Airy structures and deformations of curves in surfaces

An embedded curve in a symplectic surface $\Sigma\subset X$ defines a smooth deformation space $\mathcal{B}$ of nearby embedded curves. A key idea of Kontsevich and Soibelman arXiv:1701.09137 [math.AG], is to equip the symplectic surface $X$ with a foliation in order to study the deformation space $\mathcal{B}$. The foliation, together with a vector space $V_\Sigma$ of meromorphic differentials on $\Sigma$, endows an embedded curve $\Sigma$ with the structure of the initial data of topological recursion, which defines a collection of symmetric tensors on $V_\Sigma$. Kontsevich and Soibelman define an Airy structure on $V_\Sigma$ to be a formal quadratic Lagrangian $\mathcal{L}\subset T^*(V_\Sigma^*)$ which leads to an alternative construction of the tensors of topological recursion. In this paper we produce a formal series $\theta$ on $\mathcal{B}$ of meromorphic differentials on $\Sigma$ which takes it values in $\mathcal{L}$, and use this to produce the Donagi-Markman cubic from a natural cubic tensor on $V_\Sigma$, giving a generalisation of a result of Baraglia and Huang, arXiv:1707.04975 [math.DG].