Higgs bundles for real groups and the HitchinKostantRallis section
Abstract
We consider the moduli space of polystable $L$twisted $G$Higgs bundles over a compact Riemann surface $X$, where $G$ is a real reductive Lie group, and $L$ is a holomorphic line bundle over $X$. Evaluating the Higgs field at a basis of the ring of polynomial invariants of the isotropy representation, one defines the Hitchin map. This is a map to an affine space, whose dimension is determined by $L$ and the degrees of the polynomials in the basis. Building up on the work of KostantRallis and Hitchin, in this paper, as a first step in the study of the Hitchin map, we construct a section of this map. This generalizes the section constructed by Hitchin when $L$ is the canonical line bundle of $X$ and $G$ is complex. In this case the image of the section is related to the HitchinTeichmüller components of the moduli space of representations of the fundamental group of $X$ in $G_{\mathrm{Split}}$, a split real form of $G$. In fact, our construction is very natural in that we can start with the moduli space for $G_{\mathrm{Split}}$, instead of $G$, and construct the section for the Hitchin map for $G_{\mathrm{Split}}$ directly. The construction involves the notion of maximal split subgroup of a real reductive Lie group.
 Publication:

arXiv eprints
 Pub Date:
 November 2015
 arXiv:
 arXiv:1511.02611
 Bibcode:
 2015arXiv151102611G
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Differential Geometry;
 Primary 14H60;
 Secondary 53C07;
 58D29
 EPrint:
 50 pages, we have made minor corrections to version 1 and suppressed the last section