BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Spectral curves in surfaces Norbury, Paul


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.

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