Abstract
We define new coordinates for Fock–Goncharov’s higher Teichmüller spaces for a surface with holes, which are the moduli spaces of representations of the fundamental group into a reductive Lie group G. Some additional data on the boundary leads to two closely related moduli spaces, the \(\mathscr {X}\)-space and the \(\mathscr {A}\)-space, forming a cluster ensemble. Fock and Goncharov gave nice descriptions of the coordinates of these spaces in the cases of \(G = \mathrm{PGL}_m\) and \(G=\mathrm{SL}_m\), together with Poisson structures. We consider new coordinates for higher Teichmüller spaces given as ratios of the coordinates of the \(\mathscr {A}\)-space for \(G=\mathrm{SL}_m\), which are generalizations of Kashaev’s ratio coordinates in the case \(m=2\). Using Kashaev’s quantization for \(m=2\), we suggest a quantization of the system of these new ratio coordinates, which may lead to a new family of projective representations of mapping class groups. These ratio coordinates depend on the choice of an ideal triangulation decorated with a distinguished corner at each triangle, and the key point of the quantization is to guarantee certain consistency under a change of such choices. We prove this consistency for \(m=3\), and for completeness we also give a full proof of the presentation of Kashaev’s groupoid of decorated ideal triangulations.
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Notes
This may not be a standard terminology.
A ‘seed’ in the usual cluster theory is equipped with cluster variables, like our upcoming \(X_i\)’s and \(\Delta _i\)’s. A seed as defined here is called a ‘feed’ in [9] as a joke, to be distinguished from a usual ‘seed’.
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Acknowledgments
I thank Ivan Ip for motivation and helpful discussions. I thank Dylan Allegretti for his help on understanding the works of Fock–Goncharov. Finally, I’d like to thank the referee for the reviewing and for helpful comments.
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Kim, H.K. Ratio coordinates for higher Teichmüller spaces. Math. Z. 283, 469–513 (2016). https://doi.org/10.1007/s00209-015-1607-4
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DOI: https://doi.org/10.1007/s00209-015-1607-4