One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

Probability and Measure

Über die analytische Theorie der quadratischen Formen II

We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map ${g \mapsto R (g)}$ from ${ \rm{GL}(2, \mathbb{C}) \times \rm{GL}(1, \mathbb{C})}$ to End ${(V \otimes V)}$ , where V is a two-dimensional vector space such that if ${g, h \in G}$ then R12(g)R13(gh) R23(h) = R23(h) R13(gh)R12(g). Here Rij denotes R applied to the i, j components of ${V \otimes V \otimes V}$ . The image of this map consists of matrices whose nonzero coefficients a1, a2, b1, b2, c1, c2 are the Boltzmann weights for the non-field-free six-vertex model, constrained to satisfy a1a2 + b1b2 − c1c2 = 0. This is the exact center of the disordered regime, and is contained within the free fermionic eight-vertex models of Fan and Wu. As an application, we show that with boundary conditions corresponding to integer partitions λ, the six-vertex model is exactly solvable and equal to a Schur polynomial sλ times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of Tokuyama and Hamel and King.

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Schur Polynomials and The Yang-Baxter Equation

Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet series and their residues unify many examples that have been studied previously in a case-bycase basis, often with applications to analytic number theory. (Examples may be found in the final section of the paper.) We believe these Weyl group multiple Dirichlet series are fundamental objects. The goal of this paper is to define these series for any such Φ and F , and to indicate how to study them. We will note the following points. • The coefficients of the Weyl group multiple Dirichlet series are multiplicative, but the multiplicativity is twisted , so the Dirichlet series is not an Euler product. • Due to the multiplicativity, description of the coefficients reduces to the case where the parameters are powers of a single prime p. There are only finitely many such coefficients (for given p). • In the “stable case” where n is sufficiently large (depending on Φ), the number of nonzero coefficients in the p-part is equal to the order of the Weyl group. Indeed, these nonzero coefficients are parametrized in a natural way by the Weyl group elements. • The p-part coefficient parametrized by a Weyl group element w is a product of l(w) Gauss sums, where l is the length function on the Weyl group. We note a curious similarity between this description and the coefficients of the generalized theta series on the n-fold cover of GL(n) and GL(n − 1); these coefficients are determined in Kazhdan and Patterson [17] and discussed further in Patterson [21]. See Bump and Hoffstein [11] or Hoffstein [15] for a “classical” description of these coefficients. The noted similarity means that the complete Mellin transform of the theta function would be a multiple Dirichlet series resembling our An+1 multiple Dirichlet series. There is no a priori reason that we are

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Weyl Group Multiple Dirichlet Series I

We provide formulas for various bases of spherical Whittaker functions on the $n$-fold metaplectic cover of ${\rm GL}_r$ over a $p$-adic field and show that there is a basis of symmetric functions in the complex parameter. In addition we relate a specific spherical Whittaker function to the $p$-power part of the Weyl group multiple Dirichlet series for the root system of type $A_{r-1}$ constructed from $n$th order Gauss sums. We also show that the zonal spherical functions can be computed explicitly in terms of Hall-Littlewood polynomials as in Macdonald's formula for ${\rm GL}_r$.

A metaplectic Casselman-Shalika formula for GL r

Let Φ be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Φ is a Dirichlet series in r complex variables s1, . . . , sr, initially converging for Re(si) sufficiently large, that has meromorphic continuation to C and satisfies functional equations under the transformations of C corresponding to the Weyl group of Φ. A heuristic definition of such series was given in [BBC06], and they have been investigated in certain special cases in [BBC06,BBF06,BBFH07, BBF08, BBFa, CG07, CFG08, CG, Chi05]. In this paper we generalize results in [CG] to construct Weyl group multiple Dirichlet series by a uniform method, and show in all cases that they have the expected properties.

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Constructing Weyl group multiple Dirichlet series

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Second order modular forms

By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called the complexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study of I-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz–type zeta functions.

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Gautam Chinta

Papers

Weyl Group Multiple Dirichlet Series I

A metaplectic Casselman-Shalika formula for GL r

Constructing Weyl group multiple Dirichlet series

Second order modular forms

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori