论文标题
经典$ \ operatatorName {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
论文作者
论文摘要
经典的$ \ peratatorName {sl} _2（\ Mathbb {c}）$ - Chern-Simons理论分配了$ 3 $ -Manifold $ m $，带有表示$ρ：π_1（m）\ to \ propatatorName \ to \ propatatorName {sl} _2（sl} _2（\ mathbb {c} c}） \ Mathbb {C} /2π^2 I \ Mathbb {z} $，实际上是卷和虚构的部分，而Chern-simons不变。现有的文献专注于使用三角剖分计算$ \ permatatorname {v} $。在本文中，我们展示了如何直接从手术图中计算$ \ operatorname {v}（m，l，ρ）$，$ m $ $ m $ a comptact取向的$ 3 $ -Manifold带有圆环边界组件，嵌入式cusps $ l $，以及表示$ρ：π_1（m \ setMinus l）（m \ setMinus \ to \ operatatorName {sl} _2（\ mathbb {c}）$。当$ m $具有非发行边界$ \ operatatorName {v}（m，l，ρ）（\ mathfrak {s}）$取决于一些额外的数据$ \ mathfrak {s} $，我们称之为日志decoration。我们的方法描述了与量子组密切相关的坐标系中的$ρ$，我们认为我们的构造是Witten-Reshetikhin-Turaev的量子$ \ permatatOrName {su}（2）$ Chern-Simons理论的经典，非合作版本。
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.
