Airy structures and deformations of curves in surfaces

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