Factorization rules in quantum Teichmüller theory

Abstract. For a punctured surface S, a point of its Teichmüller space T(S) determines an irreducible representation of its quantization Tq(S). We analyze the behavior of these representations as one goes to infinity in T(S), or in the moduli space M(S) of the surface. The main result of this paper states that an irreducible representation of Tq(S) limits to a direct sum of representations of Tq(Sγ), where Sγ is obtained from S by pinching a multicurve γ to a set of nodes. The result is analogous to the factorization rule found in conformal field theory. Let S be an oriented surface of genus g obtained from a closed compact surface S by removing s punctures v1,..., vs. The Teichmüller space T(S) of S is the space of isotopy classes of complete hyperbolic metrics on S with finite area. It comes equipped with a natural Kähler metric, called the Weil–Petersson metric which is invariant under the action of the mapping class group MCG(S) onto T(S). A quantization of the Teichmüller space was successfully described by L. Chekhov and V. V. Fock in [8], and, in