On S-duality for Non-Simply-Laced Gauge Groups

TL;DR

This paper investigates S-duality in N=4 gauge theories with exceptional groups G_2 and F_4, revealing their duality groups as Hecke groups acting on the moduli space and connecting them to T-duality in string theory.

Contribution

It identifies the duality groups for these theories as Hecke groups and explains their realization via T-duality in string theory.

Findings

Duality groups are Hecke groups with elliptic elements of orders six and four.

S-duality acts as a nontrivial involution on the moduli space.

Hecke groups extend subgroups of SL(2,Z) with nontrivial actions.

Abstract

We point out that for N=4 gauge theories with exceptional gauge groups G_2 and F_4 the S-duality transformation acts on the moduli space by a nontrivial involution. We note that the duality groups of these theories are the Hecke groups with elliptic elements of order six and four, respectively. These groups extend certain subgroups of SL(2,Z) by elements with a non-trivial action on the moduli space. We show that under an embedding of these gauge theories into string theory, the Hecke duality groups are represented by T-duality transformations.