$q$-Virasoro modular double and 3d partition functions

TL;DR

This paper connects 3d $ ext{N}=2$ gauge theory partition functions on certain manifolds to free field correlators of the $q$-Virasoro modular double, revealing new algebraic structures and constraints involving Wilson loops.

Contribution

It introduces the $q$-Virasoro modular double framework for 3d gauge theories and demonstrates how Wilson loop generating functions satisfy specific $q$-Virasoro constraints.

Findings

Partition functions are free field correlators of vertex operators.

Wilson loop generating functions satisfy SL(2,Z)-related $q$-Virasoro constraints.

The construction extends to quiver theories and includes the ABJ(M) model.

Abstract

We study partition functions of 3d $\mathcal{N}=2$ U(N) gauge theories on compact manifolds which are $S^1$ fibrations over $S^2$. We show that the partition functions are free field correlators of vertex operators and screening charges of the $q$-Virasoro modular double, which we define. The inclusion of supersymmetric Wilson loops in arbitrary representations allows us to show that the generating functions of Wilson loop vacuum expectation values satisfy two SL(2,$\mathbb{Z}$)-related commuting sets of $q$-Virasoro constraints. We generalize our construction to 3d $\mathcal{N}=2$ unitary quiver gauge theories and as an example we give the free boson realization of the ABJ(M) model.