经典$\operatorname{SL}_2(\mathbb{C})$ Chern-Simons理论的手术演算
Surgery calculus for classical $\operatorname{SL}_2(\mathbb{C})$ Chern-Simons theory
摘要：
经典的$\operatorname{SL}_2(\mathbb{C})$ -Simons理论赋予一个表示为$ρ: π_1(M) \to \operatorname{SL}_2(\mathbb{C})$的$3$ -流形$M$其复体积$\operatorname{V}(M, ρ) \in \mathbb{C} / 2 π^2 i \mathbb{Z}$，实部为体积，虚部为Chern-Simons不变量。现有文献集中于使用三角测量来计算$\operatorname{V}$。在本文中，我们将展示如何计算$\operatorname{V}(M, L, ρ)$直接从手术图$M$一个紧凑的面向$3$ -流形与环面边界组件，嵌入尖$L$，并表示$ρ: π_1(M \setminus L) \to \operatorname{SL}_2(\mathbb{C})$。当$M$具有非空边界时，$\operatorname{V}(M, L, ρ)(\mathfrak{s})$依赖于一些额外的数据$\mathfrak{s}$，我们称之为对数装饰。我们的方法描述$ρ$在一个坐标系密切相关的量子群，我们认为我们的建设作为一个经典的，非紧的版本Witten-Reshetikhin-Turaev的量子$\operatorname{SU}(2)$陈-西蒙斯理论。
英文摘要：
Classical $\operatorname{SL}_2(\mathbb{C})$-Chern-Simons theory assigns a $3$-manifold $M$ with representation $ρ: π_1(M) \to \operatorname{SL}_2(\mathbb{C})$ its complex volume $\operatorname{V}(M, ρ) \in \mathbb{C} / 2 π^2 i \mathbb{Z}$, with real part the volume and imaginary part the Chern-Simons invariant. The existing literature focuses on computing $\operatorname{V}$ using a triangulation. In this paper we show how to compute $\operatorname{V}(M, L, ρ)$ directly from a surgery diagram for $M$ a compact oriented $3$-manifold with torus boundary components, embedded cusps $L$, and representation $ρ: π_1(M \setminus L) \to \operatorname{SL}_2(\mathbb{C})$. When $M$ has nonempty boundary $\operatorname{V}(M, L, ρ)(\mathfrak{s})$ depends on some extra data $\mathfrak{s}$ we call a log-decoration. Our method describes $ρ$ in a coordinate system closely related to quantum groups, and we think of our construction as a classical, noncompact version of Witten-Reshetikhin-Turaev's quantum $\operatorname{SU}(2)$ Chern-Simons theory.
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