  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.
