G3-Mini-Workshop:
Gauge theory and Geometry in Göttingen

Göttingen, March 1 - March 4, 2018

Titles and abstracts of talks

Roger Bielawski. Moduli spaces of Nahm-Schmid equations
The Nahm-Schmid equations are an "indefinite" version of the Nahm equations. Their moduli spaces come equipped with a natural hypersymplectic geometry and are related to "monopoles" on the 3-dimensional Minkowski space. In the talk I will concentrate mainly on the hypersymplectic geometry of the moduli spaces and its relation to spectral curves. This is a joint work with Markus Roeser and Nuno Romao.

Florian Beck. tt*-geometry and (parabolic) Higgs bundles
Moduli spaces arising in mirror symmetry carry an interesting structure which is referred to as tt*-geometry. The latter was discovered by Cecotti and Vafa in their more general study of supersymmetric quantum field theories. The data defining tt*-geometry satisfy the tt*-equations. In this talk, we will match the tt*-equations with Hitchin’s equations in a concrete example motivated from mirror symmetry. In particular, we will express tt*-geometry in terms of a Higgs bundle on the corresponding moduli space and translate interesting aspects of Higgs bundles to tt*-geometry (e.g. parabolic structures and the oper limit after Gaiotto and Dumitrescu et al.).

Martin Callies. Isolated singularities of special Kähler structures in dimension two
We study isolated singularities of (affine) special Kähler structures in dimension two, assuming that the holomorphic cubic form does not have an essential singularity. We construct local models and explain how the metric and connection 1-form are asymptotic to one of those models. As an application, we compute the holonomy of the flat symplectic connection around the singularity.
The talk is based on joint work with A. Haydys, cf. arxiv 1711.09118.

Vicente Cortes. Quaternionic Kähler manifolds of co-homogeneity one
Quaternionic Kähler manifolds form an important class of Riemannian manifolds of special holonomy. They provide examples of Einstein manifolds of non-zero scalar curvature. I will show how to construct explicit examples of complete quaternionic Kähler manifolds of negative scalar curvature beyond homogeneous spaces. In particular, I will present examples of co-homogeneity one in all dimensions, based on joint work with Dyckmanns, Jüngling and Lindemann, see arXiv:1701.07882 [math.DG].

Rafe Mazzeo. I: Geometry of the Hitchin moduli space; II: The Kapustin-Witten equations; III: The Extended Bogomolny equations
In these three lectures I will discuss an interconnected set of results in modern gauge theory. The starting point is a review of the structure of the Hitchin moduli space and the asymptotic geometry of its hyperKaehler metric. Much of this has been inspired by the conjectural picture of Gaiotto-Moore-Neitzke, and I will describe what is now known about this, including some new observations about the special Kaehler metric on the Hitchin base.

Next, I will describe some relatively new set of gauge-theoretic equations introduced by Kapustin and Witten, and also by Haydys. An intended application of these is to provide gauge-theoretic realizations of knot invariants. The analytic foundations of this theory have been developed by myself and Witten, as well as by Taubes and by He. There are some important open problems.

Finally, the extended Bogomolny equations are a dimensional reduction of the KW equations in which one can see the Hitchin equations in a particularly vivid manner. I will describe their geometry and some new and fairly complete existence theorems which identify the moduli space of solutions in an important special case. This is joint work with He.

Andy Neitzke. A twistorial description of hyperkahler metrics on integrable systems
I will review a conjectural twistorial description of hyperkahler metrics on integrable systems, with special emphasis on the Hitchin integrable system (moduli space of Higgs bundles). This twistorial description leads to new concrete formulas for the asymptotics of the hyperkahler metric, and to an iterative scheme for computing the exact metric. The key ingredients are

1) the special Kahler geometry of the base of the integrable system,
2) a collection of integer invariants, known as "BPS state counts" or "generalized Donaldson-Thomas invariants," obeying the wall-crossing formula of Kontsevich-Soibelman.

This scheme arose in joint work with Davide Gaiotto and Greg Moore. In the end I will describe some recent progress toward verifying that this scheme indeed gives the exact hyperkahler metric; this is joint work with David Dumas.

Andrew Swann. Special geometry, the c-map and twists
We review the construction of quaternionic manifolds from projective special Kähler manifolds (the so-called c-map), showing the naturality of the special Kähler condition and using the twist construction as a geometric replacement for T-duality.