曲面群量子表示的渐近性
Asymptotics of quantum representations of surface groups
摘要：
对于一个带状链接$L$在一个表面乘以一个圆，Witten-Reshetikhin-Turaev不变量是拓扑不变量依赖于一个序列的复杂$2p$ -th单位根$(A_p)_{p\in 2\mathbb{N}}$。我们证明了存在一个多项式$P_L$，使得当$A_p$收敛于$u$时，这些归一化不变量收敛于$P_L(u)$，对于$S^1$中除了有限个$u$之外的所有$u$。这与AMU猜想有关，AMU猜想预测非简单曲线在量子表示下具有无限阶（对于足够大的能级）。估计程度的$P_L$，我们表现出特定类型的曲线，满足这一猜想。一路上，我们证明了威滕渐近猜想的联系，在一个表面倍圆。
英文摘要：
For a banded link $L$ in a surface times a circle, the Witten-Reshetikhin-Turaev invariants are topological invariants depending on a sequence of complex $2p$-th roots of unity $(A_p)_{p\in 2\mathbb{N}}$. We show that there exists a polynomial $P_L$ such that these normalized invariants converge to $P_L(u)$ when $A_p$ converges to $u$, for all but a finite number of $u$'s in $S^1$. This is related to the AMU conjecture which predicts that non-simple curves have infinite order under quantum representations (for big enough levels). Estimating the degree of $P_L$, we exhibit particular types of curves which satisfy this conjecture. Along the way we prove the Witten asymptotic conjecture for links in a surface times a circle.
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