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

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.