$S$-duality in Vafa-Witten theory for non-simply laced gauge groups

TL;DR

This paper extends Vafa-Witten theory to non-simply laced gauge groups, proposing explicit S-duality transformations of partition functions based on theta function properties and blow-up formulae.

Contribution

It introduces a new framework for understanding S-duality in Vafa-Witten theory for non-simply laced groups, generalizing previous results for simply laced cases.

Findings

Proposes explicit S-duality transformations for non-simply laced gauge groups.

Provides consistency checks and evidence supporting the proposed transformations.

Connects theta function transformations with gauge theory dualities.

Abstract

Vafa-Witten theory is a twisted N=4 supersymmetric gauge theory whose partition functions are the generating functions of the Euler number of instanton moduli spaces. In this paper, we recall quantum gauge theory with discrete electric and magnetic fluxes and review the main results of Vafa-Witten theory when the gauge group is simply laced. Based on the transformations of theta functions and their appearance in the blow-up formulae, we propose explicit transformations of the partition functions under the Hecke group when the gauge group is non-simply laced. We provide various evidences and consistency checks.