Quantum Curve and the First Painlevé Equation

Abstract

We show that the topological recursion for the (semi-classical) spectral curve of the first Painlevé equation $P_{\mathrm{I}}$ gives a WKB solution for the isomonodromy problem for $P_{\mathrm{I}}$. In other words, the isomonodromy system is a quantum curve in the sense of [Dumitrescu O., Mulase M., Lett. Math. Phys. 104 (2014), 635-671, arXiv:1310.6022] and [Dumitrescu O., Mulase M., arXiv:1411.1023].

References in corpus (3)

• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Tracy-Widom GUE law and symplectic invariants
• Autonomous limit of 4-dimensional Painlevé-type equations and degeneration of curves of genus two

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• Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: the ${\mathfrak{sl}}_2$ case
• Quantum curves and conformal field theory
• Integrable differential systems of topological type and reconstruction by the topological recursion
• Quantum curves from refined topological recursion: the genus 0 case
• WKB solutions of difference equations and reconstruction by the topological recursion
• Topological recursion and uncoupled BPS structures II: Voros symbols and the $\tau$-function
• Quantization of classical spectral curves via topological recursion
• Voros Coefficients for the Hypergeometric Differential Equations and Eynard-Orantin's Topological Recursion - Part II : For the Confluent Family of Hypergeometric Equations
• Painlevé equations, topological type property and reconstruction by the topological recursion
• Topological recursion and uncoupled BPS structures I: BPS spectrum and free energies
• The factorial growth of topological recursion
• Painlevé Kernels and Surface Defects at Strong Coupling