# 瞬子

## 4维杨-米尔斯瞬子

${\displaystyle S_{YM}=\int tr(F\wedge *F)}$

${\displaystyle {\frac {1}{2}}{\frac {\delta S_{YM}}{\delta A}}=d_{D}F=dF+[A,F]=0}$

${\displaystyle d_{D}*F=0}$

${\displaystyle F=\pm *F}$

### 陈-西蒙斯

${\displaystyle \int _{M}c_{2}={\frac {1}{2}}({\frac {i}{2\pi }})^{2}\int _{M}tr(F^{2})={\frac {1}{2}}({\frac {i}{2\pi }})^{2}\int _{M}dCS_{3}={\frac {1}{2}}({\frac {i}{2\pi }})^{2}\int _{\partial M}CS_{3}}$

${\displaystyle A\to 0\equiv gdg^{-1}}$

${\displaystyle A\equiv g(d+A)g^{-1}}$

${\displaystyle F\to 0}$

${\displaystyle CS_{3}=tr(AF-{\frac {1}{3}}A^{3})}$

${\displaystyle CS_{3}\to -tr(A^{3})/3}$

${\displaystyle \int _{M}c_{2}=-{\frac {1}{2}}({\frac {i}{2\pi }})^{2}\int _{\partial M}tr(A^{3})/3={\frac {1}{24\pi ^{2}}}\int _{\partial M}tr(gdg^{-1})^{3}}$

${\displaystyle \int _{M}c_{2}={\frac {1}{24\pi ^{2}}}\int _{\partial M}tr(gdg^{-1})^{3}=\nu \in \mathbb {Z} }$

${\displaystyle S=S_{YM}+\theta \int c_{2}}$

${\displaystyle Z=\int dAe^{iS(A)}\to e^{i\theta \nu }\int e^{-S_{YM}}}$