New Trends in Quantum Integrable Systems

The following sections are included:

• PREFACE

• WORKSHOP PROGRAM

• LIST OF PARTICIPANTS

• CONTENTS

One of the key steps in recent work on the correlation functions of the XXZ chain was to regularize the underlying six-vertex model by a disorder parameter α. For the regularized model it was shown that all static correlation functions are polynomials in only two functions. It was further shown that these two functions can be written as contour integrals involving the solutions of a certain type of linear and non-linear integral equations. The linear integral equations depend parametrically on α and generalize linear integral equations known from the study of the bulk thermodynamic properties of the model. In this note we consider the generalized dressed charge and a generalized magnetization density. We express the generalized dressed charge as a linear combination of two quotients of Q-functions, the solutions of Baxter's t-Q-equation. With this result we give a new proof of a lemma on the asymptotics of the generalized magnetization density as a function of the spectral parameter.

We discuss some fundamental properties of the XXZ spin chain, which are important in the algebraic Bethe-ansatz derivation for the multiple-integral representations of the spin-s XXZ correlation function with an arbitrary product of elementary matrices.¹ For instance, we construct Hermitian conjugate vectors in the massless regime and introduce the spin-s Hermitian elementary matrices.

We investigate induced modules of doublet algebra in (1, p) logarithmic models. We give fermionic formulas for the characters of induced modules and coinvariants with respect to different subalgebras calculated in the irreducible modules. The characters of coinvariants give multiplicities of projective modules in fusion of induced modules.

We revisit the quantum/classical integrable model correspondence in the context of inhomogeneous finite length XXZ spin- ½ chains with periodic boundary conditions and show that the Bethe scalar product of an arbitrary state and a Bethe eigenstate is a discrete KP τ-function. The continuous Miwa variables of discrete KP are the rapidities of the arbitrary state.

This is a pedagogical review on recent progress in the exact evaluation of physical quantities in interacting quantum systems at finite temperatures. Quantum spin chains are discussed in detail as typical examples.

We study the generalized ultradiscrete periodic Toda lattice which has tropical spectral curve. We introduce a tropical analogue of Fay's trisecant identity, and apply it to construct a general solution to .

We apply the fermionic description of CFT developed in our previous work to the computation of the one-point functions of the descendent fields in the sine-Gordon model.

In this paper, we give an explicit isomorphism between the adjoint crystal 𝔹_ℓ given in Ref. 13 and the Kirillov-Reshetikhin crystals B^1,2ℓ presented in Ref. 2 for . Consequently we have a complete proof of perfectness of the crystals 𝔹_ℓ. In the sequel we also present an explicit description of the involution σ given in Ref. 2 for this special case.

We introduce partial reection K-matrices, and boundary qKZ equation with 6 parameters. We give a method to construct its solution from a solution of certain eigenvalue problem. The eigenvalue probrem is described in terms of Noumi representation of the affine Hecke algebra of type C. We also show concrete solutions in terms of non-symmetric Koornwinder polynomials. This is an anouncement of a joint work with K. Shigechi.³

This paper provides a brief review of the relations between the Feigin-Loktev conjecture on the dimension of graded tensor products of 𝔤[t]-modules, the Kirillov-Reshetikhin conjecture, the combinatorial "M = N" conjecture, their proofs for all simple Lie algebras, and a pentagon of identities which results from the proof.

We give a direct derivation of a proposal of Nekrasov-Shatashvili concerning the quantization conditions of the Toda chain. The quantization conditions are formulated in terms of solutions to a nonlinear integral equation similar to the ones coming from the thermodynamic Bethe ansatz. This is equivalent to extremizing a certain function called Yang's potential. It is shown that the Nekrasov-Shatashvili formulation of the quantization conditions follows from the solution theory of the Baxter equation, suggesting that this way of formulating the quantization conditions should indeed be applicable to large classes of quantized algebraically integrable models.

We introduce generalized energies for a class of crystals by using the piecewise linear functions that are building blocks of the combinatorial R. They include the conventional energy in the theory of affine crystals as a special case. It is shown that the generalized energies count the particles and anti-particles in a quadrant of the two dimensional lattice generated by time evolutions of an integrable cellular automaton. Explicit formulas are conjectured for some of them in the form of ultradiscrete tau functions.

We study a Fredholm determinant of the hypergeometric kernel arising in the representation theory of the infinite-dimensional unitary group. It is shown that this determinant coincides with the Palmer–Beatty–Tracy tau function of a Dirac operator on the hyperbolic disk. Solution of the connection problem for Painlevé VI equation allows to determine its asymptotic behavior up to a constant factor, for which a conjectural expression is given in terms of Barnes functions. We also present analogous asymptotic results for the Whittaker and Macdonald kernel.

Here n ∈ ℕ, k is a field, S_n is the symmetric group, δ ∈ k, and P_n(δ) is the partition algebra over k. Our aim in this note is to study the representation theory of a subalgebra of kS_n ⊕_k P_n(δ) with certain interesting combinatorial and representation theoretic properties.

In Section 1 we discuss the motivating combinatorial background. In Section 2 we define (see Proposition 1). In Section 3 we determine its complex representation theory.

We review developments made since 1959 in the search for a closed form for the susceptibility of the Ising model. The expressions for the form factors in terms of the nome q and the modulus k are compared and contrasted. The λ generalized correlations C(M, N; λ) are defined and explicitly computed in terms of theta functions for M = N = 0,1.

Consider a tensor product of finite-dimensional irreducible 𝔤𝔩_N+1-modules and its decomposition into irreducible modules. The 𝔤𝔩_N+1 Gaudin model assigns to each multiplicity space of that decomposition a commutative (Bethe) algebra of linear operators acting on the multiplicity space. The Bethe ansatz method is a method to find eigenvectors and eigenvalues of the Bethe algebra. One starts with a critical point of a suitable (master) function and constructs an eigenvector of the Bethe algebra.

In this paper we consider the algebra of functions on the critical set of the associated master function and show that the action of this algebra on itself is isomorphic to the action of the Bethe algebra on a suitable subspace of the multiplicity space.

As a byproduct we prove that the Bethe vectors corresponding to different critical points of the master function are linearly independent and, in particular, nonzero.

The T-systems and Y-systems are classes of algebraic relations originally associated with quantum affine algebras and Yangians. Recently they were generalized to quantum affinizations of quantum Kac-Moody algebras associated with a wide class of generalized Cartan matrices which we say tamely laced. Furthermore, in the simply laced case, and also in the nonsimply laced case of finite type, they were identified with relations arising from cluster algebras.In this note we generalize such an identification to any tamely laced Cartan matrices, especially to the nonsimply laced ones of nonfinite type.

We study the relation between the periodic Benjamin-Ono equation with discrete Laplacian and the two-dimensional-Toda hierarchy. We introduce the tau-functions τ_±(z) for the periodic Benjamin-Ono equation, construct two families of integrals of motion {M1,M2,⋯}, , and calculate some examples of the bilinear equations using the Hamiltonian structure. We confirmed that some of the low lying bilinear equations agree with the ones obtained from a certain reduction of the 2D-Toda hierarchy.

Recent work of Foda and his group on a connection between classical integrable hierarchies (the KP and 2D Toda hierarchies) and some quantum integrable systems (the 6-vertex model with DWBC, the finite XXZ chain of spin ½, the phase model on a finite chain, etc.) is reviewed. Some additional information on this issue is also presented.

We introduce middle convolution for systems of linear differential equations with irregular singular points, and we present a tentative definition of the index of rigidity for them. Under some assumption, we show a list of terminal patterns of irreducible systems of linear differential equations by iterated application of middle convolution when the index is positive or zero.

We give diffierential equations compatible with the rational qKZ equation with boundary reflection. The total system contains the trigonometric degeneration of the bispectral qKZ equation of type which in the case of type GL_n was studied by van Meer and Stokman. We construct an integral formula for solutions to our compatible system in a special case.

We classify and list up all the meromorphic solutions K(z) to the reflection equation associated to the critical Z_N-symmetric vertex model under two assumptions that none of the diagonal elements is constantly zero and that there is at least a pair of elements We make explicit the matrix elements of K(z), parameters they have and the relations among parameters.