Confluence on the Painlevé Monodromy Manifolds, their Poisson Structure and Quantisation

Abstract

In this paper we obtain a system of flat coordinates on the monodromy manifold of each of the Painlevé equations. This allows us to quantise such manifolds. We produce a quantum confluence procedure between cubics in such a way that quantisation and confluence commute. We also investigate the underlying cluster algebra structure and the relation to the versal deformations of singularities of type $D_4,A_3,A_2$, and $A_1$.

Version 1, 16 pages, 3 figures

References in corpus (1)

• Affine cubic surfaces and relative SL(2)-character varieties of compact surfaces

Cited by in corpus (2)

• Derivatives in noncommutative calculus and deformation property of quantum algebras
• Different linearizations of non-abelian second Painlevé systems and related monodromy surfaces