  Teichm\"uller TQFT is a unitary 3d topological theory whose Hilbert spaces
are spanned by Liouville conformal blocks. It is related but not identical to
PSL(2,R) Chern-Simons theory. To physicists, it is known in particular in the
context of 3d-3d correspondence and also in the holographic description of
Virasoro conformal blocks. We propose that this theory can be defined by an
analytically-continued Chern-Simons path-integral with an unusual integration
cycle. On hyperbolic three-manifolds, this cycle is singled out by the
requirement of invertible vielbein. Mathematically, our proposal translates a
known conjecture by Andersen and Kashaev into a conjecture about the
Kapustin-Witten equations. We further explain that Teichm\"uller TQFT is dual
to complex SL(2,C) Chern-Simons theory at integer level k=1, clarifying some
puzzles previously encountered in the 3d-3d correspondence literature. We also
present a new simple derivation of complex Chern-Simons theories from the 6d
(2,0) theory on a lens space with a transversely-holomorphic foliation.
