Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 011, 24 pages      arXiv:1507.06557      https://doi.org/10.3842/SIGMA.2016.011

Quantum Curve and the First Painlevé Equation

Kohei Iwaki a and Axel Saenz b
a) Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan
b) Department of Mathematics, University of California, Davis, CA 95616-8633, USA

Received August 04, 2015, in final form January 22, 2016; Published online January 29, 2016

Abstract
We show that the topological recursion for the (semi-classical) spectral curve of the first Painlevé equation $P_{\rm I}$ gives a WKB solution for the isomonodromy problem for $P_{\rm I}$. 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].

Key words: quantum curve; first Painlevé equation; topological recursion; isomonodoromic deformation; WKB analysis.

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