#### 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.

Notes on file formats
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

and

Dipartimento di Matematica, Universita di Pisa

Via F. Buonarroti, 2, I-56127 Pisa, ITALY

Email: baseilha@ujf-grenoble.fr, benedett@dm.unipi.it

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