Classical and quantum dilogarithmic invariants of flat $PSL(2,\mathbb{C})$–bundles over 3–manifolds

Abstract

We introduce a family of matrix dilogarithms, which are automorphisms of ${ℂ}^N\otimes {ℂ}^N$, $N$ 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 $2\to 3$ 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 $PSL\left(2,ℂ\right)$–bundle $\rho$, 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 $\left(W,L,\rho \right)$ 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 $N\to \infty$.