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

Semeon Arthamonov, Anton Pribytok

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(Σ_{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.

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

Abstract

We investigate irreducible components of the fixed point sets of -character variety of the genus two surface group under orientation preserving actions of the finite groups of the . We work in the -generator presentation of the genus two DAHA and its classical limit , 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 SCFTs.
Paper Structure (48 sections, 80 equations, 11 figures)

This paper contains 48 sections, 80 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Character varieties.
  3. Mapping Class Group Action.
  4. Classical Limit of DAHA and Mod(Σ_2) subgroups
  5. G_a subgroup
  6. G_b or G_f subgroup
  7. t-deformation.
  8. G_c subgroup
  9. $\bm{t=1}$:
  10. t-deformed:
  11. G_e subgroup
  12. t-deformed
  13. G_h or G_o subgroup
  14. $\bm{t=1}$:
  15. t-deformed:
  16. ...and 33 more sections

Figures (11)

  • Figure 1: Left Dehn twist along simple closed curve $\gamma$
  • Figure 2: Simple closed curves corresponding to generators of $\mathrm{Mod}(\Sigma_2)$
  • Figure 3: Branching data of the covering maps in Section \ref{['sec:GbGf']}.
  • Figure 4: Branching data of the covering maps in case $G_c$
  • Figure 5: Branching data of the covering maps in case $G_e$
  • ...and 6 more figures