Analytic families of quantum hyperbolic invariants

Baseilhac, Stéphane¹ ; Benedetti, Riccardo²

¹Institut de Mathématiques et de Modélisation, Université de Montpellier, Case Courrier 51, 34095 Montpellier, Cedex 5, France
²Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, I-56127 Pisa, Italy

Algebraic and Geometric Topology, Tome 15 (2015) no. 4, pp. 1983-2063

Résumé

We organize the quantum hyperbolic invariants (QHI) of 3–manifolds into sequences of rational functions indexed by the odd integers N ≥ 3 and defined on moduli spaces of geometric structures refining the character varieties. In the case of one-cusped hyperbolic 3–manifolds M we generalize the QHI and get rational functions ℋNhf,hc,kc depending on a finite set of cohomological data (hf,hc,kc) called weights. These functions are regular on a determined Abelian covering of degree N2 of a Zariski open subset, canonically associated to M, of the geometric component of the variety of augmented PSL(2, ℂ)–characters of M. New combinatorial ingredients are a weak version of branchings which exists on every triangulation, and state sums over weakly branched triangulations, including a sign correction which eventually fixes the sign ambiguity of the QHI. We describe in detail the invariants of three cusped manifolds, and present the results of numerical computations showing that the functions ℋNhf,hc,kc depend on the weights as N →∞, and recover the volume for some specific choices of the weights.

Détail

DOI : 10.2140/agt.2015.15.1983

Classification : 57M27, 57Q15, 57R56
Keywords: quantum invariants, 3–manifolds, character varieties, Chern–Simons theory, volume conjecture

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