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Asymptotic geometry of the Higgs bundle moduli space

Higgs bundles have been introduced by Hitchin as solutions of the self-duality equations on a Riemann surface. One of the key features of the finite-dimensional space of solutions $${\mathcal M}(r,d)$$ is that it carries a natural hyperkähler metric.

The nonabelian Hodge theorem of Corlette identifies the Higgs bundle moduli space with the character variety of representations $$\rho\colon \pi_1(\Sigma)\to GL(r,{\mathbb C})$$. This arguably provides the strongest motivation to study the Higgs bundle moduli space: Questions about surface group representations (and hence in particular geometric structures on surfaces) may be tackled using holomorphic techniques.

A basic question for understanding the large scale geometry of the moduli space with its hyperkähler metric is one asked by Hitchin about the dimension of the space of $$L^2$$-harmonic forms, later turned into a more precise conjecture by Hausel. The focus of this project is somewhat broader in the sense that it aims at providing concrete asymptotic models for the ends of the moduli spaces in question.

The ongoing and planned research projects concern:

• The semiflat conjecture
• Multiple zeroes, parabolic and higher rank Higgs bundles
• Parabolic Higgs bundles and hyperpolygons
• The geometry of the Hitchin component
• Higgs bundles under degenerations of the underlying Riemann surface
• Infinite-energy harmonic maps and pleated surfaces

## Publications

We study the asymptotics of the natural $$L^2$$ metric on the Hitchin moduli space with group $$G=SU(2)$$. Our main result, which addresses a detailed conjectural picture made by Gaiotto, Neitzke and Moore, is that on the regular part of the Hitchin system, this metric is well-approximated by the semiflat metric. We prove that the asymptotic rate of convergence for gauged tangent vectors to the moduli space has a precise polynomial expansion, and hence that the the difference between the two sets of metric coefficients in a certain natural coordinate system also has polynomial decay. Very recent work by Dumas and Neitzke indicates that the convergence rate for the metric is exponential, at least in certain directions.

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

We prove a gluing theorem for solutions $$(A_0, \Phi_0)$$  of Hitchin's self-duality equations with logarithmic singularities on a rank-$$2$$ vector bundle over a noded Riemann surface $$\Sigma$$ representing a boundary point of Teichmüller moduli space. We show that every nearby smooth Riemann surface $$\Sigma_1$$ carries a smooth solution $$(A_1, \Phi_1)$$ of the self-duality equations, which may be viewed as a desingularization of $$(A_0, \Phi_0)$$.

 Journal Adv. Math. Publisher Elsevier Volume 322 Pages 637-681 Link to preprint version Link to published version

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

In this note we study some analytic properties of the linearized self-duality equations on a family of smooth Riemann surfaces $$\Sigma_R$$ converging for $$R\searrow 0$$ to a surface $$\Sigma_0$$ with a finite number of nodes. It is shown that the linearization along the fibres of the Hitchin fibration $$\mathcal M_d \to \Sigma_R$$ gives rise to a graph-continuous Fredholm family, the index of it being stable when passing to the limit. We also report on similarities and differences between properties of the Hitchin fibration in this degeneration and in the limit of large Higgs fields as studied in Mazzeo et al. (Duke Math. J. 165(12):2227–2271, 2016).

 Journal Abh. Math. Semin. Univ. Hambg. Publisher Springer Berlin Heidelberg Volume 86 Pages 189--201 Link to preprint version Link to published version

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

We associate to each stable Higgs pair $$(A_0,\Phi_0)$$ on a compact Riemann surface X a singular limiting configuration $$(A_\infty,\Phi_\infty)$$, assuming that $$\det\Phi$$ has only simple zeroes. We then prove a desingularization theorem by constructing a family of solutions $$(A_t,\Phi_t)$$ to Hitchin's equations which converge to this limiting configuration as $$t\to\infty$$. This provides a new proof, via gluing methods, for elements in the ends of the Higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space.

 Journal Duke Math. J. Publisher Duke University Press Volume 165 Pages 2227-2271 Link to preprint version Link to published version

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

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## Team Members

Sven Marquardt
Doctoral student
Christian-Albrechts-Universität zu Kiel
marquardt(at)math.uni-kiel.de

Dr. Claudio Meneses Torres
Researcher
Christian-Albrechts-Universität zu Kiel
meneses(at)math.uni-kiel.de

Dr. Jan Swoboda