On the Kashaev--Luo--Vartanov Partition Function for Cusped 3-Manifolds: Asymptotics and a squared norm property

Abstract

We study the asymptotic and TQFT-type properties of the Kashaev--Luo--Vartanov (KLV) partition function. We show that the partition function depends on the prescribed angle structure only through its peripheral angular holonomies. For any geometric triangulation of a cusped 3-manifold realizing a hyperbolic cone structure, we express the exponential decay rate and the 1-loop term of the partition function in terms of, respectively, the volume and the adjoint twisted Reidemeister torsion of the corresponding hyperbolic cone structure. Consequently, these asymptotic formulas hold for any angle structure having the same peripheral angular holonomies as the geometric one. We further introduce a Reshetikhin--Turaev-type function associated with certain admissible Neumann--Zagier data, extending the Jones function in Teichmüller TQFT previously studied by Ben Aribi and the author. We show that, after a combinatorial normalization, for any triangulation, the KLV partition function can be expressed as a weighted integral of the squared norm of the Reshetikhin--Turaev-type function, where the weight is determined by the angular holonomies of peripheral curves. In particular, for FAMED triangulations, the Reshetikhin--Turaev-type function agrees with the Jones function, so that the normalized KLV partition function is obtained by integrating the squared norm of the integrand defining the Teichmüller TQFT partition function. This provides a noncompact analogue of the relationship between Turaev--Viro and Reshetikhin--Turaev invariants.