Kavli Affiliate: Yuji Tachikawa
| First 5 Authors: Yuki Kojima, Yuji Tachikawa, , ,
| Summary:
Let $N(Gamma,G)$ be the number of homomorphisms from $Gamma$ to $G$ up to
conjugation by $G$. 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 $SU(2)$, $G$ 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 couple of examples, namely $(G,tilde G)=(SU(n),PU(n))$ and
$(Sp(n),SO(2n+1))$ for arbitrary $Gamma$, and $(PSp(n),Spin(2n+1))$ 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.
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