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 is annihilated by the inhomogeneous \(\hat{A}\) -polynomial, evaluated at appropriate q-commutative operators. We obtained the latter via Geometric Quantisation on the moduli space of flat \({{\,\textrm{SL}\,}}(2,\mathbb {C})\) -connections on a genus-1 surface, by considering the holonomy functions associated with 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 the Andersen–Kashaev invariant is then defined via a trivialisation of the Hitchin–Witten connection and the Weil–Gel’Fand–Zak transform.
