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.
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BibTeX
@misc{tsiavou2026equivariant,
title = {{Equivariant Seiberg-Witten equations on Hermitian symmetric spaces}},
author = {Rafailia-Persefoni Tsiavou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14045},
eprint = {2610.14045},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14045},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Equivariant Seiberg-Witten equations on Hermitian symmetric spaces
AU - Rafailia-Persefoni Tsiavou
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.14045
UR - https://arxiv.org/abs/2610.14045
AB - 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.
ER -