On Homomorphisms from Finite Subgroups of SU(2) to Langlands Dual Pairs of Groups

Abstract

Let $N(\Gamma ,G)$ be the number of homomorphisms from $\Gamma$ to $G$ up to conjugation by $G$. The physics of four-dimensional $\mathcal{N}=4$ supersymmetric gauge theories predicts that $N(\Gamma ,G)=N(\Gamma ,\tilde{G})$ when $\Gamma$ is a finite subgroup of $\mathrm{SU}(2)$, $G$ is a connected compact simple Lie group and $\tilde{G}$ is its Langlands dual. This statement is known to be true when $\Gamma ={\mathbb{Z}}_n$, but the statement for non-Abelian $\Gamma$ is new, to the knowledge of the authors. To lend credence to this conjecture, we prove this equality in a few cases, namely $(G,\tilde{G})=(\mathrm{SU}(n),\mathrm{PU}(n))$ and $(\mathrm{Sp}(n),\mathrm{SO}(2n+1))$ for arbitrary $\Gamma$, and $(\mathrm{PSp}(n),\mathrm{Spin}(2n+1))$ for exceptional $\Gamma$.

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.