Defines metric Möbius graphs for Klein surfaces, proves a refined Norbury recursion on weighted lattice counts, derives a refined Witten-Kontsevich recursion, and explicitly computes the refined Euler characteristic of the moduli space.
2016 AMS von Neumann Symposium, Topological Recursion and its Influence in Analysis, Geometry and Topology , doi =
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abstract
We propose a new approach to the topological recursion of Eynard-Orantin based on the notion of Airy structure, which we introduce in the paper. We explain why Airy structure is a more fundamental object than the one of the spectral curve. We explain how the concept of quantization of Airy structure leads naturally to the formulas of topological recursion as well as their generalizations. The notion of spectral curve is also considered in a more general framework of Poisson surfaces endowed with foliation. We explain how the deformation theory of spectral curves is related to Airy structures. Few other topics (e.g. the Holomorphic Anomaly Equation) are also discussed from the general point of view of Airy structures.
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Lecture notes review exact WKB analysis for ODEs and its combination with topological recursion and isomonodromy to compute monodromy and resurgent structures for Painlevé equations.
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Refined lattice point counting on the moduli space of Klein surfaces
Defines metric Möbius graphs for Klein surfaces, proves a refined Norbury recursion on weighted lattice counts, derives a refined Witten-Kontsevich recursion, and explicitly computes the refined Euler characteristic of the moduli space.
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Les Houches Lectures on Exact WKB Analysis and Painlev\'e Equations
Lecture notes review exact WKB analysis for ODEs and its combination with topological recursion and isomonodromy to compute monodromy and resurgent structures for Painlevé equations.