Spinor-valued Higgs fields

TL;DR

This paper explores the geometry of holomorphic vector bundles with spinor-valued Higgs fields over Riemann surfaces, highlighting differences from traditional Higgs bundles and using mod 2 index to classify moduli spaces.

Contribution

It introduces a new class of Higgs bundles involving spinor structures and distinguishes moduli space families using the mod 2 index, with explicit low-genus examples.

Findings

Identification of two distinct families of moduli spaces

Use of mod 2 index to differentiate these families

Examples provided in low genus cases

Abstract

We investigate the geometry of holomorphic vector bundles $E$ over a Riemann surface $C$ together with a section of the endomorphism bundle tensored with $K^{1/2}$ -- a square root of the canonical bundle $K$. These parallel to some extent the various features of usual Higgs bundles, such as spectral curve constructions, but some features are radically different. We make essential use of the mod 2 index to distinguish two families of moduli spaces, and provide examples in low genus.