  We revisit the definition of the 6j-symbols from the modular double of
U_q(sl(2,R)), referred to as b-6j symbols. Our new results are (i) the
identification of particularly natural normalization conditions, and (ii) new
integral representations for this object. This is used to briefly discuss
possible applications to quantum hyperbolic geometry, and to the study of
certain supersymmetric gauge theories. We show, in particular, that the b-6j
symbol has leading semiclassical asymptotics given by the volume of a non-ideal
tetrahedron. We furthermore observe a close relation with the problem to
quantize natural Darboux coordinates for moduli spaces of flat connections on
Riemann surfaces related to the Fenchel-Nielsen coordinates. Our new integral
representations finally indicate a possible interpretation of the b-6j symbols
as partition functions of three-dimensional N=2 supersymmetric gauge theories.
