Volume 7, issue 2 (2007)

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Quantum hyperbolic geometry

Stephane Baseilhac and Riccardo Benedetti

Algebraic & Geometric Topology 7 (2007) 845–917
 arXiv: math.GT/0611504
Abstract

We construct a new family, indexed by odd integers $N\ge 1$, of $\left(2+1\right)$–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 $\left(2+1\right)$–bordisms supported by compact oriented $3$–manifolds $Y$ with a properly embedded framed tangle ${L}_{\mathsc{ℱ}}$ and an arbitrary $PSL\left(2,ℂ\right)$–character $\rho$ of $Y\setminus {L}_{\mathsc{ℱ}}$ (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 $\left(Y,{L}_{\mathsc{ℱ}},\rho \right)$ 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 invariants of $PSL\left(2,ℂ\right)$–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 $PSL\left(2,ℂ\right)$–character of a punctured surface, we produce new families of conjugacy classes of “moderately projective" representations of the mapping class groups.

Keywords
hyperbolic geometry, quantum field theory, mapping class group representations, quantum invariants, Cheeger–Chern–Simons class, dilogarithms
Mathematical Subject Classification 2000
Primary: 57M27, 57Q15
Secondary: 57R20, 20G42