Equivariant Seiberg-Witten equations on Hermitian symmetric spaces

Abstract

We formulate and solve a $G$-equivariant analogue of the Seiberg-Witten equations on any irreducible Hermitian symmetric space $G/K$. We use a canonical $K$-invariant form to define a twisted Hodge operator generalising self-duality to higher dimensions. The resulting equivariant Seiberg-Witten equations arise, for a compact $G/K$, as conditions for the absolute minima of an energy functional. Their linear part has a representation theoretic interpretation: the same Weyl group data control the highest-weight realisation of the positive Dirac kernel in the compact case and its discrete-series realisation on the non-compact dual. This space of harmonic spinors collapses almost entirely by the quadratic conditions of the monopole equation. In the compact case, the equivariant moduli set, when non-empty, is a single gauge class, identifying with the $G$-invariant part of Bradlow's trichotomy for vortices, while in the non-compact case, every irreducible solution sits outside the discrete series.