Quantum hyperbolic geometry

Abstract

We construct a new family, indexed by odd integers N ≥1, of (2+1)-dimensional quantum field theories that we call quantum hyperbolic field theories (QHFT), and we study its main structural properties. The QHFT are defined for marked (2+1)-bordisms supported by compact oriented 3-manifolds Y with a properly embedded framed tangle L F a n d a n arbitrary PSL(2, C)-character ρ of YLF (covering, for example, the case of hyperbolic cone manifolds). The marking of QHFT bordisms includes a specific set of parameters for the space of pleated hyperbolic structures on punctured surfaces. Each (QHFT) associates in a constructive way to any triple (Y,LF,ρ) with marked boundary components a tensor built on the matrix dilogarithms, which is holomorphic in the boundary parameters. When N=1 the QHFT tensors are scalar valued, and coincide with the Cheeger-Chern-Simons in variants of PSL (2, C)-characters on closed manifolds or cusped hyperbolic manifolds. We establish surgery formulas for QHFT partitions functions and describe their relations with the quantum hyperbolic invariants of Baseilhac and Benedetti (either defined for unframed links in closed manifolds and characters trivial at the link meridians, or cusped hyperbolic 3-manifolds). For every PSL (2, C)-character of a punctured surface, we produce new families of conjugacy classes of moderately projective" representations of the mapping class groups. © 2007 Mathematical Sciences Publishers.