Factorization rules in quantum Teichmüller theory

Abstract

For a punctured surface $S$, a point of its Teichmüller space $\mathcal{T}\left(S\right)$ determines an irreducible representation of its quantization ${\mathcal{T}}^q\left(S\right)$. We analyze the behavior of these representations as one goes to infinity in $\mathcal{T}\left(S\right)$, or in the moduli space $\mathcal{ℳ}\left(S\right)$ of the surface. The main result of this paper states that an irreducible representation of ${\mathcal{T}}^q\left(S\right)$ limits to a direct sum of representations of ${\mathcal{T}}^q\left(S_{\gamma }\right)$, where $S_{\gamma }$ is obtained from $S$ by pinching a multicurve $\gamma$ to a set of nodes. The result is analogous to the factorization rule found in conformal field theory.