Ratio coordinates for higher Teichmüller spaces

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 $\mathcal{X}$-space and the $\mathcal{A}$-space, forming a cluster ensemble. Fock and Goncharov gave nice descriptions of the coordinates of these spaces in the cases of $G=PG{L}_m$ and $G=S{L}_m$, together with Poisson structures. We consider new coordinates for higher Teichmüller spaces given as ratios of the coordinates of the $\mathcal{A}$-space for $G=S{L}_m$, 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.

42 pages, 6 figures

References in corpus (1)

• The quantum dilogarithm and representations quantum cluster varieties

Cited by in corpus (4)

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• Three-dimensional quantum gravity from the quantum pseudo-Kähler plane
• Irreducible self-adjoint representations of quantum Teichmüller space and the phase constants
• Finite dimensional quantum Teichmüller space from the quantum torus at root of unity