Classical and quantum dilogarithmic invariants of flat \(\text{PSL}(2,\mathbb C)\)-bundles over 3-manifolds
In a previous paper [Topology 43, No. 6, 1373--1423 (2004; Zbl 1065.57008)] the authors introduced a family of invariants of links in a closed oriented 3-manifold with a flat principal \(\text{PSL}(2,\mathbb C)\) bundle over the manifold, which they called quantum hyperbolic invariants. In this paper they study the structure of these invariants in terms of matrix dilogarithms, i.e. automorphisms of \({\mathbb C}^N\otimes {\mathbb C}^N\). These quantum matrix dilogarithms are used to define state sum invariants which generalize the hyperbolic volume and Chern-Simons invariants. The paper is divided into a ``semi-local'' part in which quantum matrix dilogarithms occur as solutions to a symmetrization problem analogous to the classical Rogers dilogarithm case, and a ``global'' part in which various volume conjectures are formulated. Both parts are detail-rich and lengthy. The paper includes one section which summarizes the main results of the balance as well as an appendix containing some background for the computations in the body of the paper.
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