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

Baseilhac, Stephane; Benedetti, Riccardo

doi:10.2140/gt.2005.9.493

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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

DOI: 10.2140/gt.2005.9.493

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 $P S L (2, ℂ)$ –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 $N \to \infty$ .

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

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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

Volume 9, issue 1 (2005)