Note on Character Varieties and Cluster Algebras

Hikami, Kazuhiro

doi:10.3842/sigma.2019.003

Cited by 2 publications

(1 citation statement)

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“…The first way is based on pants decomposition of the sphere with n punctures and the use of complex Fenchel-Nielsen coordinates which are Darboux coordinates on symplectic leaves of Goldman bracket. The second way is to use SL(2) Fock-Goncharov coordinates (which in this case were originally discovered by Thurston) (see [25,18]).…”

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Symplectic extensions of the Kirillov-Kostant and Goldman Poisson structures and Fuchsian systems

Bertola,

Korotkin

2019

Preprint

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We revisit symplectic properties of the monodromy map for Fuchsian systems on the Riemann sphere. We extend previous results of [19,3,22] where it was shown that the monodromy map is a Poisson morphism between the Kirillov-Kostant Poisson structure on the space of coefficients, on one side, and the Goldman bracket on the monodromy character variety on the other. The extension is provided by defining larger spaces on both sides which are equipped with symplectic structures naturally projecting to the canonical ones. On the coefficient side our symplectic structure corresponds to a non-degenerate quadratic Poisson structure expressed via the rational dynamical r-matrix; it reduces to the Kirillov-Kostant bracket when projected to the standard space. On the monodromy side we get a symplectic structure which induces the symplectic structure of [2] on the leaves of the Goldman Poisson bracket. We prove that the monodromy map provides a symplectomorphism using the formalism of Malgrange [24] and one of the authors [6,7]. As a corollary we prove the recent conjecture by A.Its, O.Lisovyy and A.Prokhorov in its "strong" version while the original "weak" version is derived from previously known results. We show also that the isomonodromic Jimbo-Miwa tau-function is intimately related to a generating function of such transformation. Contents 1 Introduction 1 2 Poisson and symplectic structures on the space A 8 3 Symplectomorphism between A and M via Malgrange's form 9 4 Standard monodromy map and weak version of Its-Lisovyy-Prokhorov conjecture 13

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