2001/01/29 by Stephane Baseilhac, Stéphane Baseilhac, Baseilhac, Stephane +2
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #High Energy Physics - Theory (hep-th) #Noncommutative and Quantum Gravity Theories #hep-th #math.GT
paper · pdf · doi:10.48550/arxiv.math/0101234
36 pages (including an 6-pages Appendix), 7 figures. Replacement's motivations: language improved, general clarification and minor corrections in the last section, elimination of proposition 4.4 of the previous version due to unsatisfactory proof
openalex publication_date 2001/01/29 · arxiv created 2001/02/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Any triple (W,L,ρ), where W is a compact closed oriented 3-manifold, L is a link in W and ρ is a flat principal B-bundle over W (B is the Borel subgroup of upper triangular matrices of SL(2,\mc)), can be encoded by suitable \it distinguished and \it decorated triangulations \cal T=(T,H,\cal D). For each \cal T, for each odd integer N≥ 3, one defines a state sum KN(\cal T), based on the Faddeev-Kashaev quantum dilogarithm at ω=exp(2πi/N), such that KN(W,L,ρ)=KN(\cal T) is a well-defined complex valued invariant. The purely topological, conjectural invariants KN(W,L) proposed earlier by Kashaev correspond to the special case of the \it trivial flat bundle. Moreover, we extend the definition of these invariants to the case of flat bundles on W∖ L with non necessarily trivial holonomy along the meridians of the link's components, and also to 3-manifolds endowed with a B-flat bundle and with arbitrary non-spherical parametrized boundary components. We point out some remarkable specializations of the invariants; among these, the so called \it Seifert-type invariants, when W=S3: these seem to be good candidates in orther to fully reconduct the Jones polynomials in the main stream of quantum hyperbolic invariants. Finally, we try to set our results against the heuristic backgroud of the Euclidean analytic continuation of (2+1)-quantum gravity with negative cosmological constant, regarded as a gauge theory with the \it non compact group SO(3,1) as gauge group.