# Elliptic modular double and 4d partition functions

## TL;DR

This paper links 4d supersymmetric gauge theory partition functions on specific manifolds to correlators in elliptic quantum algebras, revealing new algebraic structures underlying these physical theories.

## Contribution

It establishes a novel correspondence between 4d gauge theory partition functions and elliptic algebra correlators, introducing a new algebraic framework for these theories.

## Key findings

- Partition functions as correlators of vertex operators and screening charges
- Generation of BPS surface defect functions as coherent states in elliptic Heisenberg algebra
- Identification of algebraic structures underlying 4d supersymmetric theories

## Abstract

We consider 4d supersymmetric (special) unitary $\Gamma$ quiver gauge theories on compact manifolds which are $T^2$ fibrations over $S^2$. We show that their partition functions are correlators of vertex operators and screening charges of the modular double version of elliptic $W_{q,t;q'}(\Gamma)$ algebras. We also consider a generating function of BPS surface defects supported on $T^2$ and show that it can be identified with a particular coherent state in the Fock module over the elliptic Heisenberg algebra.