On W-representations of β- and q,t-deformed matrix models

A Morozov

doi:10.1016/j.physletb.2019.03.047

Abstract

W-representation realizes partition functions by an action of a cut-and-join operator on the vacuum state with a zero-mode background. We provide explicit formulas of this kind for β- and q,t-deformations of the simplest rectangular complex matrix model. In the latter case, instead of the complicated definition in terms of multiple Jackson integrals, we define partition functions as the weight-two series, made from Macdonald polynomials, which are evaluated at different loci in the space of time variables. Resulting expression for the Wˆ operator appears related to the problem of simple Hurwitz numbers (contributing are also the Young diagrams with all but one lines of length two and one). This problem is known to exhibit nice integrability properties. Still the answer for Wˆ can seem unexpectedly sophisticated and calls for improvements. Since matrix models lie at the very basis of all gauge- and string-theory constructions, our exercise provides a good illustration of the jump in complexity between β- and q,t-deformations – which is not always seen at the accidentally simple level of Calogero-Ruijsenaars Hamiltonians (where both deformations are equally straightforward). This complexity is, however, quite familiar in the theories of network models, topological vertices and knots.

Highlights

Eigenvalue matrix models [1] play an increasingly important role in modern theory
A variety of important applications stimulates the study of matrix models per se, which revealed a lot of important structures, widely used in modern physics, and points to the hidden matrix-model structures far beyond their traditional areal
In this paper we look for still another definition/reformulation of the Gaussian q, t-model: in terms of the W -representations [10], i.e. represent the partition functions (1) as

Summary

Introduction

Eigenvalue matrix models [1] play an increasingly important role in modern theory. Originally they appear as convenient representatives of universality classes of important statistical distributions (random matrix ensembles) [2] and as exactly solvable examples of quantum field and string theory models [3]. We work out a W representation of q, t-models, which are the first example, where the superintegability-based definition from [11] is indisputably simpler than the conventional one [12] through a matrix-like Jackson integral In this case the latter would be a network-model lifting [13, 14] of Dotsenko-Fateev matrix model [5], which is a free-field description of intertwiner (topological vertex) convolutions in DIM algebra [15], where integrals over the matrix eigenvalues (which are the arguments of the screening integrands) are substituted by sums over Young diagrams and, further, over plane partitions.

On ambiguity of W -representation

Condensed form of the W-representation

From derivatives to shifts

Conclusion