In this note, we revisit the $\Theta$-invariant as defined by R. Bott and the first author in [4]. The $\Theta$-invariant is an invariant of rational homology 3-spheres with acyclic orthogonal local systems, which is a generalization of the 2-loop term of the Chern–Simons perturbation theory. The $\Theta$-invariant can be defined when a cohomology group is vanishing. In this note, we give a slightly modified version of the $\Theta$-invariant that we can define even if the cohomology group is not vanishing.

中文翻译：

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关于三流形的$ \ Theta $不变性的一个说明

在本说明中，我们将重新讨论R. Bott和[4]中的第一作者所定义的$ \ Theta $不变式。$ \ Theta $-不变式是具有非循环正交局部系统的有理同构三层球的不变式，它是对Chern-Simons扰动理论的2环项的推广。当同调群消失时，可以定义$ \ Theta $不变量。在此注释中，我们给出了可以定义的$ \ Theta $-不变式的略微修改版本，即使同调组没有消失也是如此。