane’s conjecture fromSU(2) to SU( 𝑛 ). We begin in Section 3 by introducing the SU( 𝑛 ) Heisenberg chain, in therank- 𝑝 symmetric representation. Unlike the familiar spin chains with SU(2) symmetry, for 𝑛 >
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( 𝑛 ) chainsin other representations.Next, in Section 4, we recall various exact results that exist for these SU( 𝑛 ) Hamiltonians.Specifically, we discuss the Lieb-Shultz-Mattis Affleck theorem [166, 13], and the Affleck-Kennedy-Lieb-Tasaki construction [12].In Section 5, we extend the familiar spin-wave theory to these SU( 𝑛 ) chains, and obtainpredictions for the velocities of low lying excitations. We observe that for 𝑛 >
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 field theorydescription of the SU( 𝑛 ) chain. This extends Haldane’s original mapping of the spin chainto the CP model; now, the corresponding target space is the complete flag manifold,SU ( 𝑛 )/[ U ( )] 𝑛 − . Thus, via these sections we establish a direct link from SU( 𝑛 ) chains tothe subject matter of Chapter 1. In Section 9, we also explain how the distinct flavour wavevelocities flow 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 ‘t Hooft anomalies between the global symme-tries of the chain. After quoting the results, we offer a detailed discussion of the mathematicalstructure behind these anomalies, which involves the concept of PSU( 𝑛 ) bundles.In Section 11.3, we reinterpret the SU( 𝑛 ) Haldane conjecture in terms of fractionaltopological excitations, which generalize the notion of merons in SU(2) [8]. Finally, inSection 12, we explain how SU( 𝑛 ) representations other than the rank- 𝑝 symmetric ones mayadmit a mapping to the same flag manifold target space, SU ( 𝑛 )/[ U ( )] 𝑛 − . This leads usto non-Lagrangian embeddings of the flag 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 𝑠 , which specifiesthe irreducible representation of SU(2) that appears on each site. In SU( 𝑛 ), the most genericirrep is defined by 𝑛 − 𝑝 symmetric irreps, which have Young tableaux 𝑝 (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 𝐻 = 𝐽 ∑︁ 𝑗 Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + )) (3.2)where 𝑆 ( 𝑗 ) is an 𝑛 × 𝑛 Hermitian matrix with Tr ( 𝑆 ) = 𝑝 , whose entries correspond to the 𝑛 − 𝑛 ) and satisfy [ 𝑆 𝛼𝛽 , 𝑆 𝛾𝛿 ] = 𝛿 𝛼𝛿 𝑆 𝛾𝛿 − 𝛿 𝛾𝛽 𝑆 𝛼𝛿 . (3.3)Indeed, in SU(2), 𝑆 𝛼𝛽 = (cid:174) 𝑆 · (cid:174) 𝜎 𝛼𝛽 + 𝑝 I , and the Hamiltonian appearing in (3.2) equals theHeisenberg model with spin 𝑠 = 𝑝 (up to a constant). However, for 𝑛 >
2, this Hamiltonianpossesses local zero mode excitations that destabilize the classical ground state and inhibita low energy field theory description. To remedy this, we introduce an additional 𝑛 − 𝐻 = ∑︁ 𝑗 𝑛 − ∑︁ 𝑟 = 𝐽 𝑟 Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + 𝑟 )) , (3.4)where 𝐽 couples nearest-neighbours, 𝐽 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 𝑛 = In the large- 𝑝 limit, the commutator (3.3) is subleading in 𝑝 , allowing us to replace 𝑆 by amatrix of classical numbers. To this order in 𝑝 , the Casimir constraints of SU( 𝑛 ) completelydetermine the eigenvalues of 𝑆 . We have 𝑆 𝛼𝛽 = 𝑝𝑢 𝛼 𝑢 𝛽 (3.5) S(j) should be traceless; we have shifted it by a constant to simplify our calculations. 𝑢 ∈ C 𝑛 with 𝑢 ◦ 𝑢 =
1. Note that 𝑆 𝛼𝛽 are the components of the moment map 𝜇 from (1.10),up to an additive constant term. The interaction terms appearing in (3.2) reduce toTr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + 𝑟 )) = 𝑝 | 𝑢 ( 𝑗 ) ◦ 𝑢 ( 𝑗 + 𝑟 ) | . (3.6)Since 𝑢 lives in C 𝑛 , a classical ground state will posses local zero modes unless the Hamil-tonian gives rise to 𝑛 − 𝑛 -site ordered classical ground state, which givesrise to a Z 𝑛 symmetry in their low energy field theory description. This Z 𝑛 symmetry is alsopresent in the 𝑝 = 𝑛 -site unit cell through an “order-by-disorder”mechanism that generates effective additional couplings of order 𝑝 − that lift the local zeromodes [161, 86].Since the classical ground state minimizing (3.4) has an 𝑛 -site order, it is characterized by 𝑛 normalized vectors that mutually minimize (3.6). That is, the classical ground state givesrise to an orthonormal basis of C 𝑛 . As we recall from section 1.1, the space of 𝑛 -tuples ofmutually orthogonal vectors, defined up to a phase, is the complete flag manifold, which isthe the mechanism how flag manifolds arise in the context of spin chains. Due to this 𝑛 -foldstructure, we rewrite the Hamiltonian (3.4) as a sum over unit cells (indexed by 𝑗 ): 𝐻 = ∑︁ 𝑗 𝑛 ∑︁ 𝐴 = 𝑛 − ∑︁ 𝑟 = 𝐽 𝑟 Tr ( 𝑆 ( 𝑗 𝐴 ) 𝑆 ( 𝑗 𝐴 + 𝑟 )) 𝑗 𝐴 : = 𝑛 𝑗 + ( 𝐴 − ) . (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( 𝑛 ) Hamiltonians. Haldane’s 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 Affleck-Kennedy-Lieb-Tasaki construction [12]. Similar results also exist for chains with SU( 𝑛 )symmetry, and this is what we review in this section. The LSMA theorem is a rigorous statement about ground states in translationally invariantSU( 𝑛 ) Hamiltonians [166, 13]: 37onsider a translationally- and SU( 𝑛 )-invariant Hamiltonian of a spin chainwith symmetric rank- 𝑝 representations at each site. If 𝑝 is not a multiple of 𝑛 ,then either the ground state is unique with gapless excitations, or there is aground state degeneracy of at least 𝑛 gcd ( 𝑛,𝑝 ) .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 𝐿 sites: 𝐻 = 𝑅 ∑︁ 𝑟 = 𝐻 𝑟 𝐻 𝑟 : = 𝐿 ∑︁ 𝑗 = 𝐽 𝑟 Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + 𝑟 )) (4.1)where 𝑆 is defined as above. We assume that | 𝜓 (cid:105) is the unique ground state of 𝐻 , and istranslationally invariant: 𝑇 | 𝜓 (cid:105) = | 𝜓 (cid:105) . We then define a twist operator 𝑈 = 𝑒 𝐴 𝐴 : = 𝜋𝑖𝑛𝐿 𝐿 ∑︁ 𝑗 = 𝑗 𝑄 ( 𝑗 ) (4.2)with 𝑄 = 𝑛 − ∑︁ 𝐴 = 𝑆 𝛼𝛼 − ( 𝑛 − ) 𝑆 𝑛𝑛 = Tr ( 𝑆 ) − 𝑛𝑆 𝑛𝑛 = 𝑝 − 𝑛𝑆 𝑛𝑛 . (4.3)Using the commutation relations (3.3), it is easy to verify that (cid:104) Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + 𝑟 )) , 𝑄 ( 𝑗 ) + 𝑄 ( 𝑗 + 𝑟 ) (cid:105) = 𝑈 † Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + 𝑟 )) 𝑈 = 𝑒 − 𝑟 𝜋𝑖𝑛𝐿 ( 𝑄 ( 𝑗 + 𝑟 )− 𝑄 ( 𝑗 )) Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + 𝑟 )) 𝑒 𝑟 𝜋𝑖𝑛𝐿 ( 𝑄 ( 𝑗 + 𝑟 )− 𝑄 ( 𝑗 )) . (4.5)Using this, one can show that 𝑈 † 𝐻𝑈 = 𝐻 + [ 𝐻, 𝐴 ] + O ( 𝐿 − ) (4.6)so that 𝑈 | 𝜓 (cid:105) has energy O ( 𝐿 − ) . Now, using the translational invariance of | 𝜓 (cid:105) , we find (cid:104) 𝜓 | 𝑈 | 𝜓 (cid:105) = (cid:104) 𝜓 | 𝑇 − 𝑈𝑇 | 𝜓 (cid:105) = (cid:104) 𝜓 | 𝑈𝑒 𝜋𝑖𝑛 𝑄 ( ) 𝑒 − 𝜋𝑖𝑛𝐿 (cid:205) 𝐿𝑗 = 𝑄 ( 𝑗 ) | 𝜓 (cid:105) . (4.7)Since | 𝜓 (cid:105) is a ground state of 𝐻 , it is a SU( 𝑛 ) singlet, and so must be left unchanged by theglobal SU( 𝑛 ) transformation 𝑒 − 𝜋𝑖𝑛𝐿 (cid:205) 𝐿𝑗 = 𝑄 ( 𝑗 ) . Moreover, using (4.3), we have (cid:104) 𝜓 | 𝑈 | 𝜓 (cid:105) = 𝑒 𝜋𝑖𝑝𝑛 (cid:104) 𝜓 | 𝑈𝑒 𝜋𝑖𝑆 𝑛𝑛 | 𝜓 (cid:105) . (4.8)38s shown in section 2.3, the matrices 𝑆 can be represented in terms of Schwinger bosons;the diagonal elements are then number operators for these bosons. Thus, 𝑆 𝑛𝑛 acting on | 𝜓 (cid:105) will always return an integer, and 𝑒 𝜋𝑖𝑆 𝑛𝑛 can be dropped. Thus, we find that so long as 𝑝 isnot a multiple of 𝑛 , (cid:104) 𝜓 | 𝑈 | 𝜓 (cid:105) = 𝑈 | 𝜓 (cid:105) is a distinct, low-lying state above | 𝜓 (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) 𝜓 | 𝑈 𝑘 | 𝜓 (cid:105) = 𝑒 𝜋𝑖𝑝𝑘𝑛 (cid:104) 𝜓 | 𝑈 | 𝜓 (cid:105) . (4.10)So long as 𝑘 < 𝑟 : = 𝑛 / gcd ( 𝑛, 𝑝 ) , the family { 𝑈 𝑘 | 𝜓 (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 𝑟 -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 𝑝 =
2. Each node represents a fundamental 𝑝 = One of the first results that bolstered Haldane’s conjecture was the discovery of the so-calledAKLT model of a spin-1 chain, which exhibits a unique, translationally invariant groundstate with a finite 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( 𝑛 ) chains [126, 150, 187,78, 177, 207, 125]. Relevant to us are the symmetric representation AKLT Hamiltoniansintroduced in [126]. In particular, for 𝑝 a multiple of 𝑛 , Hamiltonians are constructed that39igure 7: Valence bond constructions for SU(6). The left subfigure corresponds to 𝑝 = 𝑝 =
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 𝑛 = 𝑝 = 𝑝 not a multiple of 𝑛 , with 𝑟 : = 𝑛 / gcd ( 𝑛, 𝑝 ) , Hamiltonians are constructedwith 𝑟 -fold degenerate ground states that are invariant under translations by 𝑟 sites (see Figures6, 7). All of these models have short range correlations, and are expected to have gappedground states, based on arguments of spinon confinement. The fact that the construction ofa gapped, nondegenerate ground state is only possible when 𝑝 is a multiple of 𝑛 is consistentwith the LSMA theorem presented above. projection onto | ψ i| ψ i| ψ i (a) p=1 (b) p=3 Figure 8: AKLT constructions in SU(3). Left: When 𝑝 ≠ 𝑛 , multiple valence bond solidscan be formed. The ground state is not translationally invariant and degenerate. Right: When 𝑝 = 𝑛 , a unique, translationally invariant ground state can be constructed, by projecting onto the symmetric- 𝑝 representation at each site. According to Coleman’s theorem [85], we do not expect spontaneous symmetry breakingof the SU( 𝑛 ) 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 sufficiently high energies theexcitations may propagate with these velocities [123]. In the familiar antiferromagnet, thisprocedure is known as spin wave theory; in SU( 𝑛 ), it is called flavour wave theory [194, 195].40o begin, we introduce 𝑛 bosons in each unit cell to reproduce the commutation relationsof the 𝑆 matrices: 𝑆 𝛼𝛽 ( 𝑗 𝐴 ) = 𝑏 † ,𝛼𝐴 𝑏 𝛽𝐴 . (5.1)The counting is 𝑛 flavours of bosons for each of the 𝑛 sites of a unit cell. The conditionTr ( 𝑆 ) = 𝑝 implies there are 𝑝 bosons at each site. The classical ground state involves only‘diagonal’ bosons of the type 𝑏 𝐴𝐴 and 𝑏 † ,𝐴𝐴 . The ‘off-diagonal’ bosons are Holstein-Primakoffbosons; in SU(2) they correspond to the operators 𝐴, 𝐴 † introduced in Section 2.4. Flavourwave theory allows for a small number of Holstein-Primakoff bosons at each site, capturedby 𝜈 ( 𝑗 𝐴 ) = ∑︁ 𝛼 ≠ 𝐴 𝑏 † ,𝛼𝐴 𝑏 𝛼𝐴 , and writes the Hamiltonian (3.4) in terms of these 𝑛 ( 𝑛 − ) bosons. In the large 𝑝 (cid:29) 𝜈 ( 𝑗 𝐴 ) limit, we expand 𝑆 𝐴𝐴 ( 𝐽 𝐴 ) = 𝑝 − 𝜈 ( 𝑗 𝐴 ) ,𝑆 𝛼𝐴 ( 𝑗 𝐴 ) ≈ √ 𝑝𝑏 † ,𝛼𝐴 ,𝑆 𝐴𝛼 ( 𝑗 𝐴 ) ≈ √ 𝑝𝑏 𝛼𝐴 , to find Tr ( 𝑆 ( 𝑗 𝐴 ) 𝑆 ( 𝑗 𝐵 )) = 𝑝 (cid:104) 𝑏 † ,𝐴𝐵 𝑏 𝐴𝐵 + 𝑏 † ,𝐴𝐵 𝑏 𝐵𝐴 + 𝑏 † ,𝐵𝐴 𝑏 † ,𝐴𝐵 + 𝑏 𝐵𝐴 𝑏 𝐴𝐵 (cid:105) + O ( 𝑝 ) . (5.2)In terms of these degrees of freedom, the Hamiltonian (3.4) decomposes into a sum 𝐻 = ∑︁ 𝐴<𝐵 𝐻 𝐴𝐵 , (5.3)where 𝐻 𝐴𝐵 is a Hamiltonian involving only the two boson flavours 𝑏 𝐴𝐵 and 𝑏 𝐵𝐴 . In momentumspace, this gives 𝑛 ( 𝑛 − ) different 2 × 𝐻 𝐴,𝐴 + 𝑡 = const. + ∑︁ 𝑘 𝜔 𝑡 ( 𝑘 ) ∑︁ 𝑚 = (cid:18) 𝑑 † ,𝑚𝑡 ( 𝑘 ) 𝑑 𝑚𝑡 ( 𝑘 ) + (cid:19) (5.4)where 𝜔 𝑡 ( 𝑘 ) = 𝑝 √︁ 𝐽 𝑡 𝐽 𝑛 − 𝑡 (cid:12)(cid:12)(cid:12)(cid:12) sin 𝑛𝑘 𝑎 (cid:12)(cid:12)(cid:12)(cid:12) . (5.5)Therefore, the corresponding flavour wave velocities are 𝑣 𝑡 = 𝑛 𝑝 √︁ 𝐽 𝑡 𝐽 𝑛 − 𝑡 𝑡 = , , · · · , 𝑛 − 𝑛 is odd, there are 𝑛 modes with each flavour wave velocity. When 𝑛 is even, this istrue except for the velocity 𝑣 𝑛 , which has only 𝑛 modes. In each case, the number of modesadds up to 𝑛 ( 𝑛 − ) . We note that for 𝑛 >
3, there is no longer a unique velocity, and theemergence of Lorentz invariance is absent. Only for a specific fine 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 different 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 different, 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. 𝑆 Let us consider in detail the case of SU(2) . We introduce the notation 𝐴 ( 𝑞, 𝑣 ) for thenormalized matrix element between coherent states of any operator ˆ 𝐴 (which for historicalreasons is called the kernel of ˆ 𝐴 ): 𝐴 ( 𝑞, 𝑣 ) = (cid:104) 𝑞 | ˆ 𝐴 | 𝑣 (cid:105)(cid:104) 𝑞 | 𝑣 (cid:105) .Once again we will consider the rank- 𝑝 symmetric representations , and for such repre-sentations the scalar product of coherent states is (cid:104) 𝑞 | 𝑣 (cid:105) = ( 𝑞 ◦ 𝑣 ) 𝑝 . (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 ) ⊗ 𝑝 , for the Hamiltonianacting on the quantum sphere we shall takeˆ 𝑇 = 𝑝 ∑︁ 𝑠 = ⊗ ... ⊗ 𝜎 ↑ 𝑠 -th position ⊗ ... ⊗ , (6.2)which may be interpreted as an external magnetic field (cid:174) 𝐻 in the 𝑧 -direction (under theassumption of a (cid:174) 𝑆 · (cid:174) 𝐻 coupling). The kernel of ˆ 𝑇 is 𝑇 ( 𝑞, 𝑣 ) = 𝑝 𝑞 𝑣 − 𝑞 𝑣 𝑞 𝑣 + 𝑞 𝑣 .We will now present the derivation of the kernel of the “evolution operator”ˆ 𝑈 = 𝑒 − 𝑖𝛼 ˆ 𝑇 . (6.3)Of course, in this finite-dimensional case this is merely a pedagogical exercise, since theaction of ˆ 𝑈 on a coherent state simply gives ˆ 𝑈 | 𝑣 (cid:105) = | 𝑒 − 𝑖𝛼𝜎 ◦ 𝑣 (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. 𝑈 ( 𝑞, 𝑣 ) = (cid:32) 𝑒 − i 𝛼 𝑞 𝑣 + 𝑒 i 𝛼 𝑞 𝑣 𝑞 𝑣 + 𝑞 𝑣 (cid:33) 𝑝 (6.4)As is standard in path integral calculations [107], in order to write a path integral repre-sentation for a matrix element 𝑈 ( 𝑞, 𝑣 ) , one first needs to know the matrix elements of thegenerator ˆ 𝑇 . Then one splits the “time” interval 𝛼 into 𝑘 subintervals of length 𝛼𝑘 and usesthe formula ˆ 𝑈 = lim 𝑘 →∞ (cid:16) − i 𝛼𝑘 ˆ 𝑇 (cid:17) 𝑘 : = lim 𝑘 →∞ ˆ 𝜏 𝑘 . Inserting the completeness relation (2.46)between every two factors of ˆ 𝜏 , we arrive at the following expression for the kernel of theevolution operator: 𝑈 ( 𝑞, 𝑣 ) = (cid:104) 𝑞 | ˆ 𝑈 | 𝑣 (cid:105)(cid:104) 𝑞 | 𝑣 (cid:105) = lim 𝑘 →∞ ∫ 𝑘 − (cid:214) 𝑎 = 𝑑𝜇 ( 𝑧 ( 𝑎 ) , 𝑧 ( 𝑎 )) × (6.5) × 𝜏 ( 𝑞, 𝑧 ( 𝑘 − )) · 𝜏 ( 𝑧 ( 𝑘 − ) , 𝑧 ( 𝑘 − )) · · · 𝜏 ( 𝑧 ( ) , 𝑧 ( )) · 𝜏 ( 𝑧 ( ) , 𝑣 ) ×× (cid:104) 𝑞 | 𝑧 ( )(cid:105)(cid:104) 𝑧 ( ) | 𝑧 ( )(cid:105) ... (cid:104) 𝑧 ( 𝑘 − ) | 𝑧 ( 𝑘 − )(cid:105)(cid:104) 𝑧 ( 𝑘 − ) | 𝑣 (cid:105)(cid:104) 𝑞 | 𝑣 (cid:105)(cid:104) 𝑧 ( ) | 𝑧 ( )(cid:105) ... (cid:104) 𝑧 ( 𝑘 − ) | 𝑧 ( 𝑘 − )(cid:105) Here 𝑧 ( 𝑎 ) is the coherent state of the 𝑎 th ‘time slice’. To complete the derivation we use theexplicit expression 𝜏 ( 𝑧 ( 𝑎 + ) , 𝑧 ( 𝑎 )) = − 𝑝 i 𝛼𝑘 𝑧 ( 𝑎 )◦ 𝜎 ◦ 𝑧 ( 𝑎 + ) 𝑧 ( 𝑎 )◦ 𝑧 ( 𝑎 + ) for 𝜏 and (6.1) for the scalarproduct of coherent states.We now want to “take the limit” in the formula (6.5), assuming that 𝑧 ( 𝑎 + ) − 𝑧 ( 𝑎 ) ∼ 𝑘 (cid:164) 𝑧 ( 𝑎 + ) (for a justification of this procedure see [268]). In order to do it we write the factors 𝑧 ( 𝑎 )◦ 𝑧 ( 𝑎 + ) 𝑧 ( 𝑎 + )◦ 𝑧 ( 𝑎 + ) in the following form: 𝑧 ( 𝑎 ) ◦ 𝑧 ( 𝑎 + ) 𝑧 ( 𝑎 + ) ◦ 𝑧 ( 𝑎 + ) = (cid:18) − ( 𝑧 ( 𝑎 + ) − 𝑧 ( 𝑎 )) ◦ 𝑧 ( 𝑎 + ) 𝑧 ( 𝑎 + ) ◦ 𝑧 ( 𝑎 + ) (cid:19) (cid:39) − 𝑘 (cid:164) 𝑧 ( 𝑎 + ) ◦ 𝑧 ( 𝑎 + ) 𝑧 ( 𝑎 + ) ◦ 𝑧 ( 𝑎 + ) for 𝑎 = , , ..., 𝑘 −
2. Then we obtain 𝑈 ( 𝑞, 𝑣 ) = ∫ (cid:214) 𝑡 ∈[ , ] 𝑑𝜇 ( 𝑧 ( 𝑡 ) , 𝑧 ( 𝑡 )) (cid:18) 𝑧 ( ) ◦ 𝑣𝑞 ◦ 𝑣 (cid:19) 𝑝 exp (cid:169)(cid:173)(cid:171) − 𝑝 ∫ 𝑑𝑡 (cid:164) 𝑧 ◦ 𝑧𝑧 ◦ 𝑧 − 𝑝 i 𝛼 ∫ 𝑑𝑡 𝑧 ◦ 𝜎 ◦ 𝑧𝑧 ◦ 𝑧 (cid:170)(cid:174)(cid:172) , (6.6)with boundary conditions 𝑧 ( ) = 𝑞, 𝑧 ( ) = 𝜈 . 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 𝐴 𝜇 (cid:164) 𝑥 𝜇 = 𝛼 𝜎 , we would arrive exactly at the action in (6.6), upon normalizing thecoordinates as | 𝑧 | = 𝑧 into 𝑢 ).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 𝑧 -axis (the one orthogonal tothe plane of the stereographic projection). Passing to the inhomogeneous coordinate 𝑧 via43 = √ +| 𝑧 | , 𝑢 = 𝑧 √ +| 𝑧 | , we find that the e.o.m. are the equations of harmonic oscillations:i (cid:164) 𝑧 = 𝛼𝑧, i (cid:164) 𝑧 = − 𝛼𝑧. (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 𝑧 ( ) = 𝑞 𝑞 : = 𝑞, 𝑧 ( ) = 𝑣 𝑣 : = 𝑣 , we obtain 𝑧 ( 𝑡 ) = 𝑞𝑒 𝛼𝑡 and 𝑧 ( 𝑡 ) = 𝑣𝑒 − 𝛼 ( 𝑡 − ) . Pluggingthis into the exponent of the path integral (6.6), we get 𝑒 − 𝑝 i 𝛼 . The term 𝑧 ( )◦ 𝑣𝑞 ◦ 𝑣 in front of theexponent produces + 𝑞 𝑦 𝑒 𝛼 + 𝑞𝑦 , and altogether we get 𝑈 ( 𝑞, 𝑣 ) = (cid:18) 𝑒 − i 𝛼 + 𝑞 𝑣 𝑒 i 𝛼 + 𝑞𝑣 (cid:19) 𝑝 , (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 ˆ U = 𝑒 𝑖𝛼 ˆ 𝐻 , ˆ 𝐻 now being a spin chain Hamiltonian. At thesame time we pass from the simple SU(2) case to the SU(3), or even SU( 𝑛 ) model. We startwith the Heisenberg Hamiltonianˆ 𝐻 Heis = 𝑝 𝐿 ∑︁ 𝑗 = ( Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + )) + const. ) , (6.9)where the spin operators are assumed to be in the symmetric powers of the fundamentalrepresentation, indexed by 𝑝 as before, and the constant may be chosen at our will. In orderto build the path integral we first need to know the matrix elements of the Hamiltonian itself,which amounts to knowing the matrix elements of P : = 𝑝 ( Tr ( 𝑆 ( 𝑗 ) · 𝑆 ( 𝑗 + )) + const. ) .This operator acts in the tensor product Sym ( C 𝑛 ) ⊗ 𝑝 ⊗ Sym ( C 𝑛 ) ⊗ 𝑝 and (for a suitable choiceof the additive constant) is a restriction of the operator acting in ( C 𝑛 ) ⊗ 𝑝 ⊗ ( C 𝑛 ) ⊗ 𝑝 as asum of permutations: P = 𝑝 𝑝 (cid:205) 𝑠,𝑡 = 𝑃 𝑠,𝑡 , where 𝑃 𝑠,𝑡 is the permutation of the 𝑠 -th and 𝑡 -th C 𝑛 -factors in the two copies of ( C 𝑛 ) ⊗ 𝑝 . The tensor product of coherent states has the form ( 𝑞 ( 𝑗 ) ◦ 𝑣 ( 𝑗 )) 𝑝 ( 𝑞 ( 𝑗 + ) ◦ 𝑣 ( 𝑗 + )) 𝑝 , andthe matrix elements of ˆ P is easily found to be P ( 𝑞 , 𝑞 ; 𝑣 , 𝑣 ) = (cid:104) 𝑞 , 𝑞 | ˆ P | 𝑣 , 𝑣 (cid:105)(cid:104) 𝑞 , 𝑞 | 𝑣 , 𝑣 (cid:105) = 𝑝 ( 𝑞 ◦ 𝑣 ) ( 𝑞 ◦ 𝑣 )( 𝑞 ◦ 𝑣 ) ( 𝑞 ◦ 𝑣 ) . (6.10)Now we can essentially repeat the steps from the previous section. The only difficulty isnotational and it comes from the fact that in this case, as opposed to the previous example,44e essentially have two “space-time” directions: one “time” or 𝛼 -direction, and a second“spatial” direction in which the spin chain is extended. As a consequence, our variables 𝑧 will now take two arguments: 𝑧 ( 𝑎, 𝑗 ) , where 𝑎 is the time index, and 𝑗 enumerates the sitesof the spin chain. The integrand will again split into two terms: the first being a geometricphase term, and the second being the Hamiltonian: 𝑈 ( 𝑞, 𝑣 ) = lim 𝐾 →∞ ∫ (cid:214) 𝑎, 𝑗 𝑑𝜇 ( 𝑧 ( 𝑎, 𝑗 ) , 𝑧 ( 𝑎, 𝑗 )) × 𝐼 geom × 𝐼 𝐻 (6.11)The geometric term is local in the spin chain index 𝑗 and has a simplest (nearest-neighbor,or first-order) nonlocality in time, which is a general feature, since in the continuum limit itshould lead to a one-form: 𝐼 geom = (cid:214) 𝑎, 𝑗 (cid:18) 𝑧 ( 𝑎, 𝑗 ) ◦ 𝑧 ( 𝑎 + , 𝑗 ) 𝑧 ( 𝑎 + , 𝑗 ) ◦ 𝑧 ( 𝑎 + , 𝑗 ) (cid:19) 𝑝 (6.12)On the other hand, the Hamiltonian term has a first-order nonlocality in the spin-chaindirection, but also has a first-order nonlocality in the time direction, since the matrix elementsof the Hamiltonian entering the integral are always of the form 𝑘 (cid:104) 𝑧 ( 𝑎 + , 𝑗 ) | ˆ 𝐻 | 𝑧 ( 𝑎, 𝑗 )(cid:105) . Thelatter nonlocality will not play a role, since the contribution of such matrix element alwayscomes with a damping factor 𝑘 , and the nonlocality being of order 𝑘 as well enters onlysubleading terms. In any case, the contribution of the Hamiltonian may be written as 𝐼 H = (cid:214) 𝑎 (cid:32) + 𝑝 𝑖𝛼𝑘 ∑︁ 𝑗 𝑧 ( 𝑎, 𝑗 ) ◦ 𝑧 ( 𝑎 + , 𝑗 + ) 𝑧 ( 𝑎, 𝑗 ) ◦ 𝑧 ( 𝑎 + , 𝑗 ) 𝑧 ( 𝑎, 𝑗 + ) ◦ 𝑧 ( 𝑎 + , 𝑗 ) 𝑧 ( 𝑎, 𝑗 + ) ◦ 𝑧 ( 𝑎 + , 𝑗 + ) (cid:33) (6.13)We may now exponentiate these expressions and take the limit 𝑘 → ∞ , thus obtaining acontinuous time variable 𝑡 : 𝑈 ( 𝑞, 𝑣 ) = ∫ (cid:214) 𝑡 ∈[ , ] (cid:214) 𝑗 𝑑𝜇 ( 𝑧 ( 𝑡, 𝑗 ) , 𝑧 ( 𝑡, 𝑗 )) (cid:18) 𝑧 ( , 𝑗 ) ◦ 𝑧 ( , 𝑗 ) 𝑧 ( , 𝑗 ) ◦ 𝑧 ( , 𝑗 ) (cid:19) 𝑝 exp ( i S) , (6.14)where S = 𝑝 ∫ 𝑑𝑡 ∑︁ 𝑗 (cid:18) 𝑖 (cid:164) 𝑧 ( 𝑗 ) ◦ 𝑧 ( 𝑗 )| 𝑧 ( 𝑗 ) | + 𝛼 | 𝑧 ( 𝑗 ) ◦ 𝑧 ( 𝑗 + ) | | 𝑧 ( 𝑗 ) | | 𝑧 ( 𝑗 + ) | (cid:19) (6.15)with boundary conditions 𝑧 ( , 𝑗 ) = 𝑞 ( 𝑗 ) , 𝑧 ( , 𝑗 ) = 𝑣 ( 𝑗 ) . 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 “ 𝑗 ”, — there are several inequivalent ways to doit. It is well-known that the isotropic (‘XXX’) spin chain has two “vacua”, 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 𝐿 — 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 𝑧 ’s at theneighboring sites are very close to each other, that is 𝑧 ( 𝑗 + ) − 𝑧 ( 𝑗 ) ∼ 𝐿 ( 𝐿 is the length ofthe spin chain, i.e. the number of sites). The first term in (6.15) then simply produces ∫ 𝑑𝑡 ∑︁ 𝑗 i (cid:164) 𝑧 ( 𝑗 ) ◦ 𝑧 ( 𝑗 ) 𝑧 ( 𝑗 ) ◦ 𝑧 ( 𝑗 ) → 𝐿 ∫ 𝑑𝑡 / ∫ − / 𝑑𝑥 i (cid:164) 𝑧 ( 𝑡, 𝑥 ) ◦ 𝑧 ( 𝑡, 𝑥 ) 𝑧 ( 𝑡, 𝑥 ) ◦ 𝑧 ( 𝑡, 𝑥 ) , (6.16)whereas the expression in the second term can be rewritten in the same spirit: ∑︁ 𝑗 | 𝑧 ( 𝑗 ) ◦ 𝑧 ( 𝑗 + ) | | 𝑧 ( 𝑗 ) | | 𝑧 ( 𝑗 + ) | = − ∑︁ 𝑗 | Δ 𝑧 ( 𝑗 ) | | 𝑧 ( 𝑗 ) | − | 𝑧 ( 𝑗 ) ◦ Δ 𝑧 ( 𝑗 ) | | 𝑧 ( 𝑗 ) | + · · · Upon taking the continuum limit and rescaling 𝑥 → 𝐿 𝑥 the full action acquires the form S = 𝑝 ∫ 𝑑𝑡 ∫ R 𝑑𝑥 (cid:20) i (cid:164) 𝑧 ( 𝑡, 𝑥 ) ◦ 𝑧 ( 𝑡, 𝑥 ) 𝑧 ( 𝑡, 𝑥 ) ◦ 𝑧 ( 𝑡, 𝑥 ) − (cid:18) 𝜕 𝑥 𝑧 ◦ 𝜕 𝑥 𝑧𝑧 ◦ 𝑧 − ( 𝑧 ◦ 𝜕 𝑥 𝑧 ) ( 𝜕 𝑥 𝑧 ◦ 𝑧 )( 𝑧 ◦ 𝑧 ) (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 𝑛 − (for example, in thesecond term in (6.17) one immediately recognizes the Fubini-Study metric). The simplestexample corresponds to 𝑛 =
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) 𝑛 instead of the complex coordinates 𝑧, 𝑧 (the two parametrizations arerelated via the stereographic projection: 𝑛 + i 𝑛 = 𝑧 + 𝑧𝑧 , 𝑛 = − 𝑧𝑧 + 𝑧𝑧 ). Then the e.o.m., whichfollows from Lagrangian (6.17), is: 𝜕 (cid:174) 𝑛𝜕𝑡 = (cid:174) 𝑛 × 𝜕 (cid:174) 𝑛𝜕𝑥 . (6.18)Expanding around a constant magnetization direction, (cid:174) 𝑛 = (cid:174) 𝑛 + 𝛿 (cid:174) 𝑛 , one obtains the linearequation 𝜕𝛿 (cid:174) 𝑛𝜕𝑡 = (cid:174) 𝑛 × 𝜕 𝛿 (cid:174) 𝑛𝜕𝑥 , which describes spin waves with quadratic dispersion. The antiferromagnetic limit is much more involved. The main difference is that in this casethe 𝑧 -variables on neighboring sites are no longer close to each other. Let us first elaborateon the case of the sphere, that is 𝑛 =
2, which was for the first 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) 𝑛 ( 𝑗 + ) (cid:39) −(cid:174) 𝑛 ( 𝑗 ) . Interms of the complex coordinates used above this may be written as 𝑧 ( 𝑗 + ) (cid:39) − 𝑧 ( 𝑗 ) , or,using homogeneous coordinates, as 𝑧 ( 𝑗 + ) = 𝑧 ( 𝑗 ) , 𝑧 ( 𝑗 + ) = − 𝑧 ( 𝑗 ) . 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 𝑛 − with 𝑛 ≥
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 first 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 configuration. For the Hamiltonian one takes the Heisenberg Hamiltonian 𝐻 = 𝐿 ∑︁ 𝑗 = Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + )) , (7.1)where it is understood that 𝑆 ( 𝑗 ) and 𝑆 ( 𝑗 + ) are in conjugate representations. For simplicitywe assume in this section that either 𝑅 or 𝑅 is the rank- 𝑝 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 𝑡 -continuum limit: S = 𝑝 ∫ 𝑑𝑡 ∑︁ 𝑗 (cid:18) 𝑖 (cid:164) 𝑧 ( 𝑗 ) ◦ 𝑧 ( 𝑗 )| 𝑧 ( 𝑗 ) | + 𝛼 | 𝑧 ( 𝑗 ) ◦ 𝑧 ( 𝑗 + ) | | 𝑧 ( 𝑗 ) | | 𝑧 ( 𝑗 + ) | (cid:19) . (7.2)The difference between the second terms in (6.15) and (7.2) precisely reflects the differencebetween the representations at adjacent sites. The minimum of the Hamiltonian H = − | 𝑧 ( 𝑗 )◦ 𝑧 ( 𝑗 + )| | 𝑧 ( 𝑗 )| | 𝑧 ( 𝑗 + )| is clearly reached for 𝑧 ( 𝑗 + ) = 𝑧 ( 𝑗 ) . The important observation is thatfor such configurations the first term in (7.2) turns into a full derivative, since on everytwo neighboring sites 𝑖 (cid:164) 𝑧 ( 𝑗 )◦ 𝑧 ( 𝑗 )| 𝑧 ( 𝑗 )| + 𝑖 (cid:164) 𝑧 ( 𝑗 + )◦ 𝑧 ( 𝑗 + )| 𝑧 ( 𝑗 + )| = 𝑖 𝑑𝑑𝑡 ( log | 𝑧 ( 𝑗 ) | ) . There is a simple butfundamental explanation of this fact. Consider the space CP 𝑛 − × CP 𝑛 − , with the symplecticform on it being the sum of two Fubini-Study forms (1.16): Ω = Ω + Ω . We then have thefollowing statement : The definition of Lagrangian submanifold was given in section 1.4. CP 𝑛 − ⊂ CP 𝑛 − × CP 𝑛 − , def. by 𝑧 → ( 𝑧, 𝑧 ) , is Lagrangian. (7.3)Indeed, since Ω = 𝑑𝜃 , where 𝜃 is the one-form entering the first term in (7.2), and therestriction of the symplectic form Ω (cid:12)(cid:12) CP 𝑛 − = 𝜃 (cid:12)(cid:12) CP 𝑛 − = 𝑑 𝑓 for somefunction 𝑓 , implying that the kinetic term is a total derivative.Let us now expand the action (7.2) around the “vacuum” 𝑧 ( 𝑗 + ) = 𝑧 ( 𝑗 ) . The variables 𝑧 ( 𝑗 + ) and 𝑧 ( 𝑗 + ) are expressed in terms of 𝑧 ( 𝑗 ) in the following fashion: 𝑧 ( 𝑗 + ) = 𝑧 ( 𝑗 ) + 𝐿 𝜏 ( 𝑗 ) , 𝑧 ( 𝑗 + ) = 𝑧 ( 𝑗 ) + 𝐿 𝑧 ( 𝑗 ) (cid:48) (7.4)For convenience we introduce the projector Π ( 𝑗 ) = − 𝑧 ( 𝑗 )⊗ 𝑧 ( 𝑗 )| 𝑧 ( 𝑗 )| onto the subspace of C 𝑛 orthogonal to the vector 𝑧 ( 𝑗 ) . Then the terms in the Hamiltonian have the following expan-sions: | 𝑧 ( 𝑗 ) ◦ 𝑧 ( 𝑗 + ) | | 𝑧 ( 𝑗 ) | | 𝑧 ( 𝑗 + ) | (cid:39) 𝐿 𝜏 ( 𝑗 ) ◦ Π ( 𝑗 ) ◦ 𝜏 ( 𝑗 )| 𝑧 ( 𝑗 ) | , | 𝑧 ( 𝑗 + ) ◦ 𝑧 ( 𝑗 + ) | | 𝑧 ( 𝑗 ) | | 𝑧 ( 𝑗 + ) | (cid:39) 𝐿 (cid:101) 𝜏 ( 𝑗 ) ◦ Π ( 𝑗 ) ◦ (cid:101) 𝜏 ( 𝑗 )| 𝑧 ( 𝑗 ) | , (7.5)where (cid:101) 𝜏 ( 𝑗 ) = 𝜏 ( 𝑗 ) − 𝑧 ( 𝑗 ) (cid:48) . The kinetic terms are expanded as follows: 𝑖 (cid:164) 𝑧 ( 𝑗 ) ◦ 𝑧 ( 𝑗 )| 𝑧 ( 𝑗 ) | + 𝑖 (cid:164) 𝑧 ( 𝑗 + ) ◦ 𝑧 ( 𝑗 + )| 𝑧 ( 𝑗 + ) | = 𝑖 𝐿 𝜏 ( 𝑗 ) ◦ Π ( 𝑗 ) ◦ (cid:164) 𝑧 ( 𝑗 ) − 𝜏 ( 𝑗 ) ◦ Π ( 𝑗 ) ◦ (cid:164) 𝑧 ( 𝑗 )| 𝑧 ( 𝑗 ) | + . . . (7.6)where . . . denotes a full derivative. Thus, the action (7.2) acquires the following form: S = 𝑝 ∫ 𝑑𝑡 ∑︁ 𝑗 | 𝑧 ( 𝑗 ) | (cid:16) 𝑖 𝐿 (cid:2) 𝜏 ( 𝑗 ) ◦ Π ( 𝑗 ) ◦ (cid:164) 𝑧 ( 𝑗 ) − 𝜏 ( 𝑗 ) ◦ Π ( 𝑗 ) ◦ (cid:164) 𝑧 ( 𝑗 ) (cid:3) + (7.7) + 𝐿 [ 𝜏 ( 𝑗 ) ◦ Π ( 𝑗 ) ◦ 𝜏 ( 𝑗 ) + ( 𝜏 ( 𝑗 ) − 𝑧 ( 𝑗 ) (cid:48) ) ◦ Π ( 𝑗 ) ◦ ( 𝜏 ( 𝑗 ) − 𝑧 ( 𝑗 ) (cid:48) )] (cid:17) . Now we simply need to “integrate out” the fields 𝜏, 𝜏 . Upon setting 𝜏, 𝜏 equal to theirstationary values we also pass to the continuum limit with respect to the “ 𝑗 ” index. Thisleads to the following expression: S = 𝑝 ∫ 𝑑𝑡 ∞ ∫ −∞ 𝑑𝑥 (cid:104) 𝜕 𝜇 𝑧 ( 𝑥, 𝑡 ) ◦ Π ( 𝑧, 𝑧 )| 𝑧 ( 𝑥, 𝑡 ) | ◦ 𝜕 𝜇 𝑧 ( 𝑥, 𝑡 ) (7.8) − i2 𝜖 𝜇𝜈 𝜕 𝜇 𝑧 ( 𝑥, 𝑡 ) ◦ Π ( 𝑧, 𝑧 )| 𝑧 ( 𝑥, 𝑡 ) | ◦ 𝜕 𝜈 𝑧 ( 𝑥, 𝑡 ) (cid:105) . CP 𝑛 − sigma model, whereas the secondterm is the pull-back to the worldsheet of the Kähler form. The second term is topologicaland corresponds to the theta-angle 𝜃 = 𝜋 𝑝 mod 2 𝜋 . 𝑛 limit A useful method for the analysis of vector-like systems (such as the CP 𝑛 − model, wherethe dynamical variable is the vector 𝑧 ) is the 𝑛 -expansion. The first step is in rewriting themodel (7.8) as a gauged linear sigma model (GLSM). This is the first 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 = ∫ 𝑑 𝑥 (cid:34) 𝑛 ∑︁ 𝛼 = | 𝐷 𝜇 𝑧 𝛼 | − 𝜆 (cid:32) 𝑛 ∑︁ 𝛼 = | 𝑧 𝛼 | − 𝑛𝑔 (cid:33) (cid:35) + 𝜃 𝜋 ∫ 𝑑𝐴. (7.9)Here 𝑧 𝛼 are the components of 𝑧 , which is normalized as | 𝑧 | = 𝑛𝑔 (we have introduced the‘t Hooft coupling constant 𝑔 of the sigma model), 𝐷 𝜇 is a U ( ) -covariant derivative, i.e. 𝐷 𝜇 𝑧 𝛼 = 𝜕 𝜇 𝑧 𝛼 − 𝑖 𝐴 𝜇 𝑧 𝛼 , and 𝜆 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 𝑧 , i.e. 𝑧 → λ 𝑧 with λ ∈ C ∗ , which is in accordance with the complex definitionof the projective space. We have used this freedom to normalize the 𝑧 vector. Even moreimportantly, we have introduced a gauge field 𝐴 𝜇 . This gauge field 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 𝜃 = 𝑝 𝜋 , 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 𝑧 fields, so that they can be integrated out.The resulting action of the 𝜆 and 𝐴 𝜇 fields is S = 𝑛 (cid:20) 𝑖 Tr Log (cid:16) − 𝐷 𝜇 − 𝜆 (cid:17) + ∫ 𝑑 𝑥 𝜆𝑔 (cid:21) + 𝜃 𝜋 ∫ 𝑑𝐴. (7.10)Since this expression appears in the exponent in the integrand of the path integral, thelarge- 𝑛 limit corresponds to a stationary phase approximation. The critical point equation,obtained by varying w.r.t. 𝜆 , – the so-called gap equation – has the form (due to Lorentz andtranslational invariance one sets 𝐴 𝜇 = 𝜆 = const . at the critical point)1 𝑔 − ∫ 𝑑 𝑘 ( 𝜋 ) 𝑘 + 𝜆 = . (7.11)As the integral is UV-divergent, one imposes a cut-off Λ , and the solution is 𝜆 = Λ 𝑒 − 𝜋𝑔 . (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 ψ 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 𝑛 massive fields 𝑧 𝛼 (with mass 𝑚 = 𝜆 ) interacting witha gauge field 𝐴 𝜇 . One can show [90, 91, 246] that the effect of the gauge field is to generatean attractive 𝜃 -dependent potential between the ‘quarks’ 𝑧 and ‘antiquarks’ 𝑧 , which confinesthem for all values of 𝜃 [246]. The mass of the lowest bound state is 2 𝑚 + . . . , where . . . arepower-like corrections in 𝑛 (which also depend on 𝜃 ). One concludes that, for large 𝑛 , themodel is massive for all values of 𝜃 . For a review of 𝜃 -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] defined by the Hamiltonian 𝐻 bi =
12 cos ψ ∑︁ 𝑗 Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + )) −
14 sin ψ ∑︁ 𝑗 ( Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + ))) , (7.13)where 𝑆 are the matrices of spin-1 generators of 𝔰𝔲 ( ) . The phases of this chain as a functionof ψ 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 fivespecial points in the diagram. They correspond to the following chains: ◦ Spin-1 Heisenberg, with Hamiltonian 𝐻 = (cid:205) 𝑗 Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + )) . By the originalargument of Haldane it is gapped, and the field theory mapping results in an 𝑆 sigmamodel with 𝜃 = ◦ AKLT (tan ( ψ ) = − ): this model lies in the Haldane phase as well. The ground statemay be calculated exactly and is translationally invariant. See Section 4.2. ◦ The critical SU ( ) spin-1 integrable point ( ψ = 𝜋 ). This is a higher-spin integrableextension of the spin-1/2 Heisenberg Hamiltonian [219, 35]. The continuum limit isdescribed by the SU ( ) 𝑘 = WZNW model.50 At ψ = − 𝜋 the symmetry is enhanced to SU ( ) and one has the SU ( ) extension [217]of the Heisenberg Hamiltonian 𝐻 = (cid:205) 𝑗 Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + )) , where 𝑆 now contain thegenerators of 𝔰𝔲 ( ) in the fundamental representation at all sites. Again the spectrumis gapless, the critical theory described by SU ( ) 𝑘 = WZNW model. ◦ At ψ = 𝜋 one again has an enhancement to SU ( ) , this time with matrices 𝑆 whoseentries generate alternating fundamental/anti-fundamental representations, i.e. 𝑆 𝛼𝛽 ( 𝑗 + ) = 𝑆 𝛼𝛽 ( 𝑗 ) . This model is integrable and gapped [5, 82, 196, 37, 156], with a two-folddegenerate ground state and broken translational invariance (the ‘dimer’).The two large dots in the diagram, which correspond to the first and last points in the abovelist, are particularly important for us. As already mentioned, the ψ = 𝑆 sigma model with vanishing 𝜃 -angle. The ψ = 𝜋 point corresponds to an alternatingSU ( ) spin chain, exactly of the type considered in the previous section, so that the resultingfield theory is a CP sigma model with 𝜃 = 𝜋 ( 𝑝 = CP 𝑛 − -models with 𝑛 > 𝜃 = 𝜋 , which makes them different from the CP model. Atthe same time this makes it consistent with the large- 𝑛 description above, which predicts amass gap for the sigma models. ( ) case We now want to move forward from the Hamiltonian (7.1) and find the sigma model whicharises upon taking the continuum limit around the antiferromagnetic “vacuum” of the SU ( ) spin chain with the Hamiltonian (3.4): H = 𝐿 ∑︁ 𝑖 = ( 𝐽 Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + )) + 𝐽 Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + ))) , (7.14)where 𝐽 > , 𝐽 > S = 𝑝 ∫ 𝑑𝑡 ∑︁ 𝑗 (cid:18) 𝑖 (cid:164) 𝑧 ( 𝑗 ) ◦ 𝑧 ( 𝑗 )| 𝑧 ( 𝑗 ) | + 𝛼 · H 𝑗 (cid:19) , where (7.15) H 𝑗 = 𝐽 | 𝑧 ( 𝑗 ) ◦ 𝑧 ( 𝑗 + ) | | 𝑧 ( 𝑗 ) | | 𝑧 ( 𝑗 + ) | + 𝐽 | 𝑧 ( 𝑗 ) ◦ 𝑧 ( 𝑗 + ) | | 𝑧 ( 𝑗 ) | | 𝑧 ( 𝑗 + ) | .
51n this formula each of the variables 𝑧 ( 𝑗 ) 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 configuration is whenthe 𝑧 -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 + 𝑧 𝑧 = 𝑧 = − 𝑧 , which isthe antipodal involution discussed above. When 𝑛 = 𝑧 -vectors 𝑧 , 𝑧 , 𝑧 sitting at these sites : 𝑧 ◦ 𝑧 = 𝑧 ◦ 𝑧 = 𝑧 ◦ 𝑧 = . (7.16)The submanifold of ( CP ) × described by (7.16) is the flag manifold F . We’re now goingto elaborate on this simplest nontrivial example.The first 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 first focus onan arbitrary set of 3 consecutive sites. Then the kinetic term in the spin chain Lagrangian isthe pull-back 𝜃 𝑡 of the following one-form: 𝜃 = i 𝑑𝑧 ◦ 𝑧 𝑧 ◦ 𝑧 + i 𝑑𝑧 ◦ 𝑧 𝑧 ◦ 𝑧 + i 𝑑𝑧 ◦ 𝑧 𝑧 ◦ 𝑧 (7.17)This is the Poincaré-Liouville one-form for the product symplectic form Ω on CP × CP × CP , so that 𝑑𝜃 = Ω . 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 ⊂ ( CP ) × , and more generally F 𝑛 ⊂ ( CP 𝑛 − ) × 𝑛 ,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 𝜃 | F = 𝑑 𝑓 , where 𝑓 is a function (in fact, 𝑓 = 𝑖 log ( 𝜖 𝛼𝛽𝛾 𝑧 𝛼 𝑧 𝛽 𝑧 𝛾 ) ),so the integral ∫ 𝜃 𝑡 𝑑𝑡 = 𝑓 ( ) − 𝑓 ( ) 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 ‘moduli space’ of Néel 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 configuration (7.16).Moreover, since 0 ≤ H 𝑗 ≤ 𝐽 + 𝐽 , 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 sufficiently long spin chain). In this casewe split the spin chain into ˆ 𝐿 segments of length 3 and focus for the moment on just one ofthese segments, which is the elementary cell number 𝑘 in the chain. On each of the three sites we have a three-dimensional complex vector 𝑧 𝐴 . Let us form a3 × 𝑍 .The antiferromagnetic configuration corresponds to the case where the three vectors aremutually orthogonal.Now we need to take the fluctuations into account, and in order to build the sought forexpansion we will employ the so-called 𝑄 𝑅 decomposition of a matrix. The
𝑄 𝑅 decomposi-tion theorem says that an arbitrary matrix 𝑍 may be decomposed into a product of a unitarymatrix 𝑈 and an upper triangular one 𝐵 + : 𝑍 = 𝑈 ◦ 𝐵 + (8.1)This statement is equivalent to the Gram-Schmidt orthogonalization theorem. Let usparametrize 𝐵 + at link 𝑘 in the following way: 𝐵 + ( 𝑘 ) = (cid:169)(cid:173)(cid:171) 𝐿 x ( 𝑘 ) 𝐿 y ( 𝑘 ) 𝐿 z ( 𝑘 ) (cid:170)(cid:174)(cid:172) (cid:169)(cid:173)(cid:171) a ( 𝑘 ) ( 𝑘 )
00 0 c ( 𝑘 ) (cid:170)(cid:174)(cid:172) . (8.2)If we denote the columns of the matrix 𝑈 as ( 𝑢 , 𝑢 , 𝑢 ) , the decomposition (8.1) saysthat 𝑧 ( 𝑘 ) = a ( 𝑘 ) 𝑢 ( 𝑘 ) , 𝑧 ( 𝑘 ) = b ( 𝑘 ) ( 𝑢 ( 𝑘 ) + 𝐿 x ( 𝑘 ) 𝑢 ( 𝑘 )) , (8.3) 𝑧 ( 𝑘 ) = c ( 𝑘 ) ( 𝑢 ( 𝑘 ) + 𝐿 y ( 𝑘 ) 𝑢 ( 𝑘 ) + 𝐿 z ( 𝑘 ) 𝑢 ( 𝑘 )) 𝐿 . A simple calculation reveals that (suppressing the index 𝑘 for the moment) 𝐽 𝑡 = (cid:18) i (cid:164) 𝑧 ◦ 𝑧 𝑧 ◦ 𝑧 + i (cid:164) 𝑧 ◦ 𝑧 𝑧 ◦ 𝑧 + i (cid:164) 𝑧 ◦ 𝑧 𝑧 ◦ 𝑧 (cid:19) x = y = z = − (8.4) − i 𝐿 (cid:16) x 𝑢 ◦ (cid:164) 𝑢 + y 𝑢 ◦ (cid:164) 𝑢 + z 𝑢 ◦ (cid:164) 𝑢 − c.c. (cid:17) + ... The hypothesis of the existence of a continuum limit implies that 𝑢 ( 𝑘 ) , 𝑢 ( 𝑘 ) , 𝑢 ( 𝑘 ) varymildly with 𝑘 , in other words we may approximate 𝑢 𝐴 ( 𝑘 + ) = 𝑢 𝐴 ( 𝑘 ) + 𝐿 𝑢 𝐴 ( 𝑘 ) (cid:48) + ... (8.5) H , ( k − , k ) H , ( k − , k ) H , ( k − , k ) H , ( k, k ) H , ( k, k ) H , ( k, k ) k − k Figure 10: Explanation of the various terms calculated in (8.7).Let us introduce the quantity H 𝐴,𝐵 ( 𝑘, 𝑘 (cid:48) ) = | 𝑧 𝐴 ( 𝑘 ) ◦ 𝑧 𝐵 ( 𝑘 (cid:48) ) | | 𝑧 𝐴 ( 𝑘 ) | | 𝑧 𝐵 ( 𝑘 (cid:48) ) | , (8.6)which is the density of the Hamiltonian H from (7.15), if the indices 𝐴, 𝐵, 𝑘, 𝑘 (cid:48) change ina particular range. Indeed, we need to calculate H 𝐴,𝐵 ( 𝑘, 𝑘 (cid:48) ) for nearest- and next-to-nearestneighbor sites, using the formulas (8.3)-(8.5) and keeping only the leading orders of 𝐿 (seeFig. 10 for an explanation of what these terms stand for): H , ( 𝑘, 𝑘 ) = | 𝑧 ( 𝑘 ) ◦ 𝑧 ( 𝑘 ) | | 𝑧 ( 𝑘 ) | | 𝑧 ( 𝑘 ) | (cid:39) 𝐿 | x ( 𝑘 ) | , H , ( 𝑘, 𝑘 ) = | 𝑧 ( 𝑘 ) ◦ 𝑧 ( 𝑘 ) | | 𝑧 ( 𝑘 ) | | 𝑧 ( 𝑘 ) | (cid:39) 𝐿 | z ( 𝑘 ) | H , ( 𝑘, 𝑘 ) = | 𝑧 ( 𝑘 ) ◦ 𝑧 ( 𝑘 ) | | 𝑧 ( 𝑘 ) | | 𝑧 ( 𝑘 ) | (cid:39) 𝐿 | y ( 𝑘 ) | H , ( 𝑘 − , 𝑘 ) = | 𝑧 ( 𝑘 − ) ◦ 𝑧 ( 𝑘 ) | | 𝑧 ( 𝑘 − ) | | 𝑧 ( 𝑘 ) | (cid:39) 𝐿 | − 𝑢 ( 𝑘 ) (cid:48) ◦ 𝑢 ( 𝑘 ) + y ( 𝑘 ) | (8.7) H , ( 𝑘 − , 𝑘 ) = | 𝑧 ( 𝑘 − ) ◦ 𝑧 ( 𝑘 ) | | 𝑧 ( 𝑘 − ) | | 𝑧 ( 𝑘 ) | (cid:39) 𝐿 | − 𝑢 (cid:48) ( 𝑘 ) ◦ 𝑢 ( 𝑘 ) + z ( 𝑘 ) | H , ( 𝑘 − , 𝑘 ) = | 𝑧 ( 𝑘 − ) ◦ 𝑧 ( 𝑘 ) | | 𝑧 ( 𝑘 − ) | | 𝑧 ,𝑘 | (cid:39) 𝐿 | − 𝑢 (cid:48) ( 𝑘 ) ◦ 𝑢 ( 𝑘 ) + x ( 𝑘 ) | 𝑥 ( 𝑘 ) , 𝑦 ( 𝑘 ) , 𝑧 ( 𝑘 ) that enter only quadratically, oneobtains the Lagrangian (we set for simplicity 𝐽 = , 𝐽 = 𝑎 ) L = + 𝑎 (| 𝑢 ◦ (cid:164) 𝑢 | − 𝑎 𝐿 | 𝑢 (cid:48) ◦ 𝑢 | ) − i 𝑎 + 𝑎 𝐿 𝜖 𝜇𝜈 ( 𝑢 ◦ 𝜕 𝜇 𝑢 ) ( 𝑢 ◦ 𝜕 𝜈 𝑢 ) + (8.8) + + 𝑎 (| 𝑢 ◦ (cid:164) 𝑢 | − 𝑎 𝐿 | 𝑢 ◦ 𝑢 (cid:48) | ) − i1 + 𝑎 𝐿 𝜖 𝜇𝜈 ( 𝑢 ◦ 𝜕 𝜇 𝑢 ) ( 𝑢 ◦ 𝜕 𝜈 𝑢 ) ++ + 𝑎 (| 𝑢 ◦ (cid:164) 𝑢 | − 𝑎 𝐿 | 𝑢 ◦ 𝑢 (cid:48) | ) − i 𝑎 + 𝑎 𝐿 𝜖 𝜇𝜈 ( 𝑢 ◦ 𝜕 𝜇 𝑢 ) ( 𝑢 ◦ 𝜕 𝜈 𝑢 ) Of course, each of the first 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 flavourwave velocities are equal in (8.8) is really a coincidence that happens for 𝑛 =
3. For 𝑛 > 𝑢 ≡ i ( 𝑢 ◦ 𝑑𝑢 ) ∧ ( 𝑢 ◦ 𝑑𝑢 ) (8.9) 𝑣 ≡ − i ( 𝑢 ◦ 𝑑𝑢 ) ∧ ( 𝑢 ◦ 𝑑𝑢 ) (8.10) 𝑤 ≡ i ( 𝑢 ◦ 𝑑𝑢 ) ∧ ( 𝑢 ◦ 𝑑𝑢 ) (8.11)Then the three epsilon-terms in (8.8) are the pull-back of the following 2-form: 𝜔 = + 𝑎 ( 𝑣 − 𝑎 𝑢 − 𝑎 𝑤 ) (8.12)The crucial fact is that 𝜔 may be split in two parts: a topological one (the 𝜃 -term) andthe non-topological one (the 𝐵 -field, or Kalb-Ramond field, in sigma model terminology).The splitting may be achieved by noting that the above Lagrangian possesses a Z ‘quasi-symmetry’, which acts on the vectors ( 𝑢 , 𝑢 , 𝑢 ) of the flag by cyclically permuting them: Z : 𝑢 → 𝑢 → 𝑢 → 𝑢 . (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: 𝜖 = It was noted in [73] that there exists a canonical geometric expression for the metric arising in this way.Suppose ( 𝚽 , 𝜔 ) is a symplectic manifold (in this case the phase space of an elementary cell), and H a functionon it – the classical Hamiltonian – that attains a minimum on a Lagrangian submanifold N ⊂ 𝚽 (the target spaceof the sigma model). In this case one can define the (inverse) metric on N by the formula 𝑔 𝑖 𝑗 = 𝜔 𝑖𝑘 (cid:16) 𝜕 H 𝜕𝑥 (cid:17) 𝑘𝑙 𝜔 𝑙 𝑗 . Z it is shifted by an integral topological term , i.e. an element of 𝐻 (F , Z ) . As a result, thepartition function, which is given by a path integral of the type Z = ∫ 𝑒 𝑖 ∫ 𝑑𝑡 𝑑𝑥 L (cid:214) 𝑑𝑈 , (8.14)is unaltered. This is the same argument that is used to prove that the path integral of Chern-Simons theory is well-defined [94, 252]. Indeed, we will now show that 𝜔 may be split in anon-topological part that is invariant w.r.t. Z and a 𝜃 -angle part that transforms non-trivially.First of all, the two-forms transform as follows under Z : Z : 𝑢 → 𝑤 → 𝑣 → 𝑢 . (8.15)As a result, the only Z -singlet is 𝑢 + 𝑣 + 𝑤 , so that one may decompose 𝜔 = − 𝑎 ( + 𝑎 ) ( 𝑢 + 𝑣 + 𝑤 ) − ( 𝑢 − 𝑣 + 𝑤 − 𝑣 ) , (8.16)Let us now show that the second term is topological. First of all, recall from section 1.4 thatevery element of 𝐻 (F , R ) is a linear combination of three forms Ω 𝐴 ( 𝐴 = , , ) , whichare the pull-backs to the flag manifold of the Fubini-Study forms corresponding to 𝑢 , 𝑢 , 𝑢 .It is easy to relate these Fubini-Study forms to 𝑢, 𝑣, 𝑤 . Recall that Ω = 𝑑𝑢 ∧ 𝑑𝑢 − ( 𝑑𝑢 ◦ 𝑢 ) ∧ ( 𝑑𝑢 ◦ 𝑢 ) (8.17)and similar expressions hold for Ω and Ω . Now let us use the identity 𝑢 ⊗ 𝑢 + 𝑢 ⊗ 𝑢 + 𝑢 ⊗ 𝑢 = . Using this, we may rewrite the restriction of Ω to the flag manifold F in thefollowing way: Ω | F = ( 𝑑𝑢 ◦ 𝑢 ) ∧ ( 𝑑𝑢 ◦ 𝑢 ) + ( 𝑑𝑢 ◦ 𝑢 ) ∧ ( 𝑑𝑢 ◦ 𝑢 ) = 𝑢 − 𝑣. (8.18)External differentiation and restriction to a submanifold are commutative operations, thereforethis restricted form is a closed 2-form on F . Analogously Ω | F = 𝑤 − 𝑢, Ω | F = 𝑣 − 𝑤 . Inparticular, we see that the sum ( Ω + Ω + Ω ) | F = 𝐻 (F , Z ) . Keeping in mind (8.19), we may write 𝜔 as 𝜔 (cid:39) − 𝑎 ( + 𝑎 ) ( 𝑢 + 𝑣 + 𝑤 ) + Ω + Ω + Ω : = 𝜔 top (8.20)Since the action stands in the exponent of the integrand in (8.14), 𝜔 is defined moduloelements of 𝐻 (F , Z ) . This is important, because under the action of Z the topological56erm 𝜔 top shifts precisely by such terms. Indeed, since the action of Z cyclically permutes Ω , Ω , Ω , under its action one has Z : 𝜔 top → 𝜔 top − Ω + Ω + Ω = + Ω ∈ 𝐻 (F , Z ) (cid:39) 𝜔 top , (8.21)where (cid:39) means ‘up to an element of 𝐻 (F , Z ) ’. This property will be essential for thediscussion of discrete ’t 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 ( 𝑛 ) case, where the Hamiltonian is given by (3.4). The Néel state inthis case is given by a direct generalization of (7.16), namely requiring that the 𝑛 consecutivevectors 𝑍 : = ( 𝑧 , . . . 𝑧 𝑛 ) are all pairwise orthogonal: 𝑧 𝐴 ◦ 𝑧 𝐵 = 𝐴 ≠ 𝐵 . (9.1)In order to derive the continuum theory, one follows the steps described in the previoussection: one first introduces deviations from the Néel configuration (9.1) and performs afactorization (8.1) 𝑍 = 𝑈 ◦ 𝐵 + , where 𝑈 ∈ U ( 𝑛 ) is unitary and 𝐵 + is strictly upper-triangular.As before, the matrix 𝐵 + describes the deviations from the Néel state in a single unit cell.One then expands the Lagrangian to quadratic order in the elements of 𝐵 + , as well as inthe cell-to-cell variations, and integrates over the 𝐵 + elements that enter algebraically. Thecalculation is rather tedious, and its details can be found either in [73] or in [237]. ( 𝑛 ) spin chain Here we write out the resulting action of the field theory describing the SU( 𝑛 ) chain in therank- 𝑝 symmetric irrep: 𝑆 = ∑︁ ≤ 𝐴<𝐵 ≤ 𝑛 ∫ 𝑑𝑥𝑑𝜏 𝑔 | 𝐴 − 𝐵 | (cid:18) 𝑣 | 𝐴 − 𝐵 | | 𝑢 𝐴 ◦ 𝜕 𝑥 𝑢 𝐵 | + 𝑣 | 𝐴 − 𝐵 | | 𝑢 𝐴 ◦ 𝜕 𝜏 𝑢 𝐵 | (cid:19) (9.2) − 𝜖 𝜇𝜈 ∑︁ ≤ 𝐴<𝐵 ≤ 𝑛 𝜆 | 𝐴 − 𝐵 | ∫ 𝑑𝑥𝑑𝜏 ( 𝑢 𝐴 ◦ 𝜕 𝜇 𝑢 𝐵 ) ( 𝑢 𝐵 ◦ 𝜕 𝜈 𝑢 𝐴 ) + 𝑆 top , where 𝑣 𝑡 = 𝑛 𝑝 √ 𝐽 𝑡 𝐽 𝑛 − 𝑡 is the flavour wave velocity associated with the pair of couplings 𝐽 𝑡 and 𝐽 𝑛 − 𝑡 of the spin chain (3.4). The 𝜆 -terms are the generalizations of the non-topologicalcontribution to 𝜔 discussed in the previous section for 𝑛 = 𝑆 top = ∫ 𝜔 top is a higher- 𝑛 generalization of the topological ( 𝜃 )-term, discussed in detail57elow. As we already mentioned earlier, the cases 𝑛 = 𝑛 = 𝑔 𝑡 = 𝑛𝑣 𝑡 ( 𝐽 𝑡 + 𝐽 𝑛 − 𝑡 ) (9.3)and 𝑛𝜆 𝑡 𝑝 = ( 𝑛 − 𝑡 ) 𝐽 𝑛 − 𝑡 − 𝑡 𝐽 𝑡 𝐽 𝑡 + 𝐽 𝑛 − 𝑡 . (9.4)Since the coupling constants and velocities satisfy 𝑔 𝑡 = 𝑔 𝑛 − 𝑡 and 𝑣 𝑡 = 𝑣 𝑛 − 𝑡 , we conclude thatthere are (cid:98) 𝑛 (cid:99) velocities and coupling constants, where (cid:98) 𝑛 (cid:99) = (cid:40) 𝑛 𝑛 even 𝑛 − 𝑛 odd (9.5)The topological term is 𝑆 top : = 𝜋𝑖 𝑝𝑛 𝑛 ∑︁ 𝐴 = 𝐴 𝑄 𝐴 , where 𝑄 𝐴 : = 𝜋 ∫ Ω 𝐴 (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) 𝑛𝐴 = Ω 𝐴 = 𝑛 ∑︁ 𝑗 = 𝑄 𝐴 = , (9.7)so that there are 𝑛 − 𝜆 -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 𝜆 -terms were absent as a result of the same fine-tuning that ensured a unique velocity.Indeed, the choice 𝐽 𝑡 = √︃ 𝑛 − 𝑡𝑡 ensures that 𝑣 𝑡 ≡ const. for all 𝑡 , and moreover that 𝜆 𝑡 = 𝑡 . Z 𝑛 symmetry Just as in the SU ( ) case, we may introduce a discrete symmetry Z 𝑛 : 𝑢 𝐴 → 𝑢 𝐴 + , 𝑢 𝑛 + ≡ 𝑢 . (9.8)It is easy to prove that the sum of 𝜆 -terms is invariant under this symmetry. Indeed, let us fix | 𝐴 − 𝐵 | = 𝑡 ≤ (cid:98) 𝑛 (cid:99) , then every form 𝐵 𝑡 : = (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) ∑︁ ≤ 𝐴<𝐵 ≤ 𝑛 | 𝐴 − 𝐵 | = 𝑡 − ∑︁ ≤ 𝐴<𝐵 ≤ 𝑛 | 𝐴 − 𝐵 | = 𝑛 − 𝑡 (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) ( 𝑢 𝐴 ◦ 𝑑𝑢 𝐵 ∧ 𝑢 𝐵 ◦ 𝑑𝑢 𝐴 ) (9.9)58s Z 𝑛 -invariant. Using the property 𝜆 𝑛 − 𝑡 = − 𝜆 𝑡 that follows from (9.4), we may write the formentering the 𝜆 -terms as ∑︁ ≤ 𝐴<𝐵 ≤ 𝑛 𝜆 | 𝐴 − 𝐵 | ( 𝑢 𝐴 ◦ 𝑑𝑢 𝐵 ) ∧ ( 𝑢 𝐵 ◦ 𝑑𝑢 𝐴 ) = (cid:98) 𝑛 (cid:99) ∑︁ 𝑡 = 𝜆 𝑡 𝐵 𝑡 , (9.10)proving that it is also invariant. In contrast, the topological part of the action is shifted underthe Z 𝑛 -transformation (9.8): 𝑆 top → 𝑆 top + 𝜋𝑖 𝑄 , just as in the SU ( ) case (8.21). Thisis the ultimate reason that allows separating the topological terms from the non-topological 𝜆 -part of the skew-symmetric tensor field. The coupling constants { 𝑔 𝑡 } and { 𝜆 𝑡 } in (9.2) correspond to the metric and torsion on the flagmanifold, respectively [190]. However, a unique metric cannot be defined, since the theory(9.2) lacks the Lorentz invariance that is often assumed for sigma models. Thus, we havea non-Lorentz invariant flag 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 flow to a single value, so that Lorentz invariance emerges.The Lorentz invariant versions of the above flag manifold sigma model were studied ingreat detail in [190]. In particular, the renormalization group flow of both the { 𝑔 𝑡 } and { 𝜆 𝑡 } were determined for general 𝑛 . Moreover, field theoretic versions of the LSMA theorem wereformulated, using the methods of ’t Hooft anomaly matching (which we review below, inSection 10). We would like to apply these results to our SU( 𝑛 ) chains which lack Lorentzinvariance in general. First, it will be useful to introduce dimensionless velocities, 𝜐 𝑡 , definedaccording to 𝜐 𝑡 : = 𝑣 𝑡 𝑣 , 𝑣 = (cid:98) 𝑛 (cid:99) (cid:205) (cid:98) 𝑛 (cid:99) 𝑡 = 𝑣 𝑡 , and introduce new spacetime coordinates by meansof a rescaling 𝑥 → 𝑥 √ 𝑣 , 𝜏 → √ 𝑣𝜏. We then consider the differences of velocities occurringin (9.2), namely Δ 𝑡𝑡 (cid:48) : = 𝜐 𝑡 − 𝜐 𝑡 (cid:48) , (9.11)and ask how they behave at low energies. More precisely, we calculate the one-loop betafunctions of these Δ 𝑡𝑡 (cid:48) , to orders O ( 𝑔 𝑡 ) and O ( 𝜆 𝑡 ) . We will find that each of the Δ 𝑡𝑡 (cid:48) flowsto zero under renormalization and we will show that this implies Lorentz invariance at ourorder of approximation. This is consistent with the fundamental SU( 𝑛 ) models with 𝑝 = { 𝑔 𝑡 } appearing in (9.2) are dimensionless, and are all proportional to 𝑝 .Since we’ve taken a large 𝑝 limit, we will expand all quantities in powers of the { 𝑔 𝑡 } . Aswe will see below, the coefficients { 𝜆 𝑡 } 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 field theories, we make the simplifying assumption that the matrices 𝑈 = ( 𝑢 , . . . , 𝑢 𝑛 ) are close to the identity matrix, and expand them in terms of the SU ( 𝑛 )generators.Recalling that the matrix 𝑈 is subject to the gauge transformations 𝑈 → 𝑈 · 𝐷 𝑔 , where 𝐷 𝑔 = Diag ( 𝑒 𝑖𝛼 , · · · , 𝑒 𝑖𝛼 𝑛 ) , we may fix this gauge invariance by the followingparametrization of 𝑈 : 𝑈 = Exp (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) 𝑖 ∑︁ 𝑎 ∈ off-diagonalgenerators √︂ (cid:101) 𝑔 𝑎 𝜙 𝑎 𝑇 𝑎 (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) = + 𝑖 √︂ (cid:101) 𝑔 𝑎 𝜙 𝑎 𝑇 𝑎 − (cid:101) 𝑔 𝑎 𝜙 𝑎 𝜙 𝑏 𝑇 𝑎 𝑇 𝑏 + O ( 𝜙 ) . (9.12)To explain how (cid:101) 𝑔 𝑎 are related to 𝑔 | 𝐴 − 𝐵 | , we will assume that the following basis of off-diagonal generators is chosen: { 𝑇 𝑎 } = { 𝐸 𝐴𝐵 + 𝐸 𝐵𝐴 , 𝑖 ( 𝐸 𝐴𝐵 − 𝐸 𝐵𝐴 ) , 𝐴 < 𝐵 } , where 𝐸 𝐴𝐵 arethe elementary matrices with 1 in 𝐴𝐵 -th position and 0 elsewhere. Whenever the generator 𝑇 𝑎 corresponds to one of these two generators, we set (cid:101) 𝑔 𝑎 : = 𝑔 | 𝐴 − 𝐵 | . As shown in [237], inthis notation the expansion of the Lagrangian to quartic order in the 𝜙 ’s has the form L = (cid:20) 𝜐 𝑎 ( 𝜕 𝜏 𝜙 𝑎 ) + 𝜐 𝑎 ( 𝜕 𝑥 𝜙 𝑎 ) (cid:21) + √︁ (cid:101) 𝑔 𝑎 (cid:101) 𝑔 𝑏 (cid:101) 𝑔 𝑐 √ ℎ 𝑎 ( 𝜇 ) (cid:101) 𝑔 𝑎 𝑓 𝑏𝑐𝑎 𝜕 𝜇 𝜙 𝑎 𝜕 𝜇 𝜙 𝑏 𝜙 𝑐 (9.13) √︁ (cid:101) 𝑔 𝑏 (cid:101) 𝑔 𝑐 (cid:101) 𝑔 𝑑 ℎ 𝑎 ( 𝜇 ) (cid:101) 𝑔 𝑎 (cid:104)√︁ (cid:101) 𝑔 𝑒 𝜕 𝜇 𝜙 𝑒 𝜕 𝜇 𝜙 𝑏 𝜙 𝑐 𝜙 𝑑 𝑓 𝑒𝑐𝑎 𝑓 𝑏𝑑𝑎 + 𝑓 𝑏𝑐𝐸 𝑓 𝐸𝑑𝑎 √︁ (cid:101) 𝑔 𝑎 𝜕 𝜇 𝜙 𝑎 𝜕 𝜇 𝜙 𝑏 𝜙 𝑐 𝜙 𝑑 (cid:105) + O ( 𝜙 ) . Here 𝑓 𝑎𝑏𝑐 , 𝑓 𝑎𝑏𝐶 (the small-letter and capital-letter indices correspond to the off-diagonalgenerators and all generators respectively) are the structure constants defined by [ 𝑇 𝑎 , 𝑇 𝑏 ] = 𝑖 𝑓 𝑎𝑏𝐶 𝑇 𝐶 . Also, ℎ 𝑎 ( 𝜇 ) = 𝜐 𝑎 for 𝜇 = 𝜏 and ℎ 𝑎 ( 𝜇 ) = 𝜐 𝑎 for 𝜇 = 𝑥 .The calculations then follow the standard procedures of renormalization theory. Werewrite the free part of the above Lagrangian in ‘renormalized variables’, i.e. L = (cid:104) 𝑍 𝜏𝑎 𝜐 𝑟𝑎 ( 𝜕 𝜏 𝜃 𝑎 ) + 𝜐 𝑟𝑎 𝑍 𝑥𝑎 ( 𝜕 𝑥 𝜃 𝑎 ) (cid:105) (and the interaction terms accordingly). The bare and renor-malized velocities are related as 𝜐 𝑎 = 𝜐 𝑟𝑎 √︃ 𝑍 𝑥𝑎 𝑍 𝜏𝑎 , so that one can define the corresponding 𝛽 -function 𝛽 𝜐 𝑎 : = 𝑑𝜐 𝑟𝑎 𝑑 log 𝑀 , where 𝑀 is the fixed energy scale. q - kk c ba Figure 11: The diagram contributing to velocity renormalization at one loop. Throughout, repeated indices will be summed over. 𝑍 𝑥𝑎 , 𝑍 𝜏𝑎 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 𝛽 𝜐 𝑡 = 𝜐 𝑡 𝑔 𝑡 𝜋 𝑛 − ∑︁ 𝐴 = 𝐴 ≠ 𝑡 𝑔 | 𝑡 − 𝐴 | 𝑔 𝐴 (cid:20) 𝜐 𝑡 𝜐 𝐴 − 𝜐 𝐴 𝜐 𝑡 (cid:21) , 𝑡 = , , · · · , 𝑞 : = (cid:98) 𝑛 (cid:99) . (9.14)The equations for 𝜐 𝑡 with 𝑡 > 𝑞 may be obtained by using the identity 𝜐 𝑡 = 𝜐 𝑛 − 𝑡 . As a resultof (9.14), the renormalization group flow equation 𝑑𝜐 𝑡 𝑑 log 𝑀 = 𝛽 𝜐 𝑡 (we drop the superscript 𝑟 to simplify the notation) is a non-linear system of ODE’s for the functions 𝜐 𝑡 ( log 𝑀 ) . Onewishes to show that the ‘point’ where all velocities are the same ( 𝜐 𝑖 = 𝜐 𝑗 for all 𝑖, 𝑗 ) 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 𝑞 − Δ 𝑖 : = Δ ,𝑖 + ( 𝑖 = , , · · · , 𝑞 − ) . The linearized equation then takes the form 𝑑𝑑 log 𝑀 𝚫 = 𝑅 𝚫 , (9.15)where 𝑅 is a ( 𝑞 − ) × ( 𝑞 − ) matrix. The spectrum of 𝑅 will reveal the low energy behaviourof the Δ 𝑡𝑡 (cid:48) : if the spectrum is strictly positive, one may conclude that all velocity differencesflow to zero in the IR. In the highly symmetric case when all of the coupling constants { 𝑔 𝑡 } are equal, one easily finds 𝑅 = 𝑔 𝜋 ( 𝑛 − ) 𝑞 − (9.16)showing that the spectrum of 𝑅 is strictly positive. In the non-symmetric case, for differ-ent choices of 𝑛 and values of the couplings, this has been checked numerically in [237],suggesting that the spectrum of 𝑅 is always positive.Up to this conjecture, we have shown that the velocity differences Δ 𝑡𝑡 (cid:48) flow to zero atlow energies. Another source of Lorentz non-invariance in the Lagrangian (9.13) are thefunctions ℎ 𝑎 ( 𝜇 ) in the interaction terms. These are however related to the velocities 𝜐 𝑎 andtherefore also flow 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 ’t Hooft anomalymatching
Based on the renormalization group analysis in the previous section, we now argue that atlow enough energies, the SU( 𝑛 ) chains in the symmetric- 𝑝 irreps (without fine-tuning), may61e described by a Lorentz invariant flag manifold sigma model L = ∑︁ 𝐴<𝐵 𝑔 | 𝐴 − 𝐵 | | 𝑢 𝐴 ◦ 𝜕 𝜇 𝑢 𝐵 | − 𝜖 𝜇𝜈 ∑︁ 𝐴<𝐵 𝜆 | 𝐴 − 𝐵 | ( 𝑢 𝐴 ◦ 𝜕 𝜇 𝑢 𝐵 ) ( 𝑢 𝐵 ◦ 𝜕 𝜈 𝑢 𝐴 ) with topological theta-term 𝑆 top = 𝑖𝜃 𝑛 − ∑︁ 𝐴 = 𝐴 𝑄 𝐴 𝜃 : = 𝜋 𝑝𝑛 . (10.1)(10.2)These sigma models have been studied in [69, 73, 190]. In [190], the renormalization groupflows of the { 𝜆 𝑡 } and the { 𝑔 𝑡 } were determined, and given a geometric interpretation. Itwas found that for 𝑛 >
4, the { 𝑔 𝑡 } flow to a common value in the IR, and that for 𝑛 > { 𝜆 𝑡 } flow to zero in the IR. Even for 𝑛 < ( 𝑛 ) WZNW model occurs. This is based on the observation that the 𝜆 term doesn’tinduce any relevant operators in the WZNW model. Thus we may expect an 𝑆 𝑛 (permutationgroup) symmetry to emerge at low enough energies, and for 𝑛 >
6. It is known that in these 𝑆 𝑛 -symmetric models, the unique coupling constant 𝑔 obeys [190] 𝑑𝑔𝑑 log 𝑀 = 𝑛 + 𝜋 𝑔. (10.3)and the theory is asymptotically free. The 𝑆 𝑛 -symmetric metric (with all 𝑔 𝑡 equal) is knownin the math literature as the ‘normal’, or reductive, metric [29]. This same metric will featurein the integrable models that we will describe in the next chapter. Interestingly, it is notKähler (unless 𝑛 =
2) but it is Einstein, with cosmological constant proportional to 𝑛 + ( 𝑛 ) Haldane conjectures
The low-energy behavior of the sigma models (10.1)-(10.2) (and of the corresponding spinchains with symmetric rank- 𝑝 representations at each site) depends drastically on the valuesof the 𝜃 -angles. In [161, 221, 190, 237] the generalizations of Haldane’s conjecture for thisclass of models were proposed. These are summarized in Table 1.Using the notion of ’t Hooft anomaly matching (which we explain below in Sections 10.2-10.3), both [190] and [221] were able to formulate a field-theoretic version of the LSMAtheorem for SU( 𝑛 ) chains. In short, the presence of an ’t Hooft anomaly signifies nontriviallow energy physics. It was shown that in the flag manifold models, an ’t Hooft anomalyis present so long as 𝑝 is not a multiple of 𝑛 . In these cases the gapped phase must havespontaneously broken translation or PSU( 𝑛 ) symmetry; the latter is ruled out by the Mermin-Wagner-Coleman theorem at any finite temperature. In the gapped phase, the ground statedegeneracy is predicted to be 𝑛 gcd ( 𝑛,𝑝 ) , 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 different62 ase Conjecture Evidence 𝑝 ≠ 𝑛 Gapless orgapped with degenerateground states (proof) ◦ LSMA theorem [166, 13, 237]gcd ( 𝑛, 𝑝 ) = ( 𝑛 ) CFT ◦ ‘t Hooft anomalies:[221, 190, 162, 260] ◦ Fractional instanton gas [8, 235]1 < gcd ( 𝑛, 𝑝 ) < 𝑛 GapDegeneracy 𝑑 : = 𝑛 gcd ( 𝑝,𝑛 ) ◦ No candidate CFT [162, 260] ◦ Fractional instanton gas 𝑝 = 𝑛 Gap ◦ Numerics ( 𝑝 = 𝑛 =
3) [124] ◦ Absence of anomalies ◦ Perturbations around theintegrable WZNW point [113] ◦ Fractional instanton gas ◦ AKLT states:[126, 150, 187, 177, 207, 125]Table 1: Generalized Haldane conjectures for SU ( 𝑛 ) spin chains with symmetric rank- 𝑝 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 𝑝 mod 𝑛 ≠ 𝑝 and 𝑛 have no nontrivialcommon divisor (in Section 10.5 below we review a simple relation between WZNW and flagmanifold models). In this case, the flow is to SU( 𝑛 ) . Otherwise, the candidate IR fixed pointis SU( 𝑛 ) 𝑞 , where 𝑞 = gcd ( 𝑛, 𝑝 ) , however we don’t expect SU ( 𝑛 ) 𝑘 low energy theories, with 𝑘 >
1, to emerge without fine-tuning. This is because they contain relevant operators allowedby symmetry which destabilize them [226, 24, 6]. Integrable spin models are known whichdo exhibit SU ( 𝑛 ) 𝑘 low energy theories but they require fine-tuned Hamiltonians [25, 142,143]. The most well-known example of this is the SU(2) case where integrable models ofspin 𝑠 have low energy SU ( ) 𝑠 critical points [219, 35, 11]. However, it has been establishedthat these critical points are unstable against infinitesimal tuning of the spin Hamiltonianand one would require fine-tuning in order for the flag manifold sigma model to flow there.This can already be seen from Fig. 9 in the example of the spin-1 integrable chain, whosecontinuum limit is described by SU ( ) 𝑘 = 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 flow from the unstable SU ( 𝑛 ) 𝑞 theory to SU ( 𝑛 ) is possible, sincethis would violate the anomaly matching conditions derived in [162, 260] for generic SU( 𝑛 )WZNW models. For SU ( ) there is another anomaly-based argument of [115], which asserts63hat a flow between an SU ( ) 𝑘 and SU ( ) 𝑘 (cid:48) theories is only possible if 𝑘 = 𝑘 (cid:48) mod 2.Based on these anomaly arguments, we conclude that the rank- 𝑝 symmetric SU( 𝑛 ) chainsmay flow to a SU( 𝑛 ) WZNW model if 𝑝 and 𝑛 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 asimplified phase diagram of the SU( 𝑛 ) chain in the case when 𝑝 and 𝑛 are coprime. Similarto the O(3) sigma model, we expect an RG flow from the flag manifold sigma model to theSU ( 𝑛 ) WZNW model. This model has an SU(n)-invariant interaction term (cid:205) 𝑎 𝐽 𝑎𝐿 𝐽 𝑎𝑅 whichis marginal. For one sign of this coupling, it is marginally irrelevant and flows to zero; forthe other sign it flows to large values [162, 237]. As in the O(3) sigma model, we expect thiscoupling to have the irrelevant sign for sufficiently weak coupling in the flag-manifold sigmamodel. If the coupling gets too large then this coupling constant changes sign and there isan RG flow 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 simplified phase diagram of the SU( 𝑛 ) chains, as a function of coupling constant 𝑔 (a collective notion for the (cid:98) 𝑛 (cid:99) different coupling constants) when 𝑝 and 𝑛 are coprime.We note that when 𝑝 and 𝑛 have a common divisor, at least one of the topological anglesoccurring in (10.2) is necessarily trivial. In the instanton gas picture of Haldane’s 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 of‘t Hooft anomaly matching. To introduce this concept, first one observes that the continuousglobal symmetry of the flag model (10.1)-(10.2) is (here Z 𝑛 ⊂ SU ( 𝑛 ) is the subgroup of thetype 𝜔 𝑘 𝑛 , where 𝜔 is an 𝑛 -th root of unity)PSU ( 𝑛 ) = SU ( 𝑛 )/ Z 𝑛 = U ( 𝑛 )/ U ( ) . (10.4)The reason is that the flag is described by 𝑛 vectors 𝑈 : = ( 𝑢 , . . . , 𝑢 𝑛 ) , up to U ( ) 𝑛 phasetransformations acting as 𝑈 → 𝑈 · 𝐷 , 𝐷 = Diag ( 𝑒 𝑖𝛼 , · · · , 𝑒 𝑖𝛼 𝑛 ) . The global symmetry isgiven by the left action 𝑈 → 𝑈 · 𝑈 , and for 𝑈 ∈ U ( 𝑛 ) 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 ( 𝑛 ) . Besides, the above64odels have a discrete Z 𝑛 -symmetry (9.8) that acts by a cyclic permutation of the vectors 𝑢 , . . . , 𝑢 𝑛 . 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: ◦ Starting with a gauged linear representation for the flag manifold models, we derivethe mixed PSU ( 𝑛 ) − Z 𝑛 anomaly, following [221]. ◦ We discuss how PSU ( 𝑛 ) -bundles are related to the LSMA theorem and to fractionalinstantons ◦ Following [190], we explain how flag manifold models may be embedded in WZNWmodels. This serves to motivate the conjecture that flag manifold models flow in theIR to those conformal points for certain values of the 𝜃 -angles ( 𝑛 ) global symmetry We start with the first point. On several occasions we already used the fact that the completeflag manifold U ( 𝑛 ) U ( ) 𝑛 may be thought of as the space of orthonormal vectors 𝑢 𝐴 ◦ 𝑢 𝐵 = 𝛿 𝐴𝐵 ,each defined up to a phase: 𝑢 𝐴 ∼ 𝑒 𝑖𝛼 𝐴 𝑢 𝐴 . It is of course standard in field theory to encodesuch equivalences by means of gauge fields, and in the present setup this leads to the so-called ‘gauged linear sigma model’ formulation, which often simplifies the calculations (inthe example of the CP 𝑛 − model we already encountered it in section 7.2). In the simplestcase 𝑔 𝑡 = const. , 𝜆 𝑡 = const. the Lagrangian (10.1) with the topological term (10.2) may berewritten as follows: 𝑆 = 𝑛 ∑︁ 𝐴 = ∫ 𝑑𝑥 𝑑𝜏 (cid:20) − 𝑔 | ( d + i 𝑎 𝐴 ) 𝑢 𝐴 | + i 𝜃 𝐴 𝜋 d 𝑎 𝐴 + 𝜆 𝜋 ( 𝑢 𝐴 + · d 𝑢 𝐴 ) ∧ ( 𝑢 𝐴 + · d 𝑢 𝐴 ) (cid:21) , (10.5)where 𝑎 𝐴 are U ( ) gauge fields. As before, the first term is the usual kinetic term of thenonlinear sigma model, and the last term is the ‘non-topological’ part of the skew-symmetricfield, 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 ’t Hooftanomaly matching. The second term in the above action is the topological theta term ofthe two-dimensional U ( ) 𝑛 gauge theory. We may set 𝑄 𝐴 = 𝜋 ∫ d 𝑎 𝐴 , moreover the 𝑄 𝐴 sodefined are equivalent to the topological charges that we encountered earlier, as one can seeby eliminating the gauge fields through their e.o.m. Indeed, solving the e.o.m. of 𝑎 𝐴 , we find 𝑎 𝐴 = i2 ( 𝑢 𝐴 · d 𝑢 𝐴 − d 𝑢 𝐴 · 𝑢 𝐴 ) = i 𝑢 𝐴 · d 𝑢 𝐴 . (10.6)We may interpret 𝑑𝑎 𝐴 as the Fubini-Study form on the 𝐴 -th copy of CP 𝑛 − (see (2.8)).Moreover, as discussed in Section 1.4 the flag manifold F 𝑛 is a Lagrangian submanifold of ( CP 𝑛 − ) × 𝑛 , which implies 𝑛 (cid:205) 𝐴 = 𝑑𝑎 𝐴 =
0. As a result, 𝑛 (cid:205) 𝐴 = 𝑄 𝐴 =
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 𝑛 ‘quasi’-symmetry (i.e. a symmetry up to anintegral of an element of 𝐻 (F , Z ) ), if the 𝜃 -angles are chosen as 𝜃 𝐴 = 𝜋𝑝 𝐴𝑛 for 𝑝 ∈ Z .Now we turn to the discussion of the mixed ’t Hooft anomaly between the PSU ( 𝑛 ) flavorsymmetry and the Z 𝑛 permutation symmetry . The anomaly manifests itself in the fact thatthe partition function of the system in a topologically non-trivial background PSU ( 𝑛 ) gaugefield is not invariant under the Z 𝑛 permutation, even at the point 𝜃 ℓ = 𝜋𝑝 ℓ𝑛 , where the systemdescribed by the action (10.5) is invariant.How do we introduce a background field for the PSU ( 𝑛 ) flavor symmetry? To answerthis question, one should first clarify the difference between SU ( 𝑛 ) and PSU ( 𝑛 ) bundles. APSU ( 𝑛 ) bundle 𝑉 over a worldsheet Σ possesses an additional Z 𝑛 -valued invariant, which isa member of the second cohomology 𝐻 ( Σ , 𝜋 ( PSU ( 𝑛 ))) – the generalized Stiefel-Whitneyclass 𝑤 ( 𝑉 ) (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 definition of the invariant.First, the PSU ( 𝑛 ) bundle 𝑉 may be lifted, in a non-unique way, to a vector bundle 𝑉 over Σ , with structure group U ( 𝑛 ) . The latter is characterized by its first Chern class 𝑐 ( 𝑉 ) (itsintegral is the ‘abelian flux’). The non-uniqueness in choosing 𝑉 has to do with the factthat we could replace it with 𝑉 ⊗ L , where L is any line bundle, since it would cancel outin the projective quotient anyway. Recalling that 𝑐 ( 𝑉 ⊗ L) = 𝑐 ( 𝑉 ) + 𝑛 𝑐 (L) , we seethat 𝑐 ( 𝑉 ) mod 𝑛 is a well-defined topological quantity. This mod 𝑛 -reduced class is calledthe generalized Stiefel-Whitney class 𝑤 ( 𝑉 ) ∈ 𝐻 ( Σ , Z 𝑛 ) = Z 𝑛 , which characterizes thetopologically non-trivial PSU ( 𝑛 ) -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 𝑉 with structuregroup U ( 𝑛 ) , we will introduce a U ( 𝑛 ) gauge field (cid:101) 𝐴 . In order to implement the quotient, onehas to postulate the additional gauge transformations 𝑎 𝐴 ↦→ 𝑎 𝐴 − 𝜉, (cid:101) 𝐴 ↦→ (cid:101) 𝐴 + 𝜉 𝑛 , (10.7)where 𝜉 is a U ( ) gauge field, which simultaneously plays a role of gauge parameter here.Now, the point is that a U ( 𝑛 ) gauge field has an integer invariant – the first Chern number– that is expressed as follows: ∫ Σ 𝑐 ( 𝑉 ) = 𝜋 ∫ 𝑑 (cid:16) Tr ( (cid:101) 𝐴 ) (cid:17) ∈ Z . Accordingly, since 𝜉 is aU ( ) gauge field, its curvature also has quantized periods, ∫ Σ 𝑐 (L) = 𝜋 ∫ 𝑑𝜉 ∈ Z . Dueto the shift symmetry (10.7) that we have imposed, the periods of (cid:101) 𝐴 are shifted by multiplesof 𝑛 : ∫ Σ 𝑐 ( 𝑉 ) ↦→ ∫ Σ 𝑐 ( 𝑉 ) + 𝑛 ∫ Σ 𝑐 (L) , and as a result we get a Z 𝑛 (cid:39) Z (cid:14) 𝑛 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 𝑛 gauge field that wecall 𝐵 . In our notation it is simply 𝐵 : = 𝑛 𝑑 (cid:16) Tr ( (cid:101) 𝐴 ) (cid:17) . Under the shift (10.7) it changes as The discussion in [221] contains also the case of a mixed PSU ( 𝑛 ) − Z 𝑛 (cid:48) anomaly, where 𝑛 (cid:48) is a divisor of 𝑛 , but we restrict here to the simpler case 𝑛 (cid:48) = 𝑛 . 𝐵 ↦→ 𝐵 + 𝑑𝜉 . (10.8)According to the above discussion, the integral12 𝜋 ∫ Σ 𝐵 ∈ Z 𝑛 (10.9)is a multiple of 𝑛 . To write a gauged version of the action (10.5), first of all we replacethe covariant derivatives ( d + i 𝑎 𝐴 ) 𝑢 𝐴 by the elongated derivatives ( d + i 𝑎 𝐴 + i (cid:101) 𝐴 ) 𝑢 𝐴 – acombination invariant under the shift (10.7). Besides, the curvatures 𝑑𝑎 𝐴 are not invariantunder the shift (10.7) but the combinations 𝑑𝑎 𝐴 + 𝐵 are. As a result, the effect of subjectingthe system (10.5) to an external PSU ( 𝑛 ) gauge field is in the following modification of theaction: 𝑆 gauged = ∑︁ 𝐴 = ∫ 𝑀 (cid:20) − 𝑔 (cid:12)(cid:12)(cid:12) ( d + i 𝑎 𝐴 + i (cid:101) 𝐴 ) 𝑢 𝐴 (cid:12)(cid:12)(cid:12) + i 𝜃 𝐴 𝜋 ( d 𝑎 𝐴 + 𝐵 ) + (10.10) + 𝜆 𝜋 { 𝑢 𝐴 + · ( d + i (cid:101) 𝐴 ) 𝑢 𝐴 } ∧ { 𝑢 𝐴 + · ( d + i (cid:101) 𝐴 ) 𝑢 𝐴 } (cid:21) . Notice that there is no need to write the 𝑎 𝐴 gauge fields in the 𝜆 -term, since their contributionsvanish due to the orthogonality constraint between 𝑢 𝐴 . Performing the path integral, 𝑍 [( 𝐴, 𝐵 )] = ∫ D 𝑎 D 𝑢 D 𝑢 exp ( 𝑆 gauged ) , (10.11)we obtain the partition function 𝑍 [( 𝐴, 𝐵 )] in the background PSU ( 𝑛 ) gauge field. 𝐴 is meantto represent the traceless part of the gauge field (cid:101) 𝐴 . One curious thing to notice about thepartition function 𝑍 is that it is no longer 2 𝜋 -periodic in the 𝜃 -angles. Indeed, a shift of oneof the angles, say 𝜃 → 𝜃 + 𝜋 , produces a phase 𝑍 [( 𝐴, 𝐵 )] ↦→ 𝑍 [( 𝐴, 𝐵 )] · 𝑒 𝑖 ∫ 𝐵 . Z 𝑛 anomaly in a PSU ( 𝑛 ) background It is instructive to notice that, in the presence of the PSU ( 𝑛 ) background gauge field, variationof the action w.r.t. 𝑎 ℓ no longer produces (10.6), and the following modified formula for thecurvatures holds instead: 𝑛 ∑︁ 𝐴 = ( 𝑑𝑎 𝐴 + 𝐵 ) = . (10.12)If we consider the modified topological charges (cid:101) 𝑄 𝐴 : = 𝜋 ∫ Σ ( 𝑑𝑎 𝐴 + 𝐵 ) , which sum tozero, then according to (10.9) they will be quantized in units of 𝑛 . 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 ( 𝑛 ) gauge field.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 ( 𝑛 ) gaugefield is not invariant under the Z 𝑛 shift symmetry 𝑢 𝐴 → 𝑢 𝐴 + (which should be obviouslysupplemented by 𝑎 𝐴 → 𝑎 𝐴 + ). First of all, the metric and 𝜆 terms are evidently invariant, soit suffices to compute the topological term. We recall that we have chosen our 𝜃 -angles as 𝜃 𝐴 = 𝜋𝑝 𝐴𝑛 , so that the topological term changes as follows under a Z 𝑛 -shift (we set 𝑎 𝑛 + ≡ 𝑎 ): 𝑆 top = i 𝑝 𝑛 ∑︁ 𝐴 = 𝐴𝑛 ∫ ( d 𝑎 𝐴 + 𝐵 ) ↦→ i 𝑝 𝑛 ∑︁ 𝐴 = 𝐴𝑛 ∫ ( d 𝑎 𝐴 + + 𝐵 ) = (10.13) = 𝑆 top − i 𝑝𝑛 ∫ 𝑛 ∑︁ 𝐴 = ( d 𝑎 𝐴 + 𝐵 ) = ( . ) + i 𝑝 ∫ ( d 𝑎 + 𝐵 ) . As ∫ d 𝑎 ∈ 𝜋 Z , this term drops off in the path-integral. However, since ∫ 𝐵 ∈ 𝜋𝑛 Z , the 𝐵 -term contributes a phase, so we have 𝑍 [( 𝐴, 𝐵 )] ↦→ 𝑍 [( 𝐴, 𝐵 )] exp (cid:18) i 𝑝 ∫ 𝐵 (cid:19) (10.14)under the Z 𝑛 permutation. This is the mixed ’t Hooft anomaly between PSU ( 𝑛 ) and Z 𝑛 .There is no local counter term that can eliminate the generation of the 𝐵 -term under the Z 𝑛 exchange symmetry. Indeed the only counter-terms allowed are i 𝑞 ∫ 𝐵 where 𝑞 ∈ Z mod 𝑛 ,and these are invariant under the Z 𝑛 symmetry.In the cases of continuous global symmetries it was argued by ’t Hooft long ago [139] thatthe anomalies should match between the UV and IR limits of the theory: even if the effectivetheory in the IR looks drastically different 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 difficult 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-called‘adiabatic continuity’ of the theory, i.e. to the smooth dependence of the physical propertiesof a theory compactified 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 𝑛 invariant point, 𝜃 𝐴 = 𝜋𝑝 𝐴𝑛 ,with 𝑝 not a multiple of 𝑛 , cannot be trivially gapped. According to [83], in 1 + ( 𝑛 ) chain (see section 4.1), and the argument reviewedhere provides its field-theoretic counterpart. 68 ( 𝑛 ) -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 𝜋 -quantized term, which is immaterial in the path integral. Secondly, at the level of the partitionfunction (10.14) the Z 𝑛 -transformation is a change of variables in the path integral (10.11),so whenever exp (cid:16) i 𝑝 ∫ 𝐵 (cid:17) ≠ 𝑇 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 ( 𝑛 ) gauge field.In fact, in several cases the results of the previous section may well be formulated in terms of flat background gauge fields, 𝑑𝐴 − 𝐴 ∧ 𝐴 =
0. We will explain this on two examples: thoseof a torus T and of a sphere 𝑆 .The torus example is somewhat easier, as there is a general statement (reviewed in [253])that, for a simple group 𝐺 , every 𝐺 -bundle over T admits a flat connection . This is alsotrue for the topologically non-trivial PSU ( 𝑛 ) bundles of the previous section. To calculate thecorresponding invariant 𝑤 , one views the gauge field as an SU ( 𝑛 ) gauge field and computesits holonomies 𝑎 and 𝑏 along two meridians of the torus. Since 𝜋 ( T ) = Z , in PSU ( 𝑛 ) theholonomies would satisfy 𝑎𝑏 = 𝑏𝑎 , which in SU ( 𝑛 ) is relaxed to 𝑎𝑏𝑎 − 𝑏 − = 𝜔 ∈ 𝑤 . Notethat these holonomies 𝑎 and 𝑏 may well be non-trivial even for a flat gauge field 𝐴 , which isthe reason that it suffices to consider flat connections.Now, the ultimate use of flat connections is that they may be completely eliminated at theexpense of imposing twisted boundary conditions on the fields. Indeed, a flat connection hasthe form 𝐴 = − 𝑔 − 𝑑𝑔 , where 𝑔 is locally a function on the worldsheet, which, when viewedglobally, encodes the holonomies 𝑎 and 𝑏 . If 𝑥 and 𝑦 are the coordinates along the meridiansof the torus, one has 𝑔 ( 𝑥 + 𝜋, 𝑦 ) = 𝑎 ◦ 𝑔 ( 𝑥, 𝑦 ) and 𝑔 ( 𝑥, 𝑦 + 𝜋 ) = 𝑏 ◦ 𝑔 ( 𝑥, 𝑦 ) . Accordingly,if 𝑢 𝐴 are the matter fields of the model charged under the PSU ( 𝑛 ) gauge group (like the unitvector fields of the flag models), we can now perform a gauge transformation 𝑢 𝐴 → 𝑔 ◦ 𝑢 𝐴 ,which completely eliminates the gauge field at the expense of imposing twisted boundaryconditions 𝑢 𝐴 ( 𝑥 + 𝜋, 𝑦 ) ∼ 𝑎 ◦ 𝑢 𝐴 ( 𝑥, 𝑦 ) , 𝑢 𝐴 ( 𝑥, 𝑦 + 𝜋 ) ∼ 𝑏 ◦ 𝑢 𝐴 ( 𝑥, 𝑦 ) ( ∼ means ‘up to aphase’, since the 𝑢 𝐴 take values in a projective space).In the language used before, eliminating the gauge field amounts to setting 𝐴 = 𝐵 = The fact that 𝐺 is simple is crucial here. For example, for a line bundle L with gauge group U ( ) one wouldhave an additional invariant – the first Chern number ∫ Σ 𝑐 (L) = 𝜋 ∫ Σ 𝐹 , expressed through the curvature 𝐹 ofthe connection. Gluing with twist 𝑈 ‘Figure 13: A schematic depiction showing that the twisted partition function Tr ( 𝑈𝑒 − 𝛽𝐻 ) ofthe spin chain leads, in the continuum limit, to a PSU ( 𝑛 ) bundle over a torus. This bundle isdescribed by the twist operator 𝑈 entering the LSMA theorem.the shifts in the topological terms: (10.13) in the presence of the 𝐴, 𝐵 gauge fields and(8.21) without them. The point is that, with the twisted boundary conditions, the shift ∫ Ω in (8.21) is no longer quantized as an integer times 2 𝜋 , but rather as an integer times 𝜋𝑛 , thusreproducing the shift by the 𝐵 -field in (10.13). One way to see this is to observe that, withthe twisted boundary conditions, the fluxes of the gauge fields 𝑎 𝐴 are quantized in multiplesof 𝜋𝑛 , and since Ω = 𝑑𝑎 , this leads to the corresponding statement for ∫ Ω . Indeed, thetwists have the form 𝑢 𝐴 ( 𝑥 + 𝜋, 𝑦 ) ∼ 𝑎 ◦ 𝑢 𝐴 ( 𝑥, 𝑦 ) , where ∼ means that actually 𝑎 ∈ PSU ( 𝑛 ) and is only defined up to a power of the root of unity 𝜔 . In the formulation with 𝑎 𝐴 gaugefields, to undo this ambiguity one performs a gauge transformation 𝑢 𝐴 → 𝑒 𝑖𝜑 𝐴 ( 𝑥,𝑦 ) · 𝑢 𝐴 , wherethe gauge parameter 𝜑 𝐴 ( 𝑥, 𝑦 ) has periodicity 𝜑 𝐴 ( 𝑥 + 𝜋, 𝑦 ) − 𝜑 𝐴 ( 𝑥, 𝑦 ) = 𝜋𝑛 . Since the gaugetransformation affects the gauge fields 𝑎 𝐴 → 𝑎 𝐴 − d 𝜑 𝐴 , we conclude that the fluxes of 𝑎 𝐴 willbe quantized as multiples of 2 𝜋 / 𝑛 . As we recall, 𝑄 𝐴 = 𝜋 ∫ 𝑑𝑎 𝐴 are the topological charges,so we come to the conclusion that these topological charges are quantized in multiples of 𝑛 .This means that the shift (8.21) that was immaterial for periodic boundary conditions (or,in general, for maps from a closed Riemann surface Σ ), now produces a non-vanishingcontribution, equal to the one of the 𝐵 -field in (10.13).In fact, we have already encountered an example of a non-trivial PSU ( 𝑛 ) -bundle over atorus (albeit in a discretized form) in the proof of the LSMA theorem in Section 4.1. Therewe defined the so-called twist operator acting on a spin chain of length 𝐿 : 𝑈 = 𝑒 𝐴 𝐴 : = 𝜋𝑖𝑛𝐿 𝐿 ∑︁ 𝑗 = 𝑗 𝑄 𝑗 , 𝑄 = 𝑛 − ∑︁ 𝛼 = 𝑆 𝛼𝛼 − ( 𝑛 − ) 𝑆 𝑛𝑛 ∈ 𝔰𝔲 ( 𝑛 ) . (10.15)Clearly, in the continuum limit we get 𝑈 = (cid:214) 𝑥 ∈[ , 𝜋 ) 𝑔 ( 𝑥 ) , 𝑔 ( 𝑥 ) = 𝑒 𝑖 𝑥𝑛 𝑄 𝑥 . (10.16)In the framework of flag 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 ( 𝑈𝑒 − 𝛽𝐻 ) 𝑈 ,as shown in Fig. 13. Topologically this produces a PSU ( 𝑛 ) -bundle over a torus, whose 𝑤 -invariant is characterized by the periodicity property of 𝑔 ( 𝑥 ) : 𝑔 ( 𝑥 + 𝜋 ) = ξ · 𝑔 ( 𝑥 ) , where ξ ∈ Z 𝑛 . Looking back at (10.16), we find ξ = 𝑒 𝜋𝑖𝑛 𝑄 . If one deals with rank- 𝑝 symmetricrepresentations, as in the LSMA theorem of section 4.1, the invariant is ξ = 𝑒 𝜋𝑖𝑝𝑛 . Thisis the ‘anomaly factor’ that appears in (4.8) upon inserting the translation operator in (4.7),which should be seen as parallel to making a cyclic permutation 𝑢 𝐴 → 𝑢 𝐴 + in the pathintegral (10.11) and obtaining a factor (10.14) as a result. ( 𝑛 ) bundles As a next step, we consider the worldsheet 𝑆 . We wish to explain that examples of sectionsof topologically non-trivial PSU ( 𝑛 ) bundles over a sphere 𝑆 are provided by the so-called‘fractional instantons’ that we will introduce shortly. For a recent discussion of these fractionalisntantons in the context of resurgence, see the recent paper [229].The fibers of the relevant bundles are the target spaces of the sigma model, i.e. the flagmanifolds. We recall that a section of a topologically trivial bundle is simply a map 𝑆 → F from the worldsheet to the flag manifold target space. One can alternatively think of it as amap R → F with ‘decay conditions at infinity’, meaning that the infinity of R is mappedto a single point in F . In general, fiber bundles over 𝑆 with structure group PSU ( 𝑛 ) maybe defined using a patching function 𝑆 → PSU ( 𝑛 ) , where 𝑆 = 𝑈 + ∩ 𝑈 − is the equator– the intersection of the two patches 𝑈 ± on 𝑆 (the northern and southern hemispheres).Topologically the bundles are determined by the class of the patching map 𝑆 → PSU ( 𝑛 ) in the homotopy group 𝜋 ( PSU ( 𝑛 )) (cid:39) Z 𝑛 . Now, suppose we want to construct a sectionof such a bundle 𝜋 : 𝐸 → 𝑆 , with fiber F and structure group PSU ( 𝑛 ) . Over either 𝑈 + or 𝑈 − one can trivialize the bundle, i.e. one identifies 𝜋 − ( 𝑈 ± ) (cid:39) 𝑈 ± × F . Constructing asection of 𝐸 is the same as specifying two maps 𝑓 ± : 𝑈 ± → F patched across the equator,i.e. 𝑓 + (cid:12)(cid:12) 𝑆 = 𝑔 ◦ 𝑓 − (cid:12)(cid:12) 𝑆 , where 𝑔 : 𝑆 → PSU ( 𝑛 ) is the patching function.Let us now explain how this construction may be used for the description of fractionalinstantons. The latter, by definition, are maps R → F with the following behavior at infinity: 𝑢 𝛼𝐴 = √ 𝑛 𝜔 𝐴𝛼 𝑒 i 𝑤 𝛼 𝜑 , 𝜔 = 𝑒 𝜋 i 𝑛 . (10.17)Here 𝜑 is the angle around a ‘circle at infinity’, and 𝑤 𝛼 ∈ Z are the winding numbers.Unless all 𝑤 𝛼 have the same value, such maps do not satisfy decay conditions at infinity(if 𝑤 𝛼 = const., the winding is undone by a gauge transformation). We will view R ,together with the circle at infinity, as the upper hemisphere 𝑈 + , and the fractional instantonwill serve to define the map 𝑓 + : 𝑈 + → F . Now, on 𝑈 − we will define a constant map,given by (cid:101) 𝑢 𝛼𝐴 = √ 𝑛 𝜔 𝐴𝛼 . Clearly, along the equator the two are related by the patchingmap 𝑔 = 𝑒 − i 𝑛 ( (cid:205) 𝑛𝛼 = 𝑤 𝛼 ) 𝜑 Diag ( 𝑒 i 𝑤 𝜑 , · · · , 𝑒 i 𝑤 𝑛 𝜑 ) ∈ PSU ( 𝑛 ) . The topology of the bundle is71haracterized by [ (cid:205) 𝑛𝛼 = 𝑤 𝛼 mod 𝑛 ] ∈ Z 𝑛 = 𝜋 ( PSU ( 𝑛 )) . If 𝑤 𝛼 = const., the correspondinginvariant vanishes, in line with the discussion above. In Section 10.3 we described the mixed PSU ( 𝑛 ) − Z 𝑛 anomalies that arise for flag manifoldsigma models. In the case when such anomalies are present, one possibility is that therenormalization group flow leads to a conformal field theory in the IR. Moreover, as it isrecorded in Table 1 and discussed in section 10.1, one conjectures that the resulting conformalfield theory is the SU ( 𝑛 ) WZNW model. To motivate this relation, we recall what is perhapsthe most vivid way to see a connection between the flag sigma model and the WZNW model.The idea is to embed the former into the latter [221, 190]. One starts with the WZNWLagrangian, defined as follows (here 𝐺 ∈ SU ( 𝑛 ) ): 𝑆 WZNW : = 𝑅 ∫ 𝑀 Tr ( 𝜕 𝜇 𝐺 𝜕 𝜇 𝐺 † ) + i12 𝜋 𝑘 ∫ 𝑀 Tr ( ( 𝐺 † 𝑑𝐺 ) ) , (10.18)where 𝑀 = Σ is the two dimensional spacetime and 𝑀 is a three-manifold whose boundaryis 𝑀 , i.e. 𝜕 𝑀 = 𝑀 . The coefficient 𝑘 is quantized to be a positive integer. In the UVthe radius 𝑅 of the target space is large and the theory comprises 𝑛 − 𝑅 = 𝑘 𝜋 in the IR [248].The main statement is that the action (10.18), when restricted to unitary matrices 𝐺 witha fixed spectrum , produces the action of a flag manifold sigma model, whose 𝜃 -angles aredictated by the spectrum of 𝐺 . In other words, we will be considering matrices 𝐺 of the form 𝐺 = 𝑈 Ω 𝑈 † , where Ω = diag ( 𝑒 i 𝜑 , 𝑒 i 𝜑 , · · · , 𝑒 i 𝜑 𝑛 ) , 𝑈 ∈ SU ( 𝑛 ) . (10.19)For simplicity we assume 𝜑 𝐴 ≠ 𝜑 𝐵 for 𝐴 ≠ 𝐵 . In this case the matrix 𝑈 is defined upto right multiplication by a diagonal matrix 𝐷 , 𝑈 ∼ 𝑈 · 𝐷 , so that 𝑈 defines a point in aflag manifold. Using (10.19), the kinetic term of the WZNW action (10.18) can be easilycomputed as: 𝑅 ( 𝜕 𝜇 𝐺 𝜕 𝜇 𝐺 † ) = 𝑅 ∑︁ 𝐴 𝜕 𝜇 𝑢 𝐴 ◦ 𝜕 𝜇 𝑢 𝐴 − 𝑅 ∑︁ 𝐴,𝐵 𝑒 i ( 𝜑 𝐴 − 𝜑 𝐵 ) | 𝑢 𝐴 ◦ 𝜕 𝜇 𝑢 𝐵 | . (10.20)To compute the WZ term one takes 𝑀 = 𝑀 × I where I = [ , ] is an interval withcoordinate 𝑦 . To make sure that 𝑀 has a single boundary 𝑀 , one effectively compactifiesthe second boundary 𝑀 × { 𝑦 = } by requiring 𝐺 (cid:12)(cid:12) 𝑦 = = . Since the WZ action (afterexponentiation) doesn’t depend on the extension of the fields to the bulk of 𝑀 , one maychoose 𝐺 ( 𝑧, 𝑧, 𝑦 ) = 𝑈 ( 𝑧, 𝑧 ) Ω ( 𝑦 ) 𝑈 ( 𝑧, 𝑧 ) † , (10.21) Ω ( 𝑦 ) = Diag ( 𝑒 i 𝜑 ( 𝑦 ) , 𝑒 i 𝜑 ( 𝑦 ) , · · · , 𝑒 i 𝜑 𝑛 ( 𝑦 ) ) , where 𝜑 𝐴 ( ) = 𝜑 𝐴 , 𝜑 𝐴 ( ) = .
72e may now substitute (10.21) into the second term of (10.18), and one finds that it splitsinto a sum of two: ∫ 𝑀 Tr ( ( 𝐺 † 𝑑𝐺 ) ) = ∫ 𝑀 (cid:0) Ω top + Ω 𝜆 (cid:1) .The first one produces the 𝜃 -terms of the flag model: ∫ 𝑀 Ω top = 𝑘 𝜋 ∫ 𝑀 ∑︁ 𝐴 𝜑 𝐴 𝑑𝑢 𝐴 ∧◦ 𝑑𝑢 𝐴 , (10.22)whereas the second one is the non-topological part of the 𝐵 -field (the ‘ 𝜆 -term’, as we referredto it earlier): ∫ 𝑀 Ω 𝜆 = − 𝑘 𝜋 ∑︁ 𝐴 ≠ 𝐵 sin ( 𝜑 𝐴 − 𝜑 𝐵 ) ∫ 𝑀 ( 𝑢 𝐵 ◦ 𝑑𝑢 𝐴 ) ∧ ( 𝑢 𝐵 ◦ 𝑑𝑢 𝐴 ) . (10.23)As was shown in section 9.2, the Z 𝑛 -invariant values correspond to the choice of angles (withan overall subtraction so that det Ω = 𝜑 𝐴 = 𝜋 𝐴𝑛 − 𝜋 ( 𝑛 + ) 𝑛 . (10.24)One also needs to have a mechanism to restrict the spectrum of the matrix 𝐺 in (10.18)to be of the form 𝑒 i 𝜑 𝐴 , with the values (10.24). The paper [190] proposes the followingscenario: one adds to (10.18) a potential 𝑉 = (cid:205) (cid:98) 𝑛 (cid:99) 𝑗 = 𝑔 𝑗 Tr ( 𝑈 𝑗 ) Tr ( ( 𝑈 † ) 𝑗 ) . In the limit whenall 𝑔 𝑛 → ∞ one restricts to the locus Tr ( 𝑈 𝑗 ) = , 𝑗 = . . . (cid:98) 𝑛 (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 ( 𝑛 ) spin chain withsymmetric rank- 𝑝 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 𝑆 nonlinear sigma model.In the case of the 𝑆 target space the idea was to arrive at the O(2) model in a speciallimit, when a large potential is added to restrict the field (cid:174) 𝑛 to the XY plane. In the absence ofthe 𝜃 -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 𝑆 model with a large potential this mass gap is stillgenerated when the potential is weakened, and (cid:174) 𝑛 can lift off 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 𝑆 sigma model which corresponds to the purely isotropic ‘Meron’ means half-instanton, and refers to the fact that these configurations have half-integer quantizedtopological charges. The word ‘instanton’ is used, in place of vortex or soliton, because one of the twodimensions in the O(2) model corresponds to Euclidean time. 𝑚 =
0. We will now review a generalization of this argument that suggests that amass gap is present in the SU( 𝑛 )/[ U ( )] 𝑛 − flag manifold sigma model (without topologicalterms), and that it is generated by topological excitations. 𝑋𝑌 -model Following [8], the strategy is to break the symmetry of the flag 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 𝑉 that breaks the SU( 𝑛 ) symmetry downto [ U ( )] 𝑛 − : 𝑉 = 𝑚 𝑛 ∑︁ 𝐴 = ∑︁ 𝛼<𝛽 (cid:0) | ( 𝑢 𝛼𝐴 | − | ( 𝑢 𝛽𝐵 | (cid:1) . (11.1)It is the SU( 𝑛 ) generalization of adding the term (cid:205) 𝑗 𝑆 𝑧 ( 𝑗 ) 𝑆 𝑧 ( 𝑗 ) to the SU(2) Hamiltonian.In the limit 𝑚 → ∞ , the potential 𝑉 restricts all of the components 𝑢 𝛼𝐴 of 𝑢 𝐴 to be ofequal absolute value (equal to √ 𝑛 due to normalization), with arbitrary phases. Taking intoaccount the gauge group U ( ) 𝑛 − , this gives 𝑛 ( 𝑛 − ) real parameters. The number of (real)orthogonality constraints is formally also 𝑛 ( 𝑛 − ) , however one should take into account theremaining U ( ) 𝑛 − global symmetry acting as 𝑢 𝛼𝐴 → 𝑒 𝑖𝜃 𝛼 𝑢 𝛼𝐴 . As a result, the configurationminimizing the potential 𝑉 is (up to permutations of the vectors 𝑢 , · · · , 𝑢 𝑛 ) 𝑢 𝛼𝐴 = √ 𝑛 𝜔 𝐴𝛼 𝑒 i 𝜎 𝛼 , 𝜔 = 𝑒 𝜋 i 𝑛 , 𝜎 𝛼 ∈ [ , 𝜋 ] . (11.2)Orthonormality of these states follows from the identity (cid:205) 𝑛𝜌 = 𝜔 𝜌 𝑗 = 𝑗 ≠ 𝑛 . The Z 𝑛 symmetry 𝑢 𝐴 → 𝑢 𝐴 + is represented on (11.2) by a shift 𝜎 𝛼 → 𝜎 𝛼 + 𝜋𝛼𝑛 . (11.3)The formula (11.2) defines an embedding ( 𝑆 ) 𝑛 − ⊂ F , and in the present setup this isthe torus of asymptotic vortex configurations away from the core. Substituting (11.2) into theLagrangian (10.1), one obtains a generalized XY-model with 𝑛 − 𝑆 -valued fields, 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 𝑉 = 𝑚 (cid:205) 𝑛𝐴 = (cid:205) 𝑛 − 𝛼 = (cid:16) Im [( 𝑢 𝐴 𝑢 𝛼𝐴 ) 𝑛 ] (cid:17) that will suppress all but one fields.The potentials 𝑉 and 𝑉 have common minima (zero locus), as one can see by rewriting 𝑉 in terms of (11.2): 𝑉 = 𝑚 (cid:205) 𝑛𝐴 = (cid:205) 𝑛 − 𝛼 = sin ( 𝑛 ( 𝜎 − 𝜎 𝛼 )) . Due to the factor of 𝑛 the set ofminima of 𝑉 is invariant under the Z 𝑛 symmetry (11.3), which corresponds to translationalinvariance in the underlying lattice model. It is clear that the effect of 𝑉 is to equate all butone of the U(1) fields (up to the shift (11.3)). Due to residual gauge symmetry, everythingthen depends only on one variable 𝜎 : = 𝜎 𝑛 − 𝜎 . This is equivalent to the O(2) model of a74ector (cid:174) 𝑛 ∈ R restricted to the XY plane. By inserting this restricted form of 𝑢 𝐴 into (10.1),it is easy to show that the 𝜆 𝐴,𝐵 terms vanish, and the resulting O(2) coupling constant is 𝑔 − = ∑︁ 𝐴,𝐵 𝑔 − 𝐴,𝐵 . (11.4) For 𝑛 = 𝑉 vanishes and the perturbation 𝑉 is equivalent to adding a massterm 𝑚 ( 𝑛 ) to the 𝑆 model Lagrangian, restricting (cid:174) 𝑛 to lie in the XY plane in the large 𝑚 limit. This follows from the equivalence 𝑛 𝑖 = 𝑢 † 𝜎 𝑖 𝑢 (we already used it earlier in (6.6),where 𝑛 = 𝑧 ◦ 𝜎 ◦ 𝑧𝑧 ◦ 𝑧 was written in stereographic coordinates).The vortices of the model, in order to be well-defined at the core, must become non-planar. They are called merons and have topological charge 𝑄 = ± (the sign depending onwhether 𝑛 = ± 𝑄 = ± 𝑆 model. More exactly, to use the notation developed for flag manifoldmodels, we will be thinking of the topological charge as a pair of charges (cid:174) 𝑄 = ( 𝑄 , 𝑄 ) ,such that 𝑄 + 𝑄 = 𝑄 = 𝑄 . As explained earlier (seesection 1.4, for example), this corresponds to the embedding 𝑆 ⊂ ( 𝑆 ) × ( 𝑆 ) mapping (cid:174) 𝑛 → ( (cid:174) 𝑛, −(cid:174) 𝑛 ) , in which case 𝑄 𝑖 = 𝜋 ∫ Σ 𝑓 ∗ Ω 𝑖 , where Ω , Ω are the two Fubini-Study forms,subject to ( Ω + Ω ) (cid:12)(cid:12) 𝑆 =
0, and 𝑓 : Σ → 𝑆 ⊂ ( 𝑆 ) × ( 𝑆 ) is a map from a worldsheet Σ .In this simplest case 𝑄 is the area on 𝑆 of the image of 𝑓 , multiplied by the number oftimes a typical point is covered. If Σ is a closed Riemann surface, such as 𝑆 , 𝑄 and 𝑄 are integers. In particular, a map 𝑓 : 𝑆 → 𝑆 may be thought of as a map R → 𝑆 witha fixed asymptotic value at infinity: (cid:174) 𝑛 (∞) = (cid:174) 𝑛 . In the case of a meron, on the contrary,the asymptotic behavior at infinity is such that 𝑓 | ∞ : 𝑆 → 𝑆 ⊂ 𝑆 , where 𝑆 ⊂ 𝑆 is theequator 𝑛 =
0. As a result, 𝑓 maps 𝐷 → 𝑆 , where Σ = 𝐷 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 𝑄 = ± (this corresponds also to a single winding of the boundary map 𝑓 | ∞ ).The setup can be generalized to the SU( 𝑛 ) case as follows. As discussed earlier, theminimum of 𝑉 is achieved at the configuration (11.2), which defines an embedding ( 𝑆 ) 𝑛 − ⊂F . We wish to compute the topological charges 𝑄 𝐴 = 𝜋 ∫ 𝐷 𝑓 ∗ Ω 𝐴 for a map 𝑓 : Σ = 𝐷 →F , such that 𝑓 (cid:12)(cid:12) ∞ : ( 𝑆 ) WS → ( 𝑆 ) 𝑛 − is a map with a fixed set of winding numbers (cid:174) 𝑤 ∈ 𝜋 ( ( 𝑆 ) 𝑛 − ) (cid:39) Z 𝑛 − . From (11.2) it is natural to think of (cid:174) 𝑤 as being the set of windingsof 𝑛 angles 𝜎 𝑗 modulo the winding vector ( , · · · , ) that can be removed by an overall U ( ) gauge transformation, i.e. (cid:174) 𝑤 = ( 𝑤 , · · · , 𝑤 𝑛 ) mod ( , · · · , ) .To compute the topological numbers, we recall the flag manifold embedding F ⊂( CP 𝑛 − ) × 𝑛 (see Section 1.4) and denote 𝜋 𝐴 the projection on the 𝐴 -th projective space.Given a map 𝑓 : Σ → F , we construct a map 𝑓 𝐴 = 𝜋 𝐴 ◦ 𝑓 to CP 𝑛 − . If ( 𝑍 : · · · : 𝑍 𝑛 ) are the homogeneous coordinates on the projective space, we choose the standard 𝑛 patches75 𝑈 𝐴 } 𝐴 = ,...,𝑛 , each one given by the condition 𝑍 𝐴 ≠ 𝐴 . Let us now considerthe special maps 𝑓 (the ‘elementary fractional instantons’) characterized by the fact that ( 𝑓 ) 𝐴 ( Σ ) ⊂ 𝑈 𝑠 ( 𝐴 ) , 𝑠 being a permutation. This means that the image of each ( 𝑓 ) 𝐴 lies in asingle patch 𝑈 𝑠 ( 𝐴 ) . In each patch 𝑈 𝐴 we set 𝑍 𝐴 = Ω (cid:12)(cid:12) 𝑈 𝐴 = 𝑑𝜃 𝐴 ,where 𝜃 𝐴 = 𝑖 (cid:205) 𝐵 𝑍 𝐵 𝑑𝑍 𝐵 + (cid:205) 𝐵 ≠ 𝐴 | 𝑍 𝐴 | is a well-defined Poincaré-Liouville one-form, so that by Stokes theorem ( 𝑄 ) 𝐴 = 𝜋 ∫ 𝑆 𝑓 ∗ 𝜃 𝑠 ( 𝐴 ) = 𝑛 ∑︁ 𝐵 (cid:0) 𝑤 𝐵 − 𝑤 𝑠 ( 𝐴 ) (cid:1) , (11.5)where we have substituted the asymptotic values (11.2). Clearly (cid:205) 𝐴 ( 𝑄 ) 𝐴 =
0, as required.A general fractional instanton may be thought of as a collection of instantons ‘on top’ of anelementary fractional instanton, resulting in the topological charge (cid:174) 𝑄 + ( 𝑠 , · · · , 𝑠 𝑛 ) , where 𝑠 𝐴 ∈ Z are integers, (cid:205) 𝐴 𝑠 𝐴 = 𝑉 , and so we would like to restrict to the configurationsthat asymptotically minimize this potential, i.e. to the special maps 𝐷 → F , such that ( 𝑆 ) WS = 𝜕 𝐷 → 𝑆 ⊂ ( 𝑆 ) 𝑛 − . In this case we may set 𝑤 = · · · = 𝑤 𝑛 − =
0. Apart fromthat, let us restrict to a single winding, 𝑤 𝑛 =
1. The topological charges of an elementaryfractional instanton then are ( 𝑠 ( 𝑘 ) = 𝑛 ) (cid:174) 𝑄 = (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) 𝑛 , · · · , 𝑛 − position 𝑘 , · · · , 𝑛 (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 configurationshas increased, the original argument from SU(2) for mass generation carries over to this moregeneral case: for each species of topological excitation in this 𝑛 -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 𝑆𝐺 = ( 𝜕 𝜇 𝜎 ) + 𝛾 cos (cid:18) 𝜋𝑔 𝜎 (cid:19) , (11.7)in the limit of large 𝛾 , 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 finite in the sG model [158]. Formallyspeaking, expanding the partition function of the sG model in 𝛾 , 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) 𝑔 is the O(2) coupling constant in (11.4), and plays the role of temperature inthe sG model. Since all of the 𝑛 species arise from the same action, each will have the same76ugacity and critical 𝑔 , so that the above model (11.7) is merely copied 𝑛 times, and the SU(2)analysis from [8] can be applied directly: for large 𝑚 , the fractional instantons are diluteand we are in a massless boson phase. As 𝑚 is lowered, the effective 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 flagmanifold sigma model (10.1), in the absence of topological angles.
We now restore the topological angles 𝜃 𝛼 , and study how the mass generating mechanismchanges. For large 𝑚 , we are in the O(2) model and the 𝜃 -terms do not play a role. However,as 𝑚 is lowered towards zero, the fugacity 𝛾 in the sine-Gordon model is modified to 𝛾 𝑛 ∑︁ 𝐴 = 𝑒 𝑖 (cid:174) 𝜃 · (cid:174) 𝑄 𝐴 , (11.8)where the sum is over the 𝑛 species of fractional instanton, and 𝜃 𝐴 = 𝐴𝜃 , with 𝜃 = 𝜋𝑝𝑛 . Using(11.6), one easily finds that this sum equals 𝛾 (cid:205) 𝐴 𝜁 𝑝 𝐴 . So long as 𝑝 is not a multiple of 𝑛 ,this sum vanishes, and the Coulomb gas is in its massless phase.At first glance, this appears to be inconsistent with the conjecture discussed in section 10,which also predicts a gap when 𝑝 is not a multiple of 𝑛 , but has a nontrivial shared divisorwith 𝑛 . This discrepancy is resolved by considering higher order topological excitations.This is summarized in the following table: Case Winding Fugacity Conclusion 𝑝 = 𝑛 𝑛𝛾 Mass generation 𝑝 and 𝑛 coprime(no common divisor) 1 , · · · , 𝑛 − 𝑛 𝑛𝛾 Masslessgcd ( 𝑝, 𝑛 ) ≠ , 𝑛 ⇒ < 𝑑 < 𝑛 , · · · , 𝑑 − 𝑑 𝑛𝛾 Mass generationHere 𝑑 : = 𝑛 gcd ( 𝑝,𝑛 ) . Winding number 𝑤 refers to the map ( 𝑆 ) WS → ( 𝑆 ) target . Forsimplest fractional instanton configurations the topological charge is 𝑤 (cid:174) 𝑄 , and thefugacity is 𝛾 (cid:205) 𝐴 𝜁 𝑝 𝐴𝑤 .While objects that have winding number greater than ± 𝑚 =
0, and itis conjectured in [235] that this holds for general 𝑛 . When 𝑝 and 𝑛 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]. 𝑛 , configurations with a smaller value of the action begin to contributeto mass generation, starting with objects that have winding number 𝑑 . As a result, the criticalvalue 𝑚 is larger than at 𝜃 = 𝜋𝑛 (although still lower than at 𝜃 =
12 More general representations: linear and quadratic dis-persion
As we already discussed earlier, depending on the sign of the coupling constant 𝐽 the groundstate of the Heisenberg chain is either ferro- or anti-ferromagnetic. The continuum limitsaround these two states lead to rather different 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 ( 𝑛 ) , 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, 𝑝 ≥ 𝑝 ≥ · · · ≥ 𝑝 𝑛 − ≥
0. Amongsuch models we will pick out those that lead to sigma models with the flag manifold targetspace U ( 𝑛 ) U ( ) 𝑛 in the continuum limit. One restriction that is imposed by this setup is that all thenonvanishing 𝑝 𝑖 ’s will be distinct (if some of the rows of the Young diagram were of the samelength, one would obtain a partial flag manifold U ( 𝑛 ) U ( 𝑛 )×···× U ( 𝑛 𝑚 ) 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 𝑆 ( 𝑗 ) of the chain should be replaced by the corresponding moment maps 𝜇 ( 𝑗 ) = 𝑈 † diag ( 𝑝 , · · · , 𝑝 𝑛 ) 𝑈 , where 𝑈 is a unitary matrix, and 𝑝 · · · 𝑝 𝑛 are the lengths of the rowsin the Young diagram of the representation R ( 𝑝 𝑛 = 𝑖 and 𝑗 becomes Tr ( 𝑆 ( 𝑖 ) 𝑆 ( 𝑗 )) → (cid:205) 𝐴,𝐵 𝑝 𝐴 𝑝 𝐵 | 𝑢 𝐴 ( 𝑖 ) ◦ 𝑢 𝐵 ( 𝑗 ) | . As a result, thesimplest SU( 𝑛 ) chain Hamiltonian, namely the nearest-neighbour model, becomes 𝐻 = 𝐽 ∑︁ 𝑗 𝑛 − ∑︁ 𝐴,𝐵 = 𝑝 𝐴 𝑝 𝐵 | 𝑢 𝐴 ( 𝑗 ) ◦ 𝑢 𝐵 ( 𝑗 + ) | , 𝐽 > . (12.1)The sums over 𝐴 and 𝐵 stop at 𝑛 −
1, since 𝑝 𝑛 = 𝑛 ) 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 flag manifold sigma model. This should already78e clear from the discussion of the rank- 𝑝 symmetric representations in sections 7.3 and 9above. Since the complete flag manifold is the space of 𝑛 -tuples of mutually orthogonalfields taking values in CP 𝑛 − , one must add ( 𝑛 − ) -neighbour interactions in order to imposeorthogonality on the 𝑛 fields. Instead, if one couples less than 𝑛 sites of the chain together,there will be leftover degrees of freedom, which manifest as local zero modes, ultimatelyprohibiting any field theory mapping.We will now explain how this construction generalizes as we increase the number 𝑘 ofnonzero 𝑝 𝑖 .For the purposes of presentation we will consider three main examples: 𝑘 = 𝑘 = 𝑛 − 𝑅 (cid:39) 𝑅 , is a generalization of the construction described in section 7.1) and the case 𝑛 = 𝜆𝑘 .We refer the reader to [234] for more general situations. We will be using some graphical notation for describing classical spin configurations ofSU( 𝑛 ) chains. First, let { (cid:174) 𝑒 𝑖 } be an orthonormal basis of C 𝑛 .We will use coloured circles to represent the first few elements of this basis, as shownin Fig 14. ~e = ~e = ~e = ~e = ~e = ~e = ~e = ~e = Figure 14: Colour dictionary for the first eight basis elements in C 𝑛 . 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 Néelstate of the SU(2) antiferromagnet is . (12.2)The benefit of these ground state pictures is that it makes it easy to read off the energy costof a term Tr ( 𝑆 ( 𝑖 ) 𝑆 ( 𝑗 )) = (cid:205) 𝐴,𝐵 𝑝 𝐴 𝑝 𝐵 | 𝑢 𝐴 ( 𝑖 ) ◦ 𝑢 𝐵 ( 𝑗 ) | . The right hand side of this expressionvanishes unless one of the complex unit vectors (i.e. one of the colours) at site 𝑖 equals oneof the complex unit vectors at site 𝑗 . In this case, the r.h.s. equals 𝑝 𝐴 𝑝 𝐵 , where 𝐴 and 𝐵 are the respective positions of the unit vector/colour in column 𝑖 and column 𝑗 .With this notation in place, we will now describe the ground state structure of SU( 𝑛 )chains with some sufficiently simple representations at each site.79 𝑘 = . We begin with recalling what occurs for the symmetric representations of SU( 𝑛 ), with Youngtableaux that have a single row of length 𝑝 (see sections 7.3-9 above). For a nearest-neighbour SU( 𝑛 ) Heisenberg Hamiltonian any configuration that has no energy cost per bondwill be a classical ground state. Since 𝑘 =
1, and only a single node is present at each site,the Néel state shown above is such an example. However, for 𝑛 >
2, the basis at each site islarger than 2 (i.e. there are other colours available), and this leads to an infinite number ofother ground states, resulting in a zero-energy mode that destabilizes any candidate groundstate above which we would like to derive a quantum field theory. As a consequence, thenearest-neighbour Hamiltonian must be modified by longer-range interactions . Since thereare 𝑛 possible colours, we require up to ( 𝑛 − ) -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 configuration (9.1). As we recall from sections 7-9, theflag manifold U ( 𝑛 ) U ( ) 𝑛 appears here as the space of 𝑛 -tuples of pairwise orthogonal complexvectors, each vector coming from a copy of CP 𝑛 − at one site of the chain. 𝑘 = 𝑛 − . The second class of representations that we consider have Young tableaux with 𝑛 − R already corresponds to the complete flag manifold, a nearest-neighbour Heisenberg interaction is sufficient to derive the associated sigma model. Letus first demonstrate this for the case of SU(4). The interaction term Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + )) = (cid:205) 𝐴,𝐵 = 𝑝 𝐴 𝑝 𝐵 | 𝑢 𝐴 ( 𝑗 ) ◦ 𝑢 𝐵 ( 𝑗 + ) | is never zero for two adjacent sites, which requires choosingthe colour for six nodes. Using the inequality 𝑝 + 𝑝 ≥ 𝑝 𝑝 , after a moment’s thoughtone finds that the ground states have the following form: (12.4)This pattern extends to general 𝑛 : the first row of nodes establishes a Néel-like state, while theremaining 𝑛 − Z translation symmetry in the sigma model. These interactions may be dynamically generated from the nearest-neighbour model [86]. 𝑛 = 𝜆𝑘 To construct the ground state, one partitions 𝑛 colors into 𝜆 sets, with 𝑘 colors in each set.We would like to place each set at one of the consecutive 𝜆 sites, so to this end we add up ( 𝜆 − ) -neighbour interactions (always with antiferromagnetic couplings) which make surethat the 𝑘 -planes at the consecutive sites are orthogonal to each other. On top of that, inorder to obtain the complete flag manifold, one still needs to orthogonalize 𝑘 vectors insideeach 𝑘 -plane, which can be achieved by adding a weaker 𝜆 -neighbour interaction that servesto reverse order within each set of the partition. For example, in SU(6) with 𝑘 =
2, theHamiltonian we should consider is 𝐻 = ∑︁ 𝑗 (cid:16) 𝐽 Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + )) + 𝐽 Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + )) + 𝐽 Tr ( 𝑆 ( 𝑗 ) 𝑆 ( 𝑗 + )) (cid:17) (12.5)with 𝐽 > 𝐽 (cid:29) 𝐽 >
0, which has, for example, the following ground state: (12.6)The 𝐽 and 𝐽 terms serve to partition the colours into three sets (the ‘2-planes’): { , } , { , } , { , } , and the 𝐽 terms serve to reverse order within each of these three sets. Clearlythe unit-cell has size 2 𝜆 for these representations. In the previous section we constructed spin chain Hamiltonians, whose classical minima lieon a complete flag manifold. This defines an embedding 𝑖 : F ⊂ M , (12.7)where M = (cid:206) 𝑑𝐴 = M 𝐴 is the phase space of an elementary cell of length 𝑑 . As we recall fromsections 6.3-7, the next step in deriving a continuum theory is in evaluating the restriction ofthe symplectic form Ω M = 𝑑 ∑︁ 𝐴 = Ω 𝐴 , (12.8)which comes from the kinetic term in the Lagrangian (the ‘Berry phase’), to the space ofminima of the Hamiltonian, i.e. to the flag 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 flag 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 Ω M (cid:12)(cid:12) F , whichdefines the number of fields with quadratic dispersion. By tuning the values of the integers81epresentation Conditions 𝑘 = 𝑘 = 𝑛 − 𝑝 𝑖 + 𝑝 𝑛 − 𝑖 + = 𝑝 𝑛 even; 𝑖 = , · · · , 𝑛 𝑝 𝑖 + 𝑝 𝑛 − 𝑖 + = 𝑝 𝑝 𝑛 + = 𝑝 𝑛 odd; 𝑖 = , · · · , 𝑛 − 𝑛 = 𝑘𝜆 𝑝 𝑖 + 𝑝 𝑘 + − 𝑖 = 𝑝 + 𝑝 𝑘 𝑘 even; 𝑖 = , · · · , 𝑘 𝑝 𝑖 + 𝑝 𝑘 + − 𝑖 = 𝑝 + 𝑝 𝑘 𝑝 𝑘 + = 𝑝 + 𝑝 𝑘 𝑘 odd; 𝑖 = , · · · , 𝑘 − Table 2: Rank reduction conditions (elimination of modes with quadratic dispersion) for thetwo-form Ω M (cid:12)(cid:12) F . 𝑝 𝑖 , i.e. by suitably choosing the representation R , one can reduce the rank down to zero, inwhich case F ⊂ M is an isotropic submanifold.At least in the linearly dispersing case (when Ω M (cid:12)(cid:12) F = 𝜔 top = 𝑑 𝑑 ∑︁ 𝐴 = 𝐴 · Ω 𝐴 (cid:12)(cid:12) F . (12.9)We recall (cf. (2.30)) that in the general case each form Ω 𝐴 may be written as Ω 𝐴 = 𝑖 𝑛 (cid:205) 𝑘 = 𝑝 𝑘 𝑑𝑢 ( 𝑘 ) 𝐴 ∧ ◦ 𝑑𝑢 ( 𝑘 ) 𝐴 , where { 𝑢 ( 𝑘 ) 𝐴 } 𝑛𝑘 = are mutually orthogonal vectors at the 𝐴 -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 ‘restriction to F ’ in (12.9) means replacingthe given vector 𝑢 ( 𝑘 ) 𝐴 by the vector of the flag corresponding to the indicated colour.Let us demonstrate how this works for 𝑘 = 𝑛 −
1. The elementary cell consists of twosites, 𝑑 =
2, so that Ω M = 𝑖 (cid:205) 𝑗 = 𝑛 (cid:205) 𝐴 = 𝑝 𝐴 𝑑𝑢 𝐴 ( 𝑗 ) ∧ ◦ 𝑑𝑢 𝐴 ( 𝑗 ) . According to the pattern ofground states (see (12.4)), two of the colours occur once (in the first position of the column),and the remaining 𝑛 − For the case of spins with rectangular Young tableau at each site, when the resulting flag manifold is themanifold of partial flags, the same expression was obtained in [73].
82f the symplectic form is1 𝑖 Ω M (cid:12)(cid:12) F = 𝑝 ( 𝑑𝑢 ∧ ◦ 𝑑𝑢 + 𝑑𝑢 ∧ ◦ 𝑑𝑢 ) + 𝑛 ∑︁ 𝐴 = ( 𝑝 𝐴 − + 𝑝 𝑛 − 𝐴 + ) 𝑑𝑢 𝐴 ∧ ◦ 𝑑𝑢 𝐴 . (12.10)Here 𝑢 𝐴 without the site label in brackets are meant to represent the 𝑛 orthogonal vectors ofthe embedded flag manifold F . We recall that (cid:205) 𝑛𝐴 = 𝑑𝑢 𝐴 ∧◦ 𝑑𝑢 𝐴 (cid:12)(cid:12) F =
0, so that the expressioncan be simplified: 1 𝑖 Ω M (cid:12)(cid:12) F = 𝑛 ∑︁ 𝐴 = ( 𝑝 𝐴 − + 𝑝 𝑛 − 𝐴 + − 𝑝 ) 𝑑𝑢 𝐴 ∧ ◦ 𝑑𝑢 𝐴 . (12.11)Now we have up to ( 𝑛 − ) fields with quadratic dispersion. The exact number will dependon how many of the conditions 𝑝 𝐴 − + 𝑝 𝑛 − 𝐴 + − 𝑝 = 𝑅 (cid:39) 𝑅 of SU( 𝑛 ). The case of SU(3) with2 𝑝 = 𝑝 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 𝜔 top (12.9), one obtains the topological term. Similarly to (12.11), it may be expanded as 𝜔 top = 𝑖 𝑛 ∑︁ 𝐴 = 𝜃 𝐴 𝑑𝑢 𝐴 ∧ ◦ 𝑑𝑢 𝐴 . (12.12)For the simple representations that we have discussed here the values of the 𝜃 -angles arerecorded in Table 3. The discrete symmetry in the general case is Z 𝑑 : it acts on 𝜔 top by shifting 𝜔 top → 𝜔 top + Ω (compare with (8.21)). One could in principle derive themixed PSU ( 𝑛 ) − Z 𝑑 anomalies in this case as well, which would provide a generalization ofHaldane-type conjectures to this type of representations.Representation Topological Angles 𝑘 = 𝜃 𝐴 = 𝜋𝑝 𝑛 ( 𝐴 − ) 𝑖 = , , · · · , 𝑛𝑘 = 𝑛 − 𝜃 𝐴 = 𝜋 𝑝 𝐴 𝑖 = , , · · · , 𝑛𝑘 = 𝑛𝜆 𝜃 𝐴,𝐵 = 𝜋 ( 𝑝 𝐴 + 𝑝 𝑘 + − 𝐵 ) 𝜆 ( 𝐵 − ) + 𝜋 𝑝 𝑘 + − 𝐴 𝐴 = , · · · , 𝑘 ; 𝐵 = , · · · , 𝜆 Table 3: Possible topological angles for some representations of SU( 𝑛 ) chains. In the lastrow the index is split as 𝐴 → ( 𝐴, 𝐵 ) . 83 hapter 3. Integrable flag manifold sigma models and beyond In the present chapter we pass to the subject of integrable sigma models with flag manifoldtarget spaces, as well as some more general models. Recall that the integrability of the 𝑆 -model [264], which predicted massive excitations over the vacuum state, was one of themotivations for Haldane’s proposal that SU ( ) integer-spin chains have a gap in the spectrum.It was subsequently shown [265] that the 𝜃 = 𝜋 model is soluble as well, this time with amassless spectrum, in line with Haldane’s treatment of the half-integer-spin chains. In the caseof SU ( 𝑛 ) chains the resulting flag manifold sigma models described in the previous chapterare apparently not integrable, and their integrable counterparts discussed below feature avery special metric and 𝐵 -field. One striking parallel between the two types of models is theimportant role played by the Z 𝑛 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 asignificant challenge. In section 16 we will argue that the integrable flag 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 𝛽 -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
Ψ = (cid:18) 𝑈𝑉 (cid:19) the two-dimensional Thirring Lagrangian reads: L = Ψ / 𝜕 Ψ + ( Ψ 𝛾 𝜇 Ψ ) = 𝑉 𝜕𝑈 + 𝑈𝜕𝑉 + | 𝑈 | | 𝑉 | . (12.13)To obtain the bosonic Thirring model, we now regard the variables 𝑈 and 𝑉 as bosonic.Eliminating 𝑉 , 𝑉 , we obtain the sigma model form of the system: L = 𝜕𝑈 𝜕𝑈𝑈𝑈 . The targetspace is a cylinder with multiplicative coordinate 𝑈 . At the quantum level, the eliminationof 𝑉 , 𝑉 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 Φ ∼ log | 𝑈 | is linear along the cylinder. This is thesystem describing the asymptotic region of Witten’s 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 𝐵 -field in Lie-algebraic terms. In this chapter we will always assume that the worldsheet Σ is a two-dimensional Riemannian manifold. As for the target-space M , in full generality wewill not require it to be a flag manifold but rather a manifold with the following properties : ◦ M is a homogeneous space 𝐺 / 𝐻, 𝐺 semi-simple and compact ◦ M has an integrable 𝐺 -invariant complex structure J (13.1) ◦ The Killing metric G on M is Hermitian w.r.t. J Let us explain what we mean by ‘Killing metric’ on a homogeneous space. To this end, wedecompose the Lie algebra 𝔤 of the Lie group 𝐺 as 𝔤 = 𝔥 ⊕ 𝔪 , (13.2)where 𝔪 is the orthogonal complement to 𝔥 with respect to the Killing metric on 𝔤 . Wemay accordingly decompose the Maurer-Cartan current 𝐽 = 𝑔 − 𝑑𝑔 = 𝐽 𝔥 ⊕ 𝐽 𝔪 . The ‘Killingmetric’ G on 𝐺𝐻 is defined by the line element 𝑑𝑠 = − Tr ( 𝐽 𝔪 ) . (13.3)The corresponding metric on 𝔪 will be called Killing as well. In the case of trivial 𝔥 thiswould then reduce to the canonical Killing metric, hence the name.For a target space M with the properties (13.1), one can define a sigma model, whoseequations of motion may be rewritten as the flatness condition for a one-parameter family of Generalizations to non-simple groups 𝐺 are also possible. 𝐴 𝑢 , 𝑢 ∈ C ∗ . This flatness 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 find Bäcklund transformations [228, 95], and it is a starting point for theconstruction of classical solutions of the models [132].Complex simply-connected homogeneous manifolds 𝐺 / 𝐻 with 𝐺 semi-simple were clas-sified long ago [238]. They are given by the following theorem: any such manifold 𝐺 / 𝐻 corresponds to a subgroup 𝐻 , whose semi-simple part coincides with the semi-simple part ofthe centralizer of a toric subgroup of 𝐺 . For the case of 𝐺 = SU ( 𝑛 ) , for example, invariantcomplex structures exist on those of the manifolds M 𝑛 ,...,𝑛 𝑚 | 𝑛 = SU ( 𝑛 ) 𝑆 ( U ( 𝑛 ) × . . . × U ( 𝑛 𝑚 )) , 𝑚 ≥ , 𝑛 𝑖 > , 𝑚 ∑︁ 𝑖 = 𝑛 𝑖 ≤ 𝑛 , (13.4)that are even-dimensional. If (cid:205) 𝑚𝑖 = 𝑛 𝑖 = 𝑛 , the manifold in (13.4) is a flag manifold. Otherwise,it is a toric bundle over a flag manifold. The fiber U ( ) 𝑠 (2 𝑠 = 𝑛 − (cid:205) 𝑚𝑖 = 𝑛 𝑖 ) of the toricbundle is even-dimensional, since the flag manifold itself is even-dimensional.The models, which will be of interest for us in the present paper, are defined by thefollowing action: S [ G , J ] : = ∫ Σ 𝑑 𝑧 (cid:107) 𝜕 𝑋 (cid:107) G + ∫ Σ 𝑋 ∗ 𝜔, (13.5)where 𝜔 is the fundamental Hermitian form corresponding to the pair ( G , J ) , defined as 𝜔 = G ◦ J . (13.6)In general, the Killing metric G is not Kähler, i.e. the fundamental Hermitian form isnot closed: 𝑑𝜔 ≠
0. Even if the manifold M admits a Kähler metric, it is in generaldifferent from G . As an example of such a phenomenon one can consider the flag manifold SU ( ) 𝑆 ( U ( ) ) . The sigma model (13.5) for this flag manifold was investigated in detail in [70, 63].Other examples of models of the class (13.1) are provided by Hermitian symmetric spaces –symmetric spaces with a complex structure. These manifolds are Kähler, and the invariantmetric is essentially unique (up to scale), thus leading to the closedness of 𝜔 : 𝑑𝜔 =
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 ] : = ∫ Σ 𝑑 𝑧 G 𝑗 𝑘 𝜕𝑈 𝑗 𝜕𝑈 𝑘 , (13.7)where we have introduced complex coordinates 𝑈 𝑗 on M . Curiously, models with the 𝐵 -fieldof 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 = 𝐺𝐻 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 𝐺 / 𝐻 possesses a 𝐺 -invariant almost complexstructure J . We are not postulating that J be integrable – this will rather follow from therequirement of the existence of a Lax connection. The almost complex structure acts on 𝔪 (the subspace featuring in the decomposition (13.2)) and may be diagonalized, its eigenvaluesbeing ± 𝑖 (see the following section for details). We denote the ± 𝑖 -eigenspaces by 𝔪 ± ⊂ 𝔪 C : 𝔤 C = 𝔥 C ⊕ 𝔪 + ⊕ 𝔪 − , J ◦ 𝔪 ± = ± 𝑖 𝔪 ± . (13.8) 𝐺 -invariance of the almost complex structure implies that [ 𝔥 , 𝔪 ± ] ⊂ 𝔪 ± . We introduce thecurrent 𝐽 = 𝑔 − 𝑑𝑔 = 𝐽 + 𝐽 + + 𝐽 − , 𝐽 ∈ 𝔥 , 𝐽 ± ∈ 𝔪 ± . (13.9)It takes values in the Lie algebra 𝔤 , 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 𝛼, 𝛽 ∈ 𝔤 in the Killing metric): S [ G , J ] : = ∫ Σ 𝑑 𝑧 (cid:104)( 𝐽 + ) 𝑧 , ( 𝐽 − ) 𝑧 (cid:105) . (13.10) Example.
Let us consider the flag manifolds of the group 𝐺 = SU ( 𝑛 ) , which are the mainsubject of this review. A typical integrable complex structure on the flag manifold defines theholomorphic/anti-holomorphic subspaces 𝔪 ± shown in Fig. 15. It is useful do decomposeFigure 15: The decomposition (13.8) of the Lie algebra. 𝐽 ± in the irreducible representations 𝑉 𝐴𝐵 of the stabilizer 𝔥 , see (1.30). For this purpose weparametrize the unitary matrix 𝑔 as follows: 𝑔 = { 𝜏 , 𝜏 , . . . , 𝜏 𝑚 − , 𝜏 𝑚 } , (13.11)87here 𝜏 . . . 𝜏 𝑚 are groups of 𝑛 . . . 𝑛 𝑚 orthonormal vectors, each group parametrizing a planeof the corresponding dimension in C 𝑛 . The projection of 𝐽 ± on 𝑉 𝐴𝐵 is given by 𝐽 𝐴𝐵 : = 𝜏 † 𝐴 𝑑𝜏 𝐵 ,and the full action (13.10) takes the form S [ G , J ] : = ∫ Σ 𝑑 𝑧 ∑︁ 𝐴<𝐵 Tr ( ( 𝐽 𝐵𝐴 ) 𝑧 ( 𝐽 𝐴𝐵 ) 𝑧 ) . (13.12)These are the integrable models with flag 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 𝐾 . It is derived by taking an infinitesimal ( 𝑧, 𝑧 ) -dependent variation 𝑔 → 𝑒 𝜖 ( 𝑧,𝑧 ) ◦ 𝑔 in the above action, which leads to 𝐾 = 𝑔 · (cid:0) ( 𝐽 + ) 𝑧 𝑑𝑧 + ( 𝐽 − ) 𝑧 𝑑𝑧 (cid:1) : = 𝑆 · 𝑔 − = 𝑔𝑆𝑔 − (13.13)Since the target space M = 𝐺 / 𝐻 is homogeneous, the equations of motion of the model areequivalent to the conservation of 𝐾 : 𝑑 ∗ 𝐾 = . (13.14)Here, ∗ denotes the Hodge star operator, whose action on one-forms is defined by ∗ 𝑑𝑧 = 𝑖 𝑑𝑧, ∗ 𝑑𝑧 = − 𝑖 𝑑𝑧 . . In order to be able to build a family of flat connections we require that 𝐾 be flat (This will be used in (13.19)-(13.20) below.): 𝑑𝐾 − 𝐾 ∧ 𝐾 = . (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 𝑆 (introduced in (13.13)) as follows: 𝑑 ∗ 𝑆 + { 𝐽, ∗ 𝑆 } = (13.16) = − 𝑖𝑑𝑧 ∧ 𝑑𝑧 (cid:0) D ( 𝐽 + ) 𝑧 − [( 𝐽 + ) 𝑧 , ( 𝐽 + ) 𝑧 ] + D ( 𝐽 − ) 𝑧 + [( 𝐽 − ) 𝑧 , ( 𝐽 − ) 𝑧 ] (cid:1) = 𝑑𝑆 + { 𝐽 − 𝑆, 𝑆 } = (13.17) = − 𝑑𝑧 ∧ 𝑑𝑧 (cid:0) D ( 𝐽 + ) 𝑧 − [( 𝐽 + ) 𝑧 , ( 𝐽 + ) 𝑧 ] − D ( 𝐽 − ) 𝑧 − [( 𝐽 − ) 𝑧 , ( 𝐽 − ) 𝑧 ] (cid:1) = D is the covariant derivative for the gauge group 𝐻 : D 𝑗 𝑀 𝑘 : = 𝜕 𝑗 𝑀 𝑘 + [( 𝐽 ) 𝑗 , 𝑀 𝑘 ] ( 𝑗 , 𝑘 = 𝑧, 𝑧 ). The conditions (13.16)-(13.17) are equivalent, if [ 𝔪 + , 𝔪 + ] ⊂ 𝔪 + , [ 𝔪 − , 𝔪 − ] ⊂ 𝔪 − . (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 𝐴 𝑢 , indexed by a parameter 𝑢 ∈ C ∗ : 𝐴 𝑢 = − 𝑢 𝐾 𝑧 𝑑𝑧 + − 𝑢 − 𝐾 𝑧 𝑑𝑧 . (13.19)88onservation and flatness of the Noether current 𝐾 , eqs. (13.14)-(13.15), imply that 𝐴 𝑢 is flat for all 𝑢 [199]: 𝑑𝐴 𝑢 − 𝐴 𝑢 ∧ 𝐴 𝑢 = 𝑢 ∈ C ∗ . (13.20)This completes the derivation of the zero-curvature representation for the class of mod-els (13.10), which includes the flag manifold models (13.12). We now turn to the general theory of complex structures on flag 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 ( 𝑛 ) U ( ) 𝑛 of complete flags in C 𝑛 thereare 2 𝑛 ( 𝑛 − ) invariant almost complex structures, with 𝑛 ! ≤ 𝑛 ( 𝑛 − ) of them being integrable. As we already saw in (13.8), the complex structure J induces a decomposition 𝔪 C =𝔪 + ⊕ 𝔪 − , where 𝔪 ± play the role of holomorphic tangent spaces to 𝐺 / 𝐻 , i.e. J ◦ 𝑎 = ± 𝑖 𝑎 for 𝑎 ∈ 𝔪 ± . In section 1.5 (formula (1.30)) we have already decomposed 𝔪 C into irreduciblecomponents. Using this decomposition, we may define an almost complex structure on F bydefining the action of J as follows: J ◦ 𝑉 𝐴𝐵 = ± 𝑖 𝑉 𝐴𝐵 for 1 ≤ 𝐴 < 𝐵 ≤ 𝑛 . (13.21)As a result, one has exactly 2 𝑛 ( 𝑛 − ) possibilities. There are several equivalent definitions ofintegrability of a complex structure: ◦ Vanishing of the Nijenhuis tensor: [ J ◦ 𝑋, J ◦ 𝑌 ] − J ◦ ( [ J ◦ 𝑋, 𝑌 ] + [ 𝑋, J ◦ 𝑌 ]) − [ 𝑋, 𝑌 ] = 𝑋, 𝑌 . ◦ Using vector fields: the commutator of two holomorphic vector fields should be holo-morphic, i.e. ( − 𝑖 J ) [( + 𝑖 J ) 𝑋, ( + 𝑖 J ) 𝑌 ] = . (13.23)(The property (13.23) may also be stated as the condition that the distribution ofholomorphic vector fields is integrable.) This is easily seen to be equivalent to (13.22). ◦ Using forms: the holomorphic forms should constitute a differential ideal in the algebraof forms, i.e. the following condition should be satisfied: 𝑑 ( 𝐽 − ) 𝑎 ∼ (cid:205) 𝑏 𝑅 𝑎𝑏 ∧ ( 𝐽 − ) 𝑏 forsome one-forms 𝑅 𝑎𝑏 . We note that for large 𝑛 , according to Stirling’s formula, 𝑒 𝑛 log ( 𝑛 ) < 𝑒 log ( ) 𝑛 .
89f the restriction to 𝔪 of the adjoint-invariant metric (cid:104)• , •(cid:105) on 𝔲 ( 𝑛 ) is Hermitian w.r.t. thechosen almost complex structure J , the last definition implies [ 𝔪 + , 𝔪 + ] ⊂ 𝔪 + , [ 𝔪 − , 𝔪 − ] ⊂ 𝔪 − . (13.24)This is proven in Appendix D. The latter will serve us as a working definition of an integrablecomplex structure.On the complete flag manifold F 𝑛 , one can define an almost complex structure by choosing 𝑛 ( 𝑛 − ) mutually non-conjugate forms 𝐽 𝐴 𝐵 , . . . , 𝐽 𝐴 𝑛 ( 𝑛 − ) 𝐵 𝑛 ( 𝑛 − ) and declaring them holomor-phic. The remaining 𝑛 ( 𝑛 − ) forms will be therefore anti-holomorphic. To determine which ofthese complex structures are integrable, it is useful to use a diagrammatic representation. Wedraw 𝑛 vertices, as well as arrows from the node 𝐴 to the node 𝐵 , from 𝐴 to 𝐵 and so on,so that all pairs of nodes are connected (such diagrams are called ‘tournaments’, see [59]).As we shall now prove, the integrability of the almost complex structure, defined 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 𝑒 𝐴 , 𝐴 = , , ( 𝑒 𝐴 ) 𝛼 = 𝛿 𝐴𝛼 ( 𝛼 = , , 𝔪 + of the Lie algebra ( 𝔰𝔲 ( )) C = 𝔰𝔩 ( ) as follows: 𝔪 + = Span ( 𝐸 𝐴 𝐵 , 𝐸 𝐴 𝐵 , 𝐸 𝐴 𝐵 ) , where 𝐸 𝐴𝐵 = 𝑒 𝐴 ⊗ 𝑒 𝐵 (13.25)Integrability of the complex structure is equivalent to the requirement that 𝔪 + is a subalgebra: [ 𝔪 + , 𝔪 + ] ⊂ 𝔪 + . On the other hand, the matrices 𝐸 𝑚𝑛 have the commutation relations [ 𝐸 𝐴𝐵 , 𝐸 𝐶𝐷 ] = 𝛿 𝐵𝐶 𝐸 𝐴𝐷 − 𝛿 𝐴𝐷 𝐸 𝐶𝐵 (13.26)In the tournament diagram, 𝐸 𝐴𝐵 is represented by an arrow from 𝐴 to 𝐵 ; thus one sees thatthe closedness of 𝔪 + under commutation is equivalent to the following statement:For any two consecutive arrows 𝐴 → 𝐵 and 𝐵 → 𝐶 (13.27)their ‘shortcut’ segment ( 𝐴, 𝐶 ) has the arrow 𝐴 → 𝐶 For the diagram with three vertices, i.e. for the 𝔰𝔲 ( ) 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 flag manifold F 𝑛 , suppose we have 𝑛 pairwise-connectedvertices, and the graph is acyclic. Then the requirement (13.27) is satisfied, since otherwisethere would be a cycle with three vertices. Reversely, suppose the graph has a cycle. Then,using (13.27), one can ‘cut corners’ to reduce again to the cycle with three vertices, which isprohibited (see Fig. 16).One can then establish that there are exactly 𝑛 ! acyclic diagrams. They correspond tothe total orderings of the set of 𝑛 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 𝑛 ! complex structures on a complete flag manifold U ( 𝑛 ) U ( ) 𝑛 . Analogously there are 𝑚 ! complex structures on a partial flag manifold U ( 𝑛 ) U ( 𝑛 )×···× U ( 𝑛 𝑚 ) .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 𝑛 𝐴 . These dimensions are therefore determined by an ordering of the set { 𝑛 𝐴 } , and there are 𝑚 ! such orderings.Now that we have described all invariant complex structures on an arbitrary flag 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 𝑚 −
1. Using the descriptionof the cohomology (1.24) based on the embedding of the flag manifold into a product ofGrassmannians (1.21), one can describe the space of closed two-forms as a hyperplane 𝑥 + . . . + 𝑥 𝑚 = R 𝑚 . (13.28)Consider a typical point in this vector space, where all 𝑥 𝑖 ’s are distinct. This correspondsto a non-degenerate two-form, i.e. a symplectic form Ω . Given any such form, one canshow that there is a unique complex structure J , such that the corresponding symmetrictensor G : = − Ω ◦ J is positive-definite. This is tantamount to saying that G defines aKähler metric on the flag manifold. In the simplest case of a Grassmannian, when 𝑚 = 𝛼 · Ω FS of the generalized Fubini-Study form. There arealso two invariant complex structures: J and − J . As a result, the real line of invariantclosed two-forms parametrized by 𝛼 = 𝑥 − 𝑥 is divided into two rays 𝛼 ≷
0, and on each ofthese rays one picks a suitable complex structure ± J to define a metric. The two rays areinterchanged by the action of 𝑆 : 𝑥 ↔ 𝑥 .Returning to the general case, one finds that the hyperplane (13.28) is divided into 𝑚 !chambers, such that the points in the interior of each chamber may be thought of as Kählerforms corresponding to the same invariant complex structure. The chambers are interchanged91y the action of the permutation group on 𝑥 , . . . , 𝑥 𝑚 , which is free provided that 𝑥 𝑖 ’s areall distinct. This action is clearly synchronized with the action of 𝑆 𝑚 that interchangesthe complex structures. The boundaries between the chambers correspond to the case whenseveral 𝑥 𝑖 ’s coincide, which leads to the degeneration of the two-forms. As already mentionedat the end of section 2.2, in this case we may find a suitable smaller flag manifold, on whichthe two-form Ω is non-degenerate. The smaller flag 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 flag, and the extreme case when all 𝑥 𝑖 = As a first 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 [ 𝔪 , 𝔪 ] ⊂ 𝔥 . The Hermitian symmetricspaces are, in turn, characterized by the relation [ 𝔪 + , 𝔪 + ] =
0. Indeed, this follows fromthe symmetric space property [ 𝔪 , 𝔪 ] ⊂ 𝔥 and the integrability of the complex structure [ 𝔪 + , 𝔪 + ] ⊂ 𝔪 + . Conversely, if [ 𝔪 + , 𝔪 + ] =
0, one shows, using ad-invariance of the Killingmetric on 𝔤 , that [ 𝔪 + , 𝔪 − ] is orthogonal to 𝔪 ± , and hence [ 𝔪 + , 𝔪 − ] ⊂ 𝔥 .The case of a symmetric target space is special in that the form 𝜔 is closed: 𝑑𝜔 =
0. Inother words, the Killing metric G is Kähler. Moreover, it is the only case when this is so:The Killing metric on F = F 𝑛 ,...,𝑛 𝑚 is Kähler if and only if F is a symmetricspace, i.e. 𝑚 = F = 𝐺𝑟 𝑛 ,𝑛 is a Grassmannian.To prove this, we note that the components 𝐽 ± of the current 𝐽 (see the decomposition(13.9)) represent holomorphic/anti-holomorphic one-forms. The Killing metric on F , whichis 𝑑𝑠 = − ( 𝐽 + 𝐽 − ) , is therefore Hermitian. The Kähler form is, accordingly, 𝜔 = 𝑖 Tr ( 𝐽 + ∧ 𝐽 − ) . (13.29)In calculating the exterior derivative of 𝜔 , we will be using the flatness equation 𝑑𝐽 − 𝐽 ∧ 𝐽 = 𝔥 ⊥ 𝔪 ± , 𝔪 ± ⊥ 𝔪 ± (isotropy of 𝔪 ± ). Simplifying the resulting expression,one gets 𝑑𝜔 = 𝑖 ( Tr ( 𝐽 + ∧ 𝐽 − ∧ 𝐽 − ) − Tr ( 𝐽 − ∧ 𝐽 + ∧ 𝐽 + )) . (13.30)The three-forms in the r.h.s. are of type ( , ) and ( , ) respectively (they are complexconjugate to each other). Therefore 𝑑𝜔 = ( 𝐽 + ∧ 𝐽 − ∧ 𝐽 − ) = . Due to thenon-degeneracy of the Killing metric, this can only hold if [ 𝔪 − , 𝔪 − ] = . (13.31)92t is easy to see that this holds if and only if 𝑚 = 𝑚 =
2, the second term inthe action (13.5) is in fact topological and therefore does not affect 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 different from the onein (13.19). Indeed, the connection usually employed in the analysis of sigma models withsymmetric target spaces has the form (cid:101) 𝐴 𝜆 = − 𝜆 (cid:101) 𝐾 𝑧 𝑑𝑧 + − 𝜆 − (cid:101) 𝐾 𝑧 𝑑𝑧, where (cid:101) 𝐾 = 𝑔 · (cid:2) 𝑔 − 𝑑𝑔 (cid:3) 𝔪 · 𝑔 − (13.32)is the Noether current derived using the canonical action S [ G ] = ∫ Σ 𝑑 𝑧 (cid:107) 𝜕 𝑋 (cid:107) G . (13.33)In the case of a Hermitian symmetric target space the difference between the two actions,(13.5) and (13.33), is a topological term: S [ G , J ] − S [ G ] = ∫ Σ 𝑋 ∗ 𝜔, (13.34)where 𝑑𝜔 = M is symmetric.Therefore the two actions lead to the same equations of motion. Nevertheless, the Noethercurrents 𝐾 and (cid:101) 𝐾 are different, although both are flat. For the current 𝐾 this was shownin (13.16)-(13.18), whereas the flatness 𝑑 (cid:101) 𝐾 − (cid:101) 𝐾 ∧ (cid:101) 𝐾 = (cid:101) 𝐾 does not, in fact, require using the equations of motion – it is purelya consequence of the structure of the Lie algebra of the symmetric space (in particular, thefact that [ 𝔪 , 𝔪 ] ⊂ 𝔥 ). Moreover, the flatness condition may be solved, in this case, in a localfashion : (cid:101) 𝐾 = − (cid:98) 𝑔 − 𝑑 (cid:98) 𝑔, where (cid:98) 𝑔 = 𝜎 ( 𝑔 ) 𝑔 − , (13.35) 𝜎 being Cartan’s involution on the Lie group 𝐺 . By definition, 𝜎 is a group homomorphism, 𝜎 ( 𝑔 𝑔 ) = 𝜎 ( 𝑔 ) 𝜎 ( 𝑔 ) , and 𝜎 ( ℎ ) = ℎ for ℎ ∈ 𝐻 . The formula (cid:98) 𝑔 = 𝜎 ( 𝑔 ) 𝑔 − , viewed as amap 𝑔 ∈ 𝐺 / 𝐻 → (cid:98) 𝑔 ∈ 𝐺 , describes the Cartan embedding 𝐺 / 𝐻 ↩ → 𝐺 . (13.36)Flatness and conservation of the current (cid:101) 𝐾 lead to the flatness of the family (cid:101) 𝐴 𝜆 . Aquestion naturally arises of what the relation between 𝐴 𝑢 and (cid:101) 𝐴 𝜆 is. The answer is that the ‘Local’ means that (cid:98) 𝑔 is a local function of the fields of the model. Cartan’s embedding is known to be totally geodesic. By definition, this means that the second fundamentalform of ˆ 𝜎 ( 𝐺𝐻 ) ⊂ 𝐺 vanishes: (∇ 𝑋 𝑌 ) ⊥ = 𝑋, 𝑌 ∈ 𝑇 ( ˆ 𝜎 ( 𝐺𝐻 )) . It is easy to check that ifˆ 𝜎 : M ⊂ N is a totally geodesic submanifold, and 𝑋 : Σ → M is a harmonic map (i.e. a solution to the sigmamodel e.o.m.), then ˆ 𝜎 ◦ 𝑋 : Σ → N is also harmonic. This means that the classical solutions of the symmetricspace 𝐺 / 𝐻 model are a subset of solutions of the principal chiral model. 𝐴 𝑢 and (cid:101) 𝐴 𝜆 are gauge-equivalent, if one makes the following identification ofspectral parameters: 𝜆 = 𝑢 / . (13.37)The gauge transformation G relating 𝐴 𝑢 and (cid:101) 𝐴 𝜆 , (cid:101) 𝐴 𝜆 = G 𝐴 𝑢 G − − G 𝑑 G − , (13.38)may be constructed explicitly. The following formula holds for the case when the target-spaceis the Grassmannian 𝐺𝑟 𝑛 ,𝑛 + 𝑛 : = SU ( 𝑛 + 𝑛 ) 𝑆 ( U ( 𝑛 )× U ( 𝑛 )) and the complex structure is chosen so thatit splits 𝔪 as 𝔪 = (cid:16) 𝔪 + 𝔪 − (cid:17) : G = 𝑔 Λ 𝑔 − , where Λ = 𝜆 𝑛 − 𝑛 𝑛 + 𝑛 Diag ( 𝜆 − / , . . . , 𝜆 − / 𝑛 , 𝜆 / , . . . , 𝜆 / 𝑛 ) . (13.39)Although the flag manifold (1.4) in general is not a symmetric space, it is a so-called Z 𝑚 -graded space. Perhaps most well-known are the Z -graded spaces, examples of whichare provided by twistor spaces of symmetric spaces [208] and nearly Kähler homogeneousspaces [61] (the latter also appear in the context of string compactifications, 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 𝑚 -graded target-spaces may be rewritten as flatness 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 𝑚 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 𝑚 -graded spaces and the models discussed so far in this chapter. Ingeneral, however, we view the approach based on complex structures as rather different fromthe one based on Z 𝑚 -gradings. This will be emphasized in the next sections, where complexstructures will be shown to play a key role through 𝛽𝛾 -systems, as well as in the formulationof the integrable sigma models as gauged chiral Gross-Neveu models. Z 𝑚 -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 different choices of complex structure. This is due to the fact, that for certain complexstructures, which we denote by J and J , the difference in the two actions may just be atopological term: S [ J ] − S [ J ] = ∫ Σ O , 𝑑 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 𝑛 , . . . C 𝑛 𝑚 constituting a flag, i.e. apoint in a flag manifold U ( 𝑛 ) U ( 𝑛 )×··· U ( 𝑛 𝑚 ) . The statement is then as follows:The actions S [ J ] and S [ J ] differ by a topological term, as in (13.40), ifand only if the corresponding sequences of spaces { C 𝑛 , . . . C 𝑛 𝑚 } differ 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 𝑚 ‘symmetry’ is very parallel to the Z 𝑛 -symmetryof sigma models arising from spin chains, which was ultimately a reflection of the transla-tional invariance of the latter and whose importance was emphasized in Sections 9.2 and 10.We write ‘symmetry’ in quotation marks, because it is really a symmetry of the theory onlyin the case 𝑛 = · · · = 𝑛 𝑚 , when it can be realized by a cyclic permutation of the groups ofvectors { 𝑢 ( ) , · · · , 𝑢 ( ) 𝑛 } , . . . , { 𝑢 ( 𝑚 ) , · · · , 𝑢 ( 𝑚 ) 𝑛 𝑚 } . In all other cases this should be seen as theequivalence of different theories, defined by the action functionals S [ J ] for different com-plex structures J . The same issue arises in the case of spin chains, when 1-site translationalinvariance (leading to Z 𝑚 -symmetry) is only present when the representations at each site areequivalent, which again leads to the condition 𝑛 = · · · = 𝑛 𝑚 . On the other hand, continuumlimits of spin chains with different representations at different sites may still be described bypartial flag 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 flag 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 𝑟 -matrix. This is a very well-known object in integrable theories, but forcompleteness we shall start by recalling its definition and providing the simplest examplesthat we will use later on. 𝑟 -matrix. The classical 𝑟 -matrix 𝑟 ( 𝑢 ) takes values in 𝔤 ⊗ 𝔤 , where 𝔤 is a semi-simple or, more generally,reductive Lie algebra and 𝑢 is a parameter taking values in a complex abelian group ( C , C ∗ orthe elliptic curve 𝐸 𝜏 , depending on whether one deals with the rational/trigonometric/elliptic95ase respectively). The 𝑟 -matrix satisfies the classical Yang-Baxter equation (CYBE), whichtakes values in 𝔤 ⊗ 𝔤 ⊗ 𝔤 and has the following form: [ 𝑟 ( 𝑢 ) , 𝑟 ( 𝑢 · 𝑣 )] + [ 𝑟 ( 𝑢 ) , 𝑟 ( 𝑣 )] + [ 𝑟 ( 𝑢 · 𝑣 ) , 𝑟 ( 𝑣 )] = . (14.1)Since we mostly have the trigonometric case in mind, we write the equation in multiplicativeform, that is to say 𝑢, 𝑣 ∈ C ∗ . The notation 𝑟 ( 𝑢 ) means 𝑟 ( 𝑢 ) = 𝑟 ( 𝑢 ) ⊗ , 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 𝑟 -matrix as a map 𝑟 ( 𝑢 ) : 𝔤 → 𝔤 , or equivalently 𝑟 ( 𝑢 ) ∈ End ( 𝔤 ) (cid:39) 𝔤 ⊗ 𝔤 ∗ . In this case we will write 𝑟 𝑢 ( 𝑎 ) ∈ 𝔤 for the 𝑟 -matrix acting on a Lie algebra element 𝑎 ∈ 𝔤 . One also often assumes the so-called‘unitarity’ property of the 𝑟 -matrix:Tr ( 𝑟 𝑢 ( 𝐴 ) 𝐵 ) = − Tr ( 𝐴 𝑟 𝑢 − ( 𝐵 )) . (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: [ 𝑟 𝑢 ( 𝑎 ) , 𝑟 𝑢𝑣 ( 𝑏 )] + 𝑟 𝑢 ( [ 𝑟 𝑣 ( 𝑏 ) , 𝑎 ]) + 𝑟 𝑢𝑣 ( [ 𝑏, 𝑟 𝑣 − ( 𝑎 )]) = . (14.3)The solution of interest has the form (for now we assume 𝔤 (cid:39) 𝔰𝔲 ( 𝑛 ) ) 𝑟 𝑢 = α 𝑢 𝜋 + + β 𝑢 𝜋 − + γ 𝑢 𝜋 , (14.4) α 𝑢 = 𝑢 − 𝑢 , β 𝑢 = − 𝑢 , γ 𝑢 =
12 1 + 𝑢 − 𝑢 , (14.5)where 𝜋 ± are projections on the upper/lower-triangular matrices, and 𝜋 is the projection onthe diagonal. The rational limit is achieved by setting 𝑢 = 𝑒 − (cid:15) and taking the limit (cid:15) →
0, inwhich case 𝑟 𝑢 → (cid:15) .The ansatz 𝑟 𝑢 =
12 1 + 𝑢 − 𝑢 Id + 𝑖 R (14.6)transforms the CYBE to an equation on R , which does not depend on the spectral parameter: [R ( 𝑎 ) , R ( 𝑏 )] + R ( [R ( 𝑏 ) , 𝑎 ] + [ 𝑏, R ( 𝑎 )]) − [ 𝑎, 𝑏 ] = . (14.7)It is known in the literature as the ‘classical modified Yang-Baxter equation’. The solu-tion (14.4) corresponds to R = 𝑖 ( 𝜋 + − 𝜋 − ) . Another option is taking an R -matrix induced by a complex structure on the Lie group 𝐺 with Liealgebra 𝔤 [64]. In this case (14.7) is the condition of vanishing of the Nijenhuis tensor (13.22). Having the right tools in place, we proceed to explain the construction of [87], which is basedon a certain ‘semi-holomorphic’ 4D Chern-Simons theory that we will now describe. In thiscase the four-dimensional ‘spacetime’ is a product Σ × C , where Σ is called the ‘topologicalplane’ and is endowed with coordinates 𝑧, 𝑧 – this will eventually be the worldsheet of thesigma model, – and C is a complex curve with coordinates 𝑤, 𝑤 (this is the spectral parametercurve). The latter is required to admit a nowhere-vanishing holomorphic differential 𝜔 = 𝑑𝑤 ≠
0, which means that its canonical class is trivial: 𝐾 C =
0. As a result, the curveis either the complex plane, a cylinder or an elliptic curve (a torus): C (cid:39) C , C ∗ , 𝐸 𝜏 . TheChern-Simons action of the model is 𝑆 CS = ℏ ∫ Σ × C 𝜔 ∧ Tr (cid:18) 𝐴 ∧ ( 𝑑𝐴 + 𝐴 ∧ 𝐴 ) (cid:19) , (14.8)where 𝐴 = 𝐴 𝑧 𝑑𝑧 + 𝐴 𝑧 𝑑𝑧 + 𝐴 𝑤 𝑑𝑤 is a gauge field corresponding to a (semi-simple) gaugegroup 𝐺 . One couples this theory to certain two-dimensional systems of a very particularsort, called 𝛽𝛾 systems. These are defined for complex symplectic target spaces, their actionin local Darboux coordinates ( 𝑝, 𝑞 ) being 𝑆 𝛽𝛾 = ∫ Σ 𝑑 𝑧 𝑝 𝑖 𝜕𝑞 𝑖 . In the context of [87] oneconsiders target spaces of the form 𝑇 ∗ M , where M is a complex manifold endowed with aholomorphic action of the group 𝐺 , and writes down a sum of two 𝛽𝛾 -system actions, oneholomorphic and the other anti-holomorphic:Figure 17: Two 𝛽𝛾 -defects located at points 𝑤 , 𝑤 on the spectral parameter curve C . 𝑆 def = ∫ Σ 𝑑 𝑧 (cid:16) 𝑝 𝑖 𝐷 ( 𝑤 ) 𝑞 𝑖 + 𝑝 𝑖 𝐷 ( 𝑤 ) 𝑞 𝑖 (cid:17) , (14.9)where 𝐷 ( 𝑤 ) 𝑞 𝑖 = 𝜕𝑞 𝑖 − ∑︁ 𝑎 (cid:16) 𝐴 ( 𝑤 ) 𝑧 (cid:17) 𝑎 𝑣 𝑖𝑎 and 𝑣 𝑎 are the holomorphic vector fields on M generating the action of 𝐺 . The full actionfunctional is the sum of two: (14.8) and (14.9). The next step is to impose the ‘light-cone’gauge 𝐴 𝑤 =
0, in which case two of the e.o.m. become linear in 𝐴 𝑧 , 𝐴 𝑧 , and the third one is97he zero-curvature constraint: 𝜕 𝐴 𝑧 − 𝜕 𝐴 𝑧 + [ 𝐴 𝑧 , 𝐴 𝑧 ] = , (14.10) 𝜕 𝑤 𝐴 𝑧 = 𝛿 ( ) ( 𝑤 − 𝑤 ) ∑︁ 𝑎 𝑝 𝑖 𝑣 𝑖𝑎 𝜏 𝑎 (14.11) 𝜕 𝑤 𝐴 𝑧 = 𝛿 ( ) ( 𝑤 − 𝑤 ) ∑︁ 𝑎 𝑝 𝑖 𝑣 𝑖𝑎 𝜏 𝑎 . (14.12)The delta-functions in the r.h.s. of the latter equations mean that 𝐴 𝑧 ( 𝑤 ) , 𝐴 𝑧 ( 𝑤 ) dependmeromorphically on 𝑤 , and the first equation is then the zero-curvature equation (on theworldsheet Σ ) for the family of connections 𝐴 = 𝐴 𝑧 𝑑𝑧 + 𝐴 𝑧 𝑑𝑧 depending on the parameter 𝑤 .In order to solve the equations (14.11)-(14.12), one needs to invert the operator 𝜕 𝑤 . Oneof the key observations in [87] is that, with suitable boundary conditions, the Green’s function 𝜕 − 𝑤 is the classical 𝑟 -matrix [41], viewed as an element of End ( 𝔤 ) . In the rational case theGreen’s function is simply the Cauchy kernel, enhanced with additional matrix structure, i.e. 𝑟 𝑤 = Id 𝑤 ∈ End ( 𝔤 ) . (14.13)As we reviewed in Section 14.0.1, the 𝑟 -matrix is sometimes written as an element of 𝔤 ⊗ 𝔤 , and the two definitions are simply related by raising/lowering an index, using theKilling metric on 𝔤 . Accordingly, the more conventional representation for (14.13) wouldbe 𝑟 ( 𝑤 ) = (cid:205) 𝜏 𝑎 ⊗ 𝜏 𝑎 𝑤 ∈ 𝔤 ⊗ 𝔤 , where 𝜏 𝑎 are the generators of 𝔤 . In the trigonometric case oneneeds to impose boundary conditions at the ends of the cylinder, i.e. at the two punctures on C = C ∗ (cid:39) CP \ { , ∞} (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 𝔤 C = 𝔤 + ⊕ 𝔤 − , where 𝔤 ± are two isotropic subspaces of 𝔤 , and requiring that 𝐴 𝑧 ∈ 𝔤 + at 𝑤 = 𝐴 𝑧 ∈ 𝔤 − at 𝑤 = ∞ . In that case the 𝑟 -matrix has the form 𝑟 𝑤 = Π + − 𝑤 − Π − − 𝑤 − ∈ End ( 𝔤 ) , (14.14)where Π ± are the projectors on 𝔤 ± , or alternatively 𝑟 ( 𝑤 ) = (cid:205) 𝜏 + 𝑎 ⊗ 𝜏 − 𝑎 − 𝑤 − (cid:205) 𝜏 − 𝑎 ⊗ 𝜏 + 𝑎 − 𝑤 − ∈ 𝔤 ⊗ 𝔤 .Once we know the 𝑟 -matrix, the solution to (14.11) is 𝐴 𝑧 = 𝑟 𝑤 (cid:0)(cid:205) 𝑎 𝑝 𝑖 𝑣 𝑖𝑎 𝜏 𝑎 (cid:1) , and analo-gously for (14.12). Substituting the solution back into the full action 𝑆 = 𝑆 CS + 𝑆 def (usingthe fact that the Chern-Simons action is quadratic in the 𝐴 -fields in the gauge 𝐴 𝑤 = 𝑝, 𝑞 -variables: 𝑆 = ∫ 𝑑 𝑧 (cid:16) 𝑝 𝑖 𝜕𝑞 𝑖 + 𝑝 𝑖 𝜕𝑞 𝑖 + 𝑟 𝑤 − 𝑤 (cid:16) 𝑝 𝑖 𝑣 𝑖𝑎 𝜏 𝑎 , 𝑝 𝑖 𝑣 𝑖𝑎 𝜏 𝑎 ) (cid:17) (cid:17) = (14.15) = ∫ 𝑑 𝑧 (cid:18) 𝑝 𝑖 𝜕𝑞 𝑖 + 𝑝 𝑖 𝜕𝑞 𝑖 + 𝑤 − 𝑤 ∑︁ | 𝑝 𝑖 𝑣 𝑖𝑎 | (cid:19) , Such decomposition is the same as picking a complex structure on 𝔤 , compatible with the metric. It is alsoknown in the literature as a Manin triple. 𝑟 -matrix. Onesees that the action is quadratic in the 𝑝 -variables, which are the coordinates in the fiber ofthe cotangent bundle 𝑇 ∗ M . Integrating out these variables as well, we get the sigma modelform of the action: 𝑆 ∼ ∫ 𝑑 𝑧 (cid:16) 𝐺 𝑖 𝑗 𝜕𝑞 𝑖 𝜕𝑞 𝑗 (cid:17) , where 𝐺 𝑖 𝑗 = (cid:32)∑︁ 𝑎 𝑣 𝑖𝑎 𝑣 𝑗𝑎 (cid:33) − (14.16)is the metric on the target space. Note that in order for the expression for the metric 𝐺 𝑖 𝑗 tomake sense the matrix (cid:205) 𝑎 𝑣 𝑖𝑎 𝑣 𝑗𝑎 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 flag manifold, the two models are equivalent (meaning that the metric 𝐺 𝑖 𝑗 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 𝑟 -matrices in (14.15). 𝛽𝛾 -systems In the present section we will prove that the flag manifold models obtained from the couplingof two 𝛽𝛾 -systems through a four-dimensional Chern-Simons field are – in the rational case– equivalent to the models that we described earlier in Section 13. To this end we thereforeeffectively set the 𝑟 -matrix to be the identity operator: 𝑟 = Id. Our main tool in identifyingthe two types of models will be the gauged linear sigma model representation that wasdeveloped in [75, 76] for flag models of type (13.12). In the case when the target space isa Grassmannian, the metric G is Kähler, and this representation is equivalent to the Kählerquotient 𝐺𝑟 𝑘,𝑛 (cid:39) Hom ( C 𝑘 , C 𝑛 ) (cid:12) U ( 𝑘 ) . In the general case our construction leads to aquotient w.r.t. a non-reductive group and to the ‘Killing’ metric G , which is not Kähler ingeneral.The construction is as follows. We introduce the field 𝑈 ∈ Hom ( C 𝑀 , C 𝑛 ) , satisfying theorthonormality condition 𝑈 † 𝑈 = 𝑀 , as well as the “gauge” field (cid:101) A = A 𝑑𝑧 + A 𝑑𝑧 of thespecial form shown in Fig. 18. The Lagrangian reads L = Tr (cid:16) (cid:107) 𝐷𝑈 (cid:107) (cid:17) , where 𝐷𝑈 = 𝜕𝑈 + 𝑖 𝑈 A . (14.18)This Lagrangian is equivalent to (13.12), as can be shown by eliminating the field A . Indeed,varying w.r.t. A one obtains ( 𝐷 𝑧 𝑈 † ◦ 𝑈 ) 𝔭 𝑀 =
0, where 𝔭 𝑀 is the space of upper-block-99 = ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗∗ ∗ (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) 𝑑 𝑑 𝑑 𝑚 − , A = (A) † . (14.17)Figure 18: The gauge field entering the GLSM description (14.18) of the flag manifold sigmamodels. Here 𝑑 , · · · , 𝑑 𝑚 − : = 𝑀 are the dimensions of the complex spaces in the flag (1.1).triangular matrices of size 𝑀 × 𝑀 . Let us parametrize 𝑈 as 𝑈 = { 𝜏 , 𝜏 , . . . , 𝜏 𝑚 − } , (14.19)where 𝜏 . . . 𝜏 𝑚 − are groups of 𝑛 . . . 𝑛 𝑚 − orthonormal vectors. Adding the last group 𝜏 𝑚 of 𝑛 𝑚 vectors orthogonal to all the rest, we obtain the matrix 𝑔 from (13.11). One then has theorthonormality condition 𝜏 † 𝐴 𝜏 𝐵 = 𝛿 𝐴𝐵 ( 𝐴, 𝐵 = , . . . , 𝑚 ) and the completeness relation 𝑚 ∑︁ 𝐴 = 𝜏 𝐴 𝜏 † 𝐴 = 𝑛 , (14.20)which is equivalent to 𝑔𝑔 † = 𝑛 . In this notation components of the gauge field A have theform (see Fig. 14.17) [A] 𝐴𝐵 = 𝑖 𝜏 † 𝐴 𝜕𝜏 𝐵 for 𝐴 ≤ 𝐵 ; [A] 𝐴𝐵 = 𝐴 > 𝐵 . 𝐴, 𝐵 = . . . 𝑚 − . In order to compute the Lagrangian (14.18), it is useful first to calculate 𝐷𝜏 𝐴 = 𝜕𝜏 𝐴 − 𝑚 ∑︁ 𝐵 = 𝐴 𝜏 𝐵 𝜏 † 𝐵 𝜕𝜏 𝐴 = using ( . ) = 𝐴 − ∑︁ 𝐵 = 𝜏 𝐵 𝜏 † 𝐵 𝜕𝜏 𝐴 𝐴 = , . . . , 𝑚 − . (14.21)Substituting into (14.18), we obtain the final expression for the Lagrangian in the ‘non-linearform’ L = ∑︁ 𝐴<𝐵 Tr (cid:16) ( 𝐽 𝐵𝐴 ) † 𝑧 ( 𝐽 𝐵𝐴 ) 𝑧 (cid:17) , where ( 𝐽 𝐵𝐴 ) 𝑧 = 𝜏 † 𝐵 𝜕𝜏 𝐴 . (14.22)This is clearly the same as (13.12), up to an exchange of complex structure J → − J .100e return to the GLSM (14.18). Due to the orthonormality condition 𝑈 † 𝑈 = 𝑀 , thegauge group of the model is U ( 𝑛 ) × · · · × U ( 𝑛 𝑚 ) . 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 𝑈 ∈ Hom ( C 𝑀 , C 𝑛 ) isan arbitrary complex matrix of rank 𝑀 . We then write down the following Lagrangian: L = Tr (cid:18) ( 𝐷𝑈 ) † 𝐷𝑈 𝑈 † 𝑈 (cid:19) . (14.23)It is easy to see that it is invariant w.r.t. complex gauge transformations 𝑈 → 𝑈𝑔 , where 𝑔 ∈ 𝑃 𝑑 ,...,𝑑 𝑚 − (a parabolic subgroup of GL ( 𝑀, 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 field 𝑉 ∈ Hom ( C 𝑛 , C 𝑀 ) and write down a new Lagrangian L = Tr (cid:16) 𝑉 𝐷𝑈 (cid:17) + Tr (cid:16) 𝑈 𝐷𝑉 (cid:17) − Tr (cid:16) 𝑉𝑉 † 𝑈 † 𝑈 (cid:17) , (14.24)that turns into the original one upon elimination of the field 𝑉 . 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 ∗ .One should also keep in mind that in the process of integration over the 𝑉 -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 𝑇 -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 field Φ ∈ End ( C 𝑛 ) and its Hermitian conjugate: Φ = ( Φ ) † .We write one more Lagrangian L = Tr (cid:16) 𝑉 D 𝑈 (cid:17) + Tr (cid:16) 𝑈 D 𝑉 (cid:17) + Tr (cid:16) ΦΦ (cid:17) , (14.25)where D is the “elongated” covariant derivative D 𝑈 = 𝜕𝑈 + 𝑖 𝑈 A + 𝑖 Φ 𝑈 . (14.26)Let us clarify the geometric meaning of the Lagrangian (14.25). The first two terms cor-respond to a sum of the so-called 𝛽𝛾 -systems on the flag manifold F , in a backgroundfield Φ [182, 256]. In the terminology of the previous section our field Φ should be viewedas the component 𝐴 𝑧 of the Chern-Simons gauge field along the “topological plane” (i.e. theworldsheet Σ ). The quadratic form in the interaction term Tr (cid:16) ΦΦ (cid:17) in (14.25) is in this contextthe inverse propagator of the field 𝐴 𝑧 , which in the present (rational) case is proportional tothe identity matrix.By definition, such a system may be defined for an arbitrary complex manifold M (dim C M = 𝐷 ) with the help of a complex fundamental ( , ) -form 𝜃 = 𝐷 (cid:205) 𝑖 = 𝑝 𝑖 𝑑𝑞 𝑖 (the101omplex analogue of the Poincaré-Liouville one-form) on 𝑇 ∗ M . Here 𝑞 𝑖 are the complexcoordinates on M and 𝑝 𝑖 are the complex coordinates in the fiber of the holomorphiccotangent bundle. The action of the 𝛽𝛾 -system is then simply 𝑆 = ∫ Σ 𝑑 𝑧 𝐷 (cid:205) 𝑖 = 𝑝 𝑖 𝜕𝑞 𝑖 . In thecase of the flag manifold this action can be most conveniently written, using two matrices 𝑈 ∈ Hom ( C 𝑀 , C 𝑛 ) , 𝑉 ∈ Hom ( C 𝑛 , C 𝑀 ) and the gauge field A . Indeed, it will be shown inthe next section that the fundamental ( , ) -form can be written as 𝜃 = Tr ( 𝑉 𝑑𝑈 ) (cid:12)(cid:12) 𝜇 C = , where ( 𝑈, 𝑉 ) satisfy the condition 𝜇 C = 𝑉𝑈 (cid:12)(cid:12) 𝔨 ∗ = , and 𝔨 = Lie ( 𝑃 𝑑 ,...,𝑑 𝑚 − ) (14.27)is the Lie algebra of the corresponding parabolic subgroup of GL ( 𝑀, C ) . It is also assumedthat the space of matrices, satisfying this condition, is factorized w.r.t. the action of 𝑃 𝑑 ,...,𝑑 𝑚 − ,i.e. one has a complex symplectic reduction. The condition (14.27) is precisely the conditionof vanishing of the moment map 𝜇 C = 𝑃 𝑑 ,...,𝑑 𝑚 − onthe space of matrices ( 𝑈, 𝑉 ) endowed with the symplectic form 𝜔 = Tr ( 𝑑𝑈 ∧ 𝑑𝑉 ) . As aresult, 𝑑𝜃 = 𝜔 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 field A ∈
Lie ( 𝑃 𝑑 ,...,𝑑 𝑚 − ) . Indeed, differentiating 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, defined by the so-called quiver. We recall that 𝑇 ∗ F is a hyper-Kähler manifoldthat may be constructed by a hyper-Kähler quotient of flat space (though we stress that thereal symplectic form – the Kähler form – 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 𝑈 𝑈 𝑚 − 𝑉 𝑉 𝑚 − 𝑈 𝑚 − 𝑉 𝑚 − C 𝑛 𝐿 𝐿 𝐿 𝑚 − 𝐿 𝑚 − · · · (14.29)Figure 19: Nakajima quiver for the cotangent bundle to the flag manifold 𝑇 ∗ 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 𝑇 ∗ F rather than the flagmanifold itself. In each node there is a vector space 𝐿 𝑘 (cid:39) C 𝑑 𝑘 , and to each arrow from node 𝑖 to node 𝑗 corresponds a field, taking values in Hom ( 𝐿 𝑖 , 𝐿 𝑗 ) . The full space of fields istherefore W : = ⊕ 𝑚 − 𝑖 = ( Hom ( 𝐿 𝑖 , 𝐿 𝑖 + ) ⊕ Hom ( 𝐿 𝑖 + , 𝐿 𝑖 )) . (14.30)In each node there is an action of a gauge group GL ( 𝐿 𝑖 ) . We then consider the GIT-quotient W 𝑓 : = W / G of the stable subset W ⊂ W w.r.t. the group G : = 𝑚 − (cid:206) 𝑖 = GL ( 𝐿 𝑖 ) . In W 𝑓 we definea submanifold given by the vanishing conditions for the moment maps ( 𝑈 = , 𝑉 = F : = { 𝜇 𝑖 = 𝑈 𝑖 − 𝑉 𝑖 − − 𝑉 𝑖 𝑈 𝑖 = , 𝑖 = , . . . , 𝑚 − } ⊂ W 𝑓 . (14.31)The (well-known) statement is that the resulting space is the flag manifold (1.3)-(1.4), whichis why we have denoted it by F . On W there is a natural complex symplectic form Ω = 𝑚 − ∑︁ 𝑖 = Tr ( 𝑑𝑈 𝑖 ∧ 𝑑𝑉 𝑖 ) . (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 Ω F . We prove thefollowing statement: Ω F = 𝜔 red , where 𝜔 red is the symplectic form (14.28) that arises as a result ofthe reduction w.r.t. a parabolic subgroup of GL ( 𝑀, C ) .To prove this, consider the fields { 𝑈 𝑖 } . 𝑈 𝑖 is a matrix with 𝑑 𝑖 columns and 𝑑 𝑖 + rows. By theaction of GL ( 𝑑 𝑖 + , C ) one can bring 𝑈 𝑖 to the form where the first 𝑑 𝑖 + − 𝑑 𝑖 rows are zero andthe last 𝑑 𝑖 rows represent a unit matrix. The stabilizer of this canonical form w.r.t. the joint(left-right) action of GL ( 𝑑 𝑖 + , C ) × GL ( 𝑑 𝑖 , C ) is the subgroup 𝑃 𝑑 𝑖 ,𝑑 𝑖 + ⊂ 𝑃 𝑑 𝑖 ,𝑑 𝑖 + × GL ( 𝑑 𝑖 , C ) ,embedded according to the rule 𝑔 → ( 𝑔, 𝜋 𝑖 ( 𝑔 )) , where 𝜋 𝑖 ( 𝑔 ) is the projection on the block ofsize 𝑑 𝑖 × 𝑑 𝑖 . Iterating this procedure, i.e. bringing all matrices 𝑈 𝑖 ( 𝑖 = , . . . , 𝑚 −
2) to canonicalform, we arrive at the situation, when one is left with a single non-trivial matrix 𝑈 𝑚 − : = 𝑈 ,and the resulting symmetry group is precisely 𝑃 𝑑 ,...,𝑑 𝑚 − . We also denote 𝑉 𝑚 − : = 𝑉 . Now,let 𝑎 ∈ 𝔨 = Lie ( 𝑃 𝑑 ,...,𝑑 𝑚 − ) . By definition of the stabilizer 𝑎𝑈 𝑚 − = 𝑈 𝑚 − 𝜋 𝑚 − ( 𝑎 ) , thereforeTr ( 𝑎𝑈 𝑚 − 𝑉 𝑚 − ) = Tr ( 𝜋 𝑚 − ( 𝑎 ) 𝑉 𝑚 − 𝑈 𝑚 − ) = Tr ( 𝜋 𝑚 − ( 𝑎 ) 𝑈 𝑚 − 𝑉 𝑚 − ) , where in the secondequality we have used the equation (14.31). Since 𝜋 𝑚 − ( 𝑎 ) ∈ Stab ( 𝑈 𝑚 − ) , we can iteratethis procedure, and at the end we will obtain Tr ( 𝑎𝑈 𝑚 − 𝑉 𝑚 − ) =
0. Due to the equation 𝑈 𝑚 − 𝑉 𝑚 − − 𝑉𝑈 = 𝑉𝑈 (cid:12)(cid:12) 𝔨 ∗ =
0, which coincides with (14.27). Besides, since thematrices 𝑈 𝑖 ( 𝑖 = , . . . , 𝑚 −
2) are constant, the restriction of the symplectic form Ω coincideswith Tr ( 𝑑𝑈 𝑚 − ∧ 𝑑𝑉 𝑚 − ) = Tr ( 𝑑𝑈 ∧ 𝑑𝑉 ) .Let us clarify the role of the field Φ . Differentiating the Lagrangian (14.25) w.r.t. Φ , weobtain Φ = − 𝑖 𝑈𝑉 . (14.33)103his coincides with the expression for the 𝑧 -component of the Noether current for the actionof the group GL ( 𝑛, C ) on the space of matrices ( 𝑈, 𝑉 ) . For the 𝛽𝛾 -system written above Φ 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 ( 𝑛 ) . In the present section, following [67], we describe the relation betweenflag manifold models and the principal chiral model. Our starting point will be the formulas (14.27)-(14.33):
Φ = − 𝑖 𝑈𝑉 , 𝜇 C = 𝑉𝑈 (cid:12)(cid:12) 𝔨 ∗ = , (15.1)where 𝔨 = Lie ( 𝑃 𝑑 ,...,𝑑 𝑚 − ) . As a warm-up we consider the case of a Grassmannian, i.e. 𝑚 =
2. Then the vanishingof the moment map is simply 𝑉𝑈 =
0. Therefore Φ =
0, which means that Φ belongsto a nilpotent orbit of the group GL ( 𝑛, C ) . From the expression for Φ it also follows thatIm ( Φ ) ⊂ Im ( 𝑈 ) ⊂ Ker ( Φ ) . As is well-known, {( 𝑈, Φ ) : rk ( 𝑈 ) = 𝑀, Φ = , Im ( Φ ) ⊂ Im ( 𝑈 ) ⊂ Ker ( Φ )} (cid:39) 𝑇 ∗ Gr 𝑀,𝑛 (15.2)is the cotangent bundle to a Grassmannian (notice the rank condition!), and the forgetful map 𝑇 ∗ Gr 𝑀,𝑛 → { Φ : Φ = } (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 Φ ,and the non-uniqueness in this factorization corresponds exactly to the gauge symmetry 𝑈 → 𝑈𝑔, 𝑉 → 𝑔 − 𝑉 , where 𝑔 ∈ GL ( 𝑀, C ) .Let us now derive the equations of motion for the field Φ . First of all, the La-grangian (14.25) implies the equations of motion D 𝑈 = , D 𝑉 = 𝑈 and 𝑉 . Therefore D Φ =
0, i.e. 𝜕 Φ + 𝑖 [ Φ , Φ ] = . (15.4)This equation is nothing but the equation of motion of the principal chiral field. Indeed,introduce a 1-form 𝑗 = 𝑖 ( Φ 𝑑𝑧 + Φ 𝑑𝑧 ) with values in the Lie algebra 𝔲 ( 𝑛 ) . In this case (15.4)together with the Hermitian conjugate equation may be written in the form of two conditions 𝑑 ∗ 𝑗 = , 𝑑 𝑗 − 𝑗 ∧ 𝑗 = , (15.5)104hich are the e.o.m. of the principal chiral field. This is consistent with the fact, reviewed insection 13.1, that the Noether current of the model (13.5) is flat.The condition Φ = Φ consists of 𝑚 cells of sizes2 × 𝑚 cells of sizes 1 ×
1. In this case 𝑛 = 𝑚 + 𝑚 and dim Ker ( Φ ) = 𝑚 + 𝑚 .Since Im ( 𝑈 ) ⊂ Ker ( Φ ) and rk ( 𝑈 ) = 𝑀 , we get the condition 𝑀 ≤ 𝑚 + 𝑚 . This easilyleads to 𝑚 ≤ 𝑛 − 𝑀 , 𝑚 ≥ 𝑀 − 𝑛 .The dynamical equation (15.4) imposes severe constraints on the way in which the Jordanstructure of the matrix Φ can change as one varies the point 𝑧, 𝑧 on the worldsheet. Indeed,it implies that Φ = 𝑘𝑄 ( 𝑧 ) 𝑘 − , where 𝑄 ( 𝑧 ) is a matrix that depends holomorphically on 𝑧 .The Jordan structure of the matrix 𝑄 ( 𝑧 ) is the same as that of Φ , and the vanishing of theJordan blocks occurs holomorphically in 𝑧 . In particular, the Jordan structure changes onlyat “special points” – isolated points on the worldsheet. As a result, “almost everywhere” thedimension of the kernel dim Ker ( Φ ) : = (cid:101) 𝑀 is the same, and the map 𝜆 : ( 𝑧, 𝑧 ) → Ker ( Φ ) isa map to the Grassmannian 𝐺𝑟 (cid:101) 𝑀,𝑛 . A more careful analysis of the behavior of Φ at a specialpoint would show that 𝜆 may be extended to these points. This may be summarized as follows:let 𝑔 ( 𝑧, 𝑧 ) be a solution of the principal chiral model, i.e. a harmonic map to the group 𝐺 ,satisfying the condition Φ =
0, where Φ : = 𝑔 − 𝜕𝑔 is a component of the Noether current,and let the dimension of the kernel of Φ at a typical point of the worlsheet be (cid:101) 𝑀 . Then onecan construct a harmonic map to the Grassmannian 𝐺𝑟 (cid:101) 𝑀,𝑛 by the rule ( 𝑧, 𝑧 ) → Ker ( Φ ) . Let us extend the results of the previous section to more general flag manifolds. We returnfirst to the equation for Φ : 𝜕 Φ + 𝑖 [ Φ , Φ ] = , (15.6)but this time we assume that the matrix Φ satisfies, in a typical point ( 𝑧, 𝑧 ) ∈ Σ , the condition Φ 𝑚 = Φ 𝑚 − ≠ . (15.7)The matrix Φ naturally defines a flag 𝑓 : = { ⊂ Ker ( Φ ) ⊂ Ker ( Φ ) ⊂ · · · ⊂ Ker ( Φ 𝑚 ) (cid:39) C 𝑛 } (15.8)The relation between the principal chiral model equations written above and the flagmanifold models is summarized by the following assertion:Given a matrix Φ satisfying (15.6)-(15.7), the map ( 𝑧, 𝑧 ) → 𝑓 is a solution tothe e.o.m. of the flag manifold sigma model (13.12). The inequalities are saturated in the case Ker ( Φ ) (cid:39) Im ( 𝑈 ) , when the number of 2 × 𝑛 − 𝑀 , and the number of cells of size 1 × 𝑀 − 𝑛 . Note that this is only possible in the case 𝑀 ≥ 𝑛 . Reduction of the number of cells of type 2 × Φ . 𝑈 of the form 𝑈 = ( 𝑈 𝑚 − | · · · | 𝑈 ) , where 𝑈 𝑖 is a matrix,whose columns are the linearly independent vectors from Ker ( Φ 𝑖 )/ Ker ( Φ 𝑖 − ) . Let us relatethe dimensions of these spaces to the dimensions of the Jordan cells of the matrix Φ . To thisend we bring Φ to the Jordan form Φ ( ) = Diag { 𝐽 𝑠 , . . . , 𝐽 𝑠 ℓ } , ℓ ∑︁ 𝑗 = 𝑠 𝑗 = 𝑛 , (15.9)where 𝐽 𝑠 is a Jordan cell of size 𝑠 × 𝑠 . We have chosen the ordering 𝑠 ≥ . . . ≥ 𝑠 ℓ , where,according to the supposition (15.7), 𝑠 = 𝑚 . We denote by 𝜅 𝑖 the number of Jordan cells ofsize at least 𝑖 ( 𝜅 = ℓ ). The following two properties are obvious: • 𝜅 𝑖 + ≤ 𝜅 𝑖 , i.e. 𝜅 , . . . , 𝜅 𝑚 is a non-increasing sequence. • dim Ker ( Φ ) = 𝜅 , dim Ker ( Φ ) = 𝜅 + 𝜅 etc.,therefore dim Ker ( Φ 𝑖 )/ Ker ( Φ 𝑖 − ) = 𝜅 𝑖 .It follows that 𝑈 𝑖 ∈ Hom ( C 𝜅 𝑖 , C 𝑛 ) and 𝑈 ∈ Hom ( C 𝑀 , C 𝑛 ) , where 𝑀 = 𝑚 − (cid:205) 𝑖 = 𝜅 𝑖 .Since, by construction, Im ( 𝑈 ) (cid:39) Ker ( Φ 𝑚 − ) and Im ( Φ ) ⊂ Ker ( Φ 𝑚 − ) , we have Im ( Φ ) ⊂ Im ( 𝑈 ) , i.e. there exists a matrix 𝑉 ∈ Hom ( C 𝑛 , C 𝑀 ) , such that Φ = − 𝑖 𝑈𝑉 . Let us now derivethe equations of motion for the matrices 𝑈 and 𝑉 . Since Φ 𝑘 𝑈 𝑘 = 𝐷 Φ =
0, onehas Φ 𝑘 𝐷𝑈 𝑘 =
0. The columns of the matrix ( 𝑈 𝑘 | · · · | 𝑈 ) span the kernel of Φ 𝑘 , hence 𝐷𝑈 𝑘 = − 𝑖 (cid:205) 𝑗 ≤ 𝑘 𝑈 𝑗 A 𝑗 𝑘 , where A 𝑗 𝑘 are matrices of relevant sizes. Out of the matrices A 𝑗 𝑘 (1 ≤ 𝑗 ≤ 𝑘 ≤ 𝑚 −
1) we form a single matrix A , which schematically looks as in (14.17).Then, clearly, the following equation is satisfied: D 𝑈 = 𝜕𝑈 + 𝑖 𝑈 A + 𝑖 Φ 𝑈 = . (15.10)As Φ = − 𝑖 𝑈𝑉 , from the non-degeneracy of 𝑈 it follows that D 𝑉 =
0. Because 𝑈 , . . . , 𝑈 𝑘 ⊂ Ker ( Φ 𝑘 ) , in the matrix Φ 𝑘 𝑈 ∼ 𝑈 ( 𝑉𝑈 ) 𝑘 the last 𝑘 (cid:205) 𝑖 = 𝜅 𝑖 columns vanish, therefore the matrix ( 𝑉𝑈 ) 𝑘 is strictly-lower-triangular and has zeros on the first 𝑘 block diagonals (the maindiagonal is counted as the first one). We denote by 𝔨 the parabolic subalgebra of 𝔤𝔩 ( 𝑀 ) thatstabilizes the subflag of (15.8) with the last element omitted. We have proven that 𝑉𝑈 (cid:12)(cid:12) 𝔨 ∗ = Φ ( 𝑧, 𝑧 ) of the system (15.6)-(15.7) produces a solution ( 𝑈, 𝑉 ) to theequations of motion of the sigma model with target space the flag manifoldU ( 𝑛 ) U ( 𝜅 ) × · · · × U ( 𝜅 𝑚 ) , where (15.11) 𝜅 𝑗 = dim Ker ( Φ 𝑗 )/ Ker ( Φ 𝑗 − ) is a non-increasing sequence.The complex structure on the flag is uniquely determined by the structure of the complexflag (15.8). 106 In the previous section we showed that the integrable flag 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 fields Φ , Φ . 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 𝑟 -matrix into the Lagrangian. The purpose of this section is to emphasize,following [71], that all of these systems – either deformed or undeformed – are really examplesof chiral gauged bosonic Gross-Neveu models. This way of formulating sigma models is notmerely a simple reformulation, but offers substantial calculational benefits. For example, aswe shall show, these methods allow to solve the renormalization group (generalized Ricciflow) 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 𝑛 ‘Dirac bosons’ Ψ 𝑎 = (cid:18) 𝑈 𝑎 𝑉 𝑎 (cid:19) , 𝑎 = , . . . , 𝑛 . (16.1)The Lagrangian (14.24) is, in this notation, L = Ψ 𝑎 / 𝐷 Ψ 𝑎 + (cid:18) Ψ 𝑎 + 𝛾 Ψ 𝑎 (cid:19) (cid:18) Ψ 𝑏 − 𝛾 Ψ 𝑏 (cid:19) . (16.2)The Dirac notations are standard: 𝜎 , are the Pauli matrices, / 𝜕 : = (cid:205) 𝑖 = 𝜎 𝑖 𝜕 𝑖 and 𝛾 : = 𝑖 𝜎 𝜎 .The Lagrangian (16.2) is the bosonic incarnation of the so-called chiral Gross-Neveu model(equivalently the SU ( 𝑛 ) Thirring model [244]) interacting with a gauge field. As in (14.24)and in the example at the very start of this Chapter, the equivalence with the sigma modelformulation (with a metric, 𝐵 -field and possibly dilaton) is established through the eliminationof 𝑉 and 𝑉 .The model (16.2) is ‘chiral’, meaning that there is a symmetry 𝑈 → λ 𝑈, 𝑉 → λ − 𝑉 , where λ ∈ C × . (16.3) In Minkowski signature this would have been the usual U ( ) chiral symmetry. This difference in chiraltransformations has been observed in [270, 174].
107 general SU ( 𝑛 ) -invariant Lagrangian would only retain a U ( ) -symmetry, | λ | =
1, and is notinvariant under the full C ∗ , which arises in (16.2) due to the chiral projectors. In other words,for a Euclidean worldsheet signature, chiral symmetry is equivalent to the complexificationof the original (non-chiral) symmetry. This chiral symmetry is of extreme importance at leastfor two reasons: ◦ Chiral symmetry ensures that the quartic interaction terms are quadratic in the 𝑉 -variables, which are the ‘momenta’ conjugate to 𝑈 . This allows integrating them outand arriving at a metric form of the sigma model. ◦ 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 ∗ , 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 flow is described in this case by a couple of elementary Feynman diagrams,which should be contrasted with the highly non-trivial generalized Ricci flow equations thatone obtains in the geometric formulation of the sigma model. This will also serve as our firststep towards a definition and solution of these models at the quantum level.We will first 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 𝑈 ∈ Hom ( C 𝑀 , C 𝑛 ) , 𝑉 ∈ Hom ( C 𝑛 , C 𝑀 ) , Φ ∈ End ( C 𝑛 ) (we willalways assume 𝑀 ≤ 𝑛 ) that we already encountered in Section 14.2, we write down thedeformed Lagrangian L = Tr (cid:16) 𝑉 D 𝑈 (cid:17) + Tr (cid:16) 𝑈 D 𝑉 (cid:17) + Tr (cid:16) 𝑟 − 𝑠 ( Φ ) Φ (cid:17) , (16.4)where 𝑟 𝑠 is the classical 𝑟 -matrix, depending on the deformation parameter 𝑠 , that we encoun-tered in Section 14.0.1. The covariant derivative is D 𝑈 = 𝜕𝑈 + 𝑖 Φ 𝑈 + 𝑖 𝑈 A , where A is agauge field, whose structure depends on the actual target space under consideration, and Φ , Φ are auxiliary fields. We recall that, from the perspective of [87], Φ , Φ are the components ofthe four-dimensional Chern-Simons gauge field along the worldsheet, and the quadratic termin Φ , Φ contains 𝑟 − 𝑠 , because, as discussed in section 14.1, the Green’s function of the gaugefield is effectively the classical 𝑟 -matrix. One can eliminate these auxiliary fields, since theyenter the Lagrangian quadratically, arriving at the following expression: L = Tr (cid:16) 𝑉 𝐷𝑈 (cid:17) + Tr (cid:16) 𝑈 𝐷𝑉 (cid:17) + Tr (cid:16) 𝑟 𝑠 ( 𝑈𝑉 ) ( 𝑈𝑉 ) † (cid:17) , (16.5)108here 𝐷𝑈 = 𝜕𝑈 + 𝑖 𝑈 A . Clearly, in Dirac notation this leads to the deformed Gross-Neveumodel of the following form: L = Ψ 𝑎 / 𝐷 Ψ 𝑎 + ( 𝑟 𝑠 ) 𝑐𝑑𝑎𝑏 (cid:18) Ψ 𝑎 + 𝛾 Ψ 𝑐 (cid:19) (cid:18) Ψ 𝑑 − 𝛾 Ψ 𝑏 (cid:19) . (16.6)To summarize, we have obtained a chiral gauged bosonic Gross-Neveu model, where thedeformation is encoded in the classical 𝑟 -matrix that defines 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 𝑟 𝑠 isproportional to the identity operator, the above Lagrangians have an SU ( 𝑛 ) global symmetryand a corresponding Noether current one-form K = 𝐾 𝑑𝑧 + 𝐾 𝑑𝑧 = 𝑈 𝑉 𝑑𝑧 + 𝑉 𝑈 𝑑𝑧 .
Usingthis one-form, we define a family of connections, following [87]: A = 𝑟 𝜅 ( 𝐾 ) 𝑑𝑧 − 𝑟 𝜅 ( 𝐾 ) 𝑑𝑧 . (16.7)Here 𝜅 , 𝜅 are complex parameters that will be related below. We wish to prove that theconnection A is flat: 𝑑 A + A ∧ A = − 𝑑𝑧 ∧ 𝑑𝑧 (cid:16) 𝑟 𝜅 ( 𝜕𝐾 ) + 𝑟 𝜅 ( 𝜕𝐾 ) + [ 𝑟 𝜅 ( 𝐾 ) , 𝑟 𝜅 ( 𝐾 )] (cid:17) ? = . (16.8)To this end, we will use the equations of motion of the model (16.5). To write them out, wedefine the ‘conjugate’ operator ˆ 𝑟 by the relation Tr ( 𝑟 𝑠 ( 𝐴 ) 𝐵 ) = − Tr ( 𝐴 ˆ 𝑟 𝑠 − ( 𝐵 )) . When theunitarity relation (14.2) holds, ˆ 𝑟 = 𝑟 . The e.o.m. for the 𝑈 and 𝑉 variables may be shown toimplythe following concise equations for the ‘Noether current’ 𝐾 : 𝜕𝐾 = [ ˆ 𝑟 𝑠 − ( 𝐾 ) , 𝐾 ] , 𝜕𝐾 = [ 𝐾, 𝑟 𝑠 ( 𝐾 )] . (16.9)These are the deformed versions of the equations (15.6) that we encountered earlier (since ineliminating Φ from (16.4) one easily sees that in the undeformed case Φ ∼ 𝐾 ). Substitutingin the equation (16.8), we see that it is satisfied if the matrix 𝑟 obeys the equation 𝑟 𝜅 ( [ 𝐾, 𝑟 𝑠 ( 𝐾 )]) + 𝑟 𝜅 ( [ ˆ 𝑟 𝑠 − ( 𝐾 ) , 𝐾 ]) + [ 𝑟 𝜅 ( 𝐾 ) , 𝑟 𝜅 ( 𝐾 )] = . (16.10)The reason why in our case ˆ 𝑟 is not necessarily equal to 𝑟 is that we will mostly be dealing witha non-simple Lie algebra 𝔤𝔩 ( 𝑛 ) = 𝔰𝔩 ( 𝑛 ) ⊕ C ( C ⊂ 𝔤𝔩 ( 𝑛 ) corresponds to matrices proportionalto the unit matrix). We will assume a block-diagonal 𝑟 -matrix, acting as follows: 𝑟 𝑠 = ( 𝑟 𝑠 ) 𝔰𝔩 ( 𝑛 ) + ( 𝑟 𝑠 ) C , ( 𝑟 𝑠 ) 𝔰𝔩 ( 𝑛 ) ∈ End ( 𝔰𝔩 ( 𝑛 )) , ( 𝑟 𝑠 ) C : = 𝑏 ( 𝑠 ) Tr . (16.11)109 𝑧 𝑧 − 𝑧 𝑖 𝑧 𝑧 − 𝑧 − 𝑧 𝑗 𝑖 𝑗𝑘𝑙 −( 𝑟 𝑠 ) 𝑘𝑙𝑖 𝑗 Figure 20: Feynman rules of the deformed model (16.5)-(16.6), with A = 𝑘𝑖𝑗 𝑙 𝑙𝑖𝑗 𝑘 Figure 21: Diagrams contributing to the 𝛽 -function at one loop.Here ( 𝑟 𝑠 ) 𝔰𝔩 ( 𝑛 ) acts on traceless matrices as in (14.6), and ( 𝑟 𝑠 ) C acts on matrices of the type 𝛼 · as multiplication by 𝑛 𝑏 ( 𝑠 ) . In this case ˆ 𝑟 𝑠 = 𝑟 𝑠 − ( 𝑏 ( 𝑠 − ) + 𝑏 ( 𝑠 )) Tr. If the unitarityrelation is satisfied, the mismatch vanishes. However, in either case 𝑏 ( 𝑠 ) completely dropsout from the equation (16.10), so we will prefer allowing an arbitrary function 𝑏 ( 𝑠 ) for themoment.Postulating the relations 𝜅 = 𝑢, 𝜅 = 𝑢𝑣, 𝑣 = 𝑠 − (implying 𝜅 = 𝜅 𝑠 ) between theparameters, we identify (16.10) with the classical Yang-Baxter equation for 𝔤 = 𝔰𝔩 ( 𝑛 ) fromSection 14.0.1. 𝛽 -function and the Ricci flow. In this section we turn to the analysis of the elementary quantum properties of the theorydefined 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 – in other words, whether it is sufficient to renormalize the parameters of the 𝑟 -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 different directions of the lines in theloops. Otherwise, the type of the divergence is the same – it is logarithmic , proportionalto ∫ 𝑑 𝑧𝑧𝑧 . As a result, the one-loop 𝛽 -function is Here we are talking about UV divergences. 𝑘𝑙𝑖 𝑗 = 𝑛 ∑︁ 𝑝,𝑞 = (cid:16) ( 𝑟 𝑠 ) 𝑘𝑞𝑖𝑝 ( 𝑟 𝑠 ) 𝑞𝑙𝑝 𝑗 − ( 𝑟 𝑠 ) 𝑞𝑙𝑖𝑝 ( 𝑟 𝑠 ) 𝑘𝑞𝑝 𝑗 (cid:17) (16.12)As already discussed earlier, we will assume a block-diagonal 𝑟 -matrix (16.11), where ( 𝑟 𝑠 ) 𝔰𝔩 ( 𝑛 ) acts as in (14.6), and ( 𝑟 𝑠 ) C acts on a unit matrix as multiplication by 𝑛 · 𝑏 ( 𝑠 ) . This is translatedinto the four-index notation as follows: ( 𝑟 𝑠 ) 𝑘𝑙𝑖 𝑗 = 𝑎 𝑘𝑙 ( 𝑠 ) (cid:18) 𝛿 𝑘𝑖 𝛿 𝑙𝑗 − 𝑛 𝛿 𝑖 𝑗 𝛿 𝑘𝑙 (cid:19) + 𝑏 ( 𝑠 ) 𝛿 𝑖 𝑗 𝛿 𝑘𝑙 . (16.13)The coefficients 𝑎 𝑘𝑙 ( 𝑠 ) are defined by the action of (14.6) in the standard basis: 𝑟 𝑠 ( 𝑒 𝑘 ⊗ 𝑒 𝑙 ) = 𝑎 𝑘𝑙 ( 𝑠 ) 𝑒 𝑘 ⊗ 𝑒 𝑙 . Concretely (see (14.5)), 𝑎 𝑖 𝑗 = 𝑠 − 𝑠 = α , 𝑖 < 𝑗 − 𝑠 = β , 𝑖 > 𝑗
12 1 + 𝑠 − 𝑠 = γ , 𝑖 = 𝑗 . (16.14)Substituting (16.13) in (16.12) and doing the summations, we obtain 𝛽 𝑘𝑙𝑖 𝑗 = (cid:20) 𝑛𝑠 ( − 𝑠 ) + ( 𝑖 − 𝑗 ) 𝑎 𝑖 𝑗 (cid:21) (cid:18) 𝛿 𝑘𝑖 𝛿 𝑙𝑗 − 𝑛 𝛿 𝑖 𝑗 𝛿 𝑘𝑙 (cid:19) : =Π 𝑘𝑙𝑖 𝑗 . (16.15)Since 𝑎 𝑖 𝑗 =
12 1 + 𝑠 − 𝑠 + 𝑖 R 𝑖 𝑗 , the one-loop result is not of the form (16.13) (due to the termproportional to 𝑖 − 𝑗 ). For this reason the straightforward Ricci flow equation 𝑑𝑟 𝑠 𝑑𝜏 = 𝛽 for 𝑠 ( 𝜏 ) does not have a solution. However, this can be easily remedied by allowing reparametrizationsof coordinates along the flow. Let us reparametrize 𝑈 → 𝜅𝑈, 𝑉 → 𝑉 𝜅 − , where 𝜅 = Diag { 𝜅 , . . . , 𝜅 𝑛 } . (16.16)The kinetic term in (16.5) is invariant, so the only effect is in the effective replacement ofthe 𝑟 -matrix by (cid:101) 𝑟 , where ( (cid:101) 𝑟 𝑠 ) 𝑘𝑙𝑖 𝑗 = 𝜅 𝑖 𝜅 𝑘 𝜅 𝑗 𝜅 𝑙 ( 𝑟 𝑠 ) 𝑘𝑙𝑖 𝑗 . One has to conjugate the 𝛽 -function tensoranalogously, and the equation we will aim to solve is 𝑑 (cid:101) 𝑟 𝑠 𝑑𝜏 = (cid:101) 𝛽 . It may be rewritten as anequation for the original 𝑟 -matrix as follows: 𝑑𝑑𝜏 ( 𝑟 𝑠 ) 𝑘𝑙𝑖 𝑗 = 𝛽 𝑘𝑙𝑖 𝑗 − 𝑑𝑑𝜏 (cid:18) log (cid:18) 𝜅 𝑖 𝜅 𝑘 𝜅 𝑗 𝜅 𝑙 (cid:19) (cid:19) ( 𝑟 𝑠 ) 𝑘𝑙𝑖 𝑗 : = (cid:98) 𝛽 𝑘𝑙𝑖 𝑗 (16.17)where (cid:98) 𝛽 𝑘𝑙𝑖 𝑗 = (cid:20) 𝑛𝑠 ( − 𝑠 ) + ( 𝑖 − 𝑗 ) 𝑎 𝑖 𝑗 − 𝑑𝑑𝜏 (cid:18) log (cid:18)(cid:12)(cid:12) 𝜅 𝑖 𝜅 𝑗 (cid:12)(cid:12) (cid:19) (cid:19) 𝑎 𝑖 𝑗 (cid:21) Π 𝑘𝑙𝑖 𝑗 The unwanted term may now be canceled by the simple substitution 𝜅 𝑗 = 𝑒 𝜏 𝑗 . (16.18)111ecalling again the expression for 𝑎 𝑖 𝑗 ( 𝑠 ) , we find that the remaining equations may be writtenas (cid:164) 𝑏 = 𝑑𝑑𝜏 (cid:16)
12 1 + 𝑠 − 𝑠 (cid:17) = 𝑛𝑠 ( − 𝑠 ) , or (cid:164) 𝑠 = 𝑛𝑠 . Therefore 𝑏 = const ., 𝑠 = 𝑒 𝑛 𝜏 (16.19)In the geometric formulation (i.e. for the sigma model defined by a metric, 𝐵 -field and dilaton)the RG-flow equations are the generalized Ricci flow equations. They look as follows [89,200]: − (cid:164) 𝑔 𝑖 𝑗 = 𝑅 𝑖 𝑗 + 𝐻 𝑖𝑚𝑛 𝐻 𝑗𝑚 (cid:48) 𝑛 (cid:48) 𝑔 𝑚𝑚 (cid:48) 𝑔 𝑛𝑛 (cid:48) + ∇ 𝑖 ∇ 𝑗 Φ , (16.20) − (cid:164) 𝐵 𝑖 𝑗 = − ∇ 𝑘 𝐻 𝑘𝑖 𝑗 + ∇ 𝑘 Φ 𝐻 𝑘𝑖 𝑗 , − (cid:164) Φ = const . − ∇ 𝑘 ∇ 𝑘 Φ + ∇ 𝑘 Φ ∇ 𝑘 Φ + 𝐻 𝑘𝑚𝑛 𝐻 𝑘𝑚𝑛 , where 𝑔 is the metric, 𝐵 is the skew-symmetric field with 𝐻 its ‘curvature’, and Φ isthe dilaton. We arrive at the important conclusion that the trigonometrically deformedsystem (16.5)-(16.6) with A = 𝛽 -functions of homogeneous models The calculation that we presented above was performed for the ungauged model ( A = 𝛽 -function.Moreover, one can easily see that, even if we replace the size- 𝑛 vectors 𝑈 and 𝑉 by 𝑀 × 𝑛 -matrices, which would be necessary for considering Grassmannian 𝐺𝑟 𝑀,𝑛 or flag 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 𝐵 -field of all homogeneousmodels satisfy the generalized Einstein condition with the same ‘cosmological constant’,equal to 𝑛 : 𝑅 𝑖 𝑗 + 𝐻 𝑖𝑚𝑛 𝐻 𝑗𝑚 (cid:48) 𝑛 (cid:48) 𝑔 𝑚𝑚 (cid:48) hom . 𝑔 𝑛𝑛 (cid:48) hom . = 𝑛 ( 𝑔 hom . ) 𝑖 𝑗 . (16.21)Here 𝑔 hom . is the homogeneous metric, and 𝐻 is the curvature of the 𝐵 -field 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 𝑔 hom . is Kähler, so that the fundamental Hermitianform is closed and 𝐻 =
0. In that case the equation (16.21) translates into the usual Einsteinequation 𝑅 𝑖 𝑗 = 𝑛 ( 𝑔 hom . ) 𝑖 𝑗 . This is the well-known fact that the one-loop 𝛽 -function forsymmetric space models is equal to the dual Coxeter number of the symmetry group 𝐺 (in112articular, it is independent of the denominator 𝐻 for symmetric spaces 𝐺𝐻 ), cf. [269] asa general reference and [178] for the case of Hermitian symmetric spaces, as well as thelectures [266]. In the case of Kähler symmetric spaces – such as Grassmannians – there is analternative explanation: since the Ricci form represents the first Chern class of the manifold,one can attribute the cosmological constant 𝑛 to the fact that 𝑐 ( 𝐺𝑟 𝑀,𝑛 ) = 𝑛 [ C ] , where C isa generator of 𝐻 ( 𝐺𝑟 𝑀,𝑛 , Z ) . For non-symmetric spaces – such as the general flag manifolds– this logic does not work, and there is no direct relation between the first Chern class and thedual Coxeter number. A related fact is that for non-symmetric spaces the metric in our sigmamodels is not Kähler, and one instead has to take into account the non-zero field 𝐻 in (16.21).The effect of including this field is that the two terms in the 𝛽 -function – the curvature andthe 𝐻 -field term – sum up in such a way that the 𝛽 -function is again proportional to 𝑛 . Before concluding this section, let us describe a simplest example, namely the deformationof the sphere 𝑆 – the so-called ‘sausage’ [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 = 𝑉 ◦ 𝐷𝑈 + 𝑈 ◦ 𝐷𝑉 + Tr (cid:0) 𝑟 𝑠 ( 𝜅𝑈 ⊗ 𝑉 𝜅 − ) ( 𝜅𝑈 ⊗ 𝑉 𝜅 − ) † (cid:1) (16.22) = 𝑉 ◦ 𝐷𝑈 − 𝐷𝑈 ◦ 𝑉 + γ (| 𝑈 | | 𝑉 | + | 𝑈 | | 𝑉 | )+ α 𝑒 − 𝜏 | 𝑈 | · | 𝑉 | + β 𝑒 𝜏 | 𝑈 | · | 𝑉 | . Next we pass to the inhomogeneous gauge 𝑈 = 𝑈 : = 𝑊 , 𝑉 : = 𝑉 . Variationw.r.t. the gauge field gives the constraint 𝑉 ◦ 𝑈 =
0, which is solved by 𝑉 = − 𝑊 · 𝑉 . TheLagrangian acquires the following form in these coordinates: L = 𝑉 · 𝐷𝑊 − 𝑉 · 𝐷𝑊 + (cid:16) γ | 𝑊 | + α 𝑒 − 𝜏 + β 𝑒 𝜏 | 𝑊 | (cid:17) | 𝑉 | (16.23)Eliminating the fields 𝑉 , 𝑉 and using the expressions (16.14) for α , β , γ with 𝑠 = 𝑒 𝜏 , wearrive at L = ( 𝑒 − 𝜏 − 𝑒 𝜏 ) | 𝐷𝑊 | (cid:0) 𝑒 𝜏 + | 𝑊 | (cid:1) (cid:0) 𝑒 − 𝜏 + | 𝑊 | (cid:1) , (16.24)which corresponds to the ‘sausage’ metric. As all models of the type (13.7), the La-grangian (16.24) features a 𝐵 -field equal to the fundamental Hermitian form of the metric,however since this is a manifold of complex dimension 1, the 𝐵 -field is closed, so that 𝐻 = Φ is constant (for details see [71]),so that the Ricci flow equation is especially simple: − 𝑑𝑔 𝑖 𝑗 𝑑𝜏 = 𝑅 𝑖 𝑗 . (16.25)113he range of the Ricci time variable is 𝜏 ∈ (−∞ , ) (accordingly 𝑠 = 𝑒 𝜏 ∈ ( , ) ), and thecorresponding solution is called ‘ancient’ in the terminology used in Ricci flow literature.Geometrically the ‘sausage’ looks exactly as its name suggests:0 < 𝑠 < ∼ | log 𝑠 | (16.26)The solution has two characteristic regimes. The first one is 𝑠 →
0, in which case 𝑑𝑠 → | 𝑑𝑊 | | 𝑊 | ,so that one obtains an infinitely long cylinder. Since the cylinder is flat, this should beinterpreted as the UV limit with asymptotic freedom. The opposite regime is the IR limit 𝑠 →
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 𝑛 − [71],where Ricci flow interpolates between a cylinder ( C ∗ ) 𝑛 − in the UV (asymptotic freedom)and a ‘round’ projective space of vanishing radius in the IR (one has a similar behavior in thecase of the so-called 𝜂 -deformed CP 𝑛 − , as shown in [74]). In the previous sections we have claimed that the sigma models of a wide class, includingthe familiar CP 𝑛 − , Grassmannian, flag, 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 𝑛 − -model. This system is described by (16.5), where 𝑈 is a column vector and 𝑉 a row vector, both of length 𝑛 , and additionally one has a C ∗ gauge field A . It turns out thegauge field may be completely eliminated by passing to the inhomogeneous C ∗ -gauge 𝑈 𝑛 = . (16.27)Variation of the Lagrangian (16.5) w.r.t. A gives 𝑉 ◦ 𝑈 =
0, so that 𝑉 𝑛 = − 𝑛 − ∑︁ 𝑘 = 𝑉 𝑘 𝑈 𝑘 . (16.28)This removes the gauge field at the expense of modifying the Feynman rules of the theory.Dropping fermionic fields for the moment, we write the Lagrangian of the model (16.5) in114his gauge: L = 𝑛 − ∑︁ 𝑘 = (cid:16) 𝑉 𝑘 𝜕𝑈 𝑘 − 𝑉 𝑘 𝜕𝑈 𝑘 + β | 𝑉 𝑘 | (cid:17) + (16.29) + 𝑛 − ∑︁ 𝑙,𝑚 = 𝑎 𝑙𝑚 | 𝑈 𝑙 | | 𝑉 𝑚 | + γ (cid:12)(cid:12) 𝑛 − ∑︁ 𝑝 = 𝑈 𝑝 𝑉 𝑝 (cid:12)(cid:12) + α (cid:32) 𝑛 − ∑︁ 𝑘 = | 𝑈 𝑘 | (cid:33) (cid:12)(cid:12) 𝑛 − ∑︁ 𝑝 = 𝑈 𝑝 𝑉 𝑝 (cid:12)(cid:12) We see that the propagators and vertices are modified, 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 ( 𝑈, 𝑉 ) -variables. In other words, instead of a nonlinear 𝜎 -model we have arrived at a different nonlinear theory – the theory of several bosonic fields(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 fields and turn the sigma models into models of multiple bosonic fieldswith 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 fields, 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 𝑆𝐿 ( , R ) 𝑆𝑂 ( ) , whereas gravity with 𝑛 − SU ( 𝑛, ) S ( U ( 𝑛 )× U ( )) – the hyperbolic analogue of projective space. These are all complex (Hermitian) symmet-ric spaces, albeit of Minkowski signature, which makes it slightly different 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 𝐾 of the sigmamodel are bilinear combinations of Ashtekar’s canonical variables ( 𝐴, 𝐸 ) : 𝐾 ∼ 𝐴 ⊗ 𝐸 . Onthe other hand, these same Noether currents, when calculated from the Lagrangian (16.2)or (16.5), have the form 𝐾 ∼ 𝑈 ⊗ 𝑉 , so that the canonical variables ( 𝑈, 𝑉 ) 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 ( 𝐴, 𝐸 ) variables are canonical, one has the Poisson brackets { 𝐴 ( 𝑥 ) , 𝐸 ( 𝑦 )} ∼ 𝛿 ( 𝑥 − 𝑦 ) , and as a result the Noether currents 𝐾 have local Poisson bracketsas well, schematically of the form { 𝐾 ( 𝑥 ) , 𝐾 ( 𝑦 )} ∼ 𝐾 ( 𝑥 ) 𝛿 ( 𝑥 − 𝑦 ) . This of course also im-mediately follows from the analogous Poisson structure of the ( 𝑈, 𝑉 ) variables, which canbe seen from the first-order Lagrangians (16.2), or (16.5). Since the flat connection (16.7)115s linear in the 𝐾 -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 𝛿 (cid:48) ( 𝑥 − 𝑦 ) . The latter cause significant difficulties 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 first 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) suffer 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 fields leads us to Schwinger’s calculation [212] of the effectiveaction of the gauge fields A : S eff . = ξ ∫ 𝑑𝑧 𝑑𝑧 𝐹 𝑧𝑧 (cid:52) 𝐹 𝑧𝑧 , 𝐹 𝑧𝑧 = 𝑖 ( 𝜕 A − 𝜕 A) . (16.30)The coefficient ξ collects some numerical factors and is proportional to the number of matterfields we have integrated over. The action is invariant w.r.t. the gauge transformations of theoriginal U ( ) , A → A + 𝜕𝛼 , A → A + 𝜕𝛼 ( 𝛼 ∈ R ), but not w.r.t. the complexified C ∗ gauge transformations A → A + 𝜕𝛼 , A → A + 𝜕𝛼 ( 𝛼 ∈ 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 effective action (16.30) suffices for mostpurposes. For example, it is clear that the anomaly may be canceled by including fermionssymmetrically with the bosons: L → Ψ 𝑎 / 𝐷 Ψ 𝑎 + Θ 𝑎 / 𝐷 Θ 𝑎 , (16.31)where Ψ are the bosons and Θ 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 first 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 defines what may be called the ‘super-phase space’ of the model.Moreover, the whole theory of generalized integrable Gross-Neveu models of the type (16.2)can be cast in pure differential-geometric terms.The relevant geometric context is as follows. Suppose we have a super phase space 𝚽 ,which is a complex symplectic (quiver) supervariety. There is a gauge (super)-group G gauge acting in the nodes of the quiver, and matter fields 𝑈 ∈ W , 𝑉 ∈ W are in representations116 ⊕ W of G gauge . We assume that the quiver is ‘doubled’, 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 define the complex moment map 𝜇 for the action of G global (cid:8) 𝚽 . In this setup one can, quite naturally, define the followingLagrangian: L = (cid:16) 𝑉 · D 𝑈 + 𝑈 · D 𝑉 (cid:17) + κ STr ( 𝜇 𝜇 ) . (16.32)The Gross-Neveu system (16.2) is a special case. The kinetic term in (16.32) corresponds tothe 𝛽𝛾 -systems – it is a pull-back of the canonical Poincaré-Liouville one-form correspondingto the complex symplectic form of the quiver. The second term provides a coupling betweenthe holomorphic and (anti)-holomorphic 𝛽𝛾 -systems and comes with an arbitrary coefficient κ 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 𝜇 satisfiesthe e.o.m. 𝜕 𝜇 = κ [ 𝜇, 𝜇 ] , (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 Poincaré-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 ( 𝑇 𝑎 𝑇 𝑏 ) = , where 𝑇 𝑎 , 𝑇 𝑏 ∈ 𝔤 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 ( 𝑇 𝑎 ) = CP 𝑛 − -model with fermions.All known models, whose bosonic part is the CP 𝑛 − sigma model, can be attributed to oneof the two cases. In both of these cases the phase spaces are complex symplectic quotientsof the form 𝚽 = ( 𝑇 ∗ C 𝑛 | 𝑛 ) (cid:12) G gauge , (16.35)where G gauge is a subgroup of GL ( | ) ⊂ GL ( 𝑛 | 𝑛 ) , the latter being the symmetry group of T ∗ C 𝑛 | 𝑛 . In this language, the two cases are distinguished by the choice of G gauge :117 The ‘minimal fermions’ phase space: G gauge = C ∗ = (cid:26) 𝑔 ∈ SL ( | ) : 𝑔 = (cid:18) λ λ (cid:19) (cid:27) ◦ The ‘supersymmetric’ phase space: G gauge = (cid:26) 𝑔 ∈ SL ( | ) : 𝑔 = (cid:18) λ ξ λ (cid:19) (cid:27) . In thiscase λ ∈ C ∗ is a bosonic element, and ξ ∈ 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: 𝑈𝑉 C 𝑛 C | (16.36)Here 𝑈 ∈ Hom ( C | , C 𝑛 ) and 𝑉 ∈ Hom ( C 𝑛 , C | ) . The difference comes from the action ofthe gauge group G gauge on C | . In fact, one can also identify the configuration spaces 𝑀 ofthe models, since in both cases 𝚽 = 𝑇 ∗ 𝑀 : 𝑀 min = CP 𝑛 − | 𝑛 , 𝑀 SUSY = Π 𝑇 ( CP 𝑛 − ) . (16.37)Here Π 𝑇 stands for the ‘fermionic tangent bundle’, i.e. the tangent bundle where the fibersare assumed fermionic. Accordingly the super-projective space CP 𝑛 − | 𝑛 may be seen as thetotal space of the super-vector bundle Π (O ( ) ⊕ · · · ⊕ O ( )) over CP 𝑛 − , which puts the twoconfiguration spaces on par with each other.Specifying the phase or configuration space is of course not sufficient to formulate thetheory. One additionally needs to choose the global symmetry group 𝐺 global , which in turndefines the Hamiltonian in (16.32) via the complex moment map 𝜇 . For 𝑀 min there are twochoices that lead to well-known models: ◦ 𝐺 global = 𝑆𝐿 ( 𝑛, C ) . In this case (16.32) defines the CP 𝑛 − model with ‘minimallycoupled fermions’ [1] (this is also the reason for the name of the phase space 𝑀 min ). ◦ 𝐺 global = 𝑃𝑆𝐿 ( 𝑛 | 𝑛, C ) . This is the sigma model with target space CP 𝑛 − | 𝑛 that hasbeen widely studied in the literature, cf. [204, 77]. In the case 𝑛 = 𝑀 SUSY .As proven in detail in [72], in order to obtain an interacting ( , ) -supersymmetric CP 𝑛 − sigma model, in this case one should choose 𝐺 global = 𝑆𝐿 ( 𝑛, C ) . This provides a newapproach to constructing worldsheet-supersymmetric models by starting from a model withtarget space supersymmetry (in this case GL ( 𝑛 | 𝑛, 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 flag manifolds in mathematics, they might not be equally familiarto the physics audience. One of the goals of this review was to fill 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 defining what flag manifolds are, and by describing theirdifferential-geometric structures. As a first application of these methods, we considered an‘almost textbook’ example of a mechanical particle interacting with a non-Abelian gaugefield. As we explained, the isotopic ‘spin’ degrees of freedom take values in a suitable flagmanifold. The symplectic form on the flag manifold is the classical analogue of the conceptof representation of the gauge group, w.r.t. which the particle is charged. The ‘Berry phase’,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 ‘quantum numbers’ 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 ‘geometric quantization’. 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-Primakoff and Dyson-Maleev representations. We alsoemphasized that the very same flag 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 flag manifold target space. The first topicis how such sigma models arise in the continuum limits of spin chains, and the second one isthe description of integrable flag manifold models.In Chapter 2, we reviewed SU( 𝑛 ) spin chains in various representations, and discussed atlength how these representations give rise to sigma models with different flag manifold targetspaces. This can be understood by considering how the Young tableau parameters 𝑝 𝛼 , whichgeneralize the notion of spin in the antiferromagnet, determine the target space of the chain’smatrix degree of freedom, 𝑆 . Mathematically speaking, 𝑆 is a moment map, and different119 𝛼 s define different co-adjoint orbits. For SU(2), the target space of 𝑆 is always 𝑆 , and leadsto the familiar CP model, but for 𝑛 >
2, the 2-sphere is promoted to some flag manifold ofSU( 𝑛 ).For most of the chapter, we focused on the totally symmetric representations of SU( 𝑛 ),which corresponded to 𝑝 = 𝑝 , and 𝑝 𝛼 = 𝛼 >
1. This defines a coadjoint orbitisomorphic to CP 𝑛 − 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 𝑛 -site order, we obtained thecomplete U ( 𝑛 )/[ U ( )] 𝑛 flag manifold sigma model. Loosely speaking, the longer-rangeinteraction terms served to couple the different CP 𝑛 − sectors together, as well as imposeorthogonality. Related to this, we found that these sigma models possessed 𝑛 topologicalterms, each of which corresponded to the pull-back to the complete flag manifold of theFubini-Study form on CP 𝑛 − . The corresponding topological angles were found to be 𝜃 𝐴 = 𝜋𝑝 𝐴𝑛 .Unless the various interaction terms in the Heisenberg Hamiltonian were tuned to specialvalues, we learned that these flag 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 flow to common value under renormalization, thus establishingthat Lorentz invariance does indeed emerge at low energies. This fact led to an SU( 𝑛 )generalization of Haldane’s conjecture: when 𝑝 and 𝑛 are coprime, the corresponding SU( 𝑛 )chain will be in a gapless phase at low energies; otherwise, a finite energy gap will persist,with ground-state degeneracy equal to 𝑛 / gcd ( 𝑛, 𝑝 ) . This conjecture was supported by variousexact results, including the LSMA theorem and AKLT constructions, as well as by ’t Hooftanomaly matching conditions. In short, ’t Hooft anomalies are present in the flag manifoldsigma model for all 𝑝 not a multiple of 𝑛 , but only when gcd ( 𝑛, 𝑝 ) = 𝑛 ) WZNW model). Theseanomalies are mixed between the PSU( 𝑛 ) symmetry of the model, and a global Z 𝑛 symmetry,which derives from the underlying 𝑛 -site order of SU( 𝑛 ) 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( 𝑛 ) bundle, and have topological charges that are multiple of 1 / 𝑛 .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 ( 𝑛, 𝑝 ) =
1, these excitations interfere, resulting in an effective fugacity of zero, and leadto a gapless phase in the sigma model. More general representations of SU( 𝑛 ), 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 different circle of questionsrelated to flag 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 flag 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 significant challenge already in classical theory.For this reason we started in Chapter 3 by describing from various angles the classicallyintegrable models with flag manifold target spaces. Technically the key new ingredient thatneeds to be included to make such models integrable is a non-topological 𝐵 -field 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 1970’s.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, 𝐵 -field and dilaton. In support of this opinion we provided acalculation of the one-loop 𝛽 -function of a wide class of trigonometrically deformed sigmamodels. This 𝛽 -function is common for all of these models and provides a far-reachinggeneralization of the so-called ‘sausage’ solution that corresponds to the 𝑆 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 𝛽 -functions ofsymmetric space models depend only on the dual Coxeter number of the symmetry groupand extends this result to the (non-symmetric) flag manifold models. Finally, we showedthat the purely bosonic gauged models suffer from chiral anomalies, which may be canceledby adding fermions. More generally, we formulated a broad differential-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 𝑛 − model withfermions, yielding all known quantum integrable models with bosonic core CP 𝑛 − , 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. Aref’eva, G. Arutyunov,A. Bourget, R. Donagi, S. Frolov, A. Hanany, E.A.Ivanov, S. Ketov, C. Klimčík, M. La-jko, G. Lopes Cardoso, D. Lüst, 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 flagmanifold sigma models. DB would also like to thank the Max-Planck-Institut für Physikin Munich (Germany), where part of this work was done, for hospitality. The research of121. Affleck 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).
References [1] E. Abdalla, M. C. B. Abdalla, and M. Gomes. “Anomaly Cancellations in the Super-symmetric CP ( 𝑁 − ) Model”.
Phys. Rev. D
25 (1982), p. 452.[2] E. Abdalla, M. Abdalla, and M. Gomes. “Anomaly in the Nonlocal Quantum Chargeof the CP ( 𝑛 − ) Model”.
Phys. Rev. D
23 (1981), p. 1800.[3] E. Abdalla, M. C. B. Abdalla, and K. D. Rothe.
Non perturbative methods in twodimensional quantum field theory . Singapore: World Scientific, 1991.[4] I. Achmed-Zade and D. Bykov. “Ricci-Flat Metrics on Vector Bundles Over FlagManifolds”.
Commun. Math. Phys. .[5] I. Affleck. “Large N limit of SU(N) quantum ’spin’ chains”.
Phys. Rev. Lett.
54 (1985),pp. 966–969.[6] I. Affleck. “Critical behaviour of SU(n) quantum chains and topological non-linear 𝜎 -models”. Nuclear Physics B 𝜎 -models at 𝜃 = 𝜋 and the quantum Hall effect”. Nuclear Physics B 𝜎 model”. Phys. Rev. Lett.
56 (5 1986), pp. 408–411.[9] I. Affleck. “Quantum Spin Chains and the Haldane Gap”.
J. Phys. C 𝜎 Models at 𝜃 = 𝜋 and Quantum Spin Chains”. Nucl. Phys. B
257 (1985), pp. 397–406.[11] I. Affleck and F. D. M. Haldane. “Critical theory of quantum spin chains”.
Phys. Rev.B
36 (10 1987), pp. 5291–5300.[12] I. Affleck, T. Kennedy, E. Lieb, and H. Tasaki. “Valence bond ground states inisotropic quantum antiferromagnets”. English.
Communications in MathematicalPhysics
115 (3 1988), pp. 477–528.[13] I. Affleck and E. H. Lieb. “A proof of part of Haldane’s conjecture on spin chains”.
Lett. Math. Phys.
12 (1986), p. 57.[14] I. Agricola, A. Borówka, and T. Friedrich. “S6 and the geometry of nearly Kähler6-manifolds”.
Differential Geometry and its Applications
57 (2018). (Non)-existenceof complex structures on S6, pp. 75 –86.12215] M. Aguado and M. Asorey. “Theta-vacuum and large N limit in C 𝑃 𝑁 − 𝜎 models”. Nucl. Phys. B
844 (2011), pp. 243–265. arXiv: .[16] O. Aharony, N. Seiberg, and Y. Tachikawa. “Reading between the lines of four-dimensional gauge theories”.
JHEP
08 (2013), p. 115. arXiv: .[17] A. Alekseev, L. Faddeev, and S. L. Shatashvili. “Quantization of symplectic orbitsof compact Lie groups by means of the functional integral”.
J.Geom.Phys.
Funktsional. Anal. i Prilozhen.
Zb. Rad. Mat. Inst. Beograd. (N.S.) 𝜃 = 𝜋 of the mass gap in thetwo-dimensional O(3) nonlinear sigma model”. Phys. Rev. B
90 (18 2014), p. 184421.[21] B. Allés and A. Papa. “Mass gap in the 2D O(3) nonlinear sigma model with a 𝜃 = 𝜋 term”. Phys. Rev. D
77 (5 2008), p. 056008.[22] Y. Amari and N. Sawado. “BPS Sphalerons in the 𝐹 Non-Linear Sigma Model”.
Phys. Rev. D
97 (6 2018), p. 065012. eprint: .[23] Y. Amari and N. Sawado. “ 𝑆𝑈 ( ) Knot Solitons: Hopfions in the 𝐹 Skyrme-Faddeev-Niemi model”.
Phys. Lett. B
784 (2018), pp. 294–300. arXiv: .[24] N. Andrei, K. Furuya, and J. Lowenstein. “Solution of the Kondo Problem”.
Rev.Mod. Phys.
55 (1983), p. 331.[25] N. Andrei and H. Johannesson. “Higher dimensional representations of the SU(N)Heisenberg model”.
Physics Letters A
Phys. Rev. B
77 (1 2008), p. 014429.[27] V. Arnold and B. Khesin.
Topological methods in hydrodynamics . Springer, 1998.[28] A. Arvanitoyeorgos.
An introduction to Lie groups and the geometry of homogeneousspaces . Vol. 22. Student Mathematical Library. American Mathematical Society,Providence, RI, 2003, 148p.[29] A. Arvanitoyeorgos. “New invariant Einstein metrics on generalized flag manifolds”.
Transactions of the American Mathematical Society
337 (2 1993), pp. 981–995.[30] A. Ashtekar. “Mathematical problems of nonperturbative quantum general relativity”.In:
Les Houches Summer School on Gravitation and Quantizations, Session 57 . Dec.1992, pp. 0181–284. arXiv: gr-qc/9302024 .[31] H. Azad, R. Kobayashi, and M. Qureshi. “Quasi-potentials and Kähler Einsteinmetrics on flag manifolds”.
Journal of Algebra
Journal of Algebra 𝜃 -vacuum systems via realaction simulations”. Physics Letters B 𝑂 ( ) nonlinear sigma model with topological term at 𝜃 = 𝜋 from numerical simulations”. Phys. Rev. D
86 (9 2012), p. 096009.12335] H. Babujian. “Exact solution of the one-dimensional isotropic Heisenberg chain witharbitrary spins S”.
Physics Letters A
Physics Reports
Phys. Rev. B
40 (7 1989), pp. 4621–4626.[38] V. Bargmann. “On a Hilbert space of analytic functions and an associated integraltransform”.
Comm. Pure Appl. Math.
14 (1961), pp. 187–214.[39] V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov. “Integrable quantumfield theories in finite volume: Excited state energies”.
Nucl. Phys. B
489 (1997),pp. 487–531. arXiv: hep-th/9607099 .[40] N. Beisert and F. Luecker. “Construction of Lax Connections by Exponentiation”.
J.Math. Phys.
53 (2012), p. 122304. arXiv: .[41] A. A. Belavin and V. G. Drinfel’d. “Solutions of the classical Yang-Baxter equationfor simple Lie algebras”.
Funktsional. Anal. i Prilozhen.
Sov. Phys. JETP
48 (1978), pp. 985–994.[43] F. Berezin. “General Concept of Quantization”.
Commun.Math.Phys.
40 (1975),pp. 153–174.[44] B. Berg, M. Karowski, P. Weisz, and V. Kurak. “Factorized U(n) Symmetric s Matricesin Two-Dimensions”.
Nucl. Phys. B
134 (1978), pp. 125–132.[45] D. Bernard. “Hidden Yangians in 2-D massive current algebras”.
Commun. Math.Phys.
137 (1991), pp. 191–208.[46] D. Bernard and A. Leclair. “Quantum group symmetries and nonlocal currents in2-D QFT”.
Commun. Math. Phys.
142 (1991), pp. 99–138.[47] H. Bethe. “Zur Theorie der Metalle”.
Zeitschrift für Physik 𝜃 vacua”. Nuclear PhysicsB 𝜃 Dependence in theO(3) 𝜎 Model”.
Phys. Rev. Lett.
53 (6 1984), pp. 519–522.[50] S. Bieri, M. Serbyn, T. Senthil, and P. A. Lee. “Paired chiral spin liquid with a Fermisurface in 𝑆 = Phys. Rev. B
86 (22 2012), p. 224409.[51] W. Bietenholz, A. Pochinsky, and U. J. Wiese. “Meron-Cluster Simulation of the 𝜃 Vacuum in the 2D O(3) Model”.
Phys. Rev. Lett.
75 (24 1995), pp. 4524–4527.[52] J. de Boer and S. L. Shatashvili. “Two-dimensional conformal field theories onAdS(2d+1) backgrounds”.
JHEP
06 (1999), p. 013. arXiv: hep-th/9905032 .[53] A. Borel and F. Hirzebruch. “Characteristic classes and homogeneous spaces. I”.
Amer. J. Math.
80 (1958), pp. 458–538.[54] R. Botet, R. Jullien, and M. Kolb. “Finite-size-scaling study of the spin-1 Heisenberg-Ising chain with uniaxial anisotropy”.
Phys. Rev. B
28 (7 1983), pp. 3914–3921.[55] L. J. Boya, A. M. Perelomov, and M. Santander. “Berry phase in homogeneous Kählermanifolds with linear Hamiltonians”.
J. Math. Phys.
Commun. Math. Phys.
209 (2000), pp. 785–810. arXiv: gr-qc/9806002 .[57] P. Breitenlohner, D. Maison, and G. W. Gibbons. “Four-Dimensional Black Holesfrom Kaluza-Klein Theories”.
Commun. Math. Phys.
120 (1988), p. 295.[58] O. Brodbeck and M. Zagermann. “Dimensionally reduced gravity, Hermitian sym-metric spaces and the Ashtekar variables”.
Class. Quant. Grav.
17 (2000), pp. 2749–2764. arXiv: gr-qc/9911118 .[59] F. E. Burstall and S. M. Salamon. “Tournaments, flags, and harmonic maps”.
Math.Ann.
Phys. Lett. B
201 (1988), pp. 466–472.[61] J.-B. Butruille. “Classification des variété approximativement kähleriennes homogénes”.
Annals of Global Analysis and Geometry
27 (3 2005), pp. 201–225. arXiv: .[62] W. J. L Buyers, R. M. Morra, R. L. Armstrong, M. J. Hogan, P. Gerlach, and K.Hirakawa. “Experimental evidence for the Haldane gap in a spin-1 nearly isotropic,antiferromagnetic chain”.
Phys. Rev. Lett.
56 (4 1986), pp. 371–374.[63] D. Bykov. “Classical solutions of a flag manifold 𝜎 -model”. Nucl. Phys. B
902 (2016),pp. 292–301. arXiv: .[64] D. Bykov. “Complex structure-induced deformations of 𝜎 -models”. JHEP .[65] D. Bykov. “Complex structures and zero-curvature equations for 𝜎 -models”. Phys.Lett. B
760 (2016), pp. 341–344. arXiv: .[66] D. Bykov. “Cyclic gradings of Lie algebras and Lax pairs for 𝜎 -models”. Theor.Math. Phys.
Proc. Steklov Inst.Math.
309 (2020), pp. 78–86. arXiv: .[68] D. Bykov. “Flag manifold 𝜎 -models: The 𝑁 -expansion and the anomaly two-form”. Nucl. Phys. B
941 (2019), pp. 316–360. arXiv: .[69] D. Bykov. “Haldane limits via Lagrangian embeddings”.
Nucl. Phys. B
855 (2012),pp. 100–127. arXiv: .[70] D. Bykov. “Integrable properties of sigma-models with non-symmetric target spaces”.
Nucl. Phys. B
894 (2015), pp. 254–267. arXiv: .[71] D. Bykov. “Quantum flag manifold 𝜎 -models and Hermitian Ricci flow” (2020).arXiv: .[72] D. Bykov. “The CP 𝑛 − -model with fermions: a new look” (Sept. 2020). arXiv: .[73] D. Bykov. “The geometry of antiferromagnetic spin chains”. Comm. Math. Phys. 𝜎 -models, Ricci flow and Toda field theories”(2020). arXiv: .[75] D. Bykov. “A gauged linear formulation for flag-manifold 𝜎 -models”. Theor. Math.Phys. 𝑁 -Expansion for Flag-Manifold 𝜎 -Models”. Theor. Math. Phys.
JHEP
02 (2010), p. 015. arXiv: .[78] S. Capponi, P. Lecheminant, and K. Totsuka. “Phases of one-dimensional SU(N) coldatomic Fermi gases-From molecular Luttinger liquids to topological phases”.
Annalsof Physics
367 (2016), pp. 50–95. arXiv: .[79] L. Castellani and D. Lust. “Superstring Compactification on Homogeneous CosetSpaces With Torsion”.
Nucl. Phys. B
296 (1988). Ed. by S. Loken, p. 143.[80] M. A. Cazalilla, A. F. Ho, and M Ueda. “Ultracold gases of ytterbium: ferromagnetismand Mott states in an SU(6) Fermi system”.
New Journal of Physics
Reports on Progress in Physics
Journal of Physics:Condensed Matter
Phys. Rev. B
83 (3 2011), p. 035107.[84] Y. Cho. “COLORED MONOPOLES”.
Phys. Rev. Lett.
44 (1980). [Erratum: PRL 44,1566 (1980)], p. 1115.[85] S. R. Coleman. “There are no Goldstone bosons in two-dimensions”.
Commun. Math.Phys.
31 (1973), pp. 259–264.[86] P. Corboz, M. Lajkó, A. M. Läuchli, K. Penc, and F. Mila. “Spin-Orbital QuantumLiquid on the Honeycomb Lattice”.
Phys. Rev. X .[88] E. Cremmer and J. Scherk. “The Supersymmetric Nonlinear Sigma Model in Four-Dimensions and Its Coupling to Supergravity”.
Phys. Lett. B
74 (1978), pp. 341–343.[89] G. Curci and G. Paffuti. “Consistency Between the String Background Field Equationof Motion and the Vanishing of the Conformal Anomaly”.
Nucl. Phys. B
286 (1987),pp. 399–408.[90] A. D’Adda, M. Lüscher, and P. Di Vecchia. “A 1/n Expandable Series of NonlinearSigma Models with Instantons”.
Nucl. Phys. B
146 (1978), pp. 63–76.[91] D’Adda, A. and Di Vecchia, P. and Lüscher, M. “Confinement and Chiral SymmetryBreaking in CP 𝑛 − Models with Quarks”.
Nucl. Phys. B
152 (1979), pp. 125–144.[92] F. Delduc, T. Kameyama, S. Lacroix, M. Magro, and B. Vicedo. “Ultralocal Laxconnection for para-complex Z 𝑇 -cosets”. Nucl. Phys. B
949 (2019), p. 114821. arXiv: .[93] F. Delduc, M. Magro, and B. Vicedo. “On classical 𝑞 -deformations of integrablesigma-models”. JHEP .12694] S. Deser, R. Jackiw, and S. Templeton. “Three-Dimensional Massive Gauge Theo-ries”.
Phys. Rev. Lett.
48 (1982), pp. 975–978.[95] C. Devchand and J. Schiff. “Hidden symmetries of the principal chiral model un-veiled”.
Commun. Math. Phys.
190 (1998), pp. 675–695. arXiv: hep-th/9611081 .[96] R. Donagi and E. Sharpe. “GLSM’s for partial flag manifolds”.
J. Geom. Phys. .[97] P. Dorey and R. Tateo. “Excited states by analytic continuation of TBA equations”.
Nucl. Phys. B
482 (1996), pp. 639–659. arXiv: hep-th/9607167 .[98] G. V. Dunne and M. Ünsal. “Continuity and Resurgence: towards a continuum defi-nition of the CP (N-1) model”. Phys. Rev. D
87 (2013), p. 025015. arXiv: .[99] G. V. Dunne and M. Ünsal. “Resurgence and Trans-series in Quantum Field Theory:The CP(N-1) Model”.
JHEP
11 (2012), p. 170. arXiv: .[100] R. Efraty and V. Nair. “Chern-Simons theory and the quark - gluon plasma”.
Phys.Rev. D
47 (1993), pp. 5601–5614. arXiv: hep-th/9212068 .[101] R. Efraty and V. Nair. “The Secret Chern-Simons action for the hot gluon plasma”.
Phys. Rev. Lett.
68 (1992), pp. 2891–2894. arXiv: hep-th/9201058 .[102] H. Eichenherr and M. Forger. “On the Dual Symmetry of the Nonlinear SigmaModels”.
Nucl. Phys.
B155 (1979), p. 381.[103] S. Elitzur. “The applicability of perturbation expansion to two-dimensional goldstonesystems”.
Nuclear Physics B
Rev. Mod. Phys.
80 (4 2008),pp. 1355–1417.[105] L. Faddeev and A. J. Niemi. “Partial duality in SU(N) Yang-Mills theory”.
Phys. Lett.B
449 (1999), pp. 214–218. arXiv: hep-th/9812090 .[106] L. Faddeev and N. Reshetikhin. “Integrability of the Principal Chiral Field Model in(1+1)-dimension”.
Annals Phys.
167 (1986), p. 227.[107] L. Faddeev and A. Slavnov. “Gauge Fields. Introduction To Quantum Theory”.
Front.Phys.
50 (1980), pp. 1–232.[108] V. A. Fateev. “The sigma model (dual) representation for a two-parameter family ofintegrable quantum field theories”.
Nucl. Phys. B
473 (1996), pp. 509–538.[109] V. Fateev, E. Onofri, and A. B. Zamolodchikov. “Integrable deformations of the 𝑂 ( ) sigma model. The sausage model”. Nucl. Phys. B
406 (1993), pp. 521–565.[110] G. Fáth and J. Sólyom. “Search for the nondimerized quantum nematic phase in thespin-1 chain”.
Phys. Rev. B
51 (6 1995), pp. 3620–3625.[111] P. de Forcrand, M. Pepe, and U. J. Wiese. “Walking near a conformal fixed point: The2-d 𝑂 ( ) model at 𝜃 ≈ 𝜋 as a test case”. Phys. Rev. D
86 (7 2012), p. 075006.[112] A. P. Fordy and J. C. Wood, eds.
Harmonic maps and integrable systems . Aspects ofMathematics, E23. Friedr. Vieweg & Sohn, Braunschweig, 1994, pp. vi+329.[113] P. Fromholz and P. Lecheminant. “Symmetry-protected topological phases in theSU(N) Heisenberg spin chain: a Majorana-fermion approach”.
Phys. Rev. B .127114] W. Fulton and J. Harris.
Representation theory. A first course . 1st edition. Springer,1991, 551p.[115] S. C. Furuya and M. Oshikawa. “Symmetry Protection of Critical Phases and aGlobal Anomaly in 1 + Phys. Rev. Lett. .[116] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett. “Generalized Global Symme-tries”.
JHEP
02 (2015), p. 172. arXiv: .[117] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett. “Generalized global symmetries”.
Journal of High Energy Physics
Phys. Rev. Lett.
19 (19 1967), pp. 1095–1097.[119] K. Gawedzki. “Noncompact WZW conformal field theories”. In:
NATO AdvancedStudy Institute: New Symmetry Principles in Quantum Field Theory . 1991, pp. 0247–274. arXiv: hep-th/9110076 .[120] R. Geroch. “A method for generating new solutions of Einstein’s equation. II”.
J.Mathematical Phys.
13 (1972), pp. 394–404.[121] Y. Goldschmidt and E. Witten. “Conservation Laws in Some Two-dimensional Mod-els”.
Phys. Lett. B
91 (1980), pp. 392–396.[122] A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, J. Ye, P. Zoller, E.Demler, M. D. Lukin, and A. M. Rey. “Two-orbital SU(N) magnetism with ultracoldalkaline-earth atoms”.
Nature Physics .[123] S. Gozel, F. Mila, and I. Affleck. “Asymptotic Freedom and Large Spin Antiferro-magnetic Chains”.
Phys. Rev. Lett.
123 (3 2019), p. 037202.[124] S. Gozel, P. Nataf, and F. Mila. “Haldane Gap of the Three-Box Symmetric SU(3)Chain”.
Phys. Rev. Lett. .[125] S. Gozel, D. Poilblanc, I. Affleck, and F. Mila. “Novel families of SU( 𝑁 ) AKLTstates with arbitrary self-conjugate edge states”. Nucl. Phys. B
945 (2019), p. 114663.arXiv: .[126] M. Greiter and S. Rachel. “Valence bond solids for SU ( 𝑛 ) spin chains: Exact models,spinon confinement, and the Haldane gap”. Phys. Rev. B
75 (18 2007), p. 184441.[127] T. Grover, D. N. Sheng, and A. Vishwanath. “Emergent Space-Time Supersymmetryat the Boundary of a Topological Phase”.
Science
Harmonic maps, loop groups, and integrable systems . CambridgeUniversity Press, 1997, 194p.[129] F. D. M. Haldane. “Nobel Lecture: Topological quantum matter”.
Rev. Mod. Phys.
89 (4 2017), p. 040502.[130] F. Haldane. “Nonlinear field theory of large spin Heisenberg antiferromagnets.Semiclassically quantized solitons of the one-dimensional easy Axis Neel state”.
Phys.Rev.Lett.
50 (1983), pp. 1153–1156.[131] D. Harland, T. A. Ivanova, O. Lechtenfeld, and A. D. Popov. “Yang-Mills flows onnearly Kahler manifolds and G(2)-instantons”.
Commun. Math. Phys.
300 (2010),pp. 185–204. arXiv: .128132] N. J. Hitchin. “Harmonic maps from a 2-torus to the 3-sphere”.
Journal of DifferentialGeometry
31 (3 1990), pp. 627–710.[133] N. J. Hitchin, A Karlhede, U Lindström, and M Roček. “Hyper-{K}ähler metrics andsupersymmetry”.
Comm. Math. Phys.
108 (4 1987), pp. 535–589.[134] B. Hoare, N. Levine, and A. A. Tseytlin. “Sigma models with local couplings: a newintegrability – RG flow connection”.
JHEP
11 (2020), p. 020. arXiv: .[135] C. Hofrichter, L. Riegger, F. Scazza, M. Höfer, D. R. Fernandes, I. Bloch, and S.Fölling. “Direct Probing of the Mott Crossover in the SU ( 𝑁 ) Fermi-Hubbard Model”.
Phys. Rev. X
Phys. Rev.
158 (2 1967), pp. 383–386.[137] T. J. Hollowood, J. L. Miramontes, and D. M. Schmidtt. “Integrable Deformationsof Strings on Symmetric Spaces”.
JHEP
11 (2014), p. 009. arXiv: .[138] C. Honerkamp and W. Hofstetter. “Ultracold Fermions and the SU ( 𝑁 ) HubbardModel”.
Phys. Rev. Lett.
92 (17 2004), p. 170403.[139] G. ’t Hooft. “Naturalness, chiral symmetry, and spontaneous chiral symmetry break-ing”.
NATO Sci. Ser. B
59 (1980). Ed. by G. ’t Hooft, C. Itzykson, A. Jaffe, H.Lehmann, P. K. Mitter, I. M. Singer, and R. Stora, pp. 135–157.[140] E. Ireson. “General Composite Non-Abelian Strings and Flag Manifold Sigma Mod-els”.
Phys. Rev. Res. .[141] E. Ireson, M. Shifman, and A. Yung. “Composite Non-Abelian Strings with Grass-mannian Models on the World Sheet”.
Phys. Rev. Research. .[142] H. Johannesson. “The integrable SU(N) Heisenberg model at finite temperature”.
Physics Letters A
Nuclear Physics B
270 (1986), pp. 235 –272.[144] J. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. I. Cirac. “Classical Simulation ofInfinite-Size Quantum Lattice Systems in Two Spatial Dimensions”.
Phys. Rev. Lett.
101 (25 2008), p. 250602.[145] A. Joseph. “Minimal realizations and spectrum generating algebras”.
Comm. Math.Phys.
36 (1974), pp. 325–338.[146] A. Joseph. “The minimal orbit in a simple Lie algebra and its associated maximalideal”.
Ann. Sci. École Norm. Sup. (4)
JHEP
05 (2014), p. 103. arXiv: .[148] V. G. Kac. “A Sketch of Lie Superalgebra Theory”.
Commun. Math. Phys.
53 (1977),pp. 31–64.[149] A. Kapustin and N. Seiberg. “Coupling a QFT to a TQFT and Duality”.
JHEP .129150] H. Katsura, T. Hirano, and V. E. Korepin. “Entanglement in an SU(n) valence-bond-solid state”.
Journal of Physics A: Mathematical and Theoretical
Journal of Physics:Condensed Matter
Bull. Amer. Math. Soc.(N.S.)
Fundamentals of quantum optics . Benjamin, New York,1968.[154] C. Klimčík. “Integrability of the bi-Yang-Baxter sigma-model”.
Lett. Math. Phys. .[155] C. Klimčík. “On integrability of the Yang-Baxter sigma-model”.
J. Math. Phys. .[156] A Klümper. “New Results for q -State Vertex Models and the Pure Biquadratic Spin-1Hamiltonian”.
Europhysics Letters (EPL)
New directions inapplied mathematics (Cleveland, Ohio, 1980) . Springer, New York-Berlin, 1982,pp. 81–84.[158] J. M. Kosterlitz and D. J. Thouless. “Ordering, metastability and phase transitionsin two-dimensional systems”.
Journal of Physics C: Solid State Physics
Lect. Notes Phys.
151 (1982). Ed. by J. Hietarintaand C. Montonen, pp. 61–119.[160] S. Lacroix. “Integrable models with twist function and affine Gaudin models”. PhDthesis. Lyon, Ecole Normale Superieure, 2018. arXiv: .[161] M. Lajkó, K. Wamer, F. Mila, and I. Affleck. “Generalization of the Haldane conjec-ture to SU(3) chains”.
Nucl. Phys. B
924 (2017). [Erratum: Nucl.Phys.B 949, 114781(2019)], pp. 508–577. arXiv: .[162] P. Lecheminant. “Massless renormalization group flow in SU(N) 𝑘 perturbed con-formal field theory”. Nucl. Phys. B
901 (2015), pp. 510–525. arXiv: .[163] S.-S. Lee. “Emergence of supersymmetry at a critical point of a lattice model”.
Phys.Rev. B
76 (7 2007), p. 075103.[164] H. Levine, S. B. Libby, and A. M. M. Pruisken. “Electron Delocalization by aMagnetic Field in Two Dimensions”.
Phys. Rev. Lett.
51 (20 1983), pp. 1915–1918.[165] E. H. Lieb and W. Liniger. “Exact analysis of an interacting Bose gas. 1. The Generalsolution and the ground state”.
Phys. Rev.
130 (1963), pp. 1605–1616.[166] E. H. Lieb, T. Schultz, and D. Mattis. “Two soluble models of an antiferromagneticchain”.
Annals Phys.
16 (1961), pp. 407–466.[167] F. Loebbert. “Lectures on Yangian Symmetry”.
J. Phys. A .[168] M. Lüscher. “Quantum Nonlocal Charges and Absence of Particle Production in theTwo-Dimensional Nonlinear Sigma Model”.
Nucl. Phys. B
135 (1978), pp. 1–19.130169] D. Lüst. “Compactification of Ten-dimensional Superstring Theories Over Ricci FlatCoset Spaces”.
Nucl. Phys. B
276 (1986), p. 220.[170] J. M. Maillet. “Hamiltonian Structures for Integrable Classical Theories From GradedKac-moody Algebras”.
Phys. Lett. B
167 (1986), pp. 401–405.[171] J. M. Maillet. “New Integrable Canonical Structures in Two-dimensional Models”.
Nucl. Phys. B
269 (1986), pp. 54–76.[172] D. Maison. “Are the Stationary, Axially Symmetric Einstein Equations CompletelyIntegrable?”
Phys. Rev. Lett.
41 (8 1978), pp. 521–522.[173] M. Mathur and D. Sen. “Coherent states for SU(3)”.
J. Math. Phys.
42 (2001),pp. 4181–4196. arXiv: quant-ph/0012099 .[174] M. R. Mehta. “Euclidean Continuation of the Dirac Fermion”.
Phys. Rev. Lett.
Phys. Rev. Lett.
17 (1966), pp. 1133–1136.[176] H.-J. Mikeska and A. K. Kolezhuk. “One-dimensional magnetism”. In:
QuantumMagnetism . Ed. by U. Schollwöck, J. Richter, D. J. J. Farnell, and R. F. Bishop.Berlin, Heidelberg: Springer Berlin Heidelberg, 2004, pp. 1–83.[177] T. Morimoto, H. Ueda, T. Momoi, and A. Furusaki. “ Z symmetry-protected topo-logical phases in the SU(3) AKLT model”. Phys. Rev. B
90 (23 2014), p. 235111.[178] A. Morozov, A. Perelomov, and M. A. Shifman. “EXACT GELL-MANN-LOWFUNCTION OF SUPERSYMMETRIC KAHLER SIGMA MODELS”.
Nucl. Phys.B
248 (1984), p. 279.[179] S. Murray and C. Sämann. “Quantization of Flag Manifolds and their Supersym-metric Extensions”.
Adv.Theor.Math.Phys.
12 (2008), pp. 641–710. arXiv: hep -th/0611328 .[180] H. Nakajima. “Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras”.
Duke Math. J. .[182] N. A. Nekrasov. “Lectures on curved beta-gamma systems, pure spinors, and anoma-lies” (2005). arXiv: hep-th/0511008 .[183] H. Nicolai. “Two-dimensional gravities and supergravities as integrable system”.
Lect. Notes Phys.
396 (1991). Ed. by H. Mitter and H. Gausterer, pp. 231–273.[184] M. P. Nightingale and H. W. J. Blöte. “Gap of the linear spin-1 Heisenberg antiferro-magnet: A Monte Carlo calculation”.
Phys. Rev. B
33 (1 1986), pp. 659–661.[185] M. den Nijs and K. Rommelse. “Preroughening transitions in crystal surfaces andvalence-bond phases in quantum spin chains”.
Phys. Rev. B
40 (7 1989), pp. 4709–4734.[186] M. Nitta. “Auxiliary field methods in supersymmetric nonlinear sigma models”.
Nucl.Phys.
B711 (2005), pp. 133–162. arXiv: hep-th/0312025 [hep-th] .[187] H. Nonne, M. Moliner, S. Capponi, P. Lecheminant, and K. Totsuka. “Symmetry-protected topological phases of alkaline-earth cold fermionic atoms in one dimen-sion”.
EPL (Europhysics Letters)
Theory of solitons .Contemporary Soviet Mathematics. The inverse scattering method, Translated fromthe Russian. Consultants Bureau [Plenum], New York, 1984, pp. xi+276.[189] S. P. Novikov. “The Hamiltonian formalism and a many-valued analogue of Morsetheory”.
Russian Mathematical Surveys
SciPost Phys. .[191] S. Östlund and S. Rommer. “Thermodynamic Limit of Density Matrix Renormaliza-tion”.
Phys. Rev. Lett.
75 (19 1995), pp. 3537–3540.[192] H. Ozawa, S. Taie, Y. Takasu, and Y. Takahashi. “Antiferromagnetic Spin Correlationof SU (N )
Fermi Gas in an Optical Superlattice”.
Phys. Rev. Lett.
121 (22 2018),p. 225303.[193] G. Pagano, M. Mancini, G. Cappellini, P. Lombardi, F. Schäfer, H. Hu, X.-J. Liu, J.Catani, C. Sias, and M. Inguscio. “A one-dimensional liquid of fermions with tunablespin”.
Nature Physics .[194] N. Papanicolaou. “Pseudospin approach for planar ferromagnets”.
Nuclear Physics B
Nuclear Physics B
Journal of Physics C: Solid State Physics ( 𝑛 ) and SU ( 𝑛 ) ”. SovietJ. Nuclear Phys.
Generalized coherent states and their applications . Springer, 1986.[199] K. Pohlmeyer. “Integrable Hamiltonian systems and interactions through quadraticconstraints”.
Communications in Mathematical Physics
46 (3 Oct. 1976), pp. 207–221.[200] J. Polchinski.
String theory. Vol. 1: An introduction to the bosonic string . CambridgeMonographs on Mathematical Physics. Cambridge University Press, Dec. 2007.[201] A. M. Polyakov. “Hidden Symmetry of the Two-Dimensional Chiral Fields”.
Phys.Lett. B
72 (1977), pp. 224–226.[202] A. M. Polyakov and P. Wiegmann. “Theory of Nonabelian Goldstone Bosons”.
Phys.Lett. B
131 (1983). Ed. by M. Stone, pp. 121–126.[203] N. Read and S. Sachdev. “Spin-Peierls, valence-bond solid, and Néel ground statesof low-dimensional quantum antiferromagnets”.
Phys. Rev. B
42 (7 1990), pp. 4568–4589.[204] N. Read and H. Saleur. “Exact spectra of conformal supersymmetric nonlinear sigmamodels in two-dimensions”.
Nucl. Phys. B
613 (2001), p. 409. arXiv: hep - th /0106124 .[205] J.-P. Renard, L.-P. Regnault, and M. Verdaguer. “Haldane Quantum Spin Chains”.In:
Magnetism: Molecules to Materials . and references therein. Wiley-VCH VerlagGmbH & Co. KGaA, 2003, pp. 49–93.[206] N. Reshetikhin and L. Faddeev. “Hamiltonian Structures For Integrable Models OfField Theory”.
Theor. Math. Phys.
56 (1983), pp. 847–862.132207] A. Roy and T. Quella. “Chiral Haldane phases of 𝑆𝑈 ( 𝑁 ) quantum spin chains”. Phys.Rev. B .[208] S. Salamon. “Harmonic and holomorphic maps”.
Lect. Notes Math.
Nature Physics
10 (2014), pp. 779–784. arXiv: .[210] U. Schollwöck, O. Golinelli, and T. Jolicœur. “ 𝑆 = Phys. Rev. B
54 (6 1996), pp. 4038–4051.[211] A. S. Schwarz and A. A. Tseytlin. “Dilaton shift under duality and torsion of ellipticcomplex”.
Nucl. Phys. B
399 (1993), pp. 691–708. arXiv: hep-th/9210015 .[212] J. S. Schwinger. “Gauge Invariance and Mass. 2.”
Phys. Rev.
128 (1962), pp. 2425–2429.[213] M. A. Semenov-Tyan-Shanski˘ı. “What a classical 𝑟 -matrix is”. Funktsional. Anal. iPrilozhen.
Nucl. Phys. B
880 (2014), pp. 225–246. arXiv: .[215] S. Sternberg. “Minimal coupling and the symplectic mechanics of a classical particlein the presence of a Yang-Mills field”.
Proc. Nat. Acad. Sci. U.S.A.
74 (12 1977),pp. 5253–5254.[216] T. Sulejmanpasic. “Global Symmetries, Volume Independence, and Continuity inQuantum Field Theories”.
Phys. Rev. Lett. .[217] B. Sutherland. “Model for a multicomponent quantum system”.
Phys. Rev. B
12 (91975), pp. 3795–3805.[218] S. Taie, R. Yamazaki, S. Sugawa, and Y. Takahashi. “An SU(6) Mott insulator of anatomic Fermi gas realized by large-spin Pomeranchuk cooling”.
Nature Physics .[219] L. Takhtajan. “The picture of low-lying excitations in the isotropic Heisenberg chainof arbitrary spins”.
Physics Letters A
JHEP
12 (2017), p. 056. arXiv: .[221] Y. Tanizaki and T. Sulejmanpasic. “Anomaly and global inconsistency matching: 𝜃 angles, 𝑆𝑈 ( )/ 𝑈 ( ) nonlinear sigma model, 𝑆𝑈 ( ) chains, and generalizations”. Phys. Rev. B
98 (11 2018), p. 115126.[222] J. Teschner. “Liouville theory revisited”.
Class. Quant. Grav.
18 (2001), R153–R222.arXiv: hep-th/0104158 .[223] S. Todo and K. Kato. “Cluster Algorithms for General- 𝑆 Quantum Spin Systems”.
Phys. Rev. Lett.
87 (4 2001), p. 047203.[224] S. Todo, H. Matsuo, and H. Shitara. “Parallel loop cluster quantum Monte Carlo sim-ulation of quantum magnets based on global union-find graph algorithm”.
ComputerPhysics Communications
239 (2019), pp. 84 –93.133225] A. A. Tseytlin. “Effective action of gauged WZW model and exact string solutions”.
Nucl. Phys. B
399 (1993), pp. 601–622. arXiv: hep-th/9301015 .[226] A. Tsvelick and P. Wiegmann. “Exact results in the theory of magnetic alloys”.
Advances in Physics https : / / doi . org / 10 .1080/00018738300101581 .[227] A. Tsvelik.
Quantum Field Theory in Condensed Matter Physics . Cambridge Univer-sity Press, 2007, 280p.[228] K. Uhlenbeck. “Harmonic maps into Lie groups: classical solutions of the chiralmodel”.
Journal of Differential Geometry
30 (1 1989), pp. 1–50.[229] M. Ünsal. “Strongly coupled QFT dynamics via TQFT coupling”. arXiv preprintarXiv:2007.03880 (2020).[230] G. Valent, C. Klimčík, and R. Squellari. “One loop renormalizability of the Poisson-Lie sigma models”.
Phys. Lett. B
678 (2009), pp. 143–148. arXiv: .[231] E. Vicari and H. Panagopoulos. “Theta dependence of SU(N) gauge theories inthe presence of a topological term”.
Phys. Rept.
470 (2009), pp. 93–150. arXiv: .[232] G. Vidal. “Class of Quantum Many-Body States That Can Be Efficiently Simulated”.
Phys. Rev. Lett.
101 (11 2008), p. 110501.[233] G. Vidal. “Efficient Classical Simulation of Slightly Entangled Quantum Computa-tions”.
Phys. Rev. Lett.
91 (14 2003), p. 147902.[234] K. Wamer and I. Affleck. “Flag manifold sigma models from SU( 𝑛 ) chains”. Nucl.Phys. B
959 (2020), p. 115156. arXiv: .[235] K. Wamer and I. Affleck. “Mass generation by fractional instantons in SU( 𝑛 ) chains”. Phys. Rev. B .[236] K. Wamer, F. H. Kim, M. Lajkó, F. Mila, and I. Affleck. “Self-conjugate represen-tation SU(3) chains”.
Phys. Rev. B .[237] K. Wamer, M. Lajkó, F. Mila, and I. Affleck. “Generalization of the Haldane con-jecture to SU( 𝑛 ) chains”. Nucl. Phys. B
952 (2020), p. 114932. arXiv: .[238] H.-C. Wang. “Closed manifolds with homogeneous complex structure”.
Amer. J.Math.
76 (1954), pp. 1–32.[239] J. Wess and B. Zumino. “Consequences of anomalous Ward identities”.
Phys. Lett. B
37 (1971), pp. 95–97.[240] S. R. White. “Density-matrix algorithms for quantum renormalization groups”.
Phys.Rev. B
48 (14 1993), pp. 10345–10356.[241] S. R. White. “Density matrix formulation for quantum renormalization groups”.
Phys.Rev. Lett.
69 (19 1992), pp. 2863–2866.[242] S. R. White and D. A. Huse. “Numerical renormalization-group study of low-lyingeigenstates of the antiferromagnetic S=1 Heisenberg chain”.
Phys. Rev. B
48 (6 1993),pp. 3844–3852.[243] E Witten. “Large N chiral dynamics”.
Annals of Physics
Nucl. Phys. B
145 (1978), pp. 110–118.[245] E. Witten. “Global Aspects of Current Algebra”.
Nucl. Phys. B
223 (1983), pp. 422–432.[246] E. Witten. “Instantons, the Quark Model, and the 1/n Expansion”.
Nucl. Phys. B .[248] E. Witten. “Nonabelian Bosonization in Two-Dimensions”.
Commun. Math. Phys.
Commun.Math. Phys.
144 (1992), pp. 189–212.[250] E. Witten. “On string theory and black holes”.
Phys. Rev. D
44 (1991), pp. 314–324.[251] E. Witten. “Perturbative gauge theory as a string theory in twistor space”.
Commun.Math. Phys.
252 (2004), pp. 189–258. arXiv: hep-th/0312171 .[252] E. Witten. “Quantum Field Theory and the Jones Polynomial”.
Commun. Math. Phys.
121 (1989). Ed. by A. N. Mitra, pp. 351–399.[253] E. Witten. “Supersymmetric index in four-dimensional gauge theories”.
Adv. Theor.Math. Phys. hep-th/0006010 .[254] E. Witten. “Theta Dependence in the Large N Limit of Four-Dimensional GaugeTheories”.
Physical Review Letters
Commun.Math.Phys.
118 (1988), p. 411.[256] E. Witten. “Two-dimensional models with (0,2) supersymmetry: Perturbative as-pects”.
Adv. Theor. Math. Phys. hep-th/0504078 .[257] S. K. Wong. “Field and particle equations for the classical Yang-Mills field andparticles with isotopic spin”.
Nuovo Cim.
A65 (1970), pp. 689–694.[258] C. Wu, J.-p. Hu, and S.-c. Zhang. “Exact SO(5) Symmetry in the Spin-3 / Phys. Rev. Lett.
91 (18 2003), p. 186402.[259] C.-N. Yang and C. Yang. “Thermodynamics of one-dimensional system of bosonswith repulsive delta function interaction”.
J. Math. Phys.
10 (1969), pp. 1115–1122.[260] Y. Yao, C.-T. Hsieh, and M. Oshikawa. “Anomaly Matching and Symmetry-ProtectedCritical Phases in SU(N) Spin Systems in 1+1 Dimensions”.
Physical Review Letters
Phys.Lett. B
632 (2006), pp. 559–565. arXiv: hep-th/0503008 .[262] V. E. Zakharov and A. V. Mikhailov. “Relativistically invariant two-dimensionalmodels of field theory which are integrable by means of the inverse scattering problemmethod”.
Soviet Physics JETP
47 (1979), pp. 1017–1027.[263] A. Zamolodchikov. “Thermodynamic Bethe Ansatz in Relativistic Models. ScalingThree State Potts and Lee-yang Models”.
Nucl. Phys. B
342 (1990), pp. 695–720.[264] A. B. Zamolodchikov and A. B. Zamolodchikov. “Factorized s Matrices in Two-Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models”.
Annals Phys.
120 (1979), pp. 253–291.135265] A. B. Zamolodchikov and A. B. Zamolodchikov. “Massless factorized scattering andsigma models with topological terms”.
Nucl. Phys. B
379 (1992), pp. 602–623.[266] K. Zarembo. “Integrability in Sigma-Models”.
Les Houches Lect. Notes
106 (2019).Ed. by P. Dorey, G. Korchemsky, N. Nekrasov, V. Schomerus, D. Serban, and L.Cugliandolo. arXiv: .[267] X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus, M. S. Safronova, P. Zoller, A. M.Rey, and J. Ye. “Spectroscopic observation of SU(N)-symmetric interactions in Srorbital magnetism”.
Science https : / /science.sciencemag.org/content/345/6203/1467.full.pdf .[268] J. Zinn-Justin.
Path Integrals in Quantum Mechanics . Oxford University Press, 2004,336p.[269] J. Zinn-Justin.
Quantum field theory and critical phenomena . Clarendon Press, 2002,1054p.[270] B. Zumino. “Euclidean Supersymmetry and the Many-Instanton Problem”.
Phys.Lett. B
69 (1977), p. 369.
A Kähler potential from the quiver quotient formulation
We showed in section 1.3 that there at least two ways to derive invariant Kähler metrics onflag 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 { 𝑈 𝐴 } shown in Fig. 2 is endowed with the standard symplecticform Ω 𝐴 = 𝑖 Tr ( 𝑑𝑈 𝐴 ∧ 𝑑𝑈 † 𝐴 ) and, accordingly, a metric ( 𝑑𝑠 ) 𝐴 = Tr ( 𝑑𝑈 𝐴 𝑑𝑈 † 𝐴 ) . The fullsymplectic form is then Ω = 𝑚 − ∑︁ 𝐴 = 𝑖 Tr ( 𝑑𝑈 𝐴 ∧ 𝑑𝑈 † 𝐴 ) . (A.1)At each circular node 𝑗 one has the action of a gauge group U ( 𝐿 𝐴 ) : = U ( 𝑑 𝐴 ) ⊂ GL ( 𝑑 𝐴 , C ) that preserves the symplectic form. Accordingly one can define the moment maps for thisaction: 𝜇 𝐴 = 𝑈 † 𝐴 𝑈 𝐴 − 𝑈 𝐴 − 𝑈 † 𝐴 − . The main statement is that the flag manifold may be definedas a quotient: F = { 𝜇 𝐴 = 𝜁 𝐴 𝑑 𝐴 , 𝐴 = , . . . , 𝑚 − } (cid:14) U ( 𝐿 ) × · · · × U ( 𝐿 𝑚 − ) , (A.2)where 𝜁 𝐴 > 𝑚 − 𝑈 † 𝐴 𝑈 𝐴 is non-degenerate, which implies rk ( 𝑈 𝐴 ) = 𝑑 𝐴 .The linear spaces 𝐿 𝐴 of the flag may be obtained as Im ( 𝑈 𝑚 − · · · 𝑈 𝐴 + 𝑈 𝐴 ) ⊂ C 𝑛 : thematrix 𝑈 𝑚 − · · · 𝑈 𝐴 + 𝑈 𝐴 has rank 𝑑 𝐴 , so that it defines 𝑑 𝐴 vectors in C 𝑛 , and the quotientw.r.t. U ( 𝐿 𝐴 ) amounts to considering the linear space spanned by these vectors (compare withthe example (1.18)-(1.19), depicted in Fig. 3). Clearly, the 𝐿 𝐴 so defined are nested in eachother: 𝐿 𝐴 − ⊂ 𝐿 𝐴 . 136he apparatus of symplectic quotient provides a symplectic form on F by restricting theoriginal symplectic form Ω to the level set of the moment maps Ω = Ω (cid:12)(cid:12) 𝜇 = 𝜁 . Since thewhole setup is Kähler, so that there is a complex structure and metric involved, the reductionalso provides a Kähler metric on the flag manifold, which should coincide with the metricgiven by the Kähler potential (1.14).Let us see how this happens. The strategy, known from the general theory of Kählerquotients (cf. [133]), is as follows: one considers generic matrices 𝑈 𝐴 , 𝑈 † 𝐴 , not necessarilysatisfying the moment map constraints, and one needs to find the complexified symmetrytransformation 𝑔 × · · · × 𝑔 𝑚 − ⊂ GL ( 𝐿 , C ) × · · · × GL ( 𝐿 𝑚 − , C ) , such that the transformedvariables would satisfy the constraints. Introducing 𝑀 𝐴 = 𝑔 † 𝐴 𝑔 𝐴 , it is easy to see that we mayrewrite this requirement as 𝑈 † 𝐴 𝑀 𝐴 + 𝑈 𝐴 − 𝑀 𝐴 𝑈 𝐴 − 𝑀 − 𝐴 − 𝑈 † 𝐴 − 𝑀 𝐴 = 𝜁 𝐴 𝑀 𝐴 , 𝑀 𝑚 = . (A.3)Given a solution 𝑀 , . . . , 𝑀 𝑚 − , we obtain the Kähler potential of the quotient manifold asfollows: K = 𝑚 − ∑︁ 𝐴 = 𝜁 𝐴 Tr ( log 𝑀 𝐴 ) = 𝑚 − ∑︁ 𝐴 = 𝜁 𝐴 log ( det 𝑀 𝐴 ) . (A.4)We proceed to compute the determinants of the matrices 𝑀 𝐴 . Denoting 𝑦 𝐴 : = 𝑀 − 𝐴 − 𝑈 † 𝐴 − 𝑀 𝐴 we may rewrite (A.3) in two equivalent forms: 𝑈 † 𝐴 𝑀 𝐴 + 𝑈 𝐴 = 𝑀 𝐴 ( 𝜁 𝐴 + 𝑈 𝐴 − 𝑦 𝐴 ) . (A.5) 𝑦 𝐴 + 𝑈 𝐴 − 𝑈 𝐴 − 𝑦 𝐴 = 𝜁 𝐴 . (A.6)Multiplying (A.5) by 𝑈 † 𝐴 − from the left and by 𝑈 𝐴 − from the right and using (A.6), we find 𝑈 † 𝐴 − 𝑈 † 𝐴 𝑀 𝐴 + 𝑈 𝐴 𝑈 𝐴 − = 𝑀 𝐴 − ( 𝜁 𝐴 − + 𝑈 𝐴 − 𝑦 𝐴 − ) ( 𝜁 𝐴 + 𝑦 𝐴 𝑈 𝐴 − ) = (A.7) = 𝑀 𝐴 − ( 𝜁 𝐴 − + 𝑈 𝐴 − 𝑦 𝐴 − ) ( 𝜁 𝐴 + 𝜁 𝐴 − + 𝑈 𝐴 − 𝑦 𝐴 − ) . Next we introduce the matrix 𝑊 𝐵 : = 𝑈 𝑚 − · · · 𝑈 𝐵 . As discussed above, Im ( 𝑊 𝐵 ) = 𝐿 𝐵 .Recalling that 𝑀 𝑚 = , we may continue (A.7) by induction to demonstrate that 𝑊 † 𝐵 𝑊 𝐵 = 𝑀 𝐵 · 𝑃 𝐵 , (A.8)where 𝑃 𝐵 is a product of matrices of the type 𝑎 + 𝑈 𝐵 − 𝑦 𝐵 ( 𝑎 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 ( 𝑎 + 𝑈 𝐵 − 𝑦 𝐵 ) 𝑈 𝐵 − = ( 𝑎 + 𝜁 𝐵 − ) 𝑈 𝐵 − + 𝑈 𝐵 − 𝑈 𝐵 − 𝑦 𝐵 − (to be continued by induction), so that the matrix 𝑎 + 𝑈 𝐵 − 𝑦 𝐵 = 𝐷 𝐵 + 𝑁 𝐵 , where 𝐷 𝐵 isdiagonal with eigenvalues 𝑎, 𝑎 + 𝜁 𝐵 − , . . . and 𝑁 𝐵 is strictly triangular, in the sense that itmaps Im ( 𝑈 𝐵 − 𝑈 𝐵 − · · · 𝑈 𝐵 − 𝐶 ) to Im ( 𝑈 𝐵 − 𝑈 𝐵 − · · · 𝑈 𝐵 − 𝐵 𝑈 𝐵 − 𝐶 − ) for all 𝐶 . It follows thatdet 𝑃 𝐵 = const . ≠
0, so that (A.8) implies det ( 𝑀 𝐵 ) ∼ det ( 𝑊 † 𝐵 𝑊 𝐵 ) , up to a constant coefficient.Substituting into (A.4) and identifying 𝜁 𝐵 = 𝛾 𝐵 , we find 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 flag manifold 𝐺𝐻 = 𝑆𝑈 ( 𝑛 ) 𝑆 ( 𝑈 ( 𝑛 )×···× 𝑈 ( 𝑛 𝑚 )) is Ω = ∑︁ 𝐴<𝐵 𝑎 𝐴𝐵 Tr ( 𝑗 𝐴𝐵 ∧ 𝑗 𝐵𝐴 ) . (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 theflatness of the Maurer-Cartan current, 𝑑 𝑗 − 𝑗 ∧ 𝑗 = . It follows that D 𝑗 𝐴𝐵 = (cid:205) 𝐶 ≠ ( 𝐴,𝐵 ) 𝑗 𝐴𝐶 ∧ 𝑗 𝐶𝐵 , where D is the 𝐻 -covariant derivative, defined as follows: D 𝑗 𝐴𝐵 : = 𝑑 𝑗 𝐴𝐵 − { 𝑗 𝔥 , 𝑗 𝐴𝐵 } . Fromthe condition that Ω is closed it follows that 𝑎 𝐴𝐵 + 𝑎 𝐵𝐶 + 𝑎 𝐶 𝐴 = ( 𝐴, 𝐵, 𝐶 ) . (B.2)The general solution to this equation is 𝑎 𝐴𝐵 = 𝑝 𝐴 − 𝑝 𝐵 . (B.3)Therefore we have a family of homogeneous symplectic forms with 𝑚 − Ω = Tr ( 𝑝 𝑗 ∧ 𝑗 ) , where 𝑝 = Diag ( 𝑝 𝑛 , . . . , 𝑝 𝑚 𝑛 𝑚 ) . (B.4)The element 𝑝 may be normalized to be traceless: Tr ( 𝑝 ) =
0. The stabilizer 𝐻 may now bethought of as the stabilizer of the matrix 𝑝 ∈ 𝔲 𝑛 , and the flag manifold itself – as an adjointorbit: F 𝑑 ,...,𝑑 𝑚 = { 𝑔 𝑝 𝑔 − , 𝑔 ∈ SU ( 𝑛 )} . (B.5) C Coherent states as polynomials
In section 2.3.4 we described the coherent states of 𝑆𝑈 ( 𝑛 ) 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 – this is the celebrated Bargmann representation [38]. The map issimple: ( 𝑎 † ) 𝑞 · · · ( 𝑎 † 𝑛 ) 𝑞 𝑛 | (cid:105) ↦→ 𝑧 𝑞 · · · 𝑧 𝑞 𝑛 𝑛 (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 ˆ 𝑓 and ˆ 𝑔 are twopolynomials): ( ˆ 𝑓 , ˆ 𝑔 ) Bargmann = ∫ ˆ 𝑓 ( 𝑧 ) ˆ 𝑔 ( 𝑧 ) 𝑒 − 𝑛 (cid:205) 𝑗 = | 𝑧 𝑗 | 𝑛 (cid:214) 𝛼 = ( 𝑖 𝑑𝑧 𝛼 ∧ 𝑑𝑧 𝛼 ) (C.2)138et us also discuss the relation to the definition of states as in homogeneous polynomialsused in the classical work [43], where coherent states were used to describe the quantizationof a sphere 𝑆 ∼ CP — the simplest homogeneous Kähler (symplectic) manifold . Tostart with, we observe that, since in our applications to representation theory the numberof oscillators is fixed, we may introduce a new Hilbert space, isomorphic to the one ofhomogeneous polynomials (C.1). Suppose we have a rank- 𝑝 symmetric representation, sothat we are dealing with homogeneous polynomials of degree 𝑝 . It is an elementary fact thatthe following two spaces are isomorphic:Homogeneous polynomials of degree 𝑝 in 𝑛 variables ↔ Polynomials of degree ≤ 𝑝 in 𝑛 − 𝑧 =
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 find the correct integration measure on the space ofpolynomials of degree ≤ 𝑝 , we first start with the homogeneous polynomials ˆ 𝑓 and ˆ 𝑔 ofdegree 𝑝 and make the change of variables { 𝑧 → 𝜆, 𝑧 → 𝜆𝑧 , · · · , 𝑧 𝑛 → 𝜆𝑧 𝑛 } . In thiscase, clearly, ˆ 𝑓 = 𝜆 𝑝 𝑓 ( 𝑧 ) and analogously for 𝑔 , where 𝑓 ( 𝑧 ) , 𝑔 ( 𝑧 ) are now inhomogeneouspolynomials of 𝑛 − ( ˆ 𝑓 , ˆ 𝑔 ) Bargmann = ∫ 𝑒 −| 𝜆 | (cid:18) + 𝑛 − (cid:205) 𝛼 = | 𝑧 𝛼 | (cid:19) (cid:16) 𝑖 | 𝜆 | ( 𝑝 + 𝑛 − ) 𝑑𝜆 ∧ 𝑑𝜆 (cid:17) 𝑓 ( 𝑧 ) 𝑔 ( 𝑧 ) 𝑛 − (cid:214) 𝛼 = ( 𝑖 𝑑𝑧 𝛼 ∧ 𝑑𝑧 𝛼 ) == integrating over 𝜆, 𝜆 ∼ ∫ 𝑓 ( 𝑧 ) 𝑔 ( 𝑧 ) (cid:18) + 𝑛 − (cid:205) 𝛼 = | 𝑧 𝛼 | (cid:19) 𝑝 + 𝑛 𝑛 − (cid:214) 𝛼 = ( 𝑖 𝑑𝑧 𝛼 ∧ 𝑑𝑧 𝛼 ) In other words, if ˆ 𝑓 , ˆ 𝑔 are homogeneous polynomials of degree 𝑝 and 𝑓 , 𝑔 are their inho-mogeneous counterparts (obtained by setting 𝑧 = ( ˆ 𝑓 , ˆ 𝑔 ) Bargmann = ( 𝑓 , 𝑔 ) providedwe define the scalar product in the space of inhomogeneous polynomials of degree ≤ 𝑝 as Coherent states, written in inhomogeneous coordinates, are also discussed in [198]. : ( 𝑓 , 𝑔 ) = ∫ 𝑓 ( 𝑧 ) 𝑔 ( 𝑧 ) (cid:18) + 𝑛 − (cid:205) 𝛼 = | 𝑧 𝛼 | (cid:19) 𝑝 𝑑𝜇 ( 𝑧, 𝑧 ) , where ( 𝑑𝜇 ) CP 𝑛 − ∼ (cid:32) + 𝑛 − ∑︁ 𝛼 = | 𝑧 𝛼 | (cid:33) − 𝑛 𝑛 − (cid:214) 𝛼 = ( 𝑖 𝑑𝑧 𝛼 ∧ 𝑑𝑧 𝛼 ) (C.3)The measure 𝑑𝜇 ∼ Ω 𝑛 − is in fact the volume form on CP 𝑛 − , 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 | 𝑣 (cid:105) , when viewed as an inhomogeneous polynomial, will be denoted 𝜙 𝑣 ( 𝑧 ) (this notation is borrowed from [43]). For example, if we take the state (2.43) with 𝑛 =
2, which in Fock space language is ( 𝑣 𝑎 † + 𝑣 𝑎 † ) | (cid:105) , the corresponding polynomialwould be 𝜙 𝑣 ( 𝑧 ) ∼ ( + 𝑣𝑧 ) , where we have set 𝑣 : = 𝑣 𝑣 and dropped an overall factor. D Integrability of the complex structure
Here we wish to prove two claims made in section 13.2. The first one is:If the restriction to 𝔪 of the adjoint-invariant metric (cid:104)• , •(cid:105) on 𝔲 ( 𝑛 ) is Hermitianw.r.t. an almost complex structure J , integrability of J (viewed as an almostcomplex structure on the flag manifold) is equivalent to [ 𝔪 + , 𝔪 + ] ⊂ 𝔪 + , [ 𝔪 − , 𝔪 − ] ⊂ 𝔪 − . (D.1)In general the integrability of an almost complex structure means that [ 𝔪 + , 𝔪 + ] ⊂ 𝔪 + ⊕ 𝔥 .To see this, note that an almost complex structure J is defined by the conditions J ◦ 𝐽 ± = ± 𝑖 𝐽 ± , where 𝐽 ± are the components of a Maurer-Cartan current: 𝐽 = − 𝑔 − 𝑑𝑔 = 𝐽 𝔥 + 𝐽 + + 𝐽 − , 𝐽 ± ∈ 𝔪 ± . (D.2)Since 𝑑𝐽 − 𝐽 ∧ 𝐽 =
0, we get 𝑑𝐽 − = (cid:2) − 𝐽 ∧ 𝐽 + ( terms with 𝐽 − ) − 𝐽 + ∧ 𝐽 + (cid:3) 𝔪 − . From a mathematical standpoint (which uses the Borel-Weil-Bott theorem briefly mentioned in sec-tion 2.3.1) 𝑓 and 𝑔 are sections of the line bundle O ( 𝑝 ) over CP 𝑛 − . The integrand may be understoodas a scalar product in the fiber at a given point 𝑧 on the base. In that case ( + (cid:205) 𝑛 − 𝛼 = | 𝑧 𝛼 | ) − 𝑝 plays the role ofa metric in the fiber, and ( 𝑓 , 𝑔 ) is obtained by integrating the fiber scalar product over all of CP 𝑛 − with thenatural measure 𝑑𝜇 . J one should have [ 𝐽 + ∧ 𝐽 + ] 𝔪 − =
0, i.e. [ 𝔪 + , 𝔪 + ] ⊂ 𝔪 + ⊕ 𝔥 .We see that the conditions (D.1) therefore define an integrable complex structure. Con-versely suppose we have an integrable complex structure on 𝐺 / 𝐻 , and 𝔪 ± are its respectiveholomorphic/anti-holomorphic subspaces. Then [ 𝑎, 𝑏 ] = 𝑐 + 𝛾 , where 𝑎, 𝑏, 𝑐 ∈ 𝔪 + and 𝛾 ∈ 𝔥 . Since (cid:104) 𝔪 + , 𝔥 (cid:105) =
0, computing the scalar product with a generic element 𝛾 (cid:48) ∈ 𝔥 ,we obtain (cid:104) 𝛾 (cid:48) , [ 𝑎, 𝑏 ](cid:105) = (cid:104) 𝛾 (cid:48) , 𝛾 (cid:105) . Using the identity (cid:104)[ 𝑎, 𝛾 (cid:48) ] , 𝑏 (cid:105) + (cid:104) 𝛾 (cid:48) , [ 𝑎, 𝑏 ](cid:105) =
0, we get (cid:104) 𝛾 (cid:48) , 𝛾 (cid:105) = − (cid:104)[ 𝑎, 𝛾 (cid:48) ] , 𝑏 (cid:105) = (cid:104) 𝑎 (cid:48) , 𝑏 (cid:105) , and 𝑎 (cid:48) = [ 𝛾 (cid:48) , 𝑎 ] ∈ 𝔪 + . The subspace 𝔪 + is isotropic, if themetric (cid:104)• , •(cid:105) is Hermitian, therefore (cid:104) 𝛾 (cid:48) , 𝛾 (cid:105) = 𝛾 (cid:48) ∈ 𝔥 , which implies 𝛾 = (cid:104)• , •(cid:105) . The result [ 𝔪 + , 𝔪 + ] ⊂ 𝔪 + follows.The second statement used in section 13.2 is:There are exactly 𝑛 ! acyclic tournament diagrams.The statement implies that there is only one combinatorial type of diagrams, and all acyclictournament diagrams (in such diagrams, by definition, all pairs of nodes are connected) maybe obtained from any one of them by the action of the permutation group 𝑆 𝑛 . Let us describethis combinatorial type. Every acyclic diagram has a ‘source’-vertex, in which all the linesare outgoing, and a ‘sink’-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 ‘source’-vertex. The diagram formed by the remaining 𝑛 − 𝑖 outgoing lines for all 𝑖 = , . . . , 𝑛 −
1. Thisstatement completely describes the combinatorial structure of the diagram. Equivalently,there is a total ordering on the set of vertices. Different diagrams differ just by a relabelingof the vertices. (A) (B)
Figure 22: (A) The ‘sink’ vertex, (B) The ‘source’ vertex .141
Proving the Z 𝑚 -‘symmetry’ of integrable models In this Appendix we prove the Z 𝑚 symmetry property of the integrable flag manifold models,introduced in section 13.4. The statement is that the e.o.m. of two models, in which thecomplex structures differ by a cyclic permutation of the subspaces C 𝑛 , · · · , C 𝑛 𝑚 , are thesame. In this case the two actions differ only by a topological term.Let us call J the standard complex structure, whose holomorphic subspace 𝔪 + is givenby upper-block-triangular matrices. Then J = 𝜎 ( J ) and J = 𝜎 ( J ) for somepermutations 𝜎 , 𝜎 ∈ 𝑆 𝑚 . We recall the notation 𝐽 𝐴𝐵 from (1.31). The correspondingKähler forms are 𝜔 = 𝑖 ∑︁ 𝐴<𝐵 Tr ( 𝐽 𝜎 ( 𝐴 ) 𝜎 ( 𝐵 ) ∧ 𝐽 𝜎 ( 𝐵 ) 𝜎 ( 𝐴 ) ) (E.1) 𝜔 = 𝑖 ∑︁ 𝐴<𝐵 Tr ( 𝐽 𝜎 ( 𝐴 ) 𝜎 ( 𝐵 ) ∧ 𝐽 𝜎 ( 𝐵 ) 𝜎 ( 𝐴 ) ) (E.2)Upon introducing the notation 𝜎 ( 𝐽 𝐴𝐵 ) : = 𝐽 𝜎 ( 𝐴 ) 𝜎 ( 𝐵 ) , we may write the difference of the twoforms as 𝜔 − 𝜔 = 𝑖 𝜎 (cid:32) ∑︁ 𝐴<𝐵 Tr ( 𝐽 𝐴𝐵 ∧ 𝐽 𝐵𝐴 ) − ∑︁ 𝐴<𝐵 Tr ( 𝐽 𝜏 − ( 𝐴 ) 𝜏 − ( 𝐵 ) ∧ 𝐽 𝜏 − ( 𝐵 ) 𝜏 − ( 𝐴 ) ) (cid:33) , (E.3)where 𝜏 − = 𝜎 − 𝜎 . This reduces the problem to that of J = J and J = 𝜏 − ( J ) .Note that the exterior derivative commutes with the permutation 𝜎 , due to the followingsimple fact following from the Maurer-Cartan equation: 𝜎 ( 𝑑𝐽 𝐴𝐵 ) = 𝜎 ( (cid:205) 𝐶 𝐽 𝐴𝐶 ∧ 𝐽 𝐶𝐵 ) = (cid:205) 𝐶 𝜎 ( 𝐽 𝐴𝐶 ) ∧ 𝜎 ( 𝐽 𝐶𝐵 ) = 𝑑𝜎 ( 𝐽 𝐴𝐵 ) , where to arrive at the last equality one has to make achange of the dummy summation index 𝐶 → 𝜎 ( 𝐶 ) .We wish to show that 𝑑 ( 𝜔 − 𝜔 ) = 𝜏 is a cyclic permutation. To this endwe rewrite the above difference as follows: 𝜔 − 𝜔 = 𝑖 𝜎 (cid:32)∑︁ 𝐴,𝐵 𝛼 𝐴𝐵 Tr ( 𝐽 𝐴𝐵 ∧ 𝐽 𝐵𝐴 ) (cid:33) , (E.4)where 𝛼 𝐴𝐵 = (cid:0) sgn ( 𝐵 − 𝐴 ) − sgn ( 𝜏 ( 𝐵 ) − 𝜏 ( 𝐴 )) (cid:1) ∈ {− , , } . (E.5)(We have made a change of dummy variables 𝐴 → 𝜏 ( 𝐴 ) and 𝐵 → 𝜏 ( 𝐵 ) in the second sumin (E.3)). Closedness of this form requires that (see (B.2)-(B.3)) 𝛼 𝐴𝐵 = (cid:0) sgn ( 𝐵 − 𝐴 ) − sgn ( 𝜏 ( 𝐵 ) − 𝜏 ( 𝐴 )) (cid:1) = 𝑝 𝐴 − 𝑝 𝐵 . (E.6)Let us consider the case 𝐵 > 𝐴 . Then 𝑝 𝐴 = 𝑝 𝐵 if 𝜏 ( 𝐵 ) > 𝜏 ( 𝐴 ) and 𝑝 𝐴 = 𝑝 𝐵 + 𝜏 ( 𝐵 ) < 𝜏 ( 𝐴 ) . This means that { 𝑝 𝐴 } 𝐴 = ...𝑚 form a non-increasing sequence, and moreover142he difference between any two elements is either zero or 1. This is only possible if the set hasthe form ( 𝑝, . . . , 𝑝 𝐾 , 𝑝 − , . . . 𝑝 − 𝑚 − 𝐾 ) . Accordingly the original sequence of 𝑚 consecutivenumbers can be split into two consecutive sets:1 . . . 𝑚 = ( 𝐼 , 𝐼 ) . (E.7)Since 𝜏 ( 𝐵 ) < 𝜏 ( 𝐴 ) for ( 𝐴 ≤ 𝐾 , 𝐵 > 𝐾 ), the permutation acts as follows: 𝜏 ( 𝐼 , 𝐼 ) = ( 𝜏 ( 𝐼 ) , 𝜏 ( 𝐼 )) . (E.8)Moreover, since 𝜏 ( 𝐴 ) < 𝜏 ( 𝐵 ) for 𝐴 < 𝐵 ≤ 𝐾 and the image 𝜏 ( 𝐼 ) is ( 𝑚 − 𝐾 + , . . . 𝑚 ) , amoment’s thought shows that 𝜏 ( 𝐴 ) = 𝑚 − 𝐾 + 𝐴 for 𝐴 = . . . 𝐾 . Analogously 𝜏 ( 𝐵 ) = 𝐵 − 𝐾 for 𝐵 = 𝐾 + . . . 𝑚 . Therefore 𝜏 is nothing but a 𝐾 -fold cyclic permutation ‘to the left’ (or 𝑚 − 𝐾 -fold to the right).Since for 𝐴 < 𝐵 the non-zero 𝛼 𝐴𝐵 are the ones, for which 𝜏 ( 𝐵 ) < 𝜏 ( 𝐴 ) , this implies 𝐵 = 𝐾 + . . . 𝑚 and 𝐴 = . . . 𝐾 . These 𝛼 𝐴𝐵 are equal to 1, therefore 𝜔 − 𝜔 = 𝑖 𝜎 (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) ∑︁ 𝐴 = ... 𝐾,𝐵 = 𝐾 + ... 𝑛 Tr ( 𝐽 𝐴𝐵 ∧ 𝐽 𝐵𝐴 ) (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 𝐺 𝐿,𝑁 , where 𝐿 = 𝐾 (cid:205) 𝐴 = 𝑛 𝐴 . Conversely, one shows that for a cyclic permutation thedifference between 𝜔 and 𝜔 is the closed form written above. F Models with Z 𝑚 -graded target spaces A homogeneous space 𝐺𝐻 is called Z 𝑚 -graded (or 𝑚 -symmetric), if the Lie algebra 𝔤 of itsisometry group admits the following decomposition: 𝔤 = ⊕ 𝑚 − 𝑖 = 𝔤 𝑖 , [ 𝔤 𝑖 , 𝔤 𝑗 ] ⊂ 𝔤 𝑖 + 𝑗 mod 𝑚 , 𝔤 = 𝔥 . (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 𝜎 -models with Z 𝑚 -gradedtarget-spaces may be written as flatness 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 : = ∫ Σ 𝑑 𝑥 (cid:107) 𝜕 𝑋 (cid:107) 𝐺 + ∫ Σ 𝑋 ∗ (cid:101) 𝜔, (F.2)where (cid:101) 𝜔 is a 2-form constructed using the Z 𝑚 -decomposition of the Lie algebra (F.1). Notethat, if (cid:101) 𝜔 were the fundamental Hermitian form, one would obtain precisely the action14313.5). Now we come to the precise definition of (cid:101) 𝜔 . Decompose the current 𝐽 = − 𝑔 − 𝑑𝑔 according to (F.1): 𝐽 = − 𝑔 − 𝑑𝑔 = 𝑚 − ∑︁ 𝑖 = 𝐽 ( 𝑖 ) , where 𝐽 ( 𝑖 ) ∈ 𝔤 𝑖 . (F.3)The form (cid:101) 𝜔 is defined as follows: (cid:101) 𝜔 = 𝑚 − ∑︁ 𝑘 = ( 𝑚 − 𝑘 ) − 𝑘𝑚 Tr ( 𝐽 ( 𝑘 ) ∧ 𝐽 ( 𝑚 − 𝑘 ) ) (F.4)This formula raises the following question. According to (F.4), the form (cid:101) 𝜔 depends on the Z 𝑚 -grading on the Lie algebra, but generally a given Lie algebra 𝔤 may have many differentgradings (with different, or same, values of 𝑚 ). The question is: are the models defined by(F.2)-(F.4), corresponding to different gradings of 𝔤 , different?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 𝔤 = 𝑠𝑢 ( 𝑛 ) . A cyclicgrading may be constructed as follows : one picks a system of 𝑛 − 𝛼 , . . . 𝛼 𝑛 − , as well as the maximal negative root 𝛼 𝑛 = − 𝛼 − . . . − 𝛼 𝑛 − . Then one assignsto these 𝑛 roots arbitrary (non-negative integer) gradings 𝑚 , . . . 𝑚 𝑛 − , 𝑚 𝑛 . The gradings ofall other roots are determined by the Lie algebra structure, and the value of 𝑚 is calculatedas 𝑚 = 𝑚 + . . . + 𝑚 𝑛 . (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) 𝑚 𝑚 . . . 𝑚 𝑛 − 𝑚 𝑛 (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 𝐴 ( ) 𝑛 − . In the paper of Kac [148] the roots 𝛼 , . . . 𝛼 𝑛 are seen as the positive simple roots of the correspondingaffine Lie algebra (cid:98) 𝐴 𝑛 − . Consider the case 𝑛 =
3. The simple positive roots of the loop algebra 𝑠𝑢 ( )( 𝑡, 𝑡 − ) may be chosen as follows: 𝛼 = (cid:169)(cid:173)(cid:171) (cid:170)(cid:174)(cid:172) , 𝛼 = (cid:169)(cid:173)(cid:171) (cid:170)(cid:174)(cid:172) , 𝛼 = 𝑡 (cid:169)(cid:173)(cid:171) (cid:170)(cid:174)(cid:172) . In this context the latter root 𝛼 – the analog of 𝛼 𝑛 – is customarily called ‘imaginary’. In fact, the whole theoryof cyclic Lie algebra gradings is formulated by Kac naturally in terms of affine Lie algebras and their Dynkindiagrams. 𝔤 , which determines the denominator 𝐻 of the quotient space 𝐺 / 𝐻 , isdetermined by those 𝑚 𝑖 ’s, which are zero. For example, if all 𝑚 𝑖 >
0, the resulting space isthe manifold of complete flags 𝑆𝑈 ( 𝑛 ) 𝑆 ( 𝑈 ( ) 𝑛 ) .In general, for a choice of grading determined by the set 𝑚 , . . . , 𝑚 𝑛 some of the subspaces 𝔤 𝑖 will be identically zero. Therefore a natural restriction to adopt is to require that 𝔤 𝑖 ≠ 𝑖 ( mod 𝑚 ) . We will call such a grading admissible . This still leaves a wide range ofpossibilities. For example, in the case of 𝑆𝑈 ( ) the following is a complete list of admissiblegradings (up to the action of the Weyl group 𝑆 ): 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 ( )/ 𝑆 ( U ( )× U ( )) = CP , and all other gradings correspond to the flag manifold F , , .We will now give an answer to the question posed above: what is the relation betweenthe 𝜎 -models with the action (F.2), taken for different gradings on the corresponding Liealgebra? Our statement is [66]:For homogeneous spaces of the unitary group, the models (F.2)-(F.4) with different 𝐴 ( ) 𝑛 − -type gradings on 𝔤 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 𝐵 -fields in the two models.To formulate it, we ‘solve’ the constraint (F.5) as follows : 𝑚 𝑘 = 𝑞 𝑘 − 𝑞 𝑘 + , (F.10)where 𝑞 𝑘 are integers and 𝑞 𝑛 + ≡ 𝑞 − 𝑚 . We then have (see [66] for a proof): (cid:101) 𝜔 = 𝜔 − 𝑛 ∑︁ 𝑖 = 𝑞 𝑖 𝑚 𝑑𝐽 𝑖𝑖 , (F.11)where 𝐽 𝑖𝑖 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 𝑞 𝑖 ), the form (cid:101) 𝜔 differs Formula (F.10) implies that the cyclic automorphism (cid:98) 𝜎 of the Lie algebra, which defines the Z 𝑚 grading,can be represented as follows: (cid:98) 𝜎 ( 𝑎 ) = 𝜎𝑎𝜎 − , where 𝜎 = diag ( 𝑒 𝜋𝑖 𝑞 𝑚 , . . . , 𝑒 𝜋𝑖 𝑞𝑛𝑚 ) . Z 𝑚 -graded spaces, the two classes of target spaces– Z 𝑚 -graded and complex homogeneous spaces – do not coincide. For example, one has thespace 𝐺 𝑆𝑈 ( ) (cid:39) 𝑆 . The stability subgroup 𝑆𝑈 ( ) acts on the tangent space 𝔪 = R via 𝑉 ⊕ 𝑉 ,where 𝑉 (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 nearlyKähler manifold and is Z -graded [61]. On the other side of the story, one has the complexmanifold 𝑆 × 𝑆 (cid:39) 𝑈 ( ) (see [66] for a discussion), which may be viewed as a T -bundleover CP (the simplest flag manifold). This manifold is not a Z 𝑚 -graded homogeneous spaceof the group 𝐺 = 𝑈 ( ) .We also note that the construction of Lax connections for models with Z 𝑚 -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 𝑚𝑚