Asymptotics of complex $b$-$6j$ symbols

We study the $b$-$6j$ symbols – an analytic extension of the $6j$-symbols for the principal series of the modular double of ${\mathrm{U}}_q\mathfrak{s}\mathfrak{l}(2;\mathbb{R})$ – with complex index $b,$ refereed to as the \emph{complex $b$-$6j$ symbols}. Then we relate their asymptotics, when the six parameters scale according to the dihedral angles of a hyperideal hyperbolic tetrahedron, to the volume and the determinant of the Gram matrix of the tetrahedron. In the case $\arg b=\pm \frac{\pi }{4},$ we believe that this work is closely related to the complex Liouville string,\cite{CEMR}.