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

Abstract

We introduce a family of matrix dilogarithms, which are automorphisms of ℂN ⊗ ℂN, 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 in [3], 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 → ∞. © Geometry & Topology Publications.