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

Abstract

In this letter we continue the development of W -representations. We propose several generalizations of the known models, such as the hypergeometric Hurwitz τ -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 β-deformation of the discussed W -representation for the Hurwitz case for two sets of times as well as for the multi-character case.

Supersymmetric partition function hierarchies and character expansions

“…The superintegrability was shown by the character expansions with respect to the Macdonald superpolynomials. When we set all the fermionic variables to zero in the supersymmetric (β and (q, t)-deformed) partition functions presented in this paper, they coincide with the known results in the literature [23,25,26]. Our results may shed new light on supereigenvalue and supermatrix models.…”

“…Setting to zero all the fermionic variables in the supersymmetric β-deformed hierarchy, we arrive at the results in Refs. [25,26]…”

“…W -representations for (β-deformed) multi-character partition functions were constructed in Ref. [26]. They involve a generic number of sets of time variables and the integral representations for such kind of partition functions are given by tensor models and multi-matrix models with multi-trace couplings in some special cases.…”

$W$-representations of two-matrix models with infinite set of variables

“…Recently, the two-matrix models with multi-set of variables were proposed [18][19][20][21], which are the superintegrable matrix models. Their W -representations and character expansions were well investigated.…”

Many-body integrable systems implied by WLZZ models

“…Description in terms of AMM/CEO topological recursion [7][8][9][10][11][12] is still at an early stage [13], but this issue should be also clarified in the near future. Various deformations of the WLZZ models are also straightforward [14,15]. Far more interesting appeared to be a matrix model representation of the WLZZ models, it is provided by a new kind of integrable 2-matrix models [16] which unify the Hermitian and Kontsevich theories.…”