A Geometric Quantisation view on the AJ-conjecture for the Teichmüller TQFT

Abstract

We provide a Geometric Quantisation formulation of the AJ-conjecture for the Teichmüller TQFT, and we prove it in detail in the case of the knot complements of $4_1$ and $5_2$. The conjecture states that the level-$N$ Andersen-Kashaev invariant, $J_{M,K}^{(\mathrm{b},N)}$, is annihilated by the non-homogeneous $\hat{A}$-polynomial, evaluated at appropriate $q$-commutative operators. We obtained the latter via Geometric Quantisation on the moduli space of flat $\mathrm{SL}(2,\mathbb{C})$-connections on a genus-$1$ surface, by considering the holonomy functions associated to a meridian and longitude. The construction depends on a parameter $\sigma$ in the Teichmüller space in a way measured by the Hitchin-Witten connection, but we show that the resulting quantum operators are covariantly constant. Their action on $J_{M,K}^{(\mathrm{b},N)}$ is then defined via a trivialisation of the Hitchin-Witten connection and the Weil-Gel'Fand-Zak transform.

27 pages (plus references), 2 figures

References in corpus (9)

• On the characteristic and deformation varieties of a knot
• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Generalized Volume Conjecture and the A-Polynomials -- the Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant
• Difference equation of the colored Jones polynomial for torus knot
• Complex Quantum Chern-Simons
• The Hitchin-Witten Connection and Complex Quantum Chern-Simons Theory
• Genus-one complex quantum Chern--Simons theory
• Asymptotic properties of the Hitchin-Witten connection
• $\mathrm{Sp}(1)$-symmetric hyperkähler quantisation

Cited by in corpus (3)

• Genus-one complex quantum Chern--Simons theory
• Asymptotic properties of the Hitchin-Witten connection
• Resurgence Analysis of Meromorphic Transforms