A Uniqueness Property for the Quantization of Teichmüller Spaces

Chekhov, Fock and Kashaev introduced a quantization of the Teichmüller space T (S) of a punctured surface S, and an exponential version of this construction was developed by Bonahon and Liu. The construction of the quantum Teichmüller space crucially depends on certain coordinate change isomorphisms between the Chekhov-Fock algebras associated to different ideal triangulations of S. We show that these coordinate change isomorphisms are essentially unique, once we require them to satisfy a certain number of natural conditions. Let S be an oriented surface of finite topological type, with at least one puncture. The Teichmüller space T (S) is the space of isotopy classes of complete hyperbolic metrics on S. A quantization of the Teichmüller space T (S) of S was introduced by L. Chekhov and V. Fock [4, 5, 7] and, independently, by R. Kashaev [10] (see also [14]) as an approach to quantum gravity in 2+1 dimensions. This is a deformation of the C-algebra of functions on the usual Teichmüller space T (S) of S, depending on a parameter ~, in such a way that the linearization of this deformation at ~ = 0 corresponds to the Weil-Petersson Poisson structure of T (S). F. Bonahon and X. Liu [2, 12] developed an exponential version of the Chekhov-Fock-Kashaev construction. This exponential version of the quantization can be formulated in terms of non-commutative algebraic geometry, and has the advantage of possessing an interesting finite dimensional representation theory [2], whereas the non-exponential version is defined in terms of self-adjoint operators of Hilbert spaces. More precisely, let S be an oriented punctured surface of finite topological type, obtained by removing a finite set {v1, v2, . . . , vp} from a closed oriented surface S. An ideal triangulation is a family λ of finitely many disjoint simple arcs λ1, λ2, . . . , λn going from puncture to puncture and decomposing S into finitely many triangles with vertices at infinity; in other words, an ideal triangulation consists of the edges of a triangulation of the closed surface S whose vertex set consists of the punctures {v1, v2, . . . , vp}. Considering q = e πi~ as an indeterminate over C, the Chekhov-Fock algebra C[X1, X2, . . . , Xn] q λ associated to the ideal triangulation λ is the algebra over C(q) defined by generatorsX 1 , X ±1 2 , . . . , X ±1 n associated to the components of λ, and by relations XiXjX −1 i X −1 j = q 2σij where the σij are integers determined by the combinatorics of the ideal triangulation and connected to the Weil-Petersson form on Teichmüller space. This algebra has a well-defined fraction division algebra C(X1, X2, . . . , Xn) q λ. In practice, the algebra C(X1, X2, . . . , Xn) q λ Date: February 2, 2008.