Painlevé equations, topological type property and reconstruction by the topological recursion

Abstract

In this article we prove that Lax pairs associated with $\hslash$-dependent six Painlevé equations satisfy the topological type property proposed by Bergère, Borot and Eynard for any generic choice of the monodromy parameters. Consequently we show that one can reconstruct the formal $\hslash$-expansion of the isomonodromic $\tau$-function and of the determinantal formulas by applying the so-called topological recursion to the spectral curve attached to the Lax pair in all six Painlevé cases. Finally we illustrate the former results with the explicit computations of the first orders of the six $\tau$-functions.

54 pages, Presentation improved. Typos corrected

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• On Painlevé/gauge theory correspondence
• Topological recursion for irregular spectral curves
• Universal scaling limits of matrix models, and (p,q) Liouville gravity
• Quantum Curve and the First Painlevé Equation
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Cited by in corpus (1)

• Topological recursion and uncoupled BPS structures I: BPS spectrum and free energies