Stéphane Baseilhac

Classical and quantum dilogarithmic invariants of flat PSL(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 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 → ∞. AMS Classification numbers Primary: 57M27, 57Q15 Secondary: 57R20, 20G42

Analytic families of quantum hyperbolic invariants

H hf;hc;kc N depending on a finite set of cohomological data.hf;hc;kc/ called weights. These functions are regular on a determined Abelian covering of degree N 2 of a Zariski open subset, canonically associated to M , of the geometric component of the variety of augmented PSL.2;C/‐characters of M . New combinatorial ingredients are a weak version of branchings which exists on every triangulation, and state sums over weakly branched triangulations, including a sign correction which eventually fixes the sign ambiguity of the QHI. We describe in detail the invariants of three cusped manifolds, and present the results of numerical computations showing that the functions H hf;hc;kc N depend on the weights as N!1, and recover the volume for some specific choices of the weights. 57M27, 57Q15; 57R56

On the quantum Teichmüller invariants of fibred cusped 3-manifolds

We show that the reduced quantum hyperbolic invariants of pseudo-Anosov diffeomorphisms of punctured surfaces are intertwiners of local representations of the quantum Teichmüller spaces. We characterize them as the only intertwiners that satisfy certain natural cut-and-paste operations of topological quantum field theories and such that their traces define invariants of mapping tori.