On homomorphisms from finite subgroups of $SU(2)$ to Langlands dual pairs of groups
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866910926343503872 |
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| author | Kojima, Yuki Tachikawa, Yuji |
| author_facet | Kojima, Yuki Tachikawa, Yuji |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01253 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On homomorphisms from finite subgroups of $SU(2)$ to Langlands dual pairs of groups Kojima, Yuki Tachikawa, Yuji Representation Theory High Energy Physics - Theory Group Theory 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. |
| title | On homomorphisms from finite subgroups of $SU(2)$ to Langlands dual pairs of groups |
| topic | Representation Theory High Energy Physics - Theory Group Theory |
| url | https://arxiv.org/abs/2505.01253 |