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

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 $Mod({\mathrm{\Sigma }}_2)$. 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.