Spinors, Lagrangians and rank 2 Higgs bundles

This article considers the Dirac operator on a Riemann surface coupled to a symplectic holomorphic vector bundle W . Each spinor in the null‐space (which is a holomorphic section of W ⊗ K 1 / 2 ) generates through the moment map a Higgs field Φ , and varying W one obtains a holomorphic Lagrangian subvariety in the moduli space of Higgs bundles. Applying this to the irreducible symplectic representations of S L ( 2 , C ) we obtain Lagrangian submanifolds of the rank 2 Higgs bundle 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.