The AJ-conjecture for the Teichmüller TQFT

We formulate the AJ-conjecture for the Teichm\"{u}ller TQFT and we prove it in the case of the figure-eight knot complement and the $5_2$-knot complement. This states that the level-$N$ Andersen-Kashaev invariant, $J^{(\mathrm{b},N)}_{M,K}$, is annihilated by the non-homogeneous $\widehat{A}$-polynomial, evaluated at appropriate $q$-commutative operators. These are obtained via geometric quantisation on the moduli space of flat $\operatorname{SL}(2,\mathbb{C})$-connections on a genus-$1$ surface. The construction depends on a parameter $\sigma$ in the Teichm\"{u}ller space in a way measured by the Hitchin-Witten connection, and results in Hitchin-Witten covariantly constant quantum operators for the holonomy functions $m$ and $\ell$ along the meridian and longitude. Their action on $J^{(\mathrm{b},N)}_{M,K}$ is then defined via a trivialisation of the Hitchin-Witten connection and the Weil-Gel'Fand-Zak transform.