Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userVarious generalizations and deformations of $PSL(2,mathbb{R})$ surface group representations and their Higgs bundles
180
0
0.0
(
0
)
Authors
Brian Collier
Ask ChatGPT about the research
No Arabic abstract
Recall that the group $PSL(2,mathbb R)$ is isomorphic to $PSp(2,mathbb R), SO_0(1,2)$ and $PU(1,1).$ The goal of this paper is to examine the various ways in which Fuchsian representations of the fundamental group of a closed surface of genus $g$ into $PSL(2,mathbb R)$ and their associated Higgs bundles generalize to the higher rank groups $PSL(n,mathbb R), PSp(2n,mathbb R), SO_0(2,n), SO_0(n,n+1)$ and $PU(n,n)$. For the $SO_0(n,n+1)$-character variety, we parameterize $n(2g-2)$ new connected components as the total space of vector bundles over appropriate symmetric powers of the surface and study how these components deform in the $SO_0(n,n+2)$-character variety. This generalizes results of Hitchin for $PSL(2,mathbb R)$.
rate research
Read More
We show that for every nonelementary representation of a surface group into $SL(2,{mathbb C})$ there is a Riemann surface structure such that the Higgs bundle associated to the representation lies outside the discriminant locus of the Hitchin fibration.
We study the character variety of representations of the fundamental group of a closed surface of genus $ggeq2$ into the Lie group SO(n,n+1) using Higgs bundles. For each integer $0<dleq n(2g-2),$ we show there is a smooth connected component of the character variety which is diffeomorphic to the product of a certain vector bundle over a symmetric product of a Riemann surface with the vector space of holomorphic differentials of degree 2,4,...,2n-2. In particular, when d=n(2g-2), this recovers Hitchins parameterization of the Hitchin component. We also exhibit $2^{2g+1}-1$ additional connected components of the SO(n,n+1)-character variety and compute their topology. Moreover, representations in all of these new components cannot be continuously deformed to representations with compact Zariski closure. Using recent work of Guichard and Wienhard on positivity, it is shown that each of the representations which define singularities (i.e. those which are not irreducible) in these $2^{2g+1}-1$ connected components are positive Anosov representations.
We develop a complete Hitchin-Kobayashi correspondence for twisted pairs on a compact Riemann surface X. The main novelty lies in a careful study of the the notion of polystability for pairs, required for having a bijective correspondence between solutions to the Hermite-Einstein equations, on one hand, and polystable pairs, on the other. Our results allow us to establish rigorously the homemomorphism between the moduli space of polystable G-Higgs bundles on X and the character variety for representations of the fundamental group of X in G. We also study in detail several interesting examples of the correspondence for particular groups and show how to significantly simplify the general stability condition in these cases.
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.
We give an overview of the work of Corlette, Donaldson, Hitchin and Simpson leading to the non-abelian Hodge theory correspondence between representations of the fundamental group of a surface and the moduli space of Higgs bundles. We then explain how this can be generalized to a correspondence between character varieties for representations of surface groups in real Lie groups G and the moduli space of G-Higgs bundles. Finally we survey recent joint work with Bradlow, Garcia-Prada and Mundet i Riera on the moduli space of maximal Sp(2n,R)-Higgs bundles.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
