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Airy structures and deformations of curves in surfaces

Wee Chaimanowong, Paul Norbury, Michael Swaddle, Mehdi Tavakol

Abstract

An embedded curve in a symplectic surface $Σ\subset X$ 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 $V_Σ$ of meromorphic differentials on $Σ$, endows an embedded curve $Σ$ with the structure of the initial data of topological recursion, which defines a collection of symmetric tensors on $V_Σ$. Kontsevich and Soibelman define an Airy structure on $V_Σ$ to be a formal quadratic Lagrangian $\mathcal{L}\subset T^*(V_Σ^*)$ which leads to an alternative construction of the tensors of topological recursion. In this paper we produce a formal series $θ$ on $\mathcal{B}$ of meromorphic differentials on $Σ$ which takes it values in $\mathcal{L}$, and use this to produce the Donagi-Markman cubic from a natural cubic tensor on $V_Σ$, giving a generalisation of a result of Baraglia and Huang, arXiv:1707.04975 [math.DG].

Airy structures and deformations of curves in surfaces

Abstract

An embedded curve in a symplectic surface defines a smooth deformation space of nearby embedded curves. A key idea of Kontsevich and Soibelman arXiv:1701.09137 [math.AG], 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 of meromorphic differentials 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, arXiv:1707.04975 [math.DG].

Paper Structure

This paper contains 27 sections, 17 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Topological recursion applied to curves in surfaces
  3. Deformation space of embedded curves
  4. Foliations and topological recursion
  5. Correlators of Sigma subset (X,Omega,F).
  6. Cohomological field theories
  7. Deformation space associated with Higgs bundles
  8. Rank one case
  9. K3 surfaces
  10. Airy structures
  11. Tate spaces
  12. Airy structures
  13. Tate spaces associated to a curve in a symplectic surface
  14. Symplectic reduction
  15. The quadratic Lagrangian L
  16. ...and 12 more sections

Key Result

Theorem 1

Define a section $\theta\in\Gamma(\widehat{\mathcal{B}}_{[\Sigma]},G_\Sigma)$ by where we sum over indices in $\{1,...,g\}$. Its cohomology class in $\Gamma(\widehat{\mathcal{B}}_{[\Sigma]},\mathcal{H})$ is analytic in $z^1,...,z^g$ and coincides with the analytic expansion of $[\theta]$ defined in theta.

Theorems & Definitions (47)

  • Theorem 1
  • Corollary 2
  • Example 1.1
  • Theorem 3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Example 2.3
  • Definition 2.1
  • Definition 2.2
  • ...and 37 more