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 $\zeta^k=1$.

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量子模块化和复陈-西蒙斯理论

Zagier 的量子模量猜想预测存在具有算术上有趣的系数的形式幂级数，这些系数出现在 Kashaev 不变量的每个单位根处的渐近线中。我们的目标是从 Neumann-Zagier 数据（即，结补的理想三角剖分和胶合方程的几何解）和复数单位根 $\zeta$ 构建幂级数。我们证明了我们级数的系数位于结的迹场中，毗邻复单位根。我们推测我们的系列是出现在 Quantum Modularity Conjecture 中的系列，并确认它们与 Zagier 和第一作者计算的 Kashaev 不变量（在不同的单位根）的数值渐近线相匹配。