[Paper Review] Painlevé equations, topological type property and reconstruction by the topological recursion

Kohei Iwaki, Olivier Marchal+1 more|arXiv (Cornell University)|Jan 11, 2016

Nonlinear Waves and Solitons28 references3 citations

TL;DR

This paper proves that Lax pairs for all six $π$-dependent Painlevé equations satisfy the topological type property, enabling reconstruction of their formal $\hbar$-expansions of the isomonodromic $\tau$-functions and determinantal formulas via topological recursion on the associated spectral curves. The key result is the explicit verification of this reconstruction for the first orders of the $\tau$-functions across all six Painlevé cases.

ABSTRACT

In this article we prove that Lax pairs associated with $\hbar$-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 $\hbar$-expansion of the isomonodromic $τ$-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 $τ$-functions.

Motivation & Objective

• To establish the topological type property for Lax pairs of all six $\hbar$-dependent Painlevé equations.
• To demonstrate that the formal $\hbar$-expansion of the isomonodromic $\tau$-function can be reconstructed via topological recursion on the spectral curve derived from the Lax pair.
• To verify that determinantal formulas associated with the Lax pairs match the correlation functions generated by topological recursion.
• To provide explicit computations of the first-order $\tau$-function expansions for all six Painlevé equations.

Proposed method

• The authors analyze $\hbar$-dependent Lax pairs for the six Painlevé equations and derive their associated spectral curves via algebraic geometry techniques.
• They verify the topological type property for generic monodromy parameters, ensuring compatibility between the Lax pair and the topological recursion framework.
• Topological recursion is applied to the spectral curves to generate free energies $F_g^{(g)}$, which are then used to reconstruct the $\tau$-function via exponentiation.
• The formal $\hbar$-expansion of the $\tau$-function is reconstructed order-by-order using the recursion-generated correlation functions.
• The method relies on the equivalence between loop equations in matrix models and those derived from Lax pairs, with determinantal formulas serving as solutions.
• Explicit computations are performed for $F_0^{(0)}$, $F_1^{(1)}$, and higher genus invariants in Painlevé VI, confirming consistency with known $\tau$-function derivatives.

Experimental results

Research questions

• RQ1 Do the Lax pairs of all six $\hbar$-dependent Painlevé equations satisfy the topological type property for generic monodromy parameters?
• RQ2 Can the formal $\hbar$-expansion of the isomonodromic $\tau$-function be reconstructed via topological recursion applied to the spectral curve of the Lax pair?
• RQ3 Do the correlation functions generated by topological recursion match the determinantal formulas derived from the Lax pair?
• RQ4 Is the topological recursion procedure consistent with the known $\tau$-function structure in the first few orders of $\hbar$ for all Painlevé equations?

Key findings

• The Lax pairs of all six $\hbar$-dependent Painlevé equations satisfy the topological type property for generic monodromy parameters, enabling the application of topological recursion.
• The formal $\hbar$-expansion of the isomonodromic $\tau$-function is successfully reconstructed via topological recursion applied to the spectral curve of the Lax pair.
• The free energy $F_{\rm VI}^{(0)}$ is explicitly computed and verified to satisfy $\frac{d}{dt}F_{\rm VI}^{(0)} = -\frac{d}{dt}\dot{\tau}_0$, confirming consistency with the $\tau$-function derivative.
• The first-order correction $F_{\rm VI}^{(1)}$ is computed and its derivative matches $-\frac{d}{dt}\tau_{\rm VI}^{(1)}$, validating the method at next order.
• Explicit expressions for $\tau$-function contributions up to $\hbar^4$ are derived for Painlevé VI, showing agreement with known asymptotic structures.
• The method successfully reproduces the $\tau$-function structure across all six Painlevé equations, including symmetric expressions in monodromy parameters and logarithmic terms involving $\theta_0, \theta_1, \theta_\infty, \theta_t$.