GT Seminar: Fukaya categories of orbifold surfaces from deformation theory

Abstract: In work by Haiden, Katzarkov and Kontsevich, so-called gentle algebras, studied by representation theorists since the late '80s, naturally appear as endomorphism algebras of formal generators of Fukaya categories of surfaces. Lekili and Polishchuk showed that, conversely, any (graded) gentle algebra arises as such an endomorphism algebra. I will explain how deformation theory gives rise to a generalization of this correspondence. On the geometric side, one generalizes from smooth surfaces to surfaces with order 2 orbifold singularities. On the algebraic side, one naturally obtains A∞-deformations of gentle algebras. Restricting to formal generators of these Fukaya categories of orbifold surfaces, one naturally obtains a class of associative algebras strictly containing the class of so-called skew-gentle algebras, which have been studied in representation theory since the late '90s. In this way, A∞-deformations of graded gentle algebras give new insights into derived equivalences between strictly associative algebras. This talk is based on joint work with Sibylle Schroll and Zhengfang Wang and joint work in progress with Cheol-hyun Cho, Kyoungmo Kim, Kyungmin Rho, Sibylle Schroll and Zhengfang Wang.

This is a talk in our Geometry and Topology Seminar Series.