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Download filePainlevé monodromy manifolds, decorated character varieties, and cluster algebras
journal contribution
posted on 2016-12-16, 14:16 authored by Leonid Chekhov, Marta Mazzocco, Vladimir RubtsovIn this paper we introduce the concept of decorated character variety for the Riemann surfaces arising in the theory of
the Painleve differential equations. Since all Painleve differential equations (apart from the sixth one) exhibit Stokes
phenomenon, we show that it is natural to consider Riemann spheres with holes and bordered cusps on such holes. The
decorated character variety is considered here as complexification of the bordered cusped Teichmuller space introduced
in arXiv:1509.07044. We show that the decorated character variety of a Riemann sphere with s holes and n 1
bordered cusps is a Poisson manifold of dimension 3s + 2n − 6 and we explicitly compute the Poisson brackets which
are naturally of cluster type. We also show how to obtain the confluence procedure of the Painleve differential equations
in geometric terms.
Funding
The work of L.O.Ch. was partially supported by the center of excellence grant “Centre for Quantum Geometry of Moduli Spaces” from the Danish National Research Foundation (DNRF95) and by the Russian Foundation for Basic Research (Grant Nos. 14-01-00860-a and 13-01-12405-ofi-m2). This research was supported by the EPSRC Research Grant EP/J007234/1, by the Hausdorff Institute, by ANR “DIADEMS”, by RFBR-12-01-00525-a, by RFBR-15-01-05990, MPIM (Bonn) and SISSA (Trieste).
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