A $(q,t)$-deformation of the 2d Toda integrable hierarchy

TL;DR

This paper introduces a $(q,t)$-deformation of the 2d Toda hierarchy by leveraging the quantum toroidal $rak{gl}(1)$ algebra, leading to new difference-differential equations and a universal refined tau function.

Contribution

It develops a novel $(q,t)$-deformation framework for the 2d Toda hierarchy using quantum algebra techniques, extending the integrable structure.

Findings

Derived new difference-differential equations for the deformed hierarchy.

Identified the deformed Casimir with screening charges of the deformed Virasoro algebra.

Defined a universal refined tau function encoding the deformed hierarchy equations.

Abstract

A $(q,t)$-deformation of the 2d Toda integrable hierarchy is introduced by enhancing the underlying symmetry algebra $\mathfrak{gl}(\infty)\simeq \text{q-W}_{1+\infty}$ to the quantum toroidal $\mathfrak{gl}(1)$ algebra. The difference-differential equations of the hierarchy are obtained from the expansion of $(q,t)$-bilinear identities, and two equations refining the 2d Toda equation are found in this way. The derivation of the bilinear identities follows from the isomorphism between the Fock representation of level $(2,0)$ of the quantum toroidal $\mathfrak{gl}(1)$ algebra and the tensor product of the q-deformed Virasoro algebra with a $u(1)$ Heisenberg algebra. It leads to identify the $(q,t)$-deformed Casimir with the screening charges of the deformed Virasoro algebra. Due to the non-trivial coproduct, equations of the hierarchy no longer involve a single tau-function, but instead relate a set of different tau functions. We then define the universal refined tau function using the $L$-matrix of the quantum toroidal $\mathfrak{gl}(1)$ algebra and interpret it as the generating function of the deformed tau functions. The equations of the hierarchy, written in terms of the universal refined tau function, combine into two-term quadratic equations similar to the $RLL$ equations.