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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. The subsequent foundational works due to him, Donaldson, Corlette, Simpson and others extended the scope of this concept to more general Kähler manifolds and revealed deep connections to neighboring subjects such as complex geometry, nonlinear PDEs on manifolds, representations of surface groups, completely integrable systems and hyperkähler geometry.

The moduli space of solutions $$\mathcal M(r,d)$$, its geometric and analytic properties and its manifold interrelations with these neighboring subjects are the central topic of the current research proposal. Our guiding motive here is to complete the study of large scale geometric properties of the spaces $$\mathcal M(r,d)$$ by means of geometric analysis. These moduli spaces are typical examples of noncompact and in some cases singular manifolds. Aiming for a better understanding of their large scale and singularity structure provides for a rich field of difficult problems.

At the same time, the very different interconnections with neighboring fields indicated above establish Higgs bundles as a multifaceted subject. In recent years a number of new and promising research directions, which involve Higgs bundles in a crucial way, have emerged. These include the various approaches to compactifications of spaces of surface group representations and, in a completely different direction, several relevant research topics in geometry influenced by supersymmetric quantum field theory. For instance, this lead to remarkably accurate, yet largely conjectural predictions concerning the hyperkähler geometry of Higgs bundle moduli spaces due to Gaiotto, Moore and Neitzke. The scope of this project is to contribute substantially to some of these directions. We also plan to take up a number of entirely new research directions. For instance, we want to embark on an in-depth study of the real four-dimensional Hitchin moduli spaces which arise in the case where the underlying Riemann surface is, for instance, a four-punctures sphere.

The ongoing and planned research projects concern:

• Asymptotic properties of Higgs bundle moduli spaces near the discriminant locus
• Moduli spaces of parabolic Higgs bundles as gravitational instantons
• The high-energy limit of the nonabelian Hodge correspondence
• Hyperpolygons and twisted Higgs bundles

## Publications

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.

 Journal International Journal of Geometric Methods in Modern Physics Publisher World Scientific Volume 17(13) Line art. no. 2050200 Link to preprint version Link to published version

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

This paper relates different approaches to the asymptotic geometry of the

Hitchin moduli space of SL(2,C) Higgs bundles on a closed Riemann surface and,

via the nonabelian Hodge theorem, the character variety of SL(2,C)

representations of a surface group. Specifically, we find an asymptotic

correspondence between the analytically defined limiting configuration of a

sequence of solutions to the self-duality equations constructed by

Mazzeo-Swoboda-Weiss-Witt, and the geometric topological shear-bend parameters

of equivariant pleated surfaces due to Bonahon and Thurston. The geometric link

comes from a study of high energy harmonic maps. As a consequence we prove: (1)

the local invariance of the partial compactification of the moduli space by

limiting configurations; (2) a refinement of the harmonic maps characterization

of the Morgan-Shalen compactification of the character variety; and (3) a

comparison between the family of complex projective structures defined by a

quadratic differential and the realizations of the corresponding flat

connections as Higgs bundles, as well as a determination of the asymptotic

shear-bend cocycle of Thurston's pleated surface.

Given a generic stable strongly parabolic $SL(2,\mathbb{C})$-Higgs bundle

$(\mathcal{E}, \varphi)$, we describe the family of harmonic metrics $h_t$ for

the ray of Higgs bundles $(\mathcal{E}, t \varphi)$ for $t\gg0$ by perturbing

from an explicitly constructed family of approximate solutions

$h_t^{\mathrm{app}}$. We then describe the natural hyperK\"ahler metric on

$\mathcal{M}$ by comparing it to a simpler "semi-flat" hyperK\"ahler metric. We

prove that $g_{L^2} - g_{\mathrm{sf}} = O(\mathrm{e}^{-\gamma t})$ along a

generic ray, proving a version of Gaiotto-Moore-Neitzke's conjecture.

Our results extend to weakly parabolic $SL(2,\mathbb{C})$-Higgs bundles as

well.

In the case of the four-puncture sphere, we describe the moduli space and

metric more explicitly. In this case, we prove that the hyperk\"ahler metric is

ALG and show that the rate of exponential decay is the conjectured optimal one,

$\gamma=4L$, where $L$ is the length of the shortest geodesic on the base curve

measured in the singular flat metric $|\mathrm{det}\, \varphi|$.

 Pages 73 pages Link to preprint version

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.

 Journal Advances in Theoretical and Mathematical Physics Publisher International Press Volume 23(8) Pages 2207 – 2254 Link to preprint version Link to published version

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

We survey several mathematical developments in the holonomy approach to gauge theory. A cornerstone of such approach is the introduction of group structures on spaces of based loops on a smooth manifold, relying on certain homotopy equivalence relations --- such as the so-called thin homotopy --- and the resulting interpretation of gauge fields as group homomorphisms to a Lie group G satisfying a suitable smoothness condition, encoding the holonomy of a gauge orbit of smooth connections on a principal G-bundle. We also prove several structural results on thin homotopy, and in particular we clarify the difference between thin equivalence and retrace equivalence for piecewise-smooth based loops on a smooth manifold, which are often used interchangeably in the physics literature. We conclude by listing a set of questions on topological and functional analytic aspects of groups of based loops, which we consider to be fundamental to establish a rigorous differential geometric foundation of the holonomy formulation of gauge theory.

 Pages 1-20 Link to preprint version

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

Parallel transport as dictated by a gauge field determines a collection of local reference systems. Comparing local reference systems in overlapping regions leads to an ensemble of algebras of relational kinematical observables for gauge theories including general relativity. Using an auxiliary cellular decomposition, we propose a discretization of the gauge field based on a decimation of the mentioned ensemble of kinematical observables. The outcome is a discrete ensemble of local subalgebras of 'macroscopic observables' characterizing a measuring scale. A set of evaluations of those macroscopic observables is called an extended lattice gauge field because it determines a G-bundle over M (and over submanifolds of M that inherit a cellular decomposition) together with a lattice gauge field over an embedded lattice. A physical observable in our algebra of macroscopic observables is constructed. An initial study of aspects of regularization and coarse graining, which are special to this description of gauge fields over a combinatorial base, is presented. The physical relevance of this extension of ordinary lattice gauge fields is discussed in the context of quantum gravity.

 Journal Classical and Quantum Gravity Publisher Inst. Phys. Volume 36, no. 23 Link to preprint version Link to published version

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

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 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