#### Volume 9, issue 1 (2005)

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

### Stephane Baseilhac and Riccardo Benedetti

Geometry & Topology 9 (2005) 493–569
 arXiv: math.GT/0306283
##### Abstract

We introduce a family of matrix dilogarithms, which are automorphisms of ${ℂ}^{N}\otimes {ℂ}^{N}$, $N$ 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 $2\to 3$ 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 $PSL\left(2,ℂ\right)$–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 $\left(W,L,\rho \right)$ 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 $N\to \infty$.

##### Keywords
dilogarithms, state sum invariants, quantum field theory, Cheeger–Chern–Simons invariants, scissors congruences, hyperbolic 3–manifolds.
##### Mathematical Subject Classification 2000
Primary: 57M27, 57Q15
Secondary: 57R20, 20G42
##### Publication
Received: 2 August 2003
Revised: 5 April 2005
Accepted: 5 April 2005
Published: 8 April 2005
Proposed: Robion Kirby
Seconded: Walter Neumann, Shigeyuki Morita
##### Authors
 Stephane Baseilhac Université de Grenoble I Institut Joseph Fourier UMR CNRS 5582 100 rue des Maths BP 74 F-38402 Saint-Martin-d’Hères Cedex France Riccardo Benedetti Dipartimento di Matematica Università di Pisa Via F. Buonarroti, 2 I-56127 Pisa Italy