Noncommutative Painlevé Equations and Systems of Calogero Type

Abstract

All Painlevé equations can be written as a time-dependent Hamiltonian system, and as such they admit a natural generalization to the case of several particles with an interaction of Calogero type (rational, trigonometric or elliptic). Recently, these systems of interacting particles have been proved to be relevant in the study of β-models. An almost two decade old open question by Takasaki asks whether these multi-particle systems can be understood as isomonodromic equations, thus extending the Painlevé correspondence. In this paper we answer in the affirmative by displaying explicitly suitable isomonodromic Lax pair formulations. As an application of the isomonodromic representation we provide a construction based on discrete Schlesinger transforms, to produce solutions for these systems for special values of the coupling constants, starting from uncoupled ones; the method is illustrated for the case of the second Painlevé equation.

Generalized model of interacting integrable tops

“…Let us also remark that the top like models with matrix-valued variables were studied in [21,35] and [6]. In contrast to these papers here we deal with the models, where the matrix variables have their own internal dynamics.…”

The Elliptic Painlevé Lax Equation vs. van Diejen's 8-Coupling Elliptic Hamiltonian

“…This correspondence has been elaborated on from several viewpoints in [11,12,19]. 1 A decade ago, one of us (S.R.) conjectured that there might exist a similar connection between van Diejen's [14] 8-parameter analytic difference operator and a Lax formulation for Sakai's [10] 8-parameter elliptic Painlevé difference equation.…”

“…This paper is a contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory. The full collection is available at https://www.emis.de/journals/SIGMA/elliptic-integrablesystems.html 1 For the higher rank models, there is also a correspondence [11], where the Painlevé side of the correspondence is a kind of multi-dimensional extension different from the Garnier systems (see [1] for recent developments).…”

Quantised Painlevé monodromy manifolds, Sklyanin and Calabi-Yau algebras

“…We conclude by mentioning the relation between the quantum algebras in Table 5 and the matrix generalisations of the Painlevé equations. Building upon work by Retakh and the third author [45], in [5] a set of non-commutative relations which are non-commutative analogues of monodromy data relations for the Painlevé II equation was constructed. The interesting feature of these non-commutative relations is that by taking the scalar degeneration of the non-commutative operator q, one obtains our quantum Painlevé II monodromy variety.…”

“…x 1 x 2 x 3 + x 5 1 + x 2 3 + x 2 3 + η 4 x 4 1 + • • • + η 1 x 1 + σx 2 + ρx 3 + ω. Following this work, P. Etingof and V. Ginzburg [15] have proposed a quantum description of del Pezzo surfaces based on the flat deformation of cubic affine cone surfaces with an isolated elliptic singularity of type E 6 , E 7 and E 8 in (weighted) projective planes: Their result gives a family of Calabi-Yau algebras parametrised by a complex number and a triple of polynomials of specifically chosen degrees.…”