Editing Chern–Simons theory (section)

==Chern–Simons terms in other theories==

The Chern–Simons term can also be added to models which aren't topological quantum field theories. In 3D, this gives rise to a massive [[photon]] if this term is added to the action of Maxwell's theory of [[electrodynamics]]. This term can be induced by integrating over a massive charged [[Fermionic field#Dirac fields|Dirac field]]. It also appears for example in the [[quantum Hall effect]]. The addition of the Chern–Simons term to various theories gives rise to vortex- or soliton-type solutions<ref>{{Cite journal|doi = 10.1063/1.1471365|title = Self-dual Chern–Simons vortices on Riemann surfaces|year = 2002|last1 = Kim|first1 = Seongtag|last2 = Kim|first2 = Yoonbai|journal = Journal of Mathematical Physics|volume = 43|issue = 5|pages = 2355–2362|arxiv = math-ph/0012045|bibcode = 2002JMP....43.2355K|s2cid = 9916364}}</ref><ref>{{Cite journal|doi = 10.1103/PhysRevD.95.085016|title = Effect of Chern-Simons dynamics on the energy of electrically charged and spinning vortices|year = 2017|last1 = Navarro-Lérida|first1 = Francisco|last2 = Radu|first2 = Eugen|last3 = Tchrakian|first3 = D. H.|journal = Physical Review D|volume = 95|issue = 8|page = 085016|arxiv = 1612.05835|bibcode = 2017PhRvD..95h5016N|s2cid = 62882649}}</ref> Ten- and eleven-dimensional generalizations of Chern–Simons terms appear in the actions of all ten- and eleven-dimensional [[supergravity]] theories.

===One-loop renormalization of the level===

If one adds matter to a Chern–Simons gauge theory then, in general it is no longer topological.  However, if one adds n [[Majorana fermion]]s then, due to the [[parity anomaly]], when integrated out they lead to a pure Chern–Simons theory with a one-loop [[renormalization]] of the Chern–Simons level by &minus;''n''/2, in other words the level k theory with n fermions is equivalent to the level ''k''&nbsp;&minus;&nbsp;''n''/2 theory without fermions.