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Real Slices of Parabolic $\mathrm{SL}(r,\mathbb{C})$-Opers

Sanjay Amrutiya, Sandipan Das

Abstract

Let $X$ be a Riemann surface equipped with an anti-holomorphic involution $σ_X$. We show that this induces a natural anti-holomorphic involution on the space of parabolic $\mathrm{SL}(r,\mathbb{C})$-opers. The fixed-point locus of this involution is defined as real slice. We further study the induced involutions on different descriptions of parabolic $\mathrm{SL}(r,\mathbb{C})$-opers, in particular differential operators, and prove that these involutions coincide.

Real Slices of Parabolic $\mathrm{SL}(r,\mathbb{C})$-Opers

Abstract

Let be a Riemann surface equipped with an anti-holomorphic involution . We show that this induces a natural anti-holomorphic involution on the space of parabolic -opers. The fixed-point locus of this involution is defined as real slice. We further study the induced involutions on different descriptions of parabolic -opers, in particular differential operators, and prove that these involutions coincide.
Paper Structure (11 sections, 18 theorems, 84 equations)

This paper contains 11 sections, 18 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parabolic Vector Bundle and Parabolic Connection
  4. Parabolic Gunning Bundle
  5. Parabolic $\mathrm{SL}(r,\mathbb{C})$-Opers
  6. Differential Operator and Symbol
  7. Orbifold Bundle
  8. Equivariant $\mathrm{SL}(r,\mathbb{C})$-opers:
  9. Correspondence between parabolic $\mathrm{SL}(r,\mathbb{C})$-opers and parabolic differential operator:
  10. Real Parabolic Vector Bundle
  11. Real Slice of Parabolic $\mathrm{SL}(r,\mathbb{C})$-Opers

Key Result

Proposition 2.1

Suppose $D$ be a parabolic connection on a parabolic vector bundle $V_*$, then $\sigma_X^*\overline{D}$ is a parabolic connection on $\sigma_X^*\overline{V_*}$.

Theorems & Definitions (45)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • ...and 35 more