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

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 P SL(2, C)-bundle , 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, ) 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 .