On the volume conjecture for the Teichm ̈uller TQFT
 Date
 May 31 (Fri) at 15:00  17:00, 2024 (JST)
 Speaker

 Soichiro Uemura (Junior Research Associate, iTHEMS / Student Trainee, iTHEMS)
 Venue
 via Zoom
 Seminar Room #359
 Language
 English
 Host
 Yuto Moriwaki
The ChernSimons theory is a topological quantum field theory (TQFT) on the principal Gbundle 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 3manifold. 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, ChernSimons theory with the noncompact G such as SL(2,C) also appears in duality in string theory called the 3d3d correspondence but has not been well formulated mathematically. Andersen and Kashaev constructed a TQFTlike 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) ChernSimons 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.
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