Volume 9 (2005)

Download This Article
with up-to-date links in citations
For screen
For printing
Recent Issues
Volume 1, 1997
Volume 2, 1998
Volume 3, 1999
Volume 4, 2000
Volume 5, 2001
Volume 6, 2002
Volume 7, 2003
Volume 8, 2004
Volume 9, 2005
Volume 10, 2006
Volume 11, 2007
Volume 12(1) 2008
Volume 12(2) 2008
Volume 12(3) 2008
Volume 12(4) 2008
Volume 12(5) 2008
Volume 13(1) 2009
Volume 13(2) 2009
Volume 13(3) 2009
Volume 13(4) 2009
Volume 13(5) 2009
Volume 14(2010) preview
G&T Monographs
The Journal
About the Journal
Editorial Board
Editorial Interests
Author Index
Editorial procedure
Submission Guidelines
Submission Page
Author copyright form
Subscriptions
Contacts
G&T Publications
GTP Author Index

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

Stephane Baseilhac and Riccardo Benedetti

Geometry & Topology 9 (2005) 493–569

DOI: 10.2140/gt.2005.9.493

Abstract

We introduce a family of matrix dilogarithms, which are automorphisms of CNCN, 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 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→ ∞.

Keywords

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

Mathematical Subject Classification

Primary: 57M27, 57Q15

Secondary: 20G42, 57R20

References
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