Geometry & Topology, Vol. 9 (2005) Paper no. 15, pages 493--569.

Classical and quantum dilogarithmic invariants of flat PSL(2,C)-bundles over 3-manifolds

Stephane Baseilhac and Riccardo Benedetti

Abstract. We introduce a family of matrix dilogarithms, which are automorphisms of C^N tensor C^N, 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 2 --> 3 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,C)-bundle rho, 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,rho) coincide with the quantum hyperbolic invariants defined in [Topology 43 (2004) 1373-1423], 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 tends to infinity.

Keywords. Dilogarithms, state sum invariants, quantum field theory, Cheeger-Chern-Simons invariants, scissors congruences, hyperbolic 3-manifolds.

AMS subject classification. Primary: 57M27, 57Q15. Secondary: 57R20, 20G42.

DOI: 10.2140/gt.2005.9.493

E-print: arXiv:math.GT/0306283

Submitted to GT on 2 August 2003. (Revised 5 April 2005.) Paper accepted 5 April 2005. Paper published 8 April 2005.

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Stephane Baseilhac and Riccardo Benedetti
Universite de Grenoble I, Institut Joseph Fourier, UMR CNRS 5582
100 rue des Maths, B.P. 74, F-38402 Saint-Martin-d'Heres Cedex, FRANCE
Dipartimento di Matematica, Universita di Pisa
Via F. Buonarroti, 2, I-56127 Pisa, ITALY


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