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

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.

Figures (6)

• Figure 1: Finite and untwisted affine Dynkin diagrams. The labels of the untwisted diagram are the same with the corresponding finite diagram. We call the extending node $\alpha_0$.
• Figure 3: Graph automorphisms of finite simply-laced Dynkin diagrams, or equivalently the outer automorphisms of the corresponding Lie algebras. See the arrow emphasized in red: only for $A_\text{even}=\mathrm{SU}(2\ell+1)$ there are two nodes which are connected both by a line in the Dynkin diagram and an arrow representing the action of the automorphism.
• Figure 4: 6d theory of type $G$ put on $T^2$ with twists on different cycles, and its interpretation as 5d theory. The orange dotted line is the place across which the $\mathbb{Z}_r$ twist is performed.
• Figure 5: 6d theory of type $G$, put on a slanted torus. The complex structure of the torus is identified with $\tau$ in \ref{['tau']}
• Figure 6: 6d theory of type $\mathrm{SU}(2\ell+1)$ put on a torus with $\mathbb{Z}_2$ twist, together with wrapped strings. Orange broken lines are the $\mathbb{Z}_2$ twist lines, across which the $\mathbb{Z}_2$ operation is performed. Strings wrapped around pink solid, blue dotted, and green chained lines produce particles with charges which match the KK W-bosons of $\mathrm{USp}(2\ell)$ theory, $\mathrm{SU}(2\ell+1)$ theory with twist $\sigma_2(\ell)$, $\mathrm{SU}(2\ell+1)$ theory with twist $\sigma_2(0)$, respectively.