SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Chern–Simons theory</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Chern–Simons theory</span></span>

</h1>

</header>

<p>The <b>Chern–Simons theory</b> is a 3-dimensional <a href="Topological_quantum_field_theory" title="Topological quantum field theory">topological quantum field theory</a> of <a href="Topological_quantum_field_theory#Schwarz-type_TQFTs" title="Topological quantum field theory">Schwarz type</a>. It was discovered first by mathematical physicist <a href="Albert_Schwarz" title="Albert Schwarz">Albert Schwarz</a>. It is named after mathematicians <a href="Shiing-Shen_Chern" title="Shiing-Shen Chern">Shiing-Shen Chern</a> and <a href="James_Harris_Simons" class="mw-redirect" title="James Harris Simons">James Harris Simons</a>, who introduced the <a href="Chern%E2%80%93Simons_3-form" class="mw-redirect" title="Chern–Simons 3-form">Chern–Simons 3-form</a>. In the Chern–Simons theory, the <a href="Action_(physics)" title="Action (physics)">action</a> is proportional to the integral of the Chern–Simons 3-form.

</p><p>In <a href="Condensed_matter_physics" title="Condensed matter physics">condensed-matter physics</a>, Chern–Simons theory describes <a href="Composite_fermion" title="Composite fermion">composite fermions</a> and the <a href="Topological_order" title="Topological order">topological order</a> in <a href="Fractional_quantum_Hall_effect" title="Fractional quantum Hall effect">fractional quantum Hall effect</a> states. In mathematics, it has been used to calculate <a href="Knot_invariants" class="mw-redirect" title="Knot invariants">knot invariants</a> and <a href="Three-manifold" class="mw-redirect" title="Three-manifold">three-manifold</a> invariants such as the <a href="Jones_polynomial" title="Jones polynomial">Jones polynomial</a>.<sup id="cite_ref-wittenjonespolynomial_1-0" class="reference"><a href="#cite_note-wittenjonespolynomial-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Particularly, Chern–Simons theory is specified by a choice of simple <a href="Lie_group" title="Lie group">Lie group</a> G known as the gauge group of the theory and also a number referred to as the <i>level</i> of the theory, which is a constant that multiplies the action. The action is gauge dependent, however the <a href="Partition_function_(quantum_field_theory)" title="Partition function (quantum field theory)">partition function</a> of the <a href="Quantum_field_theory" title="Quantum field theory">quantum</a> theory is <a href="Well-defined" class="mw-redirect" title="Well-defined">well-defined</a> when the level is an integer and the gauge <a href="Field_strength" title="Field strength">field strength</a> vanishes on all <a href="Boundary_(topology)" title="Boundary (topology)">boundaries</a> of the 3-dimensional spacetime.

</p><p>It is also the central mathematical object in theoretical models for <a href="Topological_quantum_computer" title="Topological quantum computer">topological quantum computers</a> (TQC). Specifically, an SU(2) Chern–Simons theory describes the simplest non-abelian <a href="Anyon" title="Anyon">anyonic</a> model of a TQC, the Yang–Lee–Fibonacci model.<sup id="cite_ref-FK02_2-0" class="reference"><a href="#cite_note-FK02-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-WangTQCreview_3-0" class="reference"><a href="#cite_note-WangTQCreview-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>The dynamics of Chern–Simons theory on the 2-dimensional boundary of a 3-manifold is closely related to <a href="Fusion_rules" title="Fusion rules">fusion rules</a> and <a href="Virasoro_conformal_block" title="Virasoro conformal block">conformal blocks</a> in <a href="Conformal_field_theory" title="Conformal field theory">conformal field theory</a>, and in particular <a href="Wess%E2%80%93Zumino%E2%80%93Witten_model" title="Wess–Zumino–Witten model">WZW theory</a>.<sup id="cite_ref-wittenjonespolynomial_1-1" class="reference"><a href="#cite_note-wittenjonespolynomial-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-EMSS89_4-0" class="reference"><a href="#cite_note-EMSS89-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="The_classical_theory">The classical theory</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Mathematical_origin">Mathematical origin</h3></div>

<p>In the 1940s <a href="Shiing-Shen_Chern" title="Shiing-Shen Chern">S. S. Chern</a> and <a href="Andr%C3%A9_Weil" title="André Weil">A. Weil</a> studied the global curvature properties of smooth manifolds <i>M</i> as <a href="De_Rham_cohomology" title="De Rham cohomology">de Rham cohomology</a> (<a href="Chern%E2%80%93Weil_theory" class="mw-redirect" title="Chern–Weil theory">Chern–Weil theory</a>), which is an important step in the theory of <a href="Characteristic_classes" class="mw-redirect" title="Characteristic classes">characteristic classes</a> in <a href="Differential_geometry" title="Differential geometry">differential geometry</a>. Given a flat <i>G</i>-<a href="Principal_bundle" title="Principal bundle">principal bundle</a> <i>P</i> on <i>M</i> there exists a unique homomorphism, called the <a href="Chern%E2%80%93Weil_homomorphism" title="Chern–Weil homomorphism">Chern–Weil homomorphism</a>, from the algebra of <i>G</i>-adjoint invariant polynomials on <i>g</i> (Lie algebra of <i>G</i>) to the cohomology <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{*}(M,\mathbb {R} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{*}(M,\mathbb {R} )}</annotation>

</semantics>

</math></span><img src="./f395151b51136ae59f0d10189c2d6d8b6a1a57dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.121ex; height:2.843ex;" alt="{\displaystyle H^{*}(M,\mathbb {R} )}" loading="lazy"></span>. If the invariant polynomial is homogeneous one can write down concretely any <i>k</i>-form of the closed connection <i>ω</i> as some 2<i>k</i>-form of the associated curvature form Ω of <i>ω</i>.

</p><p>In 1974 S. S. Chern and <a href="James_Harris_Simons" class="mw-redirect" title="James Harris Simons">J. H. Simons</a> had concretely constructed a (2<i>k</i> − 1)-form <i>df</i>(<i>ω</i>) such that

</p>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dTf(\omega )=f(\Omega ^{k}),}</annotation>

</semantics>

</math></span><img src="./380a460b40fd710f9ecad7119a1d37a825baa6f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.986ex; height:3.176ex;" alt="{\displaystyle dTf(\omega )=f(\Omega ^{k}),}" loading="lazy"></span></dd></dl>

<p>where <i>T</i> is the Chern–Weil homomorphism. This form is called <a href="Chern%E2%80%93Simons_form" title="Chern–Simons form">Chern–Simons form</a>. If <i>df</i>(<i>ω</i>) is closed one can integrate the above formula

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Tf(\omega )=\int _{C}f(\Omega ^{k}),}</annotation>

</semantics>

</math></span><img src="./ef5a07866e30d6406a0bb52c5be427d221005620.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.931ex; height:5.676ex;" alt="{\displaystyle Tf(\omega )=\int _{C}f(\Omega ^{k}),}" loading="lazy"></span></dd></dl>

<p>where <i>C</i> is a (2<i>k</i> − 1)-dimensional cycle on <i>M</i>. This invariant is called <b>Chern–Simons invariant</b>. As pointed out in the introduction of the Chern–Simons paper, the Chern–Simons invariant CS(<i>M</i>) is the boundary term that cannot be determined by any pure combinatorial formulation. It also can be defined as

</p>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {CS} (M)=\int _{s(M)}{\tfrac {1}{2}}Tp_{1}\in \mathbb {R} /\mathbb {Z} ,}</annotation>

</semantics>

</math></span><img src="./55a75adffcb84a741cb77335fa253a39633ace82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.406ex; height:6.009ex;" alt="{\displaystyle \operatorname {CS} (M)=\int _{s(M)}{\tfrac {1}{2}}Tp_{1}\in \mathbb {R} /\mathbb {Z} ,}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{1}}</annotation>

</semantics>

</math></span><img src="./b9b58f22283ca46dd5da309cc34303b06a797783.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{1}}" loading="lazy"></span> is the first Pontryagin number and <i>s</i>(<i>M</i>) is the section of the normal orthogonal bundle <i>P</i>. Moreover, the Chern–Simons term is described as the <a href="Eta_invariant" title="Eta invariant">eta invariant</a> defined by Atiyah, Patodi and Singer.

</p><p>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 <a href="Action_integral" class="mw-redirect" title="Action integral">action integral</a> (<a href="Path_integral_formulation" title="Path integral formulation">path integral</a>) of the <a href="Quantum_field_theory" title="Quantum field theory">field theory</a> in physics is viewed as the <a href="Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a> integral of the Chern–Simons form and Wilson loop, holonomy of vector bundle on <i>M</i>. These explain why the Chern–Simons theory is closely related to <a href="Topological_field_theory" class="mw-redirect" title="Topological field theory">topological field theory</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Configurations">Configurations</h3></div>

<p>Chern–Simons theories can be defined on any <a href="Topological_manifold" title="Topological manifold">topological</a> <a href="3-manifold" title="3-manifold">3-manifold</a> <i>M</i>, with or without boundary. As these theories are Schwarz-type topological theories, no <a href="Metric_tensor" title="Metric tensor">metric</a> needs to be introduced on <i>M</i>.

</p><p>Chern–Simons theory is a <a href="Gauge_theory" title="Gauge theory">gauge theory</a>, which means that a <a href="Classical_physics" title="Classical physics">classical</a> configuration in the Chern–Simons theory on <i>M</i> with <a href="Gauge_group" class="mw-redirect" title="Gauge group">gauge group</a> <i>G</i> is described by a <a href="Principal_bundle" title="Principal bundle">principal <i>G</i>-bundle</a> on <i>M</i>. The <a href="Connection_(principal_bundle)" title="Connection (principal bundle)">connection</a> of this bundle is characterized by a <a href="Connection_one-form" class="mw-redirect" title="Connection one-form">connection one-form</a> <i>A</i> which is <a href="Vector-valued_differential_form#Lie_algebra-valued_forms" title="Vector-valued differential form">valued</a> in the <a href="Lie_algebra" title="Lie algebra">Lie algebra</a> <b>g</b> of the <a href="Lie_group" title="Lie group">Lie group</a> <i>G</i>. In general the connection <i>A</i> is only defined on individual <a href="Coordinate_patch" class="mw-redirect" title="Coordinate patch">coordinate patches</a>, and the values of <i>A</i> on different patches are related by maps known as <a href="Gauge_symmetry" class="mw-redirect" title="Gauge symmetry">gauge transformations</a>. These are characterized by the assertion that the <a href="Gauge_covariant_derivative" title="Gauge covariant derivative">covariant derivative</a>, which is the sum of the <a href="Exterior_derivative" title="Exterior derivative">exterior derivative</a> operator <i>d</i> and the connection <i>A</i>, transforms in the <a href="Adjoint_representation_of_a_Lie_group" class="mw-redirect" title="Adjoint representation of a Lie group">adjoint representation</a> of the gauge group <i>G</i>. The square of the covariant derivative with itself can be interpreted as a <b>g</b>-valued 2-form <i>F</i> called the <a href="Curvature_form" title="Curvature form">curvature form</a> or <a href="Field_strength" title="Field strength">field strength</a>. It also transforms in the adjoint representation.

</p>

<div class="mw-heading mw-heading3"><h3 id="Dynamics">Dynamics</h3></div>

<p>The <a href="Action_(physics)" title="Action (physics)">action</a> <i>S</i> of Chern–Simons theory is proportional to the integral of the <a href="Chern%E2%80%93Simons_3-form" class="mw-redirect" title="Chern–Simons 3-form">Chern–Simons 3-form</a>

</p>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S={\frac {k}{4\pi }}\int _{M}{\text{tr}}\,(A\wedge dA+{\tfrac {2}{3}}A\wedge A\wedge A).}</annotation>

</semantics>

</math></span><img src="./c10d9d4b2d6d5e52a38c04ccbc42b434ea72941b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.791ex; height:5.843ex;" alt="{\displaystyle S={\frac {k}{4\pi }}\int _{M}{\text{tr}}\,(A\wedge dA+{\tfrac {2}{3}}A\wedge A\wedge A).}" loading="lazy"></span></dd></dl>

<p>The constant <i>k</i> is called the <i>level</i> of the theory. The classical physics of Chern–Simons theory is independent of the choice of level <i>k</i>.

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

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F=dA+A\wedge A\,}</annotation>

</semantics>

</math></span><img src="./bfa12ea3f0ebdb6372bd401b27f80e6a93ee7241.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.095ex; height:2.343ex;" alt="{\displaystyle F=dA+A\wedge A\,}" loading="lazy"></span></dd></dl>

<p>the <a href="Field_equation" title="Field equation">field equation</a> is explicitly

</p>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0={\frac {\delta S}{\delta A}}={\frac {k}{2\pi }}F.}</annotation>

</semantics>

</math></span><img src="./006225edc2bc862cdf8118267d7010789b4a0e52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.705ex; height:5.509ex;" alt="{\displaystyle 0={\frac {\delta S}{\delta A}}={\frac {k}{2\pi }}F.}" loading="lazy"></span></dd></dl>

<p>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 <i>flat</i>. Thus the classical solutions to <i>G</i> Chern–Simons theory are the flat connections of principal <i>G</i>-bundles on <i>M</i>. Flat connections are determined entirely by holonomies around noncontractible cycles on the base <i>M</i>. More precisely, they are in one-to-one correspondence with equivalence classes of homomorphisms from the <a href="Fundamental_group" title="Fundamental group">fundamental group</a> of <i>M</i> to the gauge group <i>G</i> up to conjugation.

</p><p>If <i>M</i> has a boundary <i>N</i> then there is additional data which describes a choice of trivialization of the principal <i>G</i>-bundle on <i>N</i>. Such a choice characterizes a map from <i>N</i> to <i>G</i>. The dynamics of this map is described by the <a href="Wess%E2%80%93Zumino%E2%80%93Witten_model" title="Wess–Zumino–Witten model">Wess–Zumino–Witten</a> (WZW) model on <i>N</i> at level <i>k</i>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Quantization">Quantization</h2></div>

<p>To <a href="Canonical_quantization" title="Canonical quantization">canonically quantize</a> 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 <a href="Hilbert_space" title="Hilbert space">Hilbert space</a>. There is no preferred notion of time in a Schwarz-type topological field theory and so one can require that Σ be a <a href="Cauchy_surface" title="Cauchy surface">Cauchy surface</a>, in fact, a state can be defined on any surface.

</p><p>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. <a href="Edward_Witten" title="Edward Witten">Witten</a> 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 <a href="Virasoro_conformal_block#Larger_symmetry_algebras" title="Virasoro conformal block">conformal blocks</a> of the G WZW model at level k.

</p><p>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 <a href="Group_representation" title="Group representation">representations</a> of the <a href="Affine_Lie_algebra" title="Affine Lie algebra">affine Lie algebra</a> 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.

</p>

<div class="mw-heading mw-heading2"><h2 id="Observables">Observables</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Wilson_loops">Wilson loops</h3></div>

<p>The <a href="Observable" title="Observable">observables</a> of Chern–Simons theory are the <i>n</i>-point <a href="Correlation_function" title="Correlation function">correlation functions</a> of gauge-invariant operators. The most often studied class of gauge invariant operators are <a href="Wilson_loops" class="mw-redirect" title="Wilson loops">Wilson loops</a>. A Wilson loop is the holonomy around a loop in <i>M</i>, traced in a given <a href="Representation_of_a_Lie_group" title="Representation of a Lie group">representation</a> <i>R</i> of <i>G</i>. As we will be interested in products of Wilson loops, without loss of generality we may restrict our attention to <a href="Representation_theory#Subrepresentations,_quotients,_and_irreducible_representations" title="Representation theory">irreducible representations</a> <i>R</i>.

</p><p>More concretely, given an irreducible representation <i>R</i> and a loop <i>K</i> in <i>M</i>, one may define the Wilson loop <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{R}(K)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{R}(K)}</annotation>

</semantics>

</math></span><img src="./02296556da7b787cd8934de73d18168775842a15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.549ex; height:2.843ex;" alt="{\displaystyle W_{R}(K)}" loading="lazy"></span> by

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{R}(K)=\operatorname {Tr} _{R}\,{\mathcal {P}}\exp \left(i\oint _{K}A\right)}</annotation>

</semantics>

</math></span><img src="./4356c4579bafdfacef8a1c2256946791bc6b8f1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.86ex; height:6.176ex;" alt="{\displaystyle W_{R}(K)=\operatorname {Tr} _{R}\,{\mathcal {P}}\exp \left(i\oint _{K}A\right)}" loading="lazy"></span></dd></dl>

<p>where <i>A</i> is the connection 1-form and we take the <a href="Cauchy_principal_value" title="Cauchy principal value">Cauchy principal value</a> of the <a href="Contour_integral" class="mw-redirect" title="Contour integral">contour integral</a> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}\exp }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\exp }</annotation>

</semantics>

</math></span><img src="./e615a0ebe4610661520b3eb9186d5c8d9c1364cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.643ex; height:2.509ex;" alt="{\displaystyle {\mathcal {P}}\exp }" loading="lazy"></span> is the <a href="Path-ordered_exponential" class="mw-redirect" title="Path-ordered exponential">path-ordered exponential</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="HOMFLY_and_Jones_polynomials">HOMFLY and Jones polynomials</h3></div>

<p>Consider a link <i>L</i> in <i>M</i>, which is a collection of <i>ℓ</i> disjoint loops. A particularly interesting observable is the <i>ℓ</i>-point correlation function formed from the product of the Wilson loops around each disjoint loop, each traced in the <a href="Fundamental_representation" title="Fundamental representation">fundamental representation</a> of <i>G</i>. One may form a normalized correlation function by dividing this observable by the <a href="Partition_function_(quantum_field_theory)" title="Partition function (quantum field theory)">partition function</a> <i>Z</i>(<i>M</i>), which is just the 0-point correlation function.

</p><p>In the special case in which M is the 3-sphere, Witten has shown that these normalized correlation functions are proportional to known <a href="Knot_polynomials" class="mw-redirect" title="Knot polynomials">knot polynomials</a>. For example, in <i>G</i> = <i>U</i>(<i>N</i>) Chern–Simons theory at level <i>k</i> the normalized correlation function is, up to a phase, equal to

</p>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\sin(\pi /(k+N))}{\sin(\pi N/(k+N))}}}</annotation>

</semantics>

</math></span><img src="./311d5e511a76c60de12e4877620fbcede4b431dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.984ex; height:6.509ex;" alt="{\displaystyle {\frac {\sin(\pi /(k+N))}{\sin(\pi N/(k+N))}}}" loading="lazy"></span></dd></dl>

<p>times the <a href="HOMFLY_polynomial" title="HOMFLY polynomial">HOMFLY polynomial</a>. In particular when <i>N</i> = 2 the HOMFLY polynomial reduces to the <a href="Jones_polynomial" title="Jones polynomial">Jones polynomial</a>. In the SO(<i>N</i>) case, one finds a similar expression with the <a href="Kauffman_polynomial" title="Kauffman polynomial">Kauffman polynomial</a>.

</p><p>The phase ambiguity reflects the fact that, as Witten has shown, the quantum correlation functions are not fully defined by the classical data. The <a href="Linking_number" title="Linking number">linking number</a> 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 <a href="Normal_vector" class="mw-redirect" title="Normal vector">normal vector</a> at each point along which one deforms the loop to calculate its self-linking number. This procedure is an example of the point-splitting <a href="Regularization_(physics)" title="Regularization (physics)">regularization</a> procedure introduced by <a href="Paul_Dirac" title="Paul Dirac">Paul Dirac</a> and <a href="Rudolf_Peierls" title="Rudolf Peierls">Rudolf Peierls</a> to define apparently divergent quantities in <a href="Quantum_field_theory" title="Quantum field theory">quantum field theory</a> in 1934.

</p><p><a href="Sir_Michael_Atiyah" class="mw-redirect" title="Sir Michael Atiyah">Sir Michael Atiyah</a> has shown that there exists a canonical choice of 2-framing,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> 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>i</i>/(<i>k</i> + <i>N</i>) times the linking number of <i>L</i> with itself.

</p>

<dl><dt>Problem (Extension of Jones polynomial to general 3-manifolds)　</dt></dl>

<p>"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?"

</p><p>See section 1.1 of this paper<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> 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.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> 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 <b>R</b><sup>3</sup>). This problem is also open in physics level. In the case of Alexander polynomial, this problem is solved.

</p>

<div class="mw-heading mw-heading2"><h2 id="Relationships_with_other_theories">Relationships with other theories</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Topological_string_theories">Topological string theories</h3></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Topological_string_theory" title="Topological string theory">Topological string theory</a></div>

<p>In the context of <a href="String_theory" title="String theory">string theory</a>, a <i>U</i>(<i>N</i>) Chern–Simons theory on an oriented Lagrangian 3-submanifold M of a 6-manifold <i>X</i> arises as the <a href="String_field_theory" title="String field theory">string field theory</a> of open strings ending on a <a href="D-brane" title="D-brane">D-brane</a> wrapping <i>X</i> in the <a href="Topological_string_theory#A-model" title="Topological string theory">A-model</a> topological string theory on <i>X</i>. The <a href="Topological_string_theory#B-model" title="Topological string theory">B-model</a> 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.

</p>

<div class="mw-heading mw-heading3"><h3 id="WZW_and_matrix_models">WZW and matrix models</h3></div>

<p>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 <a href="Two-dimensional_conformal_field_theory" title="Two-dimensional conformal field theory">two-dimensional conformal field theory</a> known as a G <a href="Wess%E2%80%93Zumino%E2%80%93Witten_model" title="Wess–Zumino–Witten model">Wess–Zumino–Witten model</a> on the boundary. In addition the <i>U</i>(<i>N</i>) and SO(<i>N</i>) Chern–Simons theories at large <i>N</i> are well approximated by <a href="Matrix_theory_(physics)" title="Matrix theory (physics)">matrix models</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Chern–Simons_gravity_theory">Chern–Simons gravity theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="(2%2B1)-dimensional_topological_gravity" title="(2+1)-dimensional topological gravity">(2+1)-dimensional topological gravity</a></div>

<p>In 1982, <a href="Stanley_Deser" title="Stanley Deser">S. Deser</a>, <a href="Roman_Jackiw" title="Roman Jackiw">R. Jackiw</a> and S. Templeton proposed the Chern–Simons gravity theory in three dimensions, in which the <a href="Einstein%E2%80%93Hilbert_action" title="Einstein–Hilbert action">Einstein–Hilbert action</a> in gravity theory is modified by adding the Chern–Simons term. (<a href="#CITEREFDeserJackiwTempleton1982">Deser, Jackiw & Templeton (1982)</a>)

</p><p>In 2003, R. Jackiw and S. Y. Pi extended this theory to four dimensions (<a href="#CITEREFJackiwPi2003">Jackiw & Pi (2003)</a>) and Chern–Simons gravity theory has some considerable effects not only to fundamental physics but also condensed matter theory and astronomy.

</p><p>The four-dimensional case is very analogous to the three-dimensional case. In three dimensions, the gravitational Chern–Simons term is

</p>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {CS} (\Gamma )={\frac {1}{2\pi ^{2}}}\int d^{3}x\varepsilon ^{ijk}{\biggl (}\Gamma _{iq}^{p}\partial _{j}\Gamma _{kp}^{q}+{\frac {2}{3}}\Gamma _{iq}^{p}\Gamma _{jr}^{q}\Gamma _{kp}^{r}{\biggr )}.}</annotation>

</semantics>

</math></span><img src="./a0075fd8f2a25a546317665e0498d0d921231a35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.615ex; height:6.176ex;" alt="{\displaystyle \operatorname {CS} (\Gamma )={\frac {1}{2\pi ^{2}}}\int d^{3}x\varepsilon ^{ijk}{\biggl (}\Gamma _{iq}^{p}\partial _{j}\Gamma _{kp}^{q}+{\frac {2}{3}}\Gamma _{iq}^{p}\Gamma _{jr}^{q}\Gamma _{kp}^{r}{\biggr )}.}" loading="lazy"></span></dd></dl>

<p>This variation gives the <a href="Cotton_tensor" title="Cotton tensor">Cotton tensor</a>

</p>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =-{\frac {1}{2{\sqrt {g}}}}{\bigl (}\varepsilon ^{mij}D_{i}R_{j}^{n}+\varepsilon ^{nij}D_{i}R_{j}^{m}).}</annotation>

</semantics>

</math></span><img src="./9bd75b443f9e7da2a55ae6a72a82fcd78fb30b42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.188ex; height:6.176ex;" alt="{\displaystyle =-{\frac {1}{2{\sqrt {g}}}}{\bigl (}\varepsilon ^{mij}D_{i}R_{j}^{n}+\varepsilon ^{nij}D_{i}R_{j}^{m}).}" loading="lazy"></span></dd></dl>

<p>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.

</p>

<div class="mw-heading mw-heading3"><h3 id="Chern–Simons_matter_theories">Chern–Simons matter theories</h3></div>

<p>In 2013 Kenneth A. Intriligator and <a href="Nathan_Seiberg" title="Nathan Seiberg">Nathan Seiberg</a> solved these 3d Chern–Simons gauge theories and their phases using <a href="Seiberg-Witten_monopole" class="mw-redirect" title="Seiberg-Witten monopole">monopoles</a> carrying extra degrees of freedom. The <a href="Witten_index" title="Witten index">Witten index</a> of the many <a href="Vacuum_state" class="mw-redirect" title="Vacuum state">vacua</a> discovered was computed by compactifying the space by turning on mass parameters and then computing the index. In some vacua, <a href="Supersymmetry" title="Supersymmetry">supersymmetry</a> was computed to be broken. These monopoles were related to <a href="Condensed_matter_physics" title="Condensed matter physics">condensed matter</a> <a href="Quantum_vortex" title="Quantum vortex">vortices</a>. (<a href="#CITEREFIntriligatorSeiberg2013">Intriligator & Seiberg (2013)</a>)

</p><p>The <i>N</i> = 6 Chern–Simons matter theory is the <a href="AdS/CFT_correspondence" title="AdS/CFT correspondence">holographic dual</a> of M-theory on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AdS_{4}\times S_{7}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle AdS_{4}\times S_{7}}</annotation>

</semantics>

</math></span><img src="./101afc2877d7978a4538886779d4debbe5efa115.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.758ex; height:2.509ex;" alt="{\displaystyle AdS_{4}\times S_{7}}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Four-dimensional_Chern–Simons_theory">Four-dimensional Chern–Simons theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Four-dimensional_Chern%E2%80%93Simons_theory" title="Four-dimensional Chern–Simons theory">Four-dimensional Chern–Simons theory</a></div>

<p>In 2013 <a href="Kevin_Costello" title="Kevin Costello">Kevin Costello</a> 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.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> He later studied the theory in more detail together with Witten and Masahito Yamazaki,<sup id="cite_ref-CWY1_9-0" class="reference"><a href="#cite_note-CWY1-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-CWY2_10-0" class="reference"><a href="#cite_note-CWY2-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-CY_11-0" class="reference"><a href="#cite_note-CY-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> demonstrating how the gauge theory could be related to many notions in <a href="Integrable_system" title="Integrable system">integrable systems</a> theory, including exactly solvable lattice models (like the <a href="Six-vertex_model" class="mw-redirect" title="Six-vertex model">six-vertex model</a> or the <a href="Quantum_Heisenberg_model" title="Quantum Heisenberg model">XXZ spin chain</a>), integrable quantum field theories (such as the <a href="Gross%E2%80%93Neveu_model" title="Gross–Neveu model">Gross–Neveu model</a>, <a href="Chiral_model" title="Chiral model">principal chiral model</a> and symmetric space coset <a href="Sigma_model" title="Sigma model">sigma models</a>), the <a href="Yang%E2%80%93Baxter_equation" title="Yang–Baxter equation">Yang–Baxter equation</a> and <a href="Quantum_groups" class="mw-redirect" title="Quantum groups">quantum groups</a> such as the <a href="Yangian" title="Yangian">Yangian</a> which describe symmetries underpinning the integrability of the aforementioned systems.

</p><p>The action on the 4-manifold <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\Sigma \times C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=\Sigma \times C}</annotation>

</semantics>

</math></span><img src="./5656bd4f68eec969acf1747e84dcbd58a1e5c184.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.825ex; height:2.176ex;" alt="{\displaystyle M=\Sigma \times C}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math></span><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"></span> is a two-dimensional manifold and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> is a complex curve is

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=\int _{M}\omega \wedge CS(A)}</annotation>

</semantics>

</math></span></span>

where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> is a <a href="Meromorphic" class="mw-redirect" title="Meromorphic">meromorphic</a> <a href="One-form" class="mw-redirect" title="One-form">one-form</a> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</p>

<div class="mw-heading mw-heading2"><h2 id="Chern–Simons_terms_in_other_theories">Chern–Simons terms in other theories</h2></div>

<p>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 <a href="Photon" title="Photon">photon</a> if this term is added to the action of Maxwell's theory of <a href="Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a>. This term can be induced by integrating over a massive charged <a href="Fermionic_field#Dirac_fields" title="Fermionic field">Dirac field</a>. It also appears for example in the <a href="Quantum_Hall_effect" title="Quantum Hall effect">quantum Hall effect</a>. The addition of the Chern–Simons term to various theories gives rise to vortex- or soliton-type solutions<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Ten- and eleven-dimensional generalizations of Chern–Simons terms appear in the actions of all ten- and eleven-dimensional <a href="Supergravity" title="Supergravity">supergravity</a> theories.

</p>

<div class="mw-heading mw-heading3"><h3 id="One-loop_renormalization_of_the_level">One-loop renormalization of the level</h3></div>

<p>If one adds matter to a Chern–Simons gauge theory then, in general it is no longer topological. However, if one adds n <a href="Majorana_fermion" title="Majorana fermion">Majorana fermions</a> then, due to the <a href="Parity_anomaly" title="Parity anomaly">parity anomaly</a>, when integrated out they lead to a pure Chern–Simons theory with a one-loop <a href="Renormalization" title="Renormalization">renormalization</a> of the Chern–Simons level by −<i>n</i>/2, in other words the level k theory with n fermions is equivalent to the level <i>k</i> − <i>n</i>/2 theory without fermions.

</p>

<ul><li><a href="Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory (mathematics)</a></li>

<li><a href="Chern%E2%80%93Simons_form" title="Chern–Simons form">Chern–Simons form</a></li>

<li><a href="Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li>

<li><a href="Alexander_polynomial" title="Alexander polynomial">Alexander polynomial</a></li>

<li><a href="Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li>

<li><a href="2%2B1D_topological_gravity" class="mw-redirect" title="2+1D topological gravity">2+1D topological gravity</a></li>

<li><a href="Skyrmion" title="Skyrmion">Skyrmion</a></li>

<li><a href="%E2%88%9E-Chern%E2%80%93Simons_theory" title="∞-Chern–Simons theory">∞-Chern–Simons theory</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFArthurTchrakianY.-S.1996" class="citation journal cs1">Arthur, K.; Tchrakian, D.H.; Y.-S., Yang (1996). "Topological and nontopological selfdual Chern-Simons solitons in a gauged O(3) sigma model". <i><a href="Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a></i>. <b>54</b> (8): <span class="nowrap">5245–</span>5258. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996PhRvD..54.5245A">1996PhRvD..54.5245A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.54.5245">10.1103/PhysRevD.54.5245</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10021215">10021215</a>.</cite></li>

<li><cite id="CITEREFChernSimons1974" class="citation journal cs1"><a href="S.-S._Chern" class="mw-redirect" title="S.-S. Chern">Chern, S.-S.</a> & Simons, J. (1974). "Characteristic forms and geometric invariants". <i><a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>. <b>99</b> (1): <span class="nowrap">48–</span>69. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1971013">10.2307/1971013</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1971013">1971013</a>.</cite></li>

<li><cite id="CITEREFDeserJackiwTempleton1982" class="citation journal cs1">Deser, Stanley; Jackiw, Roman; Templeton, S. (1982). <a rel="nofollow" class="external text" href="https://authors.library.caltech.edu/85895/1/PhysRevLett.48.975.pdf">"Three-Dimensional Massive Gauge Theories"</a> <span class="cs1-format">(PDF)</span>. <i><a href="Physical_Review_Letters" title="Physical Review Letters">Physical Review Letters</a></i>. <b>48</b> (15): <span class="nowrap">975–</span>978. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982PhRvL..48..975D">1982PhRvL..48..975D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.48.975">10.1103/PhysRevLett.48.975</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122537043">122537043</a>.</cite></li>

<li><cite id="CITEREFIntriligatorSeiberg2013" class="citation journal cs1">Intriligator, Kenneth; Seiberg, Nathan (2013). "Aspects of 3d <i>N</i> = 2 Chern–Simons Matter Theories". <i><a href="Journal_of_High_Energy_Physics" title="Journal of High Energy Physics">Journal of High Energy Physics</a></i>. <b>2013</b>: 79. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1305.1633">1305.1633</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013JHEP...07..079I">2013JHEP...07..079I</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FJHEP07%282013%29079">10.1007/JHEP07(2013)079</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119106931">119106931</a>.</cite></li>

<li><cite id="CITEREFJackiwPi2003" class="citation journal cs1"><a href="Roman_Jackiw" title="Roman Jackiw">Jackiw, Roman</a>; <a href="So-Young_Pi" title="So-Young Pi">Pi, S.-Y</a> (2003). "Chern–Simons modification of general relativity". <i><a href="Physical_Review_D" class="mw-redirect" title="Physical Review D">Physical Review D</a></i>. <b>68</b> (10): 104012. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0308071">gr-qc/0308071</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003PhRvD..68j4012J">2003PhRvD..68j4012J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.68.104012">10.1103/PhysRevD.68.104012</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2243511">2243511</a>.</cite></li>

<li><cite id="CITEREFKulshreshthaKulshreshthaMueller-KirstenVary2009" class="citation journal cs1">Kulshreshtha, Usha; Kulshreshtha, D.S.; Mueller-Kirsten, H. J. W.; Vary, J. P. (2009). "Hamiltonian, path integral and BRST formulations of the Chern-Simons-Higgs theory under appropriate gauge fixing". <i><a href="Physica_Scripta" title="Physica Scripta">Physica Scripta </a></i>. <b>79</b> (4): 045001. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009PhyS...79d5001K">2009PhyS...79d5001K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0031-8949%2F79%2F04%2F045001">10.1088/0031-8949/79/04/045001</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120594654">120594654</a>.</cite></li>

<li><cite id="CITEREFKulshreshthaKulshreshthaVary2010" class="citation journal cs1">Kulshreshtha, Usha; Kulshreshtha, D.S.; Vary, J. P. (2010). "Light-front Hamiltonian, path integral and BRST formulations of the Chern-Simons-Higgs theory under appropriate gauge fixing". <i><a href="Physica_Scripta" title="Physica Scripta">Physica Scripta</a></i>. <b>82</b> (5): 055101. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010PhyS...82e5101K">2010PhyS...82e5101K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0031-8949%2F82%2F05%2F055101">10.1088/0031-8949/82/05/055101</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:54602971">54602971</a>.</cite></li>

<li><cite id="CITEREFLopezFradkin1991" class="citation journal cs1">Lopez, Ana; <a href="Eduardo_Fradkin" title="Eduardo Fradkin">Fradkin, Eduardo</a> (1991). "Fractional quantum Hall effect and Chern-Simons gauge theories". <i><a href="Physical_Review_B" title="Physical Review B">Physical Review B</a></i>. <b>44</b> (10): <span class="nowrap">5246–</span>5262. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991PhRvB..44.5246L">1991PhRvB..44.5246L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.44.5246">10.1103/PhysRevB.44.5246</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9998334">9998334</a>.</cite></li>

<li><cite id="CITEREFMarino2005" class="citation journal cs1">Marino, Marcos (2005). "Chern–Simons Theory and Topological Strings". <i><a href="Reviews_of_Modern_Physics" title="Reviews of Modern Physics">Reviews of Modern Physics</a></i>. <b>77</b> (2): <span class="nowrap">675–</span>720. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/0406005">hep-th/0406005</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005RvMP...77..675M">2005RvMP...77..675M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.77.675">10.1103/RevModPhys.77.675</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6207500">6207500</a>.</cite></li>

<li><cite id="CITEREFMarino2005" class="citation book cs1">Marino, Marcos (2005). <i>Chern–Simons Theory, Matrix Models, And Topological Strings</i>. International Series of Monographs on Physics. <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</cite></li>

<li><cite id="CITEREFWitten1988" class="citation journal cs1"><a href="Edward_Witten" title="Edward Witten">Witten, Edward</a> (1988). <a rel="nofollow" class="external text" href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1104161738">"Topological Quantum Field Theory"</a>. <i><a href="Communications_in_Mathematical_Physics" title="Communications in Mathematical Physics">Communications in Mathematical Physics</a></i>. <b>117</b> (3): <span class="nowrap">353–</span>386. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1988CMaPh.117..353W">1988CMaPh.117..353W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01223371">10.1007/BF01223371</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:43230714">43230714</a>.</cite></li>

<li><cite id="CITEREFWitten1995" class="citation journal cs1">Witten, Edward (1995). "Chern–Simons Theory as a String Theory". <i>Progress in Mathematics</i>. <b>133</b>: <span class="nowrap">637–</span>678. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/9207094">hep-th/9207094</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992hep.th....7094W">1992hep.th....7094W</a>.</cite></li></ul>

<dl><dt>Specific</dt></dl>

<li id="cite_note-wittenjonespolynomial-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-wittenjonespolynomial_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-wittenjonespolynomial_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFWitten1989" class="citation journal cs1"><a href="Edward_Witten" title="Edward Witten">Witten, Edward</a> (1989). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.cmp/1104178138">"Quantum Field Theory and the Jones Polynomial"</a>. <i><a href="Communications_in_Mathematical_Physics" title="Communications in Mathematical Physics">Communications in Mathematical Physics</a></i>. <b>121</b> (3): <span class="nowrap">351–</span>399. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1989CMaPh.121..351W">1989CMaPh.121..351W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01217730">10.1007/BF01217730</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0990772">0990772</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14951363">14951363</a>.</cite></span>

</li>

<li id="cite_note-FK02-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FK02_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFFreedmanKitaevLarsenWang2002" class="citation arxiv cs1">Freedman, Michael H.; Kitaev, Alexei; Larsen, Michael J.; Wang, Zhenghan (2002-09-20). "Topological Quantum Computation". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0101025">quant-ph/0101025</a></span>.</cite></span>

</li>

<li id="cite_note-WangTQCreview-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-WangTQCreview_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFWang" class="citation web cs1">Wang, Zhenghan. <a rel="nofollow" class="external text" href="http://web.math.ucsb.edu/~zhenghwa/data/course/cbms.pdf">"Topological Quantum Computation"</a> <span class="cs1-format">(PDF)</span>.</cite></span>

</li>

<cite id="CITEREFElitzurMooreSchwimmerSeiberg1989" class="citation journal cs1">Elitzur, Shmuel; Moore, Gregory; Schwimmer, Adam; <a href="Nathan_Seiberg" title="Nathan Seiberg">Seiberg, Nathan</a> (30 October 1989). "Remarks on the canonical quantization of the Chern-Simons-Witten theory". <i><a href="Nuclear_Physics_B" class="mw-redirect" title="Nuclear Physics B">Nuclear Physics B</a></i>. <b>326</b> (1): <span class="nowrap">108–</span>134. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1989NuPhB.326..108E">1989NuPhB.326..108E</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0550-3213%2889%2990436-7">10.1016/0550-3213(89)90436-7</a>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFAtiyah1990" class="citation journal cs1">Atiyah, Michael (1990). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://doi.org/10.1016/0040-9383(90)90021-b">"On framings of 3-manifolds"</a></span>. <i>Topology</i>. <b>29</b> (1): <span class="nowrap">1–</span>7. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0040-9383%2890%2990021-b">10.1016/0040-9383(90)90021-b</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0040-9383">0040-9383</a>.</cite></span>

</li>

<cite id="CITEREFKauffmanOgasaSchneider2018" class="citation arxiv cs1">Kauffman, L.H; Ogasa, E; Schneider, J (2018). "A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1808.03023">1808.03023</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.GT">math.GT</a>].</cite> </span>

</li>

<cite id="CITEREFKauffman1998" class="citation arxiv cs1">Kauffman, L.E. (1998). "Virtual Knot Theory". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9811028">math/9811028</a></span>.</cite> </span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFCostello2013" class="citation arxiv cs1">Costello, Kevin (2013). "Supersymmetric gauge theory and the Yangian". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1303.2632">1303.2632</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/hep-th">hep-th</a>].</cite></span>

</li>

<li id="cite_note-CWY1-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-CWY1_9-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFCostelloWittenYamazaki2018" class="citation journal cs1">Costello, Kevin; Witten, Edward; Yamazaki, Masahito (2018). "Gauge Theory And Integrability, I". <i>Notices of the International Congress of Chinese Mathematicians</i>. <b>6</b> (1): <span class="nowrap">46–</span>119. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1709.09993">1709.09993</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4310%2FICCM.2018.v6.n1.a6">10.4310/ICCM.2018.v6.n1.a6</a>.</cite></span>

</li>

<li id="cite_note-CWY2-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-CWY2_10-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFCostelloWittenYamazaki2018" class="citation journal cs1">Costello, Kevin; Witten, Edward; Yamazaki, Masahito (2018). "Gauge Theory And Integrability, II". <i>Notices of the International Congress of Chinese Mathematicians</i>. <b>6</b> (1): <span class="nowrap">120–</span>146. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1802.01579">1802.01579</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4310%2FICCM.2018.v6.n1.a7">10.4310/ICCM.2018.v6.n1.a7</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119592177">119592177</a>.</cite></span>

</li>

<li id="cite_note-CY-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-CY_11-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFCostelloYamazaki2019" class="citation arxiv cs1">Costello, Kevin; Yamazaki, Masahito (2019). "Gauge Theory And Integrability, III". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1908.02289">1908.02289</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/hep-th">hep-th</a>].</cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite id="CITEREFKimKim2002" class="citation journal cs1">Kim, Seongtag; Kim, Yoonbai (2002). "Self-dual Chern–Simons vortices on Riemann surfaces". <i>Journal of Mathematical Physics</i>. <b>43</b> (5): <span class="nowrap">2355–</span>2362. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math-ph/0012045">math-ph/0012045</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002JMP....43.2355K">2002JMP....43.2355K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1471365">10.1063/1.1471365</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9916364">9916364</a>.</cite></span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><cite id="CITEREFNavarro-LéridaRaduTchrakian2017" class="citation journal cs1">Navarro-Lérida, Francisco; Radu, Eugen; Tchrakian, D. H. (2017). "Effect of Chern-Simons dynamics on the energy of electrically charged and spinning vortices". <i>Physical Review D</i>. <b>95</b> (8): 085016. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1612.05835">1612.05835</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017PhRvD..95h5016N">2017PhRvD..95h5016N</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevD.95.085016">10.1103/PhysRevD.95.085016</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:62882649">62882649</a>.</cite></span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Chern-Simons_functional">"Chern-Simons functional"</a>. <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>. <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>. 2001 [1994].</cite></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Quantum_field_theories202" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div id="Quantum_field_theories202" style="font-size:114%;margin:0 4em"><a href="Quantum_field_theory" title="Quantum field theory">Quantum field theories</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algebraic_quantum_field_theory" title="Algebraic quantum field theory">Algebraic QFT</a></li>

<li><a href="Axiomatic_quantum_field_theory" title="Axiomatic quantum field theory">Axiomatic QFT</a></li>

<li><a href="Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></li>

<li><a href="Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li>

<li><a href="Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative QFT</a></li>

<li><a href="Gauge_theory" title="Gauge theory">Gauge theory</a></li>

<li><a href="Quantum_field_theory_in_curved_spacetime" title="Quantum field theory in curved spacetime">QFT in curved spacetime</a></li>

<li><a href="String_theory" title="String theory">String theory</a></li>

<li><a href="Supergravity" title="Supergravity">Supergravity</a></li>

<li><a href="Thermal_quantum_field_theory" title="Thermal quantum field theory">Thermal QFT</a></li>

<li><a href="Topological_quantum_field_theory" title="Topological quantum field theory">Topological QFT</a></li>

<li><a href="Two-dimensional_conformal_field_theory" title="Two-dimensional conformal field theory">Two-dimensional conformal field theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Models</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Regular</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Born%E2%80%93Infeld_model" title="Born–Infeld model">Born–Infeld</a></li>

<li><a href="Euler%E2%80%93Heisenberg_Lagrangian" title="Euler–Heisenberg Lagrangian">Euler–Heisenberg</a></li>

<li><a href="Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau</a></li>

<li><a href="Non-linear_sigma_model" title="Non-linear sigma model">Non-linear sigma</a></li>

<li><a href="Proca_action" title="Proca action">Proca</a></li>

<li><a href="Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li>

<li><a href="Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li>

<li><a href="Quartic_interaction" title="Quartic interaction">Quartic interaction</a></li>

<li><a href="Scalar_electrodynamics" title="Scalar electrodynamics">Scalar electrodynamics</a></li>

<li><a href="Scalar_chromodynamics" title="Scalar chromodynamics">Scalar chromodynamics</a></li>

<li><a href="Soler_model" title="Soler model">Soler</a></li>

<li><a href="Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills</a></li>

<li><a href="Yang%E2%80%93Mills%E2%80%93Higgs_equations" title="Yang–Mills–Higgs equations">Yang–Mills–Higgs</a></li>

<li><a href="Yukawa_interaction" class="mw-redirect" title="Yukawa interaction">Yukawa</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Low dimensional</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Two-dimensional_Yang%E2%80%93Mills_theory" title="Two-dimensional Yang–Mills theory">2D Yang–Mills</a></li>

<li><a href="Bullough%E2%80%93Dodd_model" title="Bullough–Dodd model">Bullough–Dodd</a></li>

<li><a href="Gross%E2%80%93Neveu_model" title="Gross–Neveu model">Gross–Neveu</a></li>

<li><a href="Schwinger_model" title="Schwinger model">Schwinger</a></li>

<li><a href="Sine-Gordon_equation" title="Sine-Gordon equation">Sine-Gordon</a></li>

<li><a href="Thirring_model" title="Thirring model">Thirring</a></li>

<li><a href="Thirring%E2%80%93Wess_model" title="Thirring–Wess model">Thirring–Wess</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Conformal</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Massless_free_scalar_bosons_in_two_dimensions" title="Massless free scalar bosons in two dimensions">2D free massless scalar</a></li>

<li><a href="Liouville_field_theory" title="Liouville field theory">Liouville</a></li>

<li><a href="Minimal_model_(physics)" title="Minimal model (physics)">Minimal</a></li>

<li><a href="Polyakov_action" title="Polyakov action">Polyakov</a></li>

<li><a href="Wess%E2%80%93Zumino%E2%80%93Witten_model" title="Wess–Zumino–Witten model">Wess–Zumino–Witten</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Supersymmetric</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="N_%3D_1_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 1 supersymmetric Yang–Mills theory">N = 1 super Yang–Mills</a></li>

<li><a href="Seiberg%E2%80%93Witten_theory" title="Seiberg–Witten theory">Seiberg–Witten</a></li>

<li><a href="Super_QCD" title="Super QCD">Super QCD</a></li>

<li><a href="Wess%E2%80%93Zumino_model" title="Wess–Zumino model">Wess–Zumino</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Superconformal</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 4 supersymmetric Yang–Mills theory">N = 4 super Yang–Mills</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Supergravity</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Higher-dimensional_supergravity" title="Higher-dimensional supergravity">Higher dimensional</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Topological</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

</ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Particle theory</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chiral_model" title="Chiral model">Chiral</a></li>

<li><a href="Fermi's_interaction" title="Fermi's interaction">Fermi</a></li>

<li><a href="Nambu%E2%80%93Jona-Lasinio_model" title="Nambu–Jona-Lasinio model">Nambu–Jona-Lasinio</a></li>

<li><a href="Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li>

<li><a href="Standard_Model" title="Standard Model">Standard Model</a></li>

<li><a href="Stueckelberg_action" title="Stueckelberg action">Stueckelberg</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Casimir_effect" title="Casimir effect">Casimir effect</a></li>

<li><a href="Cosmic_string" title="Cosmic string">Cosmic string</a></li>

<li><a href="History_of_quantum_field_theory" title="History of quantum field theory">History</a></li>

<li><a href="Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li>

<li><a href="Loop_quantum_cosmology" title="Loop quantum cosmology">Loop quantum cosmology</a></li>

<li><a href="On_shell_and_off_shell" title="On shell and off shell">On shell and off shell</a></li>

<li><a href="Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li>

<li><a href="Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li>

<li><a href="Quantum_foam" title="Quantum foam">Quantum foam</a></li>

<li><a href="Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuations</a>

<ul><li>links</li></ul></li>

<li><a href="Quantum_gravity" title="Quantum gravity">Quantum gravity</a>

<ul><li>links</li></ul></li>

<li><a href="Quantum_hadrodynamics" title="Quantum hadrodynamics">Quantum hadrodynamics</a></li>

<li><a href="Quantum_hydrodynamics" title="Quantum hydrodynamics">Quantum hydrodynamics</a></li>

<li><a href="Quantum_information" title="Quantum information">Quantum information</a></li>

<li><a href="Quantum_information_science" title="Quantum information science">Quantum information science</a>

<ul><li>links</li></ul></li>

<li><a href="Quantum_logic" title="Quantum logic">Quantum logic</a></li>

<li><a href="Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i>See also:</i> <span class="noviewer" typeof="mw:File"><span title="Template"></span></span> Template:Quantum mechanics topics</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-05-25" href="https://en.wikipedia.org/wiki/?title=Chern%E2%80%93Simons_theory&oldid=1292148146">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>