Geometry & Topology

Classical and quantum dilogarithmic invariants of flat $PSL(2,\mathbb{C})$–bundles over 3–manifolds

Stephane Baseilhac and Riccardo Benedetti

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Abstract

We introduce a family of matrix dilogarithms, which are automorphisms of NN, N being any odd positive integer, associated to hyperbolic ideal tetrahedra equipped with an additional decoration. The matrix dilogarithms satisfy fundamental five-term identities that correspond to decorated versions of the 23 move on 3–dimensional triangulations. Together with the decoration, they arise from the solution we give of a symmetrization problem for a specific family of basic matrix dilogarithms, the classical (N=1) one being the Rogers dilogarithm, which only satisfy one special instance of five-term identity. We use the matrix dilogarithms to construct invariant state sums for closed oriented 3–manifolds W endowed with a flat principal PSL(2,)–bundle ρ, and a fixed non empty link L if N>1, and for (possibly “marked”) cusped hyperbolic 3–manifolds M. When N=1 the state sums recover known simplicial formulas for the volume and the Chern–Simons invariant. When N>1, the invariants for M are new; those for triples (W,L,ρ) coincide with the quantum hyperbolic invariants defined by the first author, though our present approach clarifies substantially their nature. We analyse the structural coincidences versus discrepancies between the cases N=1 and N>1, and we formulate “Volume Conjectures”, having geometric motivations, about the asymptotic behaviour of the invariants when N.

Article information

Source
Geom. Topol., Volume 9, Number 1 (2005), 493-569.

Dates
Received: 2 August 2003
Revised: 5 April 2005
Accepted: 5 April 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799573

Digital Object Identifier
doi:10.2140/gt.2005.9.493

Mathematical Reviews number (MathSciNet)
MR2140989

Zentralblatt MATH identifier
1093.57005

Subjects
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]

Keywords
dilogarithms state sum invariants quantum field theory Cheeger–Chern–Simons invariants scissors congruences hyperbolic 3–manifolds.

Citation

Baseilhac, Stephane; Benedetti, Riccardo. Classical and quantum dilogarithmic invariants of flat $PSL(2,\mathbb{C})$–bundles over 3–manifolds. Geom. Topol. 9 (2005), no. 1, 493--569. doi:10.2140/gt.2005.9.493. https://projecteuclid.org/euclid.gt/1513799573


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