Hecke operator andS-duality of Script N = 4 super Yang-Mills for ADE gauge group onK3

Abstract

We determine the partition functions of 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 N = 4 super Yang-Mills for ADE gauge group on K3.

GAP CONDITION AND SELF-DUALIZED ${\mathcal N}=4$ SUPER-YANG–MILLS THEORY FOR ADE GAUGE GROUP ON K3

“…We hope to find out the technique to do this and to find guiding principle like Seiberg-Witten curve or holomorphic anomaly equation in U (N ) on 1 2 K3 case. In fact we had been writing the another article concerned about N = 4 ADE partition function [20] parallel to this article. In [20] we revealed a close relation between Hecke operator and S-duality of N = 4 Super Yang-Mills on K3.…”

“…In fact we had been writing the another article concerned about N = 4 ADE partition function [20] parallel to this article. In [20] we revealed a close relation between Hecke operator and S-duality of N = 4 Super Yang-Mills on K3. There are advantages and disadvantages in either article.…”

“…To resolve this problem, we took another approach. In [22] we invent the deformed Hecke operator, which is the deformed version of Hecke operator with the dependence of the 't Hooft flux. This operator satisfies the Montonen-Olive duality automatically.…”

“…This article is one of a series of our trials to determine N = 4 ADE partition function on K3 [9,22,23]. In this article we attempt to reveal the relation between N = 4 ADE partition function on K3 and affine Lie algebra.…”

“…The second approach is to describe partition function by a kind of Hecke operator. In [22] we invented the deformed Hecke operator, which satisfies the Montonen-Olive duality automatically. We only had to determine the function so that the operated function satisfies the gap condition.…”