Quantum Hyperbolic State Sum Invariants of 3-Manifolds

Abstract

Any triple $(W,L,\rho )$, where $W$ is a compact closed oriented 3-manifold, $L$ is a link in $W$ and $\rho$ 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 $\mathcal{T}=(T,H,\mathcal{D})$. For each $\mathcal{T}$, for each odd integer $N\ge 3$, one defines a state sum $K_N(\mathcal{T})$, based on the Faddeev-Kashaev quantum dilogarithm at $\omega =\exp (2\pi i/N)$, such that $K_N(W,L,\rho )=K_N(\mathcal{T})$ is a well-defined complex valued invariant. The purely topological, conjectural invariants $K_N(W,L)$ 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 $W\setminus L$ 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 \emph{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 $W=S^3$: 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.

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

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Cited by in corpus (5)

• Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
• Quantum Teichmüller spaces and Kashaev's 6j-symbols
• Problems on invariants of knots and 3-manifolds
• Invariants of links with flat connections in their complements.II. Holonomy R-matrices related to quantized universal enveloping algebras at roots of 1
• Construction and Recognition of Hyperbolic 3-Manifolds with Geodesic Boundary