Cayley and Langlands type correspondences for orthogonal Higgs bundles

TL;DR

This paper explores geometric descriptions of moduli spaces of real orthogonal and symplectic Higgs bundles using Cayley and Langlands correspondences, revealing new components and abelianization methods.

Contribution

It provides a comprehensive geometric framework for real Higgs bundles, completing abelianization for all quasi-split real forms and explaining component emergence via spectral data.

Findings

Complete abelianization of real slices for all quasi-split real forms

Description of how extra components arise from spectral data

Geometric interpretation of moduli spaces via Cayley and Langlands correspondences

Abstract

Through Cayley and Langlands type correspondences, we give a geometric description of the moduli spaces of real orthogonal and symplectic Higgs bundles of any signature in the regular fibres of the Hitchin fibration. As applications of our methods, we complete the concrete abelianization of real slices corresponding to all quasi-split real forms, and describe how extra components emerge naturally from the spectral data point of view.