## About

Higgs bundles, Hitchin’s self-duality equations and hyper-Ka ̈hler 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.

The initial goals of these meetings 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. Due to our own thematic preferences as organizers, we would like to initially set up these meetings once per semester within the priority program.

**Applications for financial support for young participants are open until November 22nd.**

**Related project(s):**

**55**New hyperkähler spaces from the the self-duality equations

**69**Wall-crossing and hyperkähler geometry of moduli spaces

**77**Asymptotic geometry of the Higgs bundle moduli space II

#### Friday, December 03:

##### Welcome coffee (and cookies)

##### Quantum Geometry of BPS structures (Murad Alim)

##### Dinner ( t.b.a.)

#### Saturday, December 04:

##### Compactifying the GL(2,C)-Hitchin system on the space of quadratic multi-scale differentials (Johannes Horn)

##### Coffee (and cookies)

##### Twistorial construction of Hyperkähler metrics on the moduli space of weakly parabolic Higgs bundles (Maximilian Holdt)

##### Lunch ( t.b.a.)

### Quantum Geometry of BPS structures

BPS invariants of certain physical theories correspond to Donaldson-Thomas (DT) invariants of an associated Calabi-Yau geometry. BPS structures refer to the data of the DT invariants together with their wall-crossing structure. On the same Calabi-Yau geometry another set of invariants are the Gromov-Witten (GW) invariants. These are organized in the GW potential, which is an asymptotic series in a formal parameter and can be obtained from topological string theory. Bridgeland showed that the GW potential can be extracted from a Tau-function which solves a Riemann-Hilbert problem associated to BPS structures. I will show that there is also a path going in the other direction. Studying the resurgence of the GW potential leads to DT invariants. DT wall-crossing phenomena correspond to Stokes jumps of the Borel resummation of the GW potential. In an explicit example, I will furthermore discuss a corresponding hyperkähler geometry and integrable hierarchy. This is based on joint work with Arpan Saha, Iván Tulli and Jörg Teschner.

### Compactifying the GL(2,C)-Hitchin system on the space of quadratic multi-scale differentials

The complexity of the singular fibers of the Hitchin system stems from the diversity of singularities of spectral curves. Already for G=GL(2,C) all singularities of type A appear. In light of the Deligne-Mumford compactification of the moduli space of smooth projective curves, it is a natural idea to compactify the family of smooth spectral curves over the regular locus of the Hitchin base to a family of nodal curves over a modified Hitchin base. In the talk, I will compare several approaches to achieve this goal in the GL(2,C)-case. These motivate the consideration of the moduli space of quadratic multi-scale differentials with simple zeroes as modified Hitchin base. I will conclude by formulating a spectral correspondence in this context. This is joint work with Martin Möller.

### Twistorial construction of Hyperkähler metrics on the moduli space of weakly parabolic Higgs bundles

In 2009 Gaiotto, Moore and Neitzke introduced a blueprint for the twistorial construction of Hyperkähler metrics for a wide range of integrable systems. For the space of rank 2 parabolic Higgs bundles with diagonalizable residues at fixed punctures the construction becomes very concrete: One can build the metric out of sections of the Higgs bundles which are solutions of a certain flatness equation build out of the corresponding Higgs pair. The success of the procedure relies heavily on an asymptotic property of these sections, which Gaiotto, Moore and Neitzke originally conjectured based on the WKB method. I will outline the construction of the Hyperkähler metric and, using recent results of Fredrickson, Mazzeo, Swoboda and Weiß, show how one can circumvent the use of the WKB method to obtain a rigorous derivation of the necessary asymptotic properties. Finally I will explain how these properties yield the conjectured optimal decay rate of the asymptotic approximation of two different Hyperkähler metrics, by solving an associated Riemann-Hilbert problem.

- Sven-Ole Behrend | CAU Kiel
- Balázs Márk Békési | Leibniz Universität Hannover
- Jens Heber | CAU Kiel
- Sebastian Heller | Hannover
- Lynn Heller | Leibniz Universität Hannover
- Maximilian Holdt | CAU Kiel
- Sven Marquardt | CAU Kiel
- Dr. Claudio Meneses | Christian-Albrechts-Universität Kiel
- Dr. NAEEM AHMAD PUNDEER | JADAVPUR UNIVERSITY, KOLKATA, INDIA
- Thomas Raujouan | Uni Hannover
- Maximilian Rodriguez | CAU Kiel
- Lothar Schiemanowski | Uni Hannover
- Max Schult | Leibniz Universität Hannover
- Hartmut Weiß | Mathematisches Seminar CAU Kiel