Classical and quantum dilogarithmic invariants of flat \(\text{PSL}(2,\mathbb C)\)-bundles over 3-manifolds
type: Item
instance of: scholarly article
title: Classical and quantum dilogarithmic invariants of flat \(\text{PSL}(2,\mathbb C)\)-bundles over 3-manifolds
zbMATH Open document ID: 1093.57005
author: Stéphane Baseilhac
author: R. Benedetti
published in: Geometry \& Topology
publication date: 2005-09-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0306283
full work available at URL: https://eudml.org/doc/126308
full work available at URL: paper15.abs.html
review text: 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.
reviewed by: Sam Nelson
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57Q15
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 20G42
zbMATH DE Number: 2206539
zbMATH Keywords: dilogarithms
zbMATH Keywords: state sum invariants
zbMATH Keywords: quantum field theory
zbMATH Keywords: Cheeger-Chern-Simons invariants
zbMATH Keywords: scissors congruences
zbMATH Keywords: hyperbolic 3-manifolds
MaRDI profile type: Publication
OpenAlex ID: W2009145422
arXiv ID: math/0306283
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DOI: 10.2140/GT.2005.9.493
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description: scientific article