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Asymptotic geometry of the Higgs bundle moduli space (II)

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 use the theory of Gaiotto, Moore and Neitzke to construct a set of Darboux coordinates on the moduli space $$\mathcal{M}$$ of weakly parabolic $$SL(2,\mathbb{C})$$-Higgs bundles. For generic Higgs bundles$$(\mathcal{E},R\Phi)$$ with $$R\gg 0$$ the coordinates are shown to be dominated by a leading term that is given by the coordinates for a corresponding simpler space of limiting configurations and we prove that the deviation from the limiting term is given by a remainder that is exponentially suppressed in $$R$$.

We then use this result to solve an associated Riemann-Hilbert problem and construct a twistorial hyperkähler metric $$g_{\text{twist}}$$ on $$\mathcal{M}$$. Comparing this metric to the simpler semiflat metric $$g_{\text{sf}}$$, we show that their difference is $$g_{\text{twist}}-g_{\text{sf}}=O\left(e^{-\mu R}\right)$$, where $$\mu$$ is a minimal period of the determinant of the Higgs field.

Related project(s):
77Asymptotic geometry of the Higgs bundle moduli space II

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## Team Members

M.Ed. Maximilian Holdt
Researcher
Christian-Albrechts-Universität zu Kiel
holdt(at)math.uni-kiel.de

Prof. Dr. Hartmut Weiss
Christian-Albrechts-Universität zu Kiel
weiss(at)math.uni-kiel.de