Finite group actions on genus two $SL(2, \mathbb{C})$-character variety and applications to SCFTs

TL;DR

This paper studies the fixed point sets of the $SL(2, C)$-character variety for genus two surface groups under finite group actions, revealing geometric structures relevant to 4d $ =2$ SCFTs.

Contribution

It introduces a novel analysis of fixed loci in the genus two $SL(2,C)$-character variety and connects these to symmetry-reduced moduli spaces in supersymmetric quantum field theories.

Findings

Identification of irreducible components of fixed point sets.

Discovery of coincidences between fixed loci for different subgroups.

Proposal of new geometric candidates for moduli spaces in SCFTs.

Abstract

We investigate irreducible components of the fixed point sets of $ SL(2,\mathbb{C}) $-character variety of the genus two surface group under orientation preserving actions of the finite groups of the $ Mod(\Sigma_{2}) $. We work in the $ \mathcal{O} $-generator presentation of the genus two DAHA and its classical limit $ \mathcal{A}_{q=1,t} $, where we observe nontrivial coincidences between fixed loci attached to different subgroups and establish genus/irregularity transitions. The subvarieties obtained in this way provide novel geometric candidates for symmetry-reduced moduli spaces relevant to $ 4d $ $ \mathcal{N} = 2 $ SCFTs.