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Analytic families of quantum hyperbolic invariants

Stéphane Baseilhac and Riccardo Benedetti

Algebraic & Geometric Topology 15 (2015) 1983–2063

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 Nhf,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, )–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 Nhf,hc,kc depend on the weights as N , and recover the volume for some specific choices of the weights.

quantum invariants, 3–manifolds, character varieties, Chern–Simons theory, volume conjecture
Mathematical Subject Classification 2010
Primary: 57M27, 57Q15
Secondary: 57R56
Received: 1 March 2014
Revised: 25 September 2014
Accepted: 25 October 2014
Published: 10 September 2015
Stéphane Baseilhac
Institut de Mathématiques et de Modélisation
Université de Montpellier
Case Courrier 51
34095 Montpellier, Cedex 5
Riccardo Benedetti
Dipartimento di Matematica
Università di Pisa
Largo Bruno Pontecorvo 5
I-56127 Pisa