Quantum hyperbolic invariants of 3-manifolds with PSL(2,C)-characters

Abstract

We construct quantum hyperbolic invariants (QHI) for triples (W, L, ρ), where W is a compact closed oriented 3-manifold, ρ is a flat principal bundle over W with structural group P SL(2, C), and L is a non-empty link in W . These invariants are based on the Faddeev-Kashaev's quantum dilogarithms, interpreted as matrix valued functions of suitably decorated hyperbolic ideal tetrahedra. They are explicitely computed as state sums over the decorated hyperbolic ideal tetrahedra of the idealization of any fixed D-triangulation; the D-triangulations are simplicial 1-cocycle descriptions of (W, ρ) in which the link is realized as a Hamiltonian subcomplex. We also discuss how to set the Volume Conjecture for the coloured Jones invariants JN (L) of hyperbolic knots L in S 3 in the framework of the general QHI theory.

Classical and quantum dilogarithmic invariants of flatPSL(2,ℂ)–bundles over 3–manifolds

“…Studying them directly is far from a mere rephrasing of the results in [3]. As a by-product, here we show that the symmetrization is intrisically related to the algebra B ζ .…”

“…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 (W, L, ρ) coincide with the quantum hyperbolic invariants defined in [3], 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 → ∞.…”

“…The present strong version was mentioned in Section 4.5 of [3] without proof. Along with the proof of Theorem 6.8 we shall stress the main differences with respect to [3].…”

“…An I -tetrahedron (∆, b, w) (see also [3]) consists of (1) An oriented tetrahedron ∆, that we usually represent as positively embedded in R 3 (oriented by its standard basis). This gives a cross-ratio modulus w(e) to each edge e of ∆, by imposing that w(e) = w(e ′ ) for each edge e.…”

“…In our previous paper [3] we constructed quantum hyperbolic invariants (QHI) H N (W, L, ρ), well-defined possibly up to a sign and an N th root of unity phase factor. Here N > 1 is an odd integer, L is a non empty link in a closed oriented 3-manifold W , and ρ is a flat principal P SL(2, C)-bundle over W .…”