Spinors, Lagrangians and rank 2 Higgs bundles

TL;DR

This paper explores the relationship between Dirac operators, spinors, and Higgs bundles on Riemann surfaces, revealing new Lagrangian subvarieties in the moduli space of Higgs bundles with connections to Prym varieties and Brill-Noether loci.

Contribution

It introduces a novel construction of Lagrangian subvarieties in the Higgs bundle moduli space via Dirac operators and spinors, linking to spectral curves and Prym varieties.

Findings

Constructed Lagrangian subvarieties from null-spaces of Dirac operators.

Connected these subvarieties to spectral data and Prym varieties.

Detailed analysis provided for genus 2 case.

Abstract

The paper considers the Dirac operator on a Riemann surface coupled to a symplectic holomorphic vector bundle W. Each spinor in the null-space generates through the moment map a Higgs bundle, and varying W one obtains a holomorphic Lagrangian subvariety in the moduli space of Higgs bundles. Applying this to the irreducible symplectic representations of SL(2) we obtain Lagrangian submanifolds of the rank 2 moduli space which link up with m-period points on the Prym variety of the spectral curve as well as Brill-Noether loci on the moduli space of semistable bundles. The case of genus 2 is investigated in some detail.