Topology Seminar: Volume conjectures for Reshetikhin-Turaev and Turaev-Viro invariants

Seminar | September 30 | 4:10-5 p.m. | 3 Evans Hall

 Tian Yang, Stanford University

 Department of Mathematics

In a joint work with Qingtao Chen, we consider a family of Turaev-Viro type invariants for a 3-manifold $M$ with non-empty boundary, indexed by an integer $r \geq 3$, and propose a volume conjecture for hyperbolic $M$ that these invariants grow exponentially at large $r$ with a growth rate the hyperbolic volume of $M$. The crucial step is the evaluation at the root of unity $\text{exp}(2\pi\sqrt{-1}/r)$ instead of that at the usually considered root $\text{exp}(\pi\sqrt{-1}/r)$. Evaluating at the same root $\text{exp}(2\pi\sqrt{-1}/r)$, we then conjecture that, the original Turaev-Viro invariants and the Reshetikhin-Turaev invariants of a closed hyperbolic 3-manifold $M$ grow exponentially with growth rates respectively the hyperbolic and the complex volume of $M$. This uncovers a different asymptotic behavior of the values at other roots of unity than that at $\text{exp}(\pi\sqrt{-1}/r)$ predicted by Witten’s Asymptotic Expansion Conjecture, which may indicate some different geometric interpretation of the Reshetikhin-Turaev invariants than the $SU(2)$ Chern-Simons theory. Numerical evidences will be provided to support these conjectures.