$W$-representations for multi-character partition functions and their $\beta$-deformations

TL;DR

This paper advances the theory of $W$-representations by generalizing models to multi-character expansions, providing integral forms, and introducing $eta$-deformations for complex partition functions involving tensor and multi-matrix models.

Contribution

It introduces new $W$-representations for multi-character expansions and develops $eta$-deformations for these models, extending the existing framework.

Findings

Constructed $W$-representations for multi-character expansions.

Provided integral representations for tensor and multi-matrix models.

Proposed $eta$-deformations for Hurwitz and multi-character cases.

Abstract

In this letter we continue the development of $W$-representations. We propose several generalizations of the known models, such as the hypergeometric Hurwitz $\tau$-functions. We construct $W$-representations for multi-character expansions, which involve a generic number of sets of time variables. We propose integral representations for such kind of partition functions which are given by tensor models and multi-matrix models with multi-trace couplings. We further propose the $\beta$-deformation of the discussed $W$-representation for the Hurwitz case for two sets of times as well as for the multi-character case.