S-Duality in N=4 Yang-Mills Theories with General Gauge Groups,

TL;DR

This work extends the ’t Hooft flux/partition-function toolkit to $N=4$ Yang–Mills theories with arbitrary compact simple gauge groups and analyzes S-duality actions on non-Abelian electric and magnetic fluxes. By computing the leading infrared divergence of a twisted partition function, the authors derive explicit free-energy expressions $F[oldsymbol{e},oldsymbol{ ilde{m}},S]$ and demonstrate their exact transformation properties under the $SL(2, Z)$ duality group (with careful treatment of simply-laced versus non-simply-laced cases). They establish that the partition function, built as a sum over allowed flux sectors, maps correctly under S-duality to dual groups and appropriately shifts theta-parameters, providing non-perturbative evidence for S-duality in $N=4$ theories. The analysis also reveals that non-simply-laced groups realize only a subgroup of $SL(2, Z)$ due to lattice/flux constraints, offering a nuanced perspective on the string-theoretic origin of S-duality and guiding future tests via subleading terms.

Abstract

't Hooft construction of free energy, electric and magnetic fluxes, and of the partition function with twisted boundary conditions, is extended to the case of $N=4$ supersymmetric Yang-Mills theories based on arbitrary compact, simple Lie groups. The transformation of the fluxes and the free energy under S-duality is presented. We consider the partition function of $N=4$ for a particular choice of boundary conditions, and compute exactly its leading infrared divergence. We verify that this partition function obeys the transformation laws required by S-duality. This provides independent evidence in favor of S-duality in $N=4$ theories.