New Integrable Models from the Gauge/YBE Correspondence

We introduce a class of new integrable lattice models labeled by a pair of positive integers N and r. The integrable model is obtained from the Gauge/YBE correspondence, which states the equivalence of the 4d $\mathcal {N} =1$ $S^{1}\times S^{3}/ \mathbb {Z} _{r}$ index of a large class of SU(N) quiver gauge theories with the partition function of 2d classical integrable spin models. The integrability of the model (star-star relation) is equivalent with the invariance of the index under the Seiberg duality. Our solution to the Yang-Baxter equation is one of the most general known in the literature, and reproduces a number of known integrable models. Our analysis identifies the Yang-Baxter equation with a particular duality (called the Yang-Baxter duality) between two 4d $\mathcal {N} =1$ supersymmetric quiver gauge theories. This suggests that the integrability goes beyond 4d lens indices and can be extended to the full physical equivalence among the IR fixed points.     

Melting Crystal, Quantum Torus and Toda Hierarchy

Searching for the integrable structures of supersymmetric gauge theories and topological strings, we study melting crystal, which is known as random plane partition, from the viewpoint of integrable systems. We show that a series of partition functions of melting crystals gives rise to a tau function of the one-dimensional Toda hierarchy, where the models are defined by adding suitable potentials, endowed with a series of coupling constants, to the standard statistical weight. These potentials can be converted to a commutative sub-algebra of quantum torus Lie algebra. This perspective reveals a remarkable connection between random plane partition and quantum torus Lie algebra, and substantially enables to prove the statement. Based on the result, we briefly argue the integrable structures of five-dimensional supersymmetric gauge theories and A-model topological strings. The aforementioned potentials correspond to gauge theory observables analogous to the Wilson loops, and thereby the partition functions are translated in the gauge theory to generating functions of their correlators. In topological strings, we particularly comment on a possibility of topology change caused by condensation of these observables, giving a simple example.     

The Gauge–Bethe Correspondence and Geometric Representation Theory

The Gauge/Bethe correspondence of Nekrasov and Shatashvili relates the spectrum of integrable spin chains to the ground states of supersymmetric gauge theories. Up to now, this correspondence has been an observation; the underlying reason for its existence remaining elusive. We argue here that geometrical representation theory is a mathematical foundation of the Gauge/Bethe correspondence, and it provides a framework to study families of gauge theories in a unified way.     

Cluster-enriched Yang–Baxter equation from SUSY gauge theories

We propose a new generalization of the Yang–Baxter equation, where the R-matrix depends on cluster y-variables in addition to the spectral parameters. We point out that we can construct solutions to this new equation from the recently found correspondence between Yang–Baxter equations and supersymmetric gauge theories. The \(S^2\) partition function of a certain 2d \({\mathcal {N}}=(2,2)\) quiver gauge theory gives an R-matrix, whereas its FI parameters can be identified with the cluster y-variables.     

Brane tilings and reflexive polygons

Reflexive polygons have attracted great interest both in mathematics and in physics. This paper discusses a new aspect of the existing study in the context of quiver gauge theories. These theories are 4d supersymmetric worldvolume theories of D3 branes with toric Calabi‐Yau moduli spaces that are conveniently described with brane tilings. We find all 30 theories corresponding to the 16 reflexive polygons, some of the theories being toric (Seiberg) dual to each other. The mesonic generators of the moduli spaces are identified through the Hilbert series. It is shown that the lattice of generators is the dual reflexive polygon of the toric diagram. Thus, the duality forms pairs of quiver gauge theories with the lattice of generators being the toric diagram of the dual and vice versa.     

Factorization of correlation functions and the replica limit of the Toda lattice equation

Exact microscopic spectral correlation functions are derived by means of the replica limit of the Toda lattice equation. We consider both Hermitian and non-Hermitian theories in the Wigner–Dyson universality class (class A) and in the chiral universality class (class AIII). In the Hermitian case we rederive two-point correlation functions for class A and class AIII as well as several one-point correlation functions in class AIII. In the non-Hermitian case the average spectral density of non-Hermitian complex random matrices in the weak non-Hermiticity limit is obtained directly from the replica limit of the Toda lattice equation. In the case of class A, this result describes the spectral density of a disordered system in a constant imaginary vector potential (the Hatano–Nelson model) which is known from earlier work. New results are obtained for the average spectral density in the weak non-Hermiticity limit of a quenched chiral random matrix model at non-zero chemical potential. These results apply to the ergodic or ϵ domain of the quenched QCD partition function at non-zero chemical potential. Our results have been checked against numerical results obtained from a large ensemble of random matrices. The spectral density obtained is different from the result derived by Akemann for a closely related model, which is given by the leading order asymptotic expansion of our result. In all cases, the replica limit of the Toda lattice equation explains the factorization of spectral one- and two-point functions into a product of a bosonic (non-compact integral) and a fermionic (compact integral) partition function. We conclude that the fermionic partition functions, the bosonic partition functions and the supersymmetric partition function are all part of a single integrable hierarchy. This is the reason that it is possible to obtain the supersymmetric partition function, and its derivatives, from the replica limit of the Toda lattice equation.     

Curve counting,instantons and McKay correspondences

We survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kähler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting holomorphic curves. We discuss the relations of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson–Thomas theory for ideal sheaves on Calabi–Yau threefolds. For resolutions of toric singularities, an algebraic structure induced by a quiver determines the instanton moduli space through the McKay correspondence and its generalizations. The correspondence elucidates the realization of gauge theory partition functions as quasi-modular forms, and reformulates the computation of noncommutative Donaldson–Thomas invariants in terms of the enumeration of generalized instantons. New results include a general presentation of the partition functions on ALE spaces as affine characters, a rigorous treatment of equivariant partition functions on Hirzebruch surfaces, and a putative connection between the special McKay correspondence and instanton counting on Hirzebruch–Jung spaces.     

Brane tilings and their applications

We review recent developments in the theory of brane tilings and four‐dimensional 𝒩 = 1 supersymmetric quiver gauge theories. This review consists of two parts. In part I, we describe foundations of brane tilings, emphasizing the physical interpretation of brane tilings as fivebrane systems. In part II, we discuss application of brane tilings to AdS/CFT correspondence and homological mirror symmetry. More topics, such as orientifold of brane tilings, phenomenological model building, similarities with BPS solitons in supersymmetric gauge theories, are also briefly discussed. This paper is a revised version of the author's master's thesis submitted to Department of Physics, Faculty of Science, the University of Tokyo on January 2008, and is based on his several papers and some works in progress [1–7].     

A technique for analytical calculation of observables in lattice gauge theories

It is shown that the partition function for a finite lattice factorizes into terms that can be associated with each vertex in the finite lattice. This factorization property forms the basis of a well-defined and efficient technique developed to calculate partition functions to high accuracy, on finite lattices for gauge theories. This technique, along with an expansion in finite lattices, provides a powerful means for calculating observables in lattice gauge theories. This is applied to SU(2) lattice gauge theory in four dimensions. The free energy, expectation value of a plaquette and specific heat are calculated. The results are very good both in the strong coupling and the weak coupling region and describe the crossover region quite well, agreeing all the way with the Monte Carlo data.     

Gauge supersymmetric Ising model on a cubic lattice near the critical point in the free field representation

The nature of the set of free fields that represent the system at the critical point has been revealed by studying the correlation functions of the degrees of freedom of the gauge supersymmetric Ising model on the cubic lattice. The same set of free fields represents the continuous supersymmetric Abelian gauge theory. Thus, the name of the lattice system is appropriate. Comparison with the two-dimensional Ising model is given.     

Domain wall fermions with Majorana couplings

We examine the lattice boundary formulation of chiral fermions with either an explicit Majorana mass or a Higgs-Majorana coupling introduced on one of the boundaries. We demonstrate that the low-lying spectrum of the models with an explicit Majorana mass of the order of an inverse lattice spacing is chiral at tree level. Within a mean-field approximation we show that the systems with a strong Higgs-Majorana coupling have a symmetric phase, in which a Majorana mass of the order of an inverse lattice spacing is generated without spontaneous breaking of the gauge symmetry. We argue, however, that the models within such a phase have a chiral spectrum only in terms of the fermions that are singlets under the gauge group. The application of such systems to non-perturbative formulations of supersymmetric and chiral gauge theories is briefly discussed.     

Nonlattice simulation for supersymmetric gauge theories in one dimension

Lattice simulation of supersymmetric gauge theories is not straightforward. In some cases the lack of manifest supersymmetry just necessitates cumbersome fine-tuning, but in the worse cases the chiral and/or Majorana nature of fermions makes it difficult to even formulate an appropriate lattice theory. We propose circumventing all these problems inherent in the lattice approach by adopting a nonlattice approach for one-dimensional supersymmetric gauge theories, which are important in the string or M theory context. In particular, our method can be used to investigate the gauge-gravity duality from first principles, and to simulate M theory based on the matrix theory conjecture.     

New boundary conformal field theories indexed by the simply laced Lie algebras

We consider the field theory of N massless bosons which are free except for an interaction localized on the boundary of their (1+1)-dimensional world. The boundary action is the sum of two pieces: a periodic potential and a coupling to a uniform abelian gauge field. Such models arise in open-string theory and dissipative quantum mechanics, and possibly in edge state tunneling in the fractional quantized Hall effect. We explicitly show that conformal invariance is unbroken for certain special choices of the gauge field and the periodic potential. These special cases are naturally indexed by semi-simple, simply laced Lie algebras. For each such algebra, we have a discrete series of conformally invariant theories where the potential and gauge field are conveniently given in terms of the weight lattice of the algebra. We compute the exact boundary state for these theories, which explicitly shows the group structure. The partition function and correlation functions are easily computed using the boundary state result.     

New example of infinite family of quiver gauge theories

We construct a new infinite family of quiver gauge theories which blow down to the Xp,q$X^{p,q}$ quiver gauge theories found by Hanany, Kazakopoulos and Wecht. This family includes a quiver gauge theory for the third del Pezzo surface. We show, using Z-minimization, that these theories generically have irrational R-charges. The AdS/CFT correspondence implies that the dual geometries are irregular toric Sasaki–Einstein manifolds, although we do not know the explicit metrics.     

Quantum phase transitions beyond the Landau-Ginzburg paradigm and supersymmetry

We make connections between studies in the condensed matter literature on quantum phase transitions in square lattice antiferromagnets, and results in the particle theory literature on abelian supersymmetric gauge theories in 2 + 1 dimensions. In particular, we point out that supersymmetric U(1) gauge theories (with particle content similar, but not identical, to those of theories of doped antiferromagnets) provide rigorous examples of quantum phase transitions which do not obey the Landau-Ginzburg-Wilson paradigm (often referred to as transitions realizing “deconfined criticality”). We also make connections between supersymmetric mirror symmetries and condensed matter particle-vortex dualities.     

Generalized lattice gauge theory, spin foams and state sum invariants

We construct a generalization of pure lattice gauge theory (LGT) where the role of the gauge group is played by a tensor category. The type of tensor category admissible (spherical, ribbon, symmetric) depends on the dimension of the underlying manifold (≤3, ≤4, any). Ordinary LGT is recovered if the category is the (symmetric) category of representations of a compact Lie group. In the weak coupling limit we recover discretized BF-theory in terms of a coordinate-free version of the spin foam formulation. We work on general cellular decompositions of the underlying manifold.

In particular, we are able to formulate LGT as well as spin foam models of BF-type with quantum gauge group (in dimension ≤4) and with supersymmetric gauge group (in any dimension).

Technically, we express the partition function as a sum over diagrams denoting morphisms in the underlying category. On the LGT side this enables us to introduce a generalized notion of gauge fixing corresponding to a topological move between cellular decompositions of the underlying manifold. On the BF-theory side this allows a rather geometric understanding of the state sum invariants of Turaev/Viro, Barrett/Westbury and Crane/Yetter which we recover.

The construction is extended to include Wilson loop and spin network type observables as well as manifolds with boundaries. In the topological (weak coupling) case this leads to topological quantum field theories with or without embedded spin networks.     

Integrating over quiver variety and BPS/CFT correspondence

Letters in Mathematical Physics - We show the vertex operator formalism for the quiver gauge theory partition function and the qq-character of the highest weight module on quiver, both associated...     

Finite Groups and Quantum Yang–Baxter Equations

We construct integrable modifications of two-dimensional lattice gauge theories with finite gauge groups.     

Quantum hydrodynamics from large-<Emphasis Type="Italic">n</Emphasis> supersymmetric gauge theories

We study the connection between periodic finite-difference Intermediate Long Wave (\(\Delta \hbox {ILW}\)) hydrodynamical systems and integrable many-body models of Calogero and Ruijsenaars-type. The former describe quantum cohomology and quantum K-theory of the ADHM moduli space of Abelian instantons, while the latter arise in the instanton counting of four- and five-dimensional supersymmetric gauge theories with eight supercharges in the presence of defects. Using string theory dualities, we provide correspondences between hydrodynamical and many-body integrable systems. In particular, we match the energy spectra on both sides.