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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.

 Journal Comm. Math. Phys. Publisher Springer Volume 367, no. 1 Pages 151-191 Link to preprint version Link to published version

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

Moduli spaces of stable parabolic bundles of parabolic degree $$0$$ over the Riemann sphere are stratified according to the Harder-Narasimhan filtration of underlying vector bundles. Over a Zariski open subset $$\mathscr{N}_{0}$$ of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function $$\mathscr{S}$$ is defined as the regularized critical value of the non-compact Wess-Zumino-Novikov-Witten action functional. The definition of $$\mathscr{S}$$ depends on a suitable notion of parabolic bundle 'uniformization map' following from the Mehta-Seshadri and Birkhoff-Grothendieck theorems. It is shown that $$-\mathscr{S}$$ is a primitive for a (1,0)-form $$\vartheta$$ on $$\mathscr{N}_{0}$$ associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that $$-\mathscr{S}$$ is a Kähler potential for $$(\Omega-\Omega_{\mathrm{T}})|_{\mathscr{N}_{0}}$$,  where $$\Omega$$ is the Narasimhan-Atiyah-Bott Kähler form in $$\mathscr{N}$$ and $$\Omega_{\mathrm{T}}$$ is a certain linear combination of tautological $$(1,1)$$-forms associated with the marked points. These results provide an explicit relation between the cohomology class $$[\Omega]$$ and tautological classes, which holds globally over certain open chambers of parabolic weights where $$\mathscr{N}_{0} = \mathscr{N}$$.

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

Motivated by the work of Leznov-Mostovoy, we classify the linear deformations of standard $$2n$$-dimensional phase space that preserve the obvious symplectic $$\mathfrak{o}(n)$$-symmetry. As a consequence, we describe standard phase space, as well as $$T^{*}S^{n}$$ and $$T^{*}\mathbb{H}^{n}$$ with their standard symplectic forms, as degenerations of a 3-dimensional family of coadjoint orbits, which in a generic regime are identified with the Grassmannian of oriented 2-planes in $$\mathbb{R}^{n+2}$$.

 Journal Journal of Geometric Mechanics Publisher American Institute of Mathematical Sciences Volume 11(1) Pages 45-58 Link to preprint version Link to published version

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

We define a functional $${\cal J}(h)$$ for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles, and study some of its basic properties. We show that $${\cal J}(h)$$ is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite-Yang-Mills metrics. We derive a formula relating $${\cal J}(h)$$ and another functional $${\cal I}(h)$$, closely related to the Yang-Mills-Higgs functional, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of $${\cal J}(h)$$, which is expressed as a certain $$L^{2}$$-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of $${\cal J}(h)$$ if and only if the corresponding Hitchin-Simpson mean curvature is parallel with respect to the Hitchin-Simpson connection.

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

For a smooth manifold $$M$$, possibly with boundary and corners, and a Lie group $$G$$, we consider a suitable description of gauge fields in terms of parallel transport, as groupoid homomorphisms from a certain path groupoid in $$M$$ to $$G$$.  Using a cotriangulation $$\mathscr{C}$$ of $$M$$, and collections of finite-dimensional families of paths relative to $$\mathscr{C}$$, we define a homotopical equivalence relation of parallel transport maps, leading to the concept of an extended lattice gauge (ELG) field. A lattice gauge field, as used in Lattice Gauge Theory, is part of the data contained in an ELG field, but the latter contains further local topological information sufficient to reconstruct a principal $$G$$-bundle on $$M$$ up to equivalence. The space of ELG fields of a given pair $$(M,\mathscr{C})$$ is a covering for the space of fields in Lattice Gauge Theory, whose connected components parametrize equivalence classes of principal $$G$$-bundles on $$M$$. We give a criterion to determine when ELG fields over different cotriangulations define equivalent bundles.

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