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{{short description|Three-dimensional topological quantum field theory whose action is the Chern–Simons form}}
The '''Chern–Simons theory''' is a 3-dimensional [[topological quantum field theory]] of [[Topological quantum field theory#Schwarz-type TQFTs|Schwarz type]] developed by [[Edward Witten]]. It was discovered first by mathematical physicist [[Albert Schwarz]]. It is named after mathematicians [[Shiing-Shen Chern]] and [[James Harris Simons]], who introduced the [[Chern–Simons 3-form]]. In the Chern–Simons theory, the [[action (physics)|action]] is proportional to the integral of the Chern–Simons 3-form.

In [[condensed matter physics|condensed-matter physics]], Chern–Simons theory describes the [[topological order]] in [[fractional quantum Hall effect]] states. In mathematics, it has been used to calculate [[knot invariants]] and [[three-manifold]] invariants such as the [[Jones polynomial]].<ref name="wittenjonespolynomial"/>

Particularly, Chern–Simons theory is specified by a choice of simple [[Lie group]] G known as the gauge group of the theory and also a number referred to as the ''level'' of the theory, which is a constant that multiplies the action. The action is gauge dependent, however the [[partition function (quantum field theory)|partition function]] of the [[quantum field theory|quantum]] theory is [[well-defined]] when the level is an integer and the gauge [[field strength]] vanishes on all [[boundary (topology)|boundaries]] of the 3-dimensional spacetime.

It is also the central mathematical object in theoretical models for [[topological quantum computer]]s (TQC). Specifically, an SU(2) Chern–Simons theory describes the simplest non-abelian [[anyon]]ic model of a TQC, the Yang–Lee–Fibonacci model.<ref name="FK02"/><ref name="WangTQCreview"/>

The dynamics of Chern–Simons theory on the 2-dimensional boundary of a 3-manifold is closely related to [[fusion rules]] and [[Virasoro conformal block|conformal blocks]] in [[conformal field theory]], and in particular [[Wess–Zumino–Witten model|WZW theory]].<ref name="wittenjonespolynomial"/><ref name="EMSS89"/>

==The classical theory==

===Mathematical origin===

In the 1940s [[Shiing-Shen Chern|S. S. Chern]] and [[André Weil|A. Weil]] studied the global curvature properties of smooth manifolds ''M'' as [[de Rham cohomology]] ([[Chern–Weil theory]]), which is an important step in the theory of [[characteristic classes]] in [[differential geometry]]. Given a flat ''G''-[[principal bundle]] ''P'' on ''M'' there exists a unique homomorphism, called the [[Chern–Weil homomorphism]], from the algebra of ''G''-adjoint invariant polynomials on ''g'' (Lie algebra of ''G'') to the cohomology <math>H^*(M,\mathbb{R})</math>. If the invariant polynomial is homogeneous one can write down concretely any ''k''-form of the closed connection ''ω'' as some 2''k''-form of the associated curvature form Ω of ''ω''.

In 1974 S. S. Chern and [[James Harris Simons|J. H. Simons]] had concretely constructed a (2''k''&nbsp;&minus;&nbsp;1)-form ''df''(''ω'') such that
:<math>dTf(\omega)=f(\Omega^k),</math>
where ''T'' is the Chern–Weil homomorphism. This form is called [[Chern–Simons form]]. If ''df''(''ω'') is closed one can integrate the above formula
:<math>Tf(\omega)=\int_C f(\Omega^k),</math>
where ''C'' is a (2''k''&nbsp;&minus;&nbsp;1)-dimensional cycle on ''M''. This invariant is called '''Chern–Simons invariant'''. As pointed out in the introduction of the Chern–Simons paper, the Chern–Simons invariant CS(''M'') is the boundary term that cannot be determined by any pure combinatorial formulation. It also can be defined as
:<math>\operatorname{CS}(M)=\int_{s(M)}\tfrac{1}{2}Tp_1\in\mathbb{R}/\mathbb{Z},</math>
where <math>p_1</math> is the first Pontryagin number and ''s''(''M'') is the section of the normal orthogonal bundle ''P''. Moreover, the Chern–Simons term is described as the [[eta invariant]] defined by Atiyah, Patodi and Singer.

The gauge invariance and the metric invariance can be viewed as the invariance under the adjoint Lie group action in the Chern–Weil theory. The [[action integral]] ([[Path integral formulation|path integral]]) of the [[quantum field theory|field theory]] in physics is viewed as the [[Lagrangian (field theory)|Lagrangian]] integral of the Chern–Simons form and Wilson loop, holonomy of vector bundle on ''M''. These explain why the Chern–Simons theory is closely related to [[topological field theory]].

===Configurations===

Chern–Simons theories can be defined on any [[topological manifold|topological]] [[3-manifold]] ''M'', with or without boundary. As these theories are Schwarz-type topological theories, no [[metric tensor|metric]] needs to be introduced on ''M''.

Chern–Simons theory is a [[gauge theory]], which means that a [[classical physics|classical]] configuration in the Chern–Simons theory on ''M'' with [[gauge group]] ''G'' is described by a [[principal bundle|principal ''G''-bundle]] on ''M''. The [[connection (principal bundle)|connection]] of this bundle is characterized by a [[connection one-form]] ''A'' which is [[vector-valued differential form#Lie algebra-valued forms|valued]] in the [[Lie algebra]] '''g''' of the [[Lie group]] ''G''. In general the connection ''A'' is only defined on individual [[coordinate patch]]es, and the values of ''A'' on different patches are related by maps known as [[gauge symmetry|gauge transformations]]. These are characterized by the assertion that the [[gauge covariant derivative|covariant derivative]], which is the sum of the [[exterior derivative]] operator ''d'' and the connection ''A'', transforms in the [[Adjoint representation of a Lie group|adjoint representation]] of the gauge group ''G''. The square of the covariant derivative with itself can be interpreted as a '''g'''-valued 2-form ''F'' called the [[curvature form]] or [[field strength]]. It also transforms in the adjoint representation.

===Dynamics===
The [[action (physics)|action]] ''S'' of Chern–Simons theory is proportional to the integral of the [[Chern–Simons 3-form]]

:<math>S=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A).</math>

The constant ''k'' is called the ''level'' of the theory. The classical physics of Chern–Simons theory is independent of the choice of level ''k''.

Classically the system is characterized by its equations of motion which are the extrema of the action with respect to variations of the field ''A''. In terms of the field curvature

:<math>F = dA + A \wedge A \, </math>

the [[field equation]] is explicitly

:<math>0=\frac{\delta S}{\delta A}=\frac{k}{2\pi} F.</math>

The classical equations of motion are therefore satisfied if and only if the curvature vanishes everywhere, in which case the connection is said to be ''flat''. Thus the classical solutions to ''G'' Chern–Simons theory are the flat connections of principal ''G''-bundles on ''M''. Flat connections are determined entirely by holonomies around noncontractible cycles on the base ''M''. More precisely, they are in one-to-one correspondence with equivalence classes of homomorphisms from the [[fundamental group]] of ''M'' to the gauge group ''G'' up to conjugation.

If ''M'' has a boundary ''N'' then there is additional data which describes a choice of trivialization of the principal ''G''-bundle on ''N''. Such a choice characterizes a map from ''N'' to ''G''. The dynamics of this map is described by the [[Wess–Zumino–Witten model|Wess–Zumino–Witten]] (WZW) model on ''N'' at level ''k''.

==Quantization==

To [[canonical quantization|canonically quantize]] Chern–Simons theory one defines a state on each 2-dimensional surface Σ in M. As in any quantum field theory, the states correspond to rays in a [[Hilbert space]]. There is no preferred notion of time in a Schwarz-type topological field theory and so one can require that Σ be a [[Cauchy surface]], in fact, a state can be defined on any surface.

Σ is of codimension one, and so one may cut M along Σ. After such a cutting M will be a manifold with boundary and in particular classically the dynamics of Σ will be described by a WZW model. [[Edward Witten|Witten]] has shown that this correspondence holds even quantum mechanically. More precisely, he demonstrated that the Hilbert space of states is always finite-dimensional and can be canonically identified with the space of [[Virasoro conformal block#Larger symmetry algebras|conformal block]]s of the G WZW model at level k.

For example, when Σ is a 2-sphere, this Hilbert space is one-dimensional and so there is only one state. When Σ is a 2-torus the states correspond to the integrable [[group representation|representation]]s of the [[affine Lie algebra]] corresponding to g at level k. Characterizations of the conformal blocks at higher genera are not necessary for Witten's solution of Chern–Simons theory.

==Observables==

===Wilson loops===

The [[observable]]s of Chern–Simons theory are the ''n''-point [[correlation function]]s of gauge-invariant operators. The most often studied class of gauge invariant operators are [[Wilson loops]]. A Wilson loop is the holonomy around a loop in ''M'', traced in a given [[representation of a Lie group|representation]] ''R'' of ''G''. As we will be interested in products of Wilson loops, without loss of generality we may restrict our attention to [[representation theory#Subrepresentations, quotients, and irreducible representations|irreducible representation]]s ''R''.

More concretely, given an irreducible representation ''R'' and a loop ''K'' in ''M'', one may define the Wilson loop <math>W_R(K)</math> by

:<math> W_R(K) =\operatorname{Tr}_R \, \mathcal{P} \exp\left(i \oint_K A\right)</math>

where ''A'' is the connection 1-form and we take the [[Cauchy principal value]] of the [[contour integral]] and <math>\mathcal{P} \exp</math> is the [[path-ordered exponential]].

===HOMFLY and Jones polynomials===

Consider a link ''L'' in ''M'', which is a collection of ''ℓ'' disjoint loops. A particularly interesting observable is the ''ℓ''-point correlation function formed from the product of the Wilson loops around each disjoint loop, each traced in the [[fundamental representation]] of ''G''. One may form a normalized correlation function by dividing this observable by the [[partition function (quantum field theory)|partition function]] ''Z''(''M''), which is just the 0-point correlation function.

In the special case in which M is the 3-sphere, Witten has shown that these normalized correlation functions are proportional to known [[knot polynomials]]. For example, in ''G''&nbsp;=&nbsp;''U''(''N'') Chern–Simons theory at level ''k'' the normalized correlation function is, up to a phase, equal to

:<math>\frac{\sin(\pi/(k+N))}{\sin(\pi N/(k+N))}</math>

times the [[HOMFLY polynomial]]. In particular when ''N''&nbsp;=&nbsp;2 the HOMFLY polynomial reduces to the [[Jones polynomial]]. In the SO(''N'') case, one finds a similar expression with the [[Kauffman polynomial]].

The phase ambiguity reflects the fact that, as Witten has shown, the quantum correlation functions are not fully defined by the classical data. The [[linking number]] of a loop with itself enters into the calculation of the partition function, but this number is not invariant under small deformations and in particular, is not a topological invariant. This number can be rendered well defined if one chooses a framing for each loop, which is a choice of preferred nonzero [[normal vector]] at each point along which one deforms the loop to calculate its self-linking number. This procedure is an example of the [[point-splitting]] [[regularization (physics)|regularization]] procedure introduced by [[Paul Dirac]] and [[Rudolf Peierls]] to define apparently divergent quantities in [[quantum field theory]] in 1934.

[[Sir Michael Atiyah]] has shown that there exists a canonical choice of 2-framing,<ref>{{Cite journal |last=Atiyah |first=Michael |date=1990 |title=On framings of 3-manifolds |url=https://doi.org/10.1016/0040-9383(90)90021-b |journal=Topology |volume=29 |issue=1 |pages=1–7 |doi=10.1016/0040-9383(90)90021-b |issn=0040-9383}}</ref> which is generally used in the literature today and leads to a well-defined linking number. With the canonical framing the above phase is the exponential of 2π''i''/(''k''&nbsp;+&nbsp;''N'') times the linking number of ''L'' with itself.

;Problem (Extension of Jones polynomial to general 3-manifolds)　
"The original Jones polynomial was defined for 1-links in the 3-sphere (the 3-ball, the 3-space R3). Can you define the Jones polynomial for 1-links in any 3-manifold?"

See section 1.1 of this paper<ref>
{{cite arXiv|first1=L.H |last1=Kauffman 
|first2=E |last2=Ogasa 
|first3=J |last3=Schneider 
|eprint=1808.03023|
title=A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots|year=2018|class=math.GT 
}} 
</ref> for the background and the history of this problem. Kauffman submitted a solution in the case of the product manifold of closed oriented surface and the closed interval, by introducing virtual 1-knots.<ref>
{{cite arXiv|first=L.E. |last=Kauffman |eprint= math/9811028
| title=Virtual Knot Theory
|year=1998 }} 
</ref> It is open in the other cases. Witten's path integral for Jones polynomial is written for links in any compact 3-manifold formally, but the calculus is not done even in physics level in any case other than the 3-sphere (the 3-ball, the 3-space '''R'''<sup>3</sup>). This problem is also open in physics level. In the case of Alexander polynomial, this problem is solved.

==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>.

==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.

== See also ==
*[[Gauge theory (mathematics)]]
*[[Chern–Simons form]]
*[[Topological quantum field theory]]
*[[Alexander polynomial]]
*[[Jones polynomial]]
*[[2+1D topological gravity]]
*[[Skyrmion]]

== References ==
*{{Cite journal |first1= K.|last1= Arthur |author-link=K. Arthur|first2=D.H.|last2 =Tchrakian|last3 = Y.-S.|first3 = Yang |title=Topological and nontopological selfdual Chern-Simons solitons in a gauged O(3) sigma model|journal = [[Physical Review D]]|volume =54|pages = 5245–5258|year= 1996|issue= 8 |doi= 10.1103/PhysRevD.54.5245 |pmid= 10021215 |bibcode= 1996PhRvD..54.5245A }}
*{{Cite journal |first1= S.-S. |last1= Chern |author-link=S.-S. Chern |name-list-style=amp|first2=J. |last2= Simons |title=Characteristic forms and geometric invariants |journal=[[Annals of Mathematics]] |volume=99 |issue=1 |pages=48–69 |year=1974 |doi=10.2307/1971013 |jstor= 1971013 }}
*{{Cite journal |first1=Stanley|last1=Deser|first2=Roman|last2=Jackiw|first3=S.|last3=Templeton|title=Three-Dimensional Massive Gauge Theories |journal=[[Physical Review Letters]] |volume=48 |pages=975–978 |year=1982 |issue=15|doi= 10.1103/PhysRevLett.48.975|bibcode=1982PhRvL..48..975D|s2cid=122537043 |url=https://authors.library.caltech.edu/85895/1/PhysRevLett.48.975.pdf}}
*{{Cite journal |first1=Kenneth |last1=Intriligator | first2=Nathan |last2=Seiberg| title=Aspects of 3d ''N''&nbsp;=&nbsp;2 Chern–Simons Matter Theories | year=2013 | journal=[[Journal of High Energy Physics]] |volume=2013 |page=79 |doi=10.1007/JHEP07(2013)079 |arxiv = 1305.1633 |bibcode = 2013JHEP...07..079I |s2cid=119106931 }}
*{{Cite journal|author1-link=Roman Jackiw|author2-link=So-Young Pi|first1=Roman|last1=Jackiw|first2=S.-Y|last2=Pi|title=Chern–Simons modification of general relativity |journal=[[Physical Review D]] |volume=68 |pages=104012 |year=2003 |issue=10|doi=10.1103/PhysRevD.68.104012 |arxiv=gr-qc/0308071 |bibcode = 2003PhRvD..68j4012J |s2cid=2243511}}
*{{Cite journal |first1=Usha|last1=Kulshreshtha|first2=D.S.|last2=Kulshreshtha|first3=H. J. W.|last3=Mueller-Kirsten|first4=J. P.|last4=Vary|title=Hamiltonian, path integral and BRST formulations of the Chern-Simons-Higgs theory under appropriate gauge fixing |journal=[[Physica Scripta ]]|volume=79 |pages=045001|year=2009 |issue=4|doi=10.1088/0031-8949/79/04/045001 |bibcode = 2009PhyS...79d5001K |s2cid=120594654 }}
*{{Cite journal |first1=Usha|last1=Kulshreshtha|first2=D.S.|last2=Kulshreshtha|first3=J. P.|last3=Vary|title=Light-front Hamiltonian, path integral and BRST formulations of the Chern-Simons-Higgs theory under appropriate gauge fixing |journal=[[Physica Scripta]] |volume=82 |pages=055101|year=2010 |issue=5|doi=10.1088/0031-8949/82/05/055101 |bibcode = 2010PhyS...82e5101K |s2cid=54602971 }}
*{{Cite journal |author2-link=Eduardo Fradkin|first1=Ana|last1=Lopez|first2=Eduardo|last2=Fradkin|title=Fractional quantum Hall effect and Chern-Simons gauge theories |journal=[[Physical Review B]] |volume=44 |pages=5246–5262 |year=1991 |issue=10|doi= 10.1103/PhysRevB.44.5246|pmid=9998334|bibcode=1991PhRvB..44.5246L}}
*{{Cite journal |first=Marcos |last=Marino |author-link=Marcos Marino |title=Chern–Simons Theory and Topological Strings |journal=[[Reviews of Modern Physics]] |volume=77 |issue=2 |pages=675–720 |year=2005 |doi=10.1103/RevModPhys.77.675 |arxiv = hep-th/0406005 |bibcode = 2005RvMP...77..675M |s2cid=6207500 }}
*{{Cite book |first=Marcos |last=Marino |title=Chern–Simons Theory, Matrix Models, And Topological Strings |series=International Series of Monographs on Physics |publisher=[[Oxford University Press]] |year=2005 }}
*{{Cite journal |author-link=Edward Witten |first=Edward |last=Witten |title=Topological Quantum Field Theory |journal=[[Communications in Mathematical Physics]] |volume=117 |pages=353–386 |year=1988|issue=3 |bibcode = 1988CMaPh.117..353W |doi = 10.1007/BF01223371 |s2cid=43230714 |url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1104161738}}
*{{Cite journal |first=Edward |last=Witten |title=Chern–Simons Theory as a String Theory |journal=[[Progress in Mathematics]] |volume=133 |pages=637–678 |year=1995 |arxiv=hep-th/9207094 |bibcode = 1992hep.th....7094W }}

;Specific
<references>
<ref name="wittenjonespolynomial">{{Cite journal |author-link=Edward Witten |first=Edward |last=Witten |title=Quantum Field Theory and the Jones Polynomial |journal=[[Communications in Mathematical Physics]] |volume=121 |issue=3 |pages=351–399 |year=1989 |mr=0990772 |bibcode = 1989CMaPh.121..351W |doi = 10.1007/BF01217730 |s2cid=14951363 |url=http://projecteuclid.org/euclid.cmp/1104178138 }}</ref>
<ref name="FK02">{{cite arXiv|last1=Freedman|first1=Michael H.|last2=Kitaev|first2=Alexei|last3=Larsen|first3=Michael J.|last4=Wang|first4=Zhenghan|date=2002-09-20|title=Topological Quantum Computation|eprint=quant-ph/0101025}}</ref>
<ref name="WangTQCreview">{{Cite web|last=Wang|first=Zhenghan|title=Topological Quantum Computation|url=http://web.math.ucsb.edu/~zhenghwa/data/course/cbms.pdf}}</ref>
<ref name="EMSS89">
{{Cite journal 
|author1-link=Shmuel Elitzur  |first1=Shmuel |last1=Elitzur 
|author2-link=Gregory Moore (physicist) |first2=Gregory |last2=Moore
|author3-link=Adam Schwimmer  |first3=Adam |last3=Schwimmer 
|author4-link=Nathan Seiberg  |first4=Nathan |last4=Seiberg 
|title=Remarks on the canonical quantization of the Chern-Simons-Witten theory
|journal=[[Nuclear Physics B]] 
|volume=326
|issue=1
|pages=108–134
|date = 30 October 1989
|doi=10.1016/0550-3213(89)90436-7 |bibcode=1989NuPhB.326..108E }}
</ref>

</references>

==External links==
* {{springer|title=Chern-Simons functional|id=p/c120140|mode=cs1}}

{{Quantum field theories}}

{{DEFAULTSORT:Chern-Simons theory}}
[[Category:Quantum field theory]]