On S-duality of 5d super Yang-Mills on S^1

TL;DR

This paper explores a duality in 5d maximally supersymmetric Yang-Mills theory on S^1, linking Kaluza-Klein W-bosons and instantonic monopoles, and relates it to affine Lie algebra Langlands duality and 6d theories.

Contribution

It reveals a novel S-duality in 5d super Yang-Mills that connects non-simply-laced and simply-laced gauge theories via outer automorphisms, with implications for 6d theories.

Findings

Duality exchanges Kaluza-Klein modes and monopoles.

Relation to Langlands dual of affine Lie algebras.

New properties of 6d SU(2n+1) theory with Z_2 twist.

Abstract

We study a duality of 5d maximally supersymmetric Yang-Mills on S^1, which exchanges the tower of Kaluza-Klein W-bosons and the tower of instantonic monopoles. This duality maps a non-simply-laced gauge theory to a simply-laced gauge theory twisted by an outer automorphism around S^1, and is closely related to the Langlands dual of affine Lie algebras. We also discuss how this S-duality is implemented in terms of 6d N=(2,0) theory. This is straightforward except for the 6d theory of type SU(2n+1) with Z_2 outer-automorphism twist, for which a few new properties are deduced. For example, this 6d theory, when reduced on an S^1 with Z_2 twist, gives 5d USp(2n) theory with nontrivial discrete 5d theta angle.