查看“︁超對稱楊-米爾斯理論”︁的源代码

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]]、[[共形場論|共形理論]]、[[共形對稱]]、[[超对称]]有關。<ref>{{Cite web|title=Earning a PhD by studying a theory that we know is wrong|url=https://arstechnica.com/science/2013/05/earning-a-phd-by-studying-a-theory-that-we-know-is-wrong/|author=Matt von Hippel|date=2013-05-21|publisher=[[Ars Technica]]|access-date=2020-03-07|archive-date=2021-01-28|archive-url=https://web.archive.org/web/20210128105454/https://arstechnica.com/science/2013/05/earning-a-phd-by-studying-a-theory-that-we-know-is-wrong/}}</ref> 

== 四維（N=4） ==
N=4超楊米理論的[[拉格朗日量]]是<ref>{{Cite web|title=''N''&nbsp;=&nbsp;4 Super Yang–Mills theory|url=http://www-hep.physics.uiowa.edu/~vincent/courses/29276/Presentations/Luke%20Wassink.pdf|accessdate=2013-05-22|author=Luke Wassink|year=2009|archiveurl=https://web.archive.org/web/20140531090415/http://www-hep.physics.uiowa.edu/~vincent/courses/29276/Presentations/Luke%20Wassink.pdf|archivedate=2014-05-31}}</ref>

: <math>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
+g C^{ab}_i \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\},
</math>

<math>
F^k_{\mu\nu} = \partial_\mu A^k_\nu-\partial_\nu A^k_\mu+f^{klm}A^l_\mu A^m_\nu
</math>

''i'', ''j'' =1,...,6 

''a'', ''b'' =1,...,4

<math>f</math> 是楊米爾斯規範群的[[结构常数]]

<math>C_i^{ab}</math> 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[[拉格朗日量|拉氏量]]是

: <math>
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
</math>

I, J = 0, ..., 9

<math>\Gamma^I</math> 是32x32的矩阵 <math>( 32=2^{10/2} )</math> 

<math>\theta_I</math> 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]]：

: <math> \tau = \frac{\theta}{2\pi}+\frac{4\pi i}{g^2}.</math>

* [[AdS/CFT对偶]]、[[全像原理]]、[[第二型弦理論|类型IIB弦理论]]、[[量子引力]]

* [[大N展开|大N展開]]<ref>Martin Ammon, Johanna Erdmenger, ''Gauge/Gravity Duality: Foundations and Applications'', Cambridge University Press, 2015, p. 240.</ref>、平面楊米爾斯理論、 <math>N^{2-2g}</math>的[[费曼图]] <ref>{{Cite web |url=https://ncatlab.org/nlab/show/planar+limit |title=planar limit in nLab |access-date=2020-03-07 |archive-date=2020-10-01 |archive-url=https://web.archive.org/web/20201001022324/https://ncatlab.org/nlab/show/planar+limit }}</ref>
*[[楊代數|Yangian]]<ref name="Beisert">{{Cite journal|title=Review of AdS/CFT Integrability: An Overview|last=Beisert|first=Niklas|date=January 2012|journal=Letters in Mathematical Physics|doi=10.1007/s11005-011-0516-7|volume=99|page=425|arxiv=1012.4000|bibcode=2012LMaPh..99..425K}}</ref>
*[[扭量理论]]、[[格拉斯曼流形]]
*11維[[M理论|M理論]]、<math>N\rightarrow \infty</math>

== 参考文献 ==
{{Reflist}}

==  閱讀 ==
{{refbegin}}
* {{cite journal|title=Electric-magnetic duality and the geometric Langlands program|first1=Anton|last2=Witten|first2=Edward|journal=Communications in Number Theory and Physics|issue=1|doi=10.4310/cntp.2007.v1.n1.a1|year=2007|volume=1|pages=1–236|arxiv=hep-th/0604151|bibcode=2007CNTP....1....1K|last1=Kapustin}}
{{refend}}

[[Category:超对称量子场论]]
[[Category:共形場論]]
[[Category:陈省身]]