Elliptic Deformation of the Gaiotto-Rap\v{c}\'{a}k Corner VOA and the Associated Partially Symmetric Polynomials

TL;DR

This paper develops the elliptic Miura transformation to describe elliptic corner VOAs, proves a combinatorial formula for their currents, and conjectures a link to partially symmetric polynomials, including elliptic Macdonald polynomials.

Contribution

It introduces the elliptic Miura transformation, derives a combinatorial formula for currents, and proposes a conjecture connecting correlation functions to partially symmetric polynomials.

Findings

Derived the elliptic Miura transformation for corner VOAs.

Proved a new combinatorial formula for currents.

Conjectured a relation between correlation functions and partially symmetric polynomials.

Abstract

We construct the elliptic Miura transformation and use it to obtain the expression of the currents of elliptic corner VOA. We subsequently prove a novel combinatorial formula that is essential for deriving the quadratic relations of the currents. In addition, we give a conjecture that relates the correlation function of the currents of elliptic corner VOA to a certain family of partially symmetric polynomials. The elliptic Macdonald polynomials, constructed recently by Awata-Kanno-Mironov-Morozov-Zenkevich, and Fukuda-Ohkubo-Shiraishi, can be obtained as a particular case of this family.