Symplectic geometry of exact WKB analysis

Abstract: A sheaf quantization is a sheaf associated to a Lagrangian brane. This sheaf conjecturally has information as much as Floer theory of the Lagrangian. On the other hand, exact WKB analysis is an analysis of differential equations containing \hbar (the Planck constant).

In this talk, I will explain how to construct a sheaf quantization over the Novikov ring of the spectral curve of an \hbar-differential equation by using exact WKB method. In the construction, one can see how (conjecturally) the convergence in WKB analysis is related to the convergence in Fukaya category. In degree 2, there is an application to cluster theory: the sheaf quantization associates a cluster coordinate which is the same as the Voros-Iwaki-Nakanishi-Fock-Goncharov coordinate. I will also mention about some relationships to Riemann-Hilbert correspondence of D'Agnolo-Kashiwara and Kontsevich-Soibelman.

algebraic geometrydifferential geometrygeometric topologysymplectic geometry

Audience: researchers in the topic

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