On homomorphisms from finite subgroups of $SU(2)$ to Langlands dual pairs of groups

Abstract

Let $N(\hat{I}“,G)$ be the number of homomorphisms from $\hat{I}“$ to $G$ up to conjugation by $G$. Physics of four-dimensional $\mathcal{N}=4$ supersymmetric gauge theories predicts that $N(\hat{I}“,G)=N(\hat{I}“,\tilde{G})$ when $\hat{I}“$ is a finite subgroup of $SU(2)$, $G$ is a connected compact simple Lie group and $\tilde{G}$ is its Langlands dual. This statement is known to be true when $\hat{I}“={\mathbb{Z}}_n$, but the statement for non-Abelian $\hat{I}“$ is new, to the knowledge of the authors. To lend credence to this conjecture, we prove this equality in a couple of examples, namely $(G,\tilde{G})=(SU(n),PU(n))$ and $(Sp(n),SO(2n+1))$ for arbitrary $\hat{I}“$, and $(PSp(n),Spin(2n+1))$ for exceptional $\hat{I}“$. A more refined version of the conjecture, together with proofs of some concrete cases, will also be presented. The authors would like to ask mathematicians to provide a more uniform proof applicable to all cases.

31 pages