Anti-holomorphic involutions of the moduli spaces of Higgs bundles

Indranil Biswas,Oscar García-Prada

doi:10.5802/jep.16

Abstract

We study anti-holomorphic involutions of the moduli space of principal $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a complex semisimple Lie group. These involutions are defined by fixing anti-holomorphic involutions on both $X$ and $G$. We analyze the fixed point locus in the moduli space and their relation with representations of the orbifold fundamental group of $X$ equipped with the anti-holomorphic involution. We also study the relation with branes. This generalizes work by Biswas--Garc\'{\i}a-Prada--Hurtubise and Baraglia--Schaposnik.

Highlights

Let G be a complex semisimple affine algebraic group with Lie algebra g
A G–Higgs bundle over X is a pair (E, φ), where E is a holomorphic principal G-bundle over X and φ is a holomorphic section of E(g) ⊗ K with E(g) being the vector bundle associated to E for the adjoint action of G on g and K being the canonical line bundle on X
J.É.P. — M., 2015, tome 2 defined on the moduli space of Higgs bundles for the Riemann surface X

Summary

Introduction

Let G be a complex semisimple affine algebraic group with Lie algebra g. — M., 2015, tome 2 defined on the moduli space of Higgs bundles for the Riemann surface X The study of these branes is of great interest in connection with mirror symmetry and the Langlands correspondence in the theory of Higgs bundles (see [20, 18, 3, 2]). We identify these involutions in the moduli space of representations of the fundamental group of X in G, and describe the fixed points corresponding to the (α, σ, c, ±)-pseudo-real G-Higgs bundles in terms of representations of the orbifold fundamental group of (X, α) in a group whose underlying set is G × Z/2Z. We want to thank Steve Bradlow and Laura Schaposnik for useful discussions

G-Higgs bundles and representations of the fundamental group

Real G-Higgs bundles

Findings

Involutions of moduli spaces