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Register a new userCritical loci for Higgs bundles
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Authors
Nigel Hitchin
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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.
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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.
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 $D,=, 0$.
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.
By studying the Higgs bundle equations with the gauge group replaced by the group of symplectic diffeomorphisms of the 2-sphere we encounter the notion of a folded hyperkaehler 4-manifold and conjecture the existence of a family of such metrics parametrised by an infinite-dimensional analogue of Teichmueller space.
We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the bundle to admit a holomorphic structure. We study compatibility under the Chern isomorphism of these filtrations with the Hodge filtration on cohomology.
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