Airy structures and deformations of curves in surfaces

Abstract

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 $\mathcal{L}\subset T^{\ast }(V_{\Sigma }^{\ast })$ 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].

50 pages

References in corpus (2)

• A new cohomology class on the moduli space of curves
• Topological Recursion, Airy structures in the space of cycles

Cited by in corpus (2)

• Taking limits in topological recursion
• Seiberg-Witten Theory and Topological Recursion