Chern-Simons

若$M^3$为可定向紧三维流形，即其可平行化，取其上平凡丛$F(M)=M\times \text{SO}(3)$，$A=A_\mu\dd x^\mu$和$F=\dd A+A\wedge A=\displaystyle\frac{1}{2}F_{\mu\nu}\dd x^\mu\wedge\dd x^\nu$分别为丛上斜对称联络和曲率形式，可知$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+[A_\mu,A_\nu]$，构造以下作用量

$$
\begin{aligned}
S_\text{CS}&=\frac{k}{4\pi}\int_{M^3}\left(\dd A\wedge A+\frac{2}{3}A\wedge A\wedge A\right)\\
&=\frac{k}{8\pi}\int_{M^3}\epsilon^{\mu\nu\rho}\left(A^\mu(\partial_\nu A_\rho-\partial_\rho A_\nu)+\frac{2}{3}A_\mu[A_\nu,A_\rho]\right)
\end{aligned}
$$