Takano’s Theory of Quantum Painlevé Equations
نویسنده
چکیده
Recently, a quantum version of Painlevé equations from the point of view of their symmetries was proposed by H. Nagoya. These quantum Painlevé equations can be written as Hamiltonian systems with a (noncommutative) polynomial Hamiltonian HJ. We give a characterization of the quantum Painlevé equations by certain holomorphic properties. Namely, we introduce canonical transformations such that the Painlevé Hamiltonian system is again transformed into a polynomial Hamiltonian system, and we show that the Hamiltonian can be uniquely characterized through this holomorphic property.
منابع مشابه
Painlevé transcendents in two-dimensional topological field theory
Introduction Lecture 1. Algebraic properties of correlators in 2D topological field theory. Moduli of a 2D TFT and WDVV equations of associativity. Lecture 2. Equations of associativity and Frobenius manifolds. Deformed flat connection and its monodromy at the origin. Lecture 3. Semisimplicity and canonical coordinates. Lecture 4. Classification of semisimple Frobenius manifolds. Lecture 5. Mon...
متن کاملQuantizing the Bäcklund transformations of Painlevé equations and the quantum discrete Painlevé VI equation
Based on the works by Kajiwara, Noumi and Yamada, we propose a canonically quantized version of the rational Weyl group representation which originally arose as “symmetries” or the Bäcklund transformations in Painlevé equations. We thereby propose a quantization of discrete Painlevé VI equation as a discrete Hamiltonian flow commuting with the action of W (D (1) 4 ). 1
متن کاملPainlevé transcendents and two-dimensional topological field theory
Introduction Lecture 1. Algebraic properties of correlators in 2D topological field theory. Moduli of a 2D TFT and WDVV equations of associativity. Lecture 2. Equations of associativity and Frobenius manifolds. Deformed flat connection and its monodromy at the origin. Lecture 3. Semisimplicity and canonical coordinates. Lecture 4. Classification of semisimple Frobenius manifolds. Lecture 5. Mon...
متن کاملSpectral curves and discrete Painlevé equations
It is well known that isomonodromic deformations admit a Hamiltonian description. These Hamiltonians appear as coefficients of the characteristic equations of their Lax matrices, which define spectral curves for linear systems of differential and difference systems. The characteristic equations in the case of the associated linear problems for various discrete Painlevé equations is biquadratic ...
متن کاملLocal Cohomology of Generalized Okamoto–painlevé Pairs and Painlevé Equations
In the theory of deformation of Okamoto-Painlevé pair (S, Y ), a local cohomology group H D (ΘS(− logD)) plays an important role. In this paper, we estimate the local cohomology group of pair (S, Y ) for several types, and obtain the following results. For a pair (S, Y ) corresponding to the space of initial conditions of the Painlevé equations, we show that the local cohomology group H D (ΘS(−...
متن کامل