Airy structures and deformations of curves in surfaces
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| Format: | Preprint |
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2020
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| _version_ | 1866910336881262592 |
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| author | Chaimanowong, Wee Norbury, Paul Swaddle, Michael Tavakol, Mehdi |
| author_facet | Chaimanowong, Wee Norbury, Paul Swaddle, Michael Tavakol, Mehdi |
| contents | 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]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_00254 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Airy structures and deformations of curves in surfaces Chaimanowong, Wee Norbury, Paul Swaddle, Michael Tavakol, Mehdi Algebraic Geometry Mathematical Physics 14H70, 14H15, 14H60 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]. |
| title | Airy structures and deformations of curves in surfaces |
| topic | Algebraic Geometry Mathematical Physics 14H70, 14H15, 14H60 |
| url | https://arxiv.org/abs/2012.00254 |