# 超对称杨-米尔斯理论

## 四维（N=4）

N=4超杨米理论的拉格朗日量[2]

${\displaystyle L=\operatorname {tr} \left\{-{\frac {1}{2g^{2))}F_{\mu \nu }F^{\mu \nu }+{\frac {\theta _{I)){8\pi ^{2))}F_{\mu \nu }{\bar {F))^{\mu \nu }-i{\overline {\lambda ))^{a}{\overline {\sigma ))^{\mu }D_{\mu }\lambda _{a}-D_{\mu }X^{i}D^{\mu }X^{i}+gC_{i}^{ab}\lambda _{a}[X^{i},\lambda _{b}]+g{\overline {C))_{iab}{\overline {\lambda ))^{a}[X^{i},{\overline {\lambda ))^{b}]+{\frac {g^{2)){2))[X^{i},X^{j}]^{2}\right\},}$

${\displaystyle F_{\mu \nu }^{k}=\partial _{\mu }A_{\nu }^{k}-\partial _{\nu }A_{\mu }^{k}+f^{klm}A_{\mu }^{l}A_{\nu }^{m))$

i, j =1,...,6

a, b =1,...,4

${\displaystyle f}$ 是杨米尔斯规范群的结构常数

${\displaystyle C_{i}^{ab))$ 是SU(4)的结构常数

## 十维

N=10的拉氏量

${\displaystyle L=\operatorname {tr} \left\((\frac {1}{g^{2))}F_{IJ}F^{IJ}-i{\bar {\lambda ))\Gamma ^{I}D_{I}\lambda \right\}+\theta _{I}dCS_{9}^{I))$

I, J = 0, ..., 9

${\displaystyle \Gamma ^{I))$ 是32x32的矩阵 ${\displaystyle (32=2^{10/2})}$

${\displaystyle \theta _{I))$陈类，CS9是9维的陈-西蒙斯形式陈-西蒙斯理论）。

## 应用

${\displaystyle \tau ={\frac {\theta }{2\pi ))+{\frac {4\pi i}{g^{2))}.}$

## 参考文献

1. ^ Matt von Hippel. Earning a PhD by studying a theory that we know is wrong. Ars Technica. 2013-05-21.
2. ^ Luke Wassink. N = 4 Super Yang–Mills theory (PDF). 2009 [2013-05-22]. （原始内容 (PDF)存档于2014-05-31）.
3. ^ Martin Ammon, Johanna Erdmenger, Gauge/Gravity Duality: Foundations and Applications, Cambridge University Press, 2015, p. 240.
4. ^ planar limit in nLab
5. ^ Beisert, Niklas. Review of AdS/CFT Integrability: An Overview. Letters in Mathematical Physics. January 2012, 99: 425. Bibcode:2012LMaPh..99..425K. arXiv:1012.4000. doi:10.1007/s11005-011-0516-7.