# On $W$-representations of $\beta$- and $q,t$-deformed matrix models

## TL;DR

This paper develops explicit $W$-representations for $eta$- and $q,t$-deformed matrix models, revealing increased complexity in the deformed cases and connecting to integrability and Hurwitz numbers.

## Contribution

It provides explicit formulas for $W$-representations of $eta$- and $q,t$-deformed matrix models, highlighting the complexity differences and their relation to integrability and Hurwitz numbers.

## Key findings

- Explicit $W$-operator formulas for deformed models
- Connection to simple Hurwitz numbers and integrability
- Increased complexity in $q,t$-deformations compared to $eta$-deformations

## Abstract

$W$-representation realizes partition functions by an action of a cut-and-join-like operator on the vacuum state with a zero-mode background. We provide explicit formulas of this kind for $\beta$- 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 $\hat W$ 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 $\hat W$ 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 $\beta$- and $q,t$-deformations -- which is not always seen at the accidently 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.