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Register a new userHiggs bundles and the real symplectic group
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Authors
Peter B. Gothen
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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.
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Given a compact Riemann surface $Sigma$ of genus $g_Sigma, geq, 2$, and an effective divisor $D, =, sum_i n_i x_i$ on $Sigma$ with $text{degree}(D), <, 2(g_Sigma -1)$, there is a unique cone metric on $Sigma$ of constant negative curvature $-4$ such that the cone angle at each $x_i$ is $2pi n_i$ (see McOwen and Troyanov [McO,Tr]). We describe the Higgs bundle corresponding to this uniformization associated to the above conical metric. We also give a family of Higgs bundles on $Sigma$ parametrized by a nonempty open subset of $H^0(Sigma,,K_Sigma^{otimes 2}otimes{mathcal O}_Sigma(-2D))$ that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchins results in [Hi1], for the case $D,=, 0$.
The paper studies the locus in the rank 2 Higgs bundle moduli space corresponding to points which are critical for d of the Poisson commuting functions. These correspond to the Higgs field vanishing on a divisor of degree D. The degree D critical locus has an induced integrable system related to K(-D)-twisted Higgs bundles. Topological and differential-geometric properties of the critical loci are addressed.
Using Hitchins parameterization of the Hitchin-Teichmuller component of the $SL(n,mathbb{R})$ representation variety, we study the asymptotics of certain families of representations. In fact, for certain Higgs bundles in the $SL(n,mathbb{R})$-Hitchin component, we study the asymptotics of the Hermitian metric solving the Higgs bundle equations. This analysis is used to estimate the asymptotics of the corresponding family of flat connections as we scale the differentials by a real parameter. We consider Higgs fields that have only one holomorphic differential $q_n$ of degree $n$ or $q_{n-1}$ of degree $n-1.$ We also study the asymptotics of the associated family of equivariant harmonic maps to the symmetric space $SL(n,mathbb{R})/SO(n,mathbb{R})$ and relate it to recent work of Katzarkov, Noll, Pandit and Simpson.
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.
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.
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