Towards elliptic deformation of $q,t$-matrix models

TL;DR

This paper proposes an elliptic deformation of specific loci in $q,t$-matrix models to develop elliptic matrix models that maintain superintegrability, aiming to extend the algebraic structure of these models.

Contribution

It introduces a novel elliptic deformation of the loci $p_k^{ riangle_n}$, crucial for constructing elliptic matrix models with superintegrability.

Findings

Proposes elliptic functions for $p_k^{ riangle_n}$-loci with correct asymptotics.

Lays groundwork for replacing Schur and Macdonald functions with elliptic GNS.

Suggests studying behavior at deformed topological and $ riangle$ loci.

Abstract

As a necessary step in constructing elliptic matrix models, which preserve the superintegrability property $<char>\sim {\rm char}$, we suggest an elliptic deformation of the peculiar loci $p_k^{\Delta_n}$, which play an important role in precise formulation of this property. The suggestion is to define the $p_k^{\Delta_n}$-loci as elliptic functions with the right asymptotics at $\tau\to i\infty$. If this hypothesis is correct, one can move to substituting the Schur and Macdonald functions in the role of characters by the elliptic GNS and study their behavior at the deformed topological and $\Delta$ loci.