The Automorphism Equivariant Hitchin Index

TL;DR

The paper defines the automorphism equivariant Hitchin index $\chi_{\mathbb{T}}(\mathcal{M},\mathcal{L}^k,f)(t)$ for rank two Higgs bundles with odd determinant over a Riemann surface $X$ equipped with a finite-order automorphism $f$. It develops a fixed-locus description in terms of parabolic Higgs moduli on the quotient $\tilde{X}$ and proves a localization formula that expresses the index as a sum of cohomological pairings on fixed components $\tilde{\mathcal{M}}^{\mathbb{T}}_{i,j}$ with canonical classes $\tilde{\chi}_{i,j}$ and $\tilde{\omega}_{i}$. In the Galois case, the index reduces to the base curve's quantization via a $t^p$-rescaling, and the index is shown to be determined by the Seifert invariants of the mapping torus $M_f$. The work connects non-abelian Hodge theory, geometric quantization, and quantum topology by relating the Hitchin index to WRT-type invariants and complex Chern-Simons theory, offering a path toward explicit computation and a deeper topological interpretation of Higgs-bundle quantization.

Abstract

Let T be the one-dimensional complex torus. We consider the action of an automorphism of a Riemann surface X on the cohomology of the T-equivariant determinant line bundle over the moduli space of rank two Higgs bundles on X with fixed determinant of odd degree. We define and study the automorphism equivariant Hitchin index. We prove a formula for it in terms of cohomological pairings of canonical T-equivariant classes of certain moduli spaces of parabolic Higgs bundles over the quotient Riemann surface.

Theorems & Definitions (47)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Corollary 1.1
• Definition 1.2
• Corollary 1.2
• Corollary 1.3
• Remark 1
• Definition 2.1
• Definition 2.2