Symmetries of Quantum Lax Equations for the Painleve Equations

Based on the fact that the Painleve equations can be written as Hamiltonian systems with affine Weyl group symmetries, a canonical quantization of the Painlev, equations preserving such symmetries has been studied recently. On the other hand, since the Painleve equations can also be described as isomonodromic deformations of certain second-order linear differential equations, a quantization of such Lax formalism is also a natural problem. In this paper, we introduce a canonical quantization of Lax equations for the Painleve equations and study their symmetries. We also show that our quantum Lax equations are derived from Virasoro conformal field theory.