Algebraic & Geometric Topology

Quantum hyperbolic geometry

Stephane Baseilhac and Riccardo Benedetti

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We construct a new family, indexed by odd integers N1, 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 and an arbitrary PSL(2,)–character ρ of YL (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,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 invariants of PSL(2,)–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,)–character of a punctured surface, we produce new families of conjugacy classes of “moderately projective" representations of the mapping class groups.

Article information

Algebr. Geom. Topol., Volume 7, Number 2 (2007), 845-917.

Received: 16 November 2006
Accepted: 21 March 2007
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 57Q15: Triangulating manifolds
Secondary: 57R20: Characteristic classes and numbers 20G42: Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50]

hyperbolic geometry quantum field theory mapping class group representations quantum invariants Cheeger–Chern–Simons class dilogarithms


Baseilhac, Stephane; Benedetti, Riccardo. Quantum hyperbolic geometry. Algebr. Geom. Topol. 7 (2007), no. 2, 845--917. doi:10.2140/agt.2007.7.845.

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