Activities
13/05/2022
to 14/05/2022
Higgs bundles at infinity II

Higgs bundles, Hitchin’s self-duality equations and hyper-Kähler manifolds have played a prominent role in the work of many geometers, physicists and algebraists in the last 3 decades. Within these broad and active research areas, there are three recent developments of special interest and directly related to the scope of the Schwerpunktprogramm:

• Motivated by the work of Gaiotto-Moore-Neitzke, the hyper-Kähler geometry of the moduli space of solutions of Hitchin’s equations near infinity, i.e., for large Higgs fields, has been investigated in detail. In particular, complex 2-dimensional examples have recently been shown to be ALG spaces.

• Solutions of the self-duality equations with special singularities along curves have been investigated, and some examples have been constructed. The corresponding equivariant harmonic maps intersect the boundary at infinity perpendicularly. It was shown that certain spaces of singular solutions are equipped with hyper-Kähler structures, such that a normalized energy serves as the Kähler potential.

• In connection to recent classifications of gravitational instantons, it is of paramount importance to clarify the dependance of the moduli space’s hyper-Kähler geometry on more general parabolic structures. This problem provides a natural link between its differential and algebro-geometric facets.

There has been further important progress and interesting applications, including, but not limited to, higher Teichmüller theory and the investigation of geometric structures on surfaces. Since the number of people in the field working at German universities has recently increased, we foresee the necessity to establish a regular workshop in Germany focused on these subjects.

This is the second installment of the workshop series "Higgs bundles at infinity",  whose goals are to achieve a continuous exchange between the different working groups and related colleagues, which could also serve as a platform supporting young scientists and under-represented groups. We intend to continue hosting these meetings once per semester within the priority program.

Applications for financial support for young participants are open until May 2nd.

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### Tame parahoric nonabelian Hodge correspondence

Pengfei Huang, Mathematisches Institut, Universität Heidelberg

Parabolic structures appear in Mazzeo–Swoboda–Weiss–Witt’s study of the limiting configuration of the ends of the Dolbeault moduli space of Higgs bundles. Historically, parabolic structures were introduced by Seshadri and naturally appear in the nonabelian Hodge correspondence of Simpson for the group GL(n, C) over a noncompact algebraic curve. The full problem involving an arbitrary complex reductive group G is much more subtle and the generalization of a weighted flag in this case does not seem to provide an one-to-one correspondence between the Dolbeault and the Betti spaces. The correct set up for groups beyond GL(n, C) is in the language of parahoric group schemes and a nonabelian Hodge correspondence can be fully described in this case. Joint work in progress with G. Kydonakis, H. Sun and L. Zhao.

### Partial compactifications of ALE hyperkähler four manifolds of type A

Graeme Wilkin, Department of Mathematics, University of York

In this talk I will describe a number of different ways to view ALE hyperkähler four manifolds of type A. From one point of view we can see an analog of the Hitchin system (with affine fibres rather than projective), and by compactifying the fibres we arrive at a partial compactification which (for type A_n with n=3 or n=4) has the structure of a holomorphic symplectic manifold. In particular, we can see continuous families of complex Lagrangian submanifolds inside this space, where one family appears as fibres of the Hitchin map and the other families can be viewed as rotations of these fibres. The whole picture has many features in common with general moduli of Higgs bundles, and I will focus on these analogies during the talk. This is joint work with Rafe Mazzeo.

### Constructing Higgs bundles in higher Teichmüller spaces by gluing

Georgios Kydonakis, Mathematisches Institut, Universität Heidelberg

We will see an application of standard gluing techniques from gauge theory in order to obtain specific models of Higgs bundles lying inside certain connected components of the moduli space. The parameters involved in the procedure are the genus of the underlying Riemann surface and the holonomy of a surface group representation along the boundary components of the surface. Such gauge theoretic techniques are also used to understand the family of Teichmüller moduli spaces in the limit when one approaches a boundary point of the Deligne-Mumford compactification.

### Instanton-corrected hyperkähler geometries

Ivan Tulli, Universität Hamburg

Abstract: Given a 4d N=2 supersymmetric field theory, one can obtain an associated hyperkähler (HK) manifold by considering its dimensional reduction on a circle. Such HK metrics were described in the physics literature in twistorial terms by Gaiotto-Moore-Neitzke (arXiv:0807.4723), and are an instance of instanton-corrected HK metrics". When we restrict to the special case of 4d N=2 theories of class S, the corresponding HK manifold is expected to be a Hitchin moduli space. The main goal of this talk is to present a mathematical treatment of the constructions of such instanton corrected HK metrics in the simpler case of mutually-local" instanton corrections. Namely, given the structure of an affine special Kähler (ASK) manifold together with a (compatible) variation of mutually local BPS structures, we construct an instanton corrected HK manifold. The simplifying assumption of mutual-locality allows for an explicit description of the HK structure which does not require dealing with the twistor space. For this I will mostly follow section 3 of the joint work with V. Cortés arXiv:2105.09011. I will close with some comments on possible relations between the Hitchin moduli space HK metric and the above mutually local instanton-corrected HK metrics. P.S: no physics knowledge will be required for the talk.

1. Renan Assimos | Hannover
2. Sven-Ole Behrend | CAU Kiel
3. Balázs Márk Békési | Leibniz Universität Hannover
4. Roger Bielawski | Leibniz University Hannover
5. Sebastian Heller | Hannover
6. Lynn Heller | Leibniz Universität Hannover
7. Maximilian Holdt | CAU Kiel
8. Johannes Horn | Frankfurt
9. Georgios Kydonakis | Alexander von Humboldt-Stiftung & Universität Heidelberg
10. Dr. Jørgen Olsen Lye | Leibniz Universität Hannover
11. Claudio Meneses | Christian-Albrechts-Universität Kiel
12. Thomas Raujouan | University of Hannover
13. Lothar Schiemanowski | Uni Hannover
14. Hartmut Weiß
15. Yuguang Zhang |