超對稱楊-米爾斯理論

在物理學中，超对称杨-米爾斯 (SYM) 理论跟楊-米爾斯理論、弦理論、共形理論、共形對稱、超对称有關。^[1]

四維（N=4）

N=4超楊米理論的拉格朗日量是^[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

${\displaystyle f}$ 是楊米爾斯規範群的结构常数

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

由於非重整化定理，這是一個超共形場論。

十维

N=10的拉氏量是

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

I, J = 0, ..., 9

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

${\displaystyle \theta _{I}}$ 是陈类，CS9是9維的陳-西蒙斯形式（陳-西蒙斯理論）。

應用

• S對偶的耦合常數：

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

• AdS/CFT对偶、全像原理、类型IIB弦理论、量子引力

• 大N展開^[3]、平面楊米爾斯理論、 ${\displaystyle N^{2-2g}}$的费曼图 ^[4]
• Yangian^[5]
• 扭量理论、格拉斯曼流形
• 11維M理論、${\displaystyle N\rightarrow \infty }$

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 [2020-03-07]. （原始内容存档于2021-01-28）. 
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. [2020-03-07]. （原始内容存档于2020-10-01）. 
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. 

 閱讀

• Kapustin, Anton; Witten, Edward. Electric-magnetic duality and the geometric Langlands program. Communications in Number Theory and Physics. 2007, 1 (1): 1–236. Bibcode:2007CNTP....1....1K. arXiv:hep-th/0604151 . doi:10.4310/cntp.2007.v1.n1.a1.