$q$-Virasoro modular triple

TL;DR

This paper introduces the modular triple, an algebraic structure for the $q$-Virasoro algebra inspired by 5d supersymmetric theories, revealing an $ ext{SL}(3, ext{Z})$ symmetry and proposing a novel Lagrangian formulation.

Contribution

It constructs the $q$-Virasoro modular triple, demonstrating its $ ext{SL}(3, ext{Z})$ symmetry and providing a 2d CFT-like and Lagrangian perspective, a first in the field.

Findings

The modular triple exhibits an $ ext{SL}(3, ext{Z})$ structure.

It aligns with the triple factorization of supersymmetric partition functions.

A conjectured non-local Lagrangian for the $q$-Virasoro model is proposed.

Abstract

Inspired by 5d supersymmetric Yang-Mills theories placed on the compact space $\mathbb{S}^5$, we propose an intriguing algebraic construction for the $q$-Virasoro algebra. We show that, when multiple $q$-Virasoro "chiral" sectors have to be fused together, a natural $\mathrm{SL}(3,\mathbb{Z})$ structure arises. This construction, which we call the modular triple, is consistent with the observed triple factorization properties of supersymmetric partition functions derived from localization arguments. We also give a 2d CFT-like construction of the modular triple, and conjecture for the first time a (non-local) Lagrangian formulation for a $q$-Virasoro model, resembling ordinary Liouville theory.