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Register a new userSO(p,q)-Higgs bundles and higher Teichmuller components
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Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this paper we describe new examples of such `exotic components in moduli spaces of SO(p,q)-Higgs bundles on closed Riemann surfaces or, equivalently, moduli spaces of surface group representations into the Lie group SO(p,q). Furthermore, we discuss how these exotic components are related to the notion of positive Anosov representations recently developed by Guichard and Wienhard. We also provide a complete count of the connected components of these moduli spaces (except for SO(2,q), with q> 3).
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We introduce a new class of $mathfrak{sl}_2$-triples in a complex simple Lie algebra $mathfrak{g}$, which we call magical. Such an $mathfrak{sl}_2$-triple canonically defines a real form and various decompositions of $mathfrak{g}$. Using this decomposition data, we explicitly parameterize special connected components of the moduli space of Higgs bundles on a compact Riemann surface $X$ for an associated real Lie group, hence also of the corresponding character variety of representations of $pi_1X$ in the associated real Lie group. This recovers known components when the real group is split, Hermitian of tube type, or $mathrm{SO}_{p,q}$ with $1<pleq q$, and also constructs previously unknown components for the quaternionic real forms of $mathrm{E}_6$, $mathrm{E}_7$, $mathrm{E}_8$ and $mathrm{F}_4$. The classification of magical $mathfrak{sl}_2$-triples is shown to be in bijection with the set of $Theta$-positive structures in the sense of Guichard--Wienhard, thus the mentioned parameterization conjecturally detects all examples of higher Teichmuller spaces. Indeed, we discuss properties of the surface group representations obtained from these Higgs bundle components and their relation to $Theta$-positive Anosov representations, which indicate that this conjecture holds.
For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups, are the only previously know cases where natural invariants do not fully distinguish connected components. In this note we announce the existence of new such exotic components in the moduli spaces for the groups $mathrm{SO}(p,q)$ with $2<p<q$. These groups lie outside formerly know classes of groups associated with exotic components.
These are the lecture notes from my course in the January 2011 School on Moduli Spaces at the Newton Institute. I give an introduction to Higgs bundles and their application to the study of character varieties for surface group representations.
A $mathrm{U}(p,q)$-Higgs bundle on a Riemann surface (twisted by a line bundle) consists of a pair of holomorphic vector bundles, together with a pair of (twisted) maps between them. Their moduli spaces depend on a real parameter $alpha$. In this paper we study wall crossing for the moduli spaces of $alpha$-polystable twisted $mathrm{U}(p,q)$-Higgs bundles. Our main result is that the moduli spaces are birational for a certain range of the parameter and we deduce irreducibility results using known results on Higgs bundles. Quiver bundles and the Hitchin-Kobayashi correspondence play an essential role.
This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic bundles and generalised parabolic $L$-twisted Hitchin pairs on its normalisation as well as on an analogue of the Hitchin map for generalised parabolic $L$-twisted Hitchin pairs.
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