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Spectral curves in surfaces

Banff International Research Station for Mathematical Innovation and Discovery

An embedded curve in a Poisson surface $\Sigma\subset X$ defines a smooth deformation space $\mathcal{B}$ of nearby embedded curves. In this talk we will describe a key idea of Kontsevich and Soibelman to equip the Poisson surface $X$ with a foliation in order to study the deformation space $\mathcal{B}$. For example, $X=TP^1\to P^1$ is a Poisson surface surface foliated by its fibres. 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 $V_\Sigma$. These tensors produce a formal series, which turns out to be a formal Seiberg-Witten differential, that descends under a quotient to an analytic series.

Mathematics; Algebraic Geometry; Differential Geometry

video/mp4

Author affiliation: University of Melbourne

2021-08-02

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/79173

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/