The perturbative expansion of Chern–Simons gauge theory leads to invariants of knots and links, the so-called finite type invariants or Vassiliev invariants. It has been proved that at any order in perturbation theory the superposition of certain amplitudes is an invariant of that order. Bott–Taubes integrals on configuration spaces are introduced in the present context to write Feynman diagrams at a given order in perturbation theory in a geometrical and topological framework. One of the consequences of this formalism is that the resulting amplitudes are rewritten in cohomological terms in configuration spaces. This cohomological structure can be used to translate Bott–Taubes integrals into Chern–Simons perturbative amplitudes and vice versa. In this paper, this program is performed up to third order in the coupling constant. This expands some work previously worked out by Thurston. Finally we take advantage of these results to incorporate in the formalism a smooth and divergenceless vector field on the 3-manifold. The Bott–Taubes integrals obtained are used for constructing higher-order average asymptotic Vassiliev invariants extending the work of Komendarczyk and Volić.

中文翻译：

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通过 Chern-Simons 微扰理论的流的 Vasiliev 不变量

Chern-Simons 规范理论的微扰扩展导致了结和链接的不变量，即所谓的有限类型不变量或 Vasiliev 不变量。已经证明，在微扰理论的任何阶下，某些振幅的叠加都是该阶的不变量。在本文中介绍了配置空间上的 Bott-Taubes 积分，以在几何和拓扑框架中的微扰理论中以给定顺序编写费曼图。这种形式主义的后果之一是，所产生的幅度在配置空间中被重写为上同调项。这种上同调结构可用于将 Bott-Taubes 积分转换为 Chern-Simons 微扰幅度，反之亦然。在本文中，该程序在耦合常数中执行到三阶。这扩展了瑟斯顿之前完成的一些工作。最后，我们利用这些结果在形式主义中加入了一个平滑且无散的向量场3-歧管。获得的 Bott-Taubes 积分用于构建扩展 Komendarczyk 和 Volić 工作的高阶平均渐近 Vasiliev 不变量。