Quantum hyperbolic geometry

Abstract

We construct a new family, indexed by the odd integers $N\geq 1$, of $(2+1)$-dimensional quantum field theories called {\it 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 $L_\Ff$ and an {\it arbitrary} $PSL(2,\C)$-character $\rho$ of $Y \setminus L_\Ff$ (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 $(Y,L_\Ff,\rho)$ with marked boundary components a tensor built on the matrix dilogarithms, which is holomorphic in the boundary parameters. We establish {\it surgery formulas} for QHFT partitions functions and describe their relations with the {\it quantum hyperbolic invariants} of \cite{BB1,BB2} (either defined for unframed links in closed manifolds and characters trivial at the link meridians, or hyperbolic {\it cusped} 3-manifolds). For every $PSL(2,\mc)$-character of a punctured surface, we produce new families of conjugacy classes of "moderately projective" representations of the mapping class groups.