On homomorphisms from finite subgroups of $SU(2)$ to Langlands dual pairs of groups
Yuki Kojima, Yuji Tachikawa
TL;DR
The paper investigates a physics-inspired counting problem: for finite subgroups $\Gamma\subset SU(2)$ and Langlands dual pairs $(G,\tilde G)$, do the numbers of homomorphisms $N(\Gamma,G)$ and $N(\Gamma,\tilde G)$ agree? It develops a refined framework using twisted homomorphisms and central data to compare the $G$- and Langlands-dual sides via representations $V_Z(\Gamma,G)$ and the swap isomorphism $s$, with a conjectural equivalence of these representations. The authors prove the basic and refined conjectures in several key cases: $(G,\tilde G)=(SU(n),PU(n))$ and $(Sp(n),SO(2n+1))$ for arbitrary $\Gamma$, plus $(PSp(n),Spin(2n+1))$ for exceptional $\Gamma$, employing the McKay correspondence, Weyl-group orbit counts, and generating functions. A central theme is translating the problem into discrete Fourier-type transforms on representation spaces and using cohomological data to organize twisted sectors, thereby providing substantial evidence for a broad duality between $G$ and its Langlands dual at the level of finite subgroup representations. The results illuminate connections between 4d $\mathcal{N}=4$ physics, algebraic group theory, and ADE/McKay structures, and pave the way toward a uniform, all-cases proof.
Abstract
Let $N(Γ,G)$ be the number of homomorphisms from $Γ$ to $G$ up to conjugation by $G$. Physics of four-dimensional $\mathcal{N}=4$ supersymmetric gauge theories predicts that $N(Γ,G)=N(Γ, \tilde G)$ when $Γ$ 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 $Γ=\mathbb{Z}_n$, but the statement for non-Abelian $Γ$ 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 $Γ$, and $(PSp(n),Spin(2n+1))$ for exceptional $Γ$. 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.