Quantum modularity and complex Chern-Simons theory

TL;DR

This paper constructs a power series from knot data and roots of unity, linking quantum modularity conjectures with complex Chern-Simons theory, and confirms its relation to Kashaev invariant asymptotics.

Contribution

It introduces a new series derived from Neumann-Zagier data that aligns with the Quantum Modularity Conjecture and matches numerical asymptotics of the Kashaev invariant.

Findings

Coefficients lie in the trace field of the knot extended by a root of unity

Series conjectured to match the Quantum Modularity Conjecture

Confirmed numerical asymptotics of the Kashaev invariant at roots of unity

Abstract

The Quantum Modularity Conjecture of Zagier predicts the existence of a formal power series with arithmetically interesting coefficients that appears in the asymptotics of the Kashaev invariant at each root of unity. Our goal is to construct a power series from a Neumann-Zagier datum (i.e., an ideal triangulation of the knot complement and a geometric solution to the gluing equations) and a complex root of unity $\zeta$. We prove that the coefficients of our series lie in the trace field of the knot, adjoined a complex root of unity. We conjecture that our series are those that appear in the Quantum Modularity Conjecture and confirm that they match the numerical asymptotics of the Kashaev invariant (at various roots of unity) computed by Zagier and the first author. Our construction is motivated by the analysis of singular limits in Chern-Simons theory with gauge group $SL(2,C)$ at fixed level $k$, where $\zeta^k=1$.