Quadratic differentials, stability conditions, and DT invariants, lecture 2

Abstract: Lecture 2: Calabi-Yau categories of surfaces

Combining ideas from string theory and geometric invariant theory, Bridgeland introduced the notion of a stability condition on a triangulated category. Since then, the problem of determining the structure of spaces of all stability conditions on triangulated categories, which are complex analytic manifolds, has proven to be quite challenging and has only been solved in a handful of examples. In some cases though, spaces of stability conditions turn out to have a very concrete and geometric interpretation as spaces of quadratic differentials, or equivalently flat surfaces - objects of intense study in ergodic theory. In these cases, the triangulated category is the (partially wrapped) Fukaya category of a surface. However, one can also instead consider certain 3-d Calabi-Yau triangulated categories, and this is necessary to make contact with the theory of motivic Donaldson-Thomas invariants of Kontsevich-Soibelman. As an application one obtains wall-crossing formulas for counts of finite-length geodesics on flat surfaces. This lecture series will be based on arXiv:1409.8611, arXiv:2104.06018, as well as unpublished work. The aim will be to present a unified picture of existing results as well as indicate open questions and future directions of research.

algebraic geometrysymplectic geometry

Audience: researchers in the topic

M-seminar