Editing Chern–Simons theory (section)

===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>.