Symmetries of quantum Lax equations for the Painlev\'e equations

TL;DR

This paper introduces a canonical quantization of Lax equations for Painlevé equations, preserving affine Weyl group symmetries and connecting to Virasoro conformal field theory, advancing the understanding of quantum integrable systems.

Contribution

It presents a new canonical quantization of Lax equations for Painlevé equations that maintains their symmetries and links to conformal field theory, which was not previously established.

Findings

Constructed symmetries of quantum Lax equations.

Derived quantum Lax equations from Virasoro conformal field theory.

Preserved affine Weyl group symmetries in quantization.

Abstract

The Painlev\'e equations can be written as Hamiltonian systems with affine Weyl group symmetries. A canonical quantization of the Painlev\'e equations preserving the affine Weyl group symmetries has been studied. While, the Painlev\'e equations are isomonodromic equations for certain second-order linear differential equations. In this paper, we introduce a canonical quantization of Lax equations for the Painlev\'e equations and construct symmetries of the quantum Lax equations. We also show that our quantum Lax equations are derived from Virasoro conformal field theory.