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| The '''Chern–Simons theory''', named after [[Shiing-Shen Chern]] and [[James Harris Simons]], is a 3-dimensional [[topological quantum field theory]] of [[Topological quantum field theory#Schwarz-type TQFTs|Schwarz type]], developed by [[Edward Witten]]. It is so named because its [[action (physics)|action]] is proportional to the integral of the [[Chern–Simons 3-form]].
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| In [[condensed matter physics]], Chern–Simons theory describes the [[topological order]]
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| 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]].
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| 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.
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| ==The classical theory==
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| ===Mathematical origin===
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| In the 1940s [[Shiing-Shen Chern|S. S. Chern]] and [[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 [[Chern–Weil homomorphism]], from the algebra of ''G''-adjoint invariant polynomial 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''.
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| In 1974 S. S. Chern and [[James Harris Simons|J. H. Simons]] had concretely constructed a (2''k'' − 1)-form ''df''(''ω'') such that
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| :<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
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| :<math>Tf(\omega)=\int_C f(\Omega^k)</math>,
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| where ''C'' is a (2''k'' − 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 been determined by any pure combinatorial formulation. It also can be defined as
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| :<math>CS(M)=\int_{s(M)}\tfrac{1}{2}Tp_1\in\mathbb{R}/\mathbb{Z}</math>,
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| 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.
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| 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]] 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]].
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| ===Configurations===
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| 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''.
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| 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.
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| ===Dynamics===
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| The [[action (physics)|action]] ''S'' of Chern–Simons theory is proportional to the integral of the [[Chern–Simons 3-form]]
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| :<math>S=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A).</math>
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| 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''.
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| 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
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| :<math>F = dA + A \wedge A \, </math>
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| the [[field equation]] is explicitly | |
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| :<math>0=\frac{\delta S}{\delta A}=\frac{k}{2\pi} F.</math>
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| 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.
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| 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''.
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| ==Quantization==
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| 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 impose that Σ be [[Cauchy surface]]s, in fact a state can be defined on any surface.
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| Σ is 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 [[conformal block]]s of the G WZW model at level k. Conformal blocks are locally [[holomorphic]] and antiholomorphic factors whose products sum to the [[correlation function]]s of a 2-dimensional conformal field theory.
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| 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.
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| ==Observables==
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| ===Wilson loops===
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| 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 represention]]s R.
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| 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
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| :<math> W_R(K) =\text{Tr}_R \, \mathcal{P} \, \exp{i \oint_K A}</math>
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| where A is the connection 1-form and we take the [[Cauchy principal value]] of the [[contour integral]]
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| and <math>\mathcal{P} \, \exp</math> is the [[path-ordered exponential]].
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| ===HOMFLY and Jones polynomials===
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| 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.
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| 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=U(N) Chern–Simons theory at level ''k'' the normalized correlation function is, up to a phase, equal to
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| :<math>\frac{\sin(\pi/(k+N))}{\sin(\pi N/(k+N))}</math> | |
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| times the HOMFLY polynomial. In particular when ''N'' = 2 the HOMFLY polynomial reduces to the Jones polynomial. In the SO(''N'') case one finds a similar expression with the [[Kauffman polynomial]].
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| 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.
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| [[Sir Michael Atiyah]] has shown that there exists a canonical choice of framing, 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'' + ''N'') times the linking number of ''L'' with itself.
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| ==Relationships with other theories==
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| ===[[Topological string theory|Topological string theories]]===
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| 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.
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| ===WZW and matrix models===
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| 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 2-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 model]]s.
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| ===Chern–Simons, the Kodama wavefunction and loop quantum gravity===
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| {{main|Kodama state}}
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| Edward Witten argued that the Kodama state in [[loop quantum gravity]] is unphysical due to an analogy to Chern–Simons state resulting in negative [[helicity]] and energy. {{harvtxt|Witten|2003}}
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| ===Chern–Simons gravity theory===
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| 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}}
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| 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 affects not only to fundamental physics but also condensed matter theory and astronomy.
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| The four-dimensional case is very analogous to the three-dimensional case. In three dimensions, the gravitational Chern–Simons term is
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| :<math>CS(\Gamma)=\frac{1}{2\pi^2}\int d^3x\epsilon^{ijk}\biggl(\Gamma^p_{iq}\partial_j\Gamma^q_{kp}+\frac{2}{3}\Gamma^p_{iq}\Gamma^q_{jr}\Gamma^r_{kp}\biggr).</math>
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| This variation gives the [[Cotton tensor]]
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| :<math>=-\frac{1}{2\sqrt{g}}\bigl(\epsilon^{mij}D_i R^n_j+\epsilon^{nij}D_i R^m_j).</math>
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| 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.
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| See also [[(2+1)–dimensional topological gravity]].
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| ===Chern–Simons matter theories===
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| 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}})
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| ==Chern–Simons terms in other theories==
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| 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]]. Ten and eleven dimensional generalizations of Chern–Simons terms appear in the actions of all ten and eleven dimensional [[supergravity]] theories.
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| ===One-loop renormalization of the level===
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| 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 −''n''/2, in other words the level k theory with n fermions is equivalent to the level ''k'' − ''n''/2 theory without fermions.
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| == See also ==
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| *[[Chern–Simons form]]
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| *[[Topological quantum field theory]]
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| *[[Alexander polynomial]]
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| *[[Jones polynomial]]
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| *[[2+1D topological gravity]]
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| == References ==
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| *{{Cite journal |first= S.-S. |last= Chern |lastauthoramp=yes |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 }}
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| *{{Cite journal |authorlink=Edward Witten |first=Edward |last=Witten |title=Topological Quantum Field Theory |journal=[[Communications in Mathematical Physics|Commun. Math. Phys.]] |volume=117 |pages=353 |year=1988|bibcode = 1988CMaPh.117..353W |doi = 10.1007/BF01223371 }}http=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1104161738
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| *{{Cite journal |authorlink=Edward Witten |first=Edward |last=Witten |title=Quantum Field Theory and the Jones Polynomial |journal=[[Communications in Mathematical Physics|Commun. Math. Phys.]] |volume=121 |issue=3 |pages=351–399 |year=1989 |mr=0990772 |bibcode = 1989CMaPh.121..351W |doi = 10.1007/BF01217730 }}
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| *{{Cite journal |first=Edward |last=Witten |title=Chern–Simons Theory as a String Theory |journal=Prog. Math. |volume=133 |issue= |pages=637–678 |year=1995 |arxiv=hep-th/9207094 |bibcode = 1992hep.th....7094W }}
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| *{{Cite journal |first=Edward |last=Witten |title=A Note On The Chern-Simons And Kodama Wavefunctions |year=2003 |arxiv=gr-qc/0306083 }}
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| *{{Cite journal |first=Marcos |last=Marino |authorlink=Marcos Marino |title=Chern–Simons Theory and Topological Strings |journal=[[Reviews of Modern Physics|Rev. Mod. Phys.]] |volume=77 |issue=2 |pages=675–720 |year=2005 |doi=10.1103/RevModPhys.77.675 |arxiv = hep-th/0406005 |bibcode = 2005RvMP...77..675M }}
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| *{{Cite book |first=Marcos |last=Marino |title=Chern–Simons Theory, Matrix Models, And Topological Strings |series=International Series of Monographs on Physics |publisher=OUP |year=2005 }}
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| *{{Cite journal |first1=Stanley|last1=Deser|first2=Roman|last2=Jackiw|first3=S.|last3=Templeton|title=Three-Dimensional Massive Gauge Theories |series=Phys. Rev. Lett. 48, 975–978 |publisher=American Physical Society |year=1982|ref=harv }}
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| *{{Cite journal |first1=Roman|last1=Jackiw|first2=S.-Y|last2=Pi|title=Chern–Simons modification of general relativity |series=Phys.Rev. D68|publisher=American Physical Society |year=2003|ref=harv }}
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| *{{Cite journal |first1=Kenneth |last1=Intriligator | first2=Nathan |last2=Seiberg| title=Aspects of 3d ''N'' = 2 Chern–Simons-Matter Theories | year=2013 | journal=[[JHEP]] | url=http://inspirehep.net/record/1232411 | ref=harv }}
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| ==External links==
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| * {{springer|title=Chern-Simons functional|id=p/c120140}}
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| {{Quantum field theories}}
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| {{DEFAULTSORT:Chern-Simons theory}}
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| [[Category:Quantum field theory]]
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