Editing Chern–Simons theory (section)

==Relationships with other theories==

===Topological string theories===
{{further|Topological string theory}}

In the context of [[string theory]], a ''U''(''N'') Chern–Simons theory on an oriented Lagrangian 3-submanifold M of a 6-manifold ''X'' arises as the [[string field theory]] of open strings ending on a [[D-brane]] wrapping ''X'' in the [[topological string theory#A-model|A-model]] topological string theory on ''X''. The [[topological string theory#B-model|B-model]] topological open string field theory on the spacefilling worldvolume of a stack of D5-branes is a 6-dimensional variant of Chern–Simons theory known as holomorphic Chern–Simons theory.

===WZW and matrix models===

Chern–Simons theories are related to many other field theories. For example, if one considers a Chern–Simons theory with gauge group G on a manifold with boundary then all of the 3-dimensional propagating degrees of freedom may be gauged away, leaving a [[two-dimensional conformal field theory]] known as a G [[Wess–Zumino–Witten model]] on the boundary. In addition the ''U''(''N'') and SO(''N'') Chern–Simons theories at large ''N'' are well approximated by [[matrix theory (physics)|matrix models]].

===Chern–Simons gravity theory===
{{See also|(2+1)-dimensional topological gravity}}
In 1982, [[Stanley Deser|S. Deser]], [[Roman Jackiw|R. Jackiw]] and S. Templeton proposed the Chern–Simons gravity theory in three dimensions, in which the [[Einstein–Hilbert action]] in gravity theory is modified by adding the Chern–Simons term. ({{harvtxt|Deser|Jackiw|Templeton|1982}})

In 2003, R. Jackiw and S. Y. Pi extended this theory to four dimensions ({{harvtxt|Jackiw|Pi|2003}}) and Chern–Simons gravity theory has some considerable effects not only to fundamental physics but also condensed matter theory and astronomy.

The four-dimensional case is very analogous to the three-dimensional case. In three dimensions, the gravitational Chern–Simons term is
:<math>\operatorname{CS}(\Gamma)=\frac{1}{2\pi^2}\int d^3x\varepsilon^{ijk}\biggl(\Gamma^p_{iq}\partial_j\Gamma^q_{kp}+\frac{2}{3}\Gamma^p_{iq}\Gamma^q_{jr}\Gamma^r_{kp}\biggr).</math>
This variation gives the [[Cotton tensor]]
:<math>=-\frac{1}{2\sqrt{g}}\bigl(\varepsilon^{mij}D_i R^n_j+\varepsilon^{nij}D_i R^m_j).</math>

Then, Chern–Simons modification of three-dimensional gravity is made by adding the above Cotton tensor to the field equation, which can be obtained as the vacuum solution by varying the Einstein–Hilbert action.

===Chern–Simons matter theories===
In 2013 Kenneth A. Intriligator and [[Nathan Seiberg]] solved these 3d Chern–Simons gauge theories and their phases using [[Seiberg-Witten monopole|monopole]]s carrying extra degrees of freedom. The [[Witten index]] of the many [[vacuum state|vacua]] discovered was computed by compactifying the space by turning on mass parameters and then computing the index. In some vacua, [[supersymmetry]] was computed to be broken. These monopoles were related to [[condensed matter physics|condensed matter]] [[quantum vortex|vortices]]. ({{harvtxt|Intriligator|Seiberg|2013}})

The ''N''&nbsp;=&nbsp;6 Chern–Simons matter theory is the [[AdS/CFT correspondence|holographic dual]] of M-theory on <math>AdS_4\times S_7</math>.

===Four-dimensional Chern–Simons theory===
{{See also|Four-dimensional Chern–Simons theory}}
In 2013 [[Kevin Costello]] defined a closely related theory defined on a four-dimensional manifold consisting of the product of a two-dimensional 'topological plane' and a two-dimensional (or one complex dimensional) complex curve.<ref>{{cite arXiv |last1=Costello |first1=Kevin |title=Supersymmetric gauge theory and the Yangian |date=2013 |class=hep-th |eprint=1303.2632 }}</ref> He later studied the theory in more detail together with Witten and Masahito Yamazaki,<ref name="CWY1">{{cite journal |last1=Costello |first1=Kevin |last2=Witten |first2=Edward |last3=Yamazaki |first3=Masahito |title=Gauge Theory And Integrability, I |journal=Notices of the International Congress of Chinese Mathematicians |date=2018 |volume=6 |issue=1 |pages=46–119 |doi=10.4310/ICCM.2018.v6.n1.a6 |arxiv=1709.09993 }}</ref><ref name="CWY2">{{cite journal |last1=Costello |first1=Kevin |last2=Witten |first2=Edward |last3=Yamazaki |first3=Masahito |title=Gauge Theory And Integrability, II |journal=Notices of the International Congress of Chinese Mathematicians |date=2018 |volume=6 |issue=1 |pages=120–146 |doi=10.4310/ICCM.2018.v6.n1.a7 |arxiv=1802.01579 |s2cid=119592177 }}</ref><ref name="CY">{{cite arXiv|last1=Costello |first1=Kevin |last2=Yamazaki |first2=Masahito |title=Gauge Theory And Integrability, III |date=2019 |class=hep-th |eprint=1908.02289 }}</ref> demonstrating how the gauge theory could be related to many notions in [[integrable system]]s theory, including exactly solvable lattice models (like the [[six-vertex model]] or the [[Quantum Heisenberg model|XXZ spin chain]]), integrable quantum field theories (such as the [[Gross–Neveu model]], [[Chiral model|principal chiral model]] and symmetric space coset [[sigma model]]s), the [[Yang–Baxter equation]] and [[quantum groups]] such as the [[Yangian]] which describe symmetries underpinning the integrability of the aforementioned systems.

The action on the 4-manifold <math>M = \Sigma \times C</math> where <math>\Sigma</math> is a two-dimensional manifold and <math>C</math> is a complex curve is
<math display = block>S = \int_M \omega \wedge CS(A)</math>
where <math>\omega</math> is a [[meromorphic]] [[one-form]] on <math>C</math>.