Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 111, 21 pages      arXiv:2403.07258      https://doi.org/10.3842/SIGMA.2024.111

Harmonic Metrics for Higgs Bundles of Rank 3 in the Hitchin Section

Hitoshi Fujioka
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Received April 21, 2024, in final form November 29, 2024; Published online December 11, 2024

Abstract
Given a tuple of holomorphic differentials on a Riemann surface, one can define a Higgs bundle in the Hitchin section and a natural symmetric pairing of the Higgs bundle. We study whether a Higgs bundle of rank 3 in the Hitchin section has a compatible harmonic metric when the spectral curve is a 2-sheeted branched covering of the Riemann surface. In particular, we give a condition for Higgs bundles in the Hitchin section on $\mathbb{C}$ or $\mathbb{C}^*$ to have compatible harmonic metrics.

Key words: Higgs bundles; Hitchin section; harmonic metrics.

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