# 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 $A_1$.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-00776844
Submitted on : Wednesday, January 16, 2013 - 12:47:22 PM
Last modification on : Monday, March 9, 2020 - 6:16:02 PM

### Identifiers

• HAL Id : hal-00776844, version 1
• ARXIV : 1212.6723

### Citation

Marta Mazzocco, Vladimir Rubtsov. Confluence on the Painlevé Monodromy Manifolds, their Poisson Structure and Quantisation. 2012. ⟨hal-00776844⟩

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