Airy structures and deformations of curves in surfaces

Abstract

AbstractAn embedded curve in a symplectic surface defines a smooth deformation space of nearby embedded curves. A key idea of Kontsevich and Soibelman is to equip the symplectic surface with a foliation in order to study the deformation space . The foliation, together with a vector space of meromorphic differentials on , endows an embedded curve with the structure of the initial data of topological recursion, which defines a collection of symmetric tensors on . Kontsevich and Soibelman define an Airy structure on to be a formal quadratic Lagrangian which leads to an alternative construction of the tensors of topological recursion. In this paper, we produce a formal series on which takes it values in , and use this to produce the Donagi–Markman cubic from a natural cubic tensor on , giving a generalisation of a result of Baraglia and Huang.