Hecke Operator and S-Duality of N=4 Super Yang-Mills for ADE Gauge Group on K3

TL;DR

This paper computes partition functions of N=4 super Yang-Mills theories with ADE gauge groups on K3 surfaces, revealing a connection between Hecke operators and S-duality, consistent with Montonen-Olive duality.

Contribution

It extends the understanding of S-duality and Hecke operators to ADE gauge groups on K3, building on prior SU(N) results by Vafa and Witten.

Findings

Partition functions satisfy the gap condition.

Partition functions obey Montonen-Olive duality.

Identifies a relation between Hecke operators and S-duality.

Abstract

We determine the partition functions of ${\cal N}=4$ super Yang-Mills gauge theory for some $ADE$ gauge groups on $K3$, under the assumption that they are holomorphic. Our partition functions satisfy the gap condition and Montonen-Olive duality at the same time, like the SU(N) partition functions of Vafa and Witten. As a result we find a close relation between Hecke operator and $S$-duality of ${\cal N}=4$ super Yang-Mills for $ADE$ gauge group on $K3$.