Research projects

Gauge theory

The aim of this project is a construction of invariants of compact G2-manifolds via gauge-theoretic methods. A program for constructing such invariants was pioneered by Donaldson and Thomas. Roughly speaking, one would like to define an invariant of G2-manifolds which would remain constant along isotopies of G2-metrics by counting certain G2-instantons. One of the main problems of this approach is that the number of G2-instantons does not need to be constant along isotopies of G2-metrics. Conjecturally, one should be able to obtain an invariant of a G2-manifold that remains constant along isotopies of G2-structures by counting G2-instantons together with certain Seiberg-Witten monopoles on associative submanifolds. Some details of this approach can be found for example in Haydys' paper G2 instantons and the Seiberg-Witten monopoles.

Selected publications:

A. Haydys. Topology of the blow-up set for the Seiberg-Witten equation with multiple spinors. arXiv:1607.01763
A. Haydys, Th.Walpuski. A compactness theorem for the Seiberg-Witten equation with multiple spinors in dimension three. Geom. Funct. Anal. (GAFA), 25(6):1799-1821, 2015; arXiv:1406.5683
A. Haydys. Gauge theory, calibrated geometry and harmonic spinors, J. Lond. Math. Soc. (2), 86(2):482-498, 2012. arXiv:0902.3738

HyperKähler manifolds and related geometries

A mid-term goal of this project is to construct examples of hyperKähler metrics admitting the structure of a complex Lagrangian fibration via the so called c-map construction, which takes an (affine) special Kähler manifold as input and gives a hyperKähler manifold as output. The resulting Riemannian metric is sometimes called semi-flat. The main challenge concerned with the c-map construction is to extend semi-flat metrics over the singularities after a suitable modification.

Since a complete special Kähler metric is necessarily flat, a singular special Kähler metric is a relevant input for the c-map construction. Haydys' research in the framework of this project has been concerned so far with a description of isolated singularities of affine special Kähler structures.

Selected publications:

M. Callies and A. Haydys. Local models of isolated singularities for affine special Kähler structures in dimension two, arXiv:1711.09118
M. Callies and A. Haydys. Affine special Kaehler structures in real dimension two, arXiv:1710.05211
A. Haydys. Isolated singularities of affine special Kaehler metrics in two dimensions. Commun. Math. Phys., 340(3):1231-1237, 2015; arXiv:1505.00462

Some Recorded Talks