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⊗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 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

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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