On the volume conjecture for the Teichm ̈uller TQFT

The Chern-Simons theory is a topological quantum field theory (TQFT) on the principal G-bundle and has been studied in both mathematics and physics. When G is SU(2), which is compact, Witten conjectured that its path integral gives the topological invariant of the base 3-manifold. This invariant was formulated rigorously and is known as the WRT invariant. In addition, it is known that the expectation value of the Wilson loop along the hyperbolic knot in S3 gives the invariant of knots, which is called the colored Jones polynomial. Invariants of knots and manifolds derived from the path integral are called quantum invariants. There is an open conjecture called the volume conjecture, which states that the complete hyperbolic volume of the knot complement appears in the asymptotic expansion of the colored Jones polynomial. The volume conjecture suggests a close connection between quantum invariants and hyperbolic geometry.
On the other hand, Chern-Simons theory with the non-compact G such as SL(2,C) also appears in duality in string theory called the 3d-3d correspondence but has not been well formulated mathematically. Andersen and Kashaev constructed a TQFT-like theory called the Teichm ̈uller TQFT by quantizing the Teichm ̈uller space, which is the deformation space of the hyperbolic structures on a surface. The Teichm ̈uller TQFT is expected to correspond to the SL(2,C) Chern-Simons theory. In this theory, a conjecture similar to the volume conjecture has been proposed and proven for several hyperbolic knots.
In this talk, I will introduce the outline of the Teichm ̈uller TQFT and explain our results on the volume conjecture and its proof using techniques in hyperbolic geometry by Thurston, Casson, Rivin, and others.