Teichmuller TQFT vs Chern-Simons Theory

Understand

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.