Teichmuller TQFT vs Chern-Simons Theory

TL;DR

This paper explores Teichmüller TQFT, relating it to Chern-Simons theories, proposing a novel path-integral definition, and clarifying its duality and mathematical structure within 3d-3d correspondence.

Contribution

It introduces a new path-integral formulation of Teichmüller TQFT and establishes its duality with complex SL(2,C) Chern-Simons theory at level 1.

Findings

Proposes an analytically-continued Chern-Simons path-integral with a special cycle.

Shows Teichmüller TQFT is dual to complex SL(2,C) Chern-Simons theory at k=1.

Provides a derivation of complex Chern-Simons theories from 6d (2,0) theory.

Abstract

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.