# Wall-crossing structures and Chern-Simons theory.

In 2008 jointly with Maxim Kontsevich we introduced the notion of stability data on graded Lie algebras. In the case of the Lie algebra of vector fields on a symplectic torus it underlies the wall-crossing formulas for Donaldson-Thomas invariants of 3-dimensional Calabi-Yau categories. In 2013 we introduced the notion of wall-crossing structure, which is a locally-constant sheaf of stability data. Wall-crossing structures naturally appear in complex integrable systems, Homological Mirror Symmetry and many other topics not necessarily related to Donaldson-Thomas theory. Recently, in 2020 we introduced a sublass of analytic wall-crossing structures. We formulated a general conjecture that analytic wall-crossing structure gives rise to resurgent (i.e. Borel resummable) series.

Many wall-crossing structures have geometric origin, and moreover they naturally appear in our Holomorphic Floer Theory program. Aim of my talk is to discuss wall-crossing structures associated with a pair of holomorphic Lagrangian submanifolds of a complex symplectic manifold (in most cases it will be the cotangent bundle). These wall-crossing structures underly Cecotti-Vafa wall-crossing formulas, and as such they appear naturally in the study of exponential integrals in finite and infinite dimensions. I am going to explain our conjectural approach to Chern-Simons theory which is based on the idea of wall-crossing structure. In some aspects this approach is related to the work of Witten on analytic continuation of Chern-Simons theory.

Zoom link: https://pitp.zoom.us/j/99446428842?pwd=aDRzbFJoNytDNURDUVFMNGQzNjBFQT09