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</li> <li id="toc-Chern–Simons_gravity_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Chern–Simons_gravity_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Chern–Simons gravity theory</span> </div> </a> <ul id="toc-Chern–Simons_gravity_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Chern–Simons_matter_theories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Chern–Simons_matter_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Chern–Simons matter theories</span> </div> </a> <ul id="toc-Chern–Simons_matter_theories-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Four-dimensional_Chern–Simons_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Four-dimensional_Chern–Simons_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Four-dimensional Chern–Simons theory</span> </div> </a> <ul id="toc-Four-dimensional_Chern–Simons_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Chern–Simons_terms_in_other_theories" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Chern–Simons_terms_in_other_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Chern–Simons terms in other theories</span> </div> </a> <button aria-controls="toc-Chern–Simons_terms_in_other_theories-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Chern–Simons terms in other theories subsection</span> </button> <ul id="toc-Chern–Simons_terms_in_other_theories-sublist" class="vector-toc-list"> <li id="toc-One-loop_renormalization_of_the_level" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#One-loop_renormalization_of_the_level"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>One-loop renormalization of the level</span> </div> </a> <ul id="toc-One-loop_renormalization_of_the_level-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" 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href="https://es.wikipedia.org/wiki/Teor%C3%ADa_de_Chern-Simons" title="Teoría de Chern-Simons – Spanish" lang="es" hreflang="es" data-title="Teoría de Chern-Simons" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_de_Chern-Simons" title="Théorie de Chern-Simons – French" lang="fr" hreflang="fr" data-title="Théorie de Chern-Simons" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B2%9C-%EC%82%AC%EC%9D%B4%EB%A8%BC%EC%8A%A4_%EC%9D%B4%EB%A1%A0" title="천-사이먼스 이론 – Korean" lang="ko" hreflang="ko" data-title="천-사이먼스 이론" data-language-autonym="한국어" data-language-local-name="Korean" 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href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">topological quantum field theory</a> of <a href="/wiki/Topological_quantum_field_theory#Schwarz-type_TQFTs" title="Topological quantum field theory">Schwarz type</a> developed by <a href="/wiki/Edward_Witten" title="Edward Witten">Edward Witten</a>. It was discovered first by mathematical physicist <a href="/wiki/Albert_Schwarz" title="Albert Schwarz">Albert Schwarz</a>. It is named after mathematicians <a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Shiing-Shen Chern</a> and <a href="/wiki/James_Harris_Simons" class="mw-redirect" title="James Harris Simons">James Harris Simons</a>, who introduced the <a href="/wiki/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="/wiki/Action_(physics)" title="Action (physics)">action</a> is proportional to the integral of the Chern–Simons 3-form. </p><p>In <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed-matter physics</a>, Chern–Simons theory describes the <a href="/wiki/Topological_order" title="Topological order">topological order</a> in <a href="/wiki/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="/wiki/Knot_invariants" class="mw-redirect" title="Knot invariants">knot invariants</a> and <a href="/wiki/Three-manifold" class="mw-redirect" title="Three-manifold">three-manifold</a> invariants such as the <a href="/wiki/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="/wiki/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="/wiki/Partition_function_(quantum_field_theory)" title="Partition function (quantum field theory)">partition function</a> of the <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum</a> theory is <a href="/wiki/Well-defined" class="mw-redirect" title="Well-defined">well-defined</a> when the level is an integer and the gauge <a href="/wiki/Field_strength" title="Field strength">field strength</a> vanishes on all <a href="/wiki/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="/wiki/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="/wiki/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="/wiki/Fusion_rules" title="Fusion rules">fusion rules</a> and <a href="/wiki/Virasoro_conformal_block" title="Virasoro conformal block">conformal blocks</a> in <a href="/wiki/Conformal_field_theory" title="Conformal field theory">conformal field theory</a>, and in particular <a href="/wiki/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> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="The_classical_theory">The classical theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=1" title="Edit section: The classical theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Mathematical_origin">Mathematical origin</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=2" title="Edit section: Mathematical origin"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the 1940s <a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">S. S. Chern</a> and <a href="/wiki/Andr%C3%A9_Weil" title="André Weil">A. Weil</a> studied the global curvature properties of smooth manifolds <i>M</i> as <a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">de Rham cohomology</a> (<a href="/wiki/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="/wiki/Characteristic_classes" class="mw-redirect" title="Characteristic classes">characteristic classes</a> in <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>. Given a flat <i>G</i>-<a href="/wiki/Principal_bundle" title="Principal bundle">principal bundle</a> <i>P</i> on <i>M</i> there exists a unique homomorphism, called the <a href="/wiki/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"><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} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{*}(M,\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f395151b51136ae59f0d10189c2d6d8b6a1a57dd" 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} )}"></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="/wiki/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> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dTf(\omega )=f(\Omega ^{k}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>T</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dTf(\omega )=f(\Omega ^{k}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/380a460b40fd710f9ecad7119a1d37a825baa6f9" 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}),}"></span></dd></dl> <p>where <i>T</i> is the Chern–Weil homomorphism. This form is called <a href="/wiki/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> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Tf(\omega )=\int _{C}f(\Omega ^{k}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Tf(\omega )=\int _{C}f(\Omega ^{k}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5a07866e30d6406a0bb52c5be427d221005620" 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}),}"></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> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {CS} (M)=\int _{s(M)}{\tfrac {1}{2}}Tp_{1}\in \mathbb {R} /\mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>CS</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>T</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </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="https://wikimedia.org/api/rest_v1/media/math/render/svg/55a75adffcb84a741cb77335fa253a39633ace82" 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} ,}"></span></dd></dl> <p>where <span class="mwe-math-element"><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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b58f22283ca46dd5da309cc34303b06a797783" 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}}"></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="/wiki/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="/wiki/Action_integral" class="mw-redirect" title="Action integral">action integral</a> (<a href="/wiki/Path_integral_formulation" title="Path integral formulation">path integral</a>) of the <a href="/wiki/Quantum_field_theory" title="Quantum field theory">field theory</a> in physics is viewed as the <a href="/wiki/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="/wiki/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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=3" title="Edit section: Configurations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Chern–Simons theories can be defined on any <a href="/wiki/Topological_manifold" title="Topological manifold">topological</a> <a href="/wiki/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="/wiki/Metric_tensor" title="Metric tensor">metric</a> needs to be introduced on <i>M</i>. </p><p>Chern–Simons theory is a <a href="/wiki/Gauge_theory" title="Gauge theory">gauge theory</a>, which means that a <a href="/wiki/Classical_physics" title="Classical physics">classical</a> configuration in the Chern–Simons theory on <i>M</i> with <a href="/wiki/Gauge_group" class="mw-redirect" title="Gauge group">gauge group</a> <i>G</i> is described by a <a href="/wiki/Principal_bundle" title="Principal bundle">principal <i>G</i>-bundle</a> on <i>M</i>. The <a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">connection</a> of this bundle is characterized by a <a href="/wiki/Connection_one-form" class="mw-redirect" title="Connection one-form">connection one-form</a> <i>A</i> which is <a href="/wiki/Vector-valued_differential_form#Lie_algebra-valued_forms" title="Vector-valued differential form">valued</a> in the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> <b>g</b> of the <a href="/wiki/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="/wiki/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="/wiki/Gauge_symmetry" class="mw-redirect" title="Gauge symmetry">gauge transformations</a>. These are characterized by the assertion that the <a href="/wiki/Gauge_covariant_derivative" title="Gauge covariant derivative">covariant derivative</a>, which is the sum of the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> operator <i>d</i> and the connection <i>A</i>, transforms in the <a href="/wiki/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="/wiki/Curvature_form" title="Curvature form">curvature form</a> or <a href="/wiki/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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=4" title="Edit section: Dynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Action_(physics)" title="Action (physics)">action</a> <i>S</i> of Chern–Simons theory is proportional to the integral of the <a href="/wiki/Chern%E2%80%93Simons_3-form" class="mw-redirect" title="Chern–Simons 3-form">Chern–Simons 3-form</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={\frac {k}{4\pi }}\int _{M}{\text{tr}}\,(A\wedge dA+{\tfrac {2}{3}}A\wedge A\wedge A).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>tr</mtext> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>d</mi> <mi>A</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>A</mi> <mo>∧</mo> <mi>A</mi> <mo>∧</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>.</mo> </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="https://wikimedia.org/api/rest_v1/media/math/render/svg/c10d9d4b2d6d5e52a38c04ccbc42b434ea72941b" 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).}"></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> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=dA+A\wedge A\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>d</mi> <mi>A</mi> <mo>+</mo> <mi>A</mi> <mo>∧</mo> <mi>A</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=dA+A\wedge A\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfa12ea3f0ebdb6372bd401b27f80e6a93ee7241" 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\,}"></span></dd></dl> <p>the <a href="/wiki/Field_equation" title="Field equation">field equation</a> is explicitly </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0={\frac {\delta S}{\delta A}}={\frac {k}{2\pi }}F.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>S</mi> </mrow> <mrow> <mi>δ</mi> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mi>F</mi> <mo>.</mo> </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="https://wikimedia.org/api/rest_v1/media/math/render/svg/006225edc2bc862cdf8118267d7010789b4a0e52" 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.}"></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="/wiki/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="/wiki/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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=5" title="Edit section: Quantization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To <a href="/wiki/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="/wiki/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="/wiki/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="/wiki/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="/wiki/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="/wiki/Group_representation" title="Group representation">representations</a> of the <a href="/wiki/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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=6" title="Edit section: Observables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Wilson_loops">Wilson loops</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=7" title="Edit section: Wilson loops"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Observable" title="Observable">observables</a> of Chern–Simons theory are the <i>n</i>-point <a href="/wiki/Correlation_function" title="Correlation function">correlation functions</a> of gauge-invariant operators. The most often studied class of gauge invariant operators are <a href="/wiki/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="/wiki/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="/wiki/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"><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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{R}(K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02296556da7b787cd8934de73d18168775842a15" 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)}"></span> by </p> <dl><dd><span class="mwe-math-element"><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)=\operatorname {Tr} _{R}\,{\mathcal {P}}\exp \left(i\oint _{K}A\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>Tr</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mi>A</mi> </mrow> <mo>)</mo> </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="https://wikimedia.org/api/rest_v1/media/math/render/svg/4356c4579bafdfacef8a1c2256946791bc6b8f1e" 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)}"></span></dd></dl> <p>where <i>A</i> is the connection 1-form and we take the <a href="/wiki/Cauchy_principal_value" title="Cauchy principal value">Cauchy principal value</a> of the <a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">contour integral</a> and <span class="mwe-math-element"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mi>exp</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\exp }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e615a0ebe4610661520b3eb9186d5c8d9c1364cd" 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 }"></span> is the <a href="/wiki/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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=8" title="Edit section: HOMFLY and Jones polynomials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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="/wiki/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="/wiki/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="/wiki/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> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sin(\pi /(k+N))}{\sin(\pi N/(k+N))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </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="https://wikimedia.org/api/rest_v1/media/math/render/svg/311d5e511a76c60de12e4877620fbcede4b431dc" 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))}}}"></span></dd></dl> <p>times the <a href="/wiki/HOMFLY_polynomial" title="HOMFLY polynomial">HOMFLY polynomial</a>. In particular when <i>N</i> = 2 the HOMFLY polynomial reduces to the <a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a>. In the SO(<i>N</i>) case, one finds a similar expression with the <a href="/wiki/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="/wiki/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="/wiki/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 <a href="/w/index.php?title=Point-splitting&action=edit&redlink=1" class="new" title="Point-splitting (page does not exist)">point-splitting</a> <a href="/wiki/Regularization_(physics)" title="Regularization (physics)">regularization</a> procedure introduced by <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a> and <a href="/wiki/Rudolf_Peierls" title="Rudolf Peierls">Rudolf Peierls</a> to define apparently divergent quantities in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> in 1934. </p><p><a href="/wiki/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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=9" title="Edit section: Relationships with other theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Topological_string_theories">Topological string theories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=10" title="Edit section: Topological string theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.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}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Topological_string_theory" title="Topological string theory">Topological string theory</a></div> <p>In the context of <a href="/wiki/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="/wiki/String_field_theory" title="String field theory">string field theory</a> of open strings ending on a <a href="/wiki/D-brane" title="D-brane">D-brane</a> wrapping <i>X</i> in the <a href="/wiki/Topological_string_theory#A-model" title="Topological string theory">A-model</a> topological string theory on <i>X</i>. The <a href="/wiki/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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=11" title="Edit section: WZW and matrix models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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="/wiki/Two-dimensional_conformal_field_theory" title="Two-dimensional conformal field theory">two-dimensional conformal field theory</a> known as a G <a href="/wiki/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="/wiki/Matrix_theory_(physics)" title="Matrix theory (physics)">matrix models</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Chern–Simons_gravity_theory"><span id="Chern.E2.80.93Simons_gravity_theory"></span>Chern–Simons gravity theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=12" title="Edit section: Chern–Simons gravity theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/(2%2B1)-dimensional_topological_gravity" title="(2+1)-dimensional topological gravity">(2+1)-dimensional topological gravity</a></div> <p>In 1982, <a href="/wiki/Stanley_Deser" title="Stanley Deser">S. Deser</a>, <a href="/wiki/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="/wiki/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> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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 )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>CS</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>x</mi> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msubsup> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>.</mo> </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="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0075fd8f2a25a546317665e0498d0d921231a35" 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 )}.}"></span></dd></dl> <p>This variation gives the <a href="/wiki/Cotton_tensor" title="Cotton tensor">Cotton tensor</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =-{\frac {1}{2{\sqrt {g}}}}{\bigl (}\varepsilon ^{mij}D_{i}R_{j}^{n}+\varepsilon ^{nij}D_{i}R_{j}^{m}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>g</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>i</mi> <mi>j</mi> </mrow> </msup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>+</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>i</mi> <mi>j</mi> </mrow> </msup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>.</mo> </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="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bd75b443f9e7da2a55ae6a72a82fcd78fb30b42" 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}).}"></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"><span id="Chern.E2.80.93Simons_matter_theories"></span>Chern–Simons matter theories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=13" title="Edit section: Chern–Simons matter theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 2013 Kenneth A. Intriligator and <a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Nathan Seiberg</a> solved these 3d Chern–Simons gauge theories and their phases using <a href="/wiki/Seiberg-Witten_monopole" class="mw-redirect" title="Seiberg-Witten monopole">monopoles</a> carrying extra degrees of freedom. The <a href="/wiki/Witten_index" title="Witten index">Witten index</a> of the many <a href="/wiki/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="/wiki/Supersymmetry" title="Supersymmetry">supersymmetry</a> was computed to be broken. These monopoles were related to <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter</a> <a href="/wiki/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="/wiki/AdS/CFT_correspondence" title="AdS/CFT correspondence">holographic dual</a> of M-theory on <span class="mwe-math-element"><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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>d</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AdS_{4}\times S_{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/101afc2877d7978a4538886779d4debbe5efa115" 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}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Four-dimensional_Chern–Simons_theory"><span id="Four-dimensional_Chern.E2.80.93Simons_theory"></span>Four-dimensional Chern–Simons theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=14" title="Edit section: Four-dimensional Chern–Simons theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/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="/wiki/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="/wiki/Integrable_system" title="Integrable system">integrable systems</a> theory, including exactly solvable lattice models (like the <a href="/wiki/Six-vertex_model" class="mw-redirect" title="Six-vertex model">six-vertex model</a> or the <a href="/wiki/Quantum_Heisenberg_model" title="Quantum Heisenberg model">XXZ spin chain</a>), integrable quantum field theories (such as the <a href="/wiki/Gross%E2%80%93Neveu_model" title="Gross–Neveu model">Gross–Neveu model</a>, <a href="/wiki/Chiral_model" title="Chiral model">principal chiral model</a> and symmetric space coset <a href="/wiki/Sigma_model" title="Sigma model">sigma models</a>), the <a href="/wiki/Yang%E2%80%93Baxter_equation" title="Yang–Baxter equation">Yang–Baxter equation</a> and <a href="/wiki/Quantum_groups" class="mw-redirect" title="Quantum groups">quantum groups</a> such as the <a href="/wiki/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"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mi mathvariant="normal">Σ</mi> <mo>×</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\Sigma \times C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5656bd4f68eec969acf1747e84dcbd58a1e5c184" 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}"></span> where <span class="mwe-math-element"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" 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 }"></span> is a two-dimensional manifold and <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" 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}"></span> is a complex curve is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\int _{M}\omega \wedge CS(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mi>ω</mi> <mo>∧</mo> <mi>C</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\int _{M}\omega \wedge CS(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaad5c75890a03104e3730e1ee7ccf8bd46fced3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.083ex; height:5.676ex;" alt="{\displaystyle S=\int _{M}\omega \wedge CS(A)}"></span> where <span class="mwe-math-element"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" 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 }"></span> is a <a href="/wiki/Meromorphic" class="mw-redirect" title="Meromorphic">meromorphic</a> <a href="/wiki/One-form" class="mw-redirect" title="One-form">one-form</a> on <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" 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}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Chern–Simons_terms_in_other_theories"><span id="Chern.E2.80.93Simons_terms_in_other_theories"></span>Chern–Simons terms in other theories</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=15" title="Edit section: Chern–Simons terms in other theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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="/wiki/Photon" title="Photon">photon</a> if this term is added to the action of Maxwell's theory of <a href="/wiki/Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a>. This term can be induced by integrating over a massive charged <a href="/wiki/Fermionic_field#Dirac_fields" title="Fermionic field">Dirac field</a>. It also appears for example in the <a href="/wiki/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="/wiki/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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=16" title="Edit section: One-loop renormalization of the level"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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="/wiki/Majorana_fermion" title="Majorana fermion">Majorana fermions</a> then, due to the <a href="/wiki/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="/wiki/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> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=17" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory (mathematics)</a></li> <li><a href="/wiki/Chern%E2%80%93Simons_form" title="Chern–Simons form">Chern–Simons form</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li> <li><a href="/wiki/Alexander_polynomial" title="Alexander polynomial">Alexander polynomial</a></li> <li><a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li> <li><a href="/wiki/2%2B1D_topological_gravity" class="mw-redirect" title="2+1D topological gravity">2+1D topological gravity</a></li> <li><a href="/wiki/Skyrmion" title="Skyrmion">Skyrmion</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Chern%E2%80%93Simons_theory&action=edit&section=18" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.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("//upload.wikimedia.org/wikipedia/commons/6/65/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("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/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}}</style><cite id="CITEREFArthurTchrakianY.-S.1996" class="citation journal cs1"><a href="/w/index.php?title=K._Arthur&action=edit&redlink=1" class="new" title="K. 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"Self-dual Chern–Simons vortices on Riemann surfaces". <i>Journal of Mathematical Physics</i>. <b>43</b> (5): 2355–2362. <a href="/wiki/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="/wiki/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="/wiki/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="/wiki/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 title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Physics&rft.atitle=Self-dual+Chern%E2%80%93Simons+vortices+on+Riemann+surfaces&rft.volume=43&rft.issue=5&rft.pages=2355-2362&rft.date=2002&rft_id=info%3Aarxiv%2Fmath-ph%2F0012045&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9916364%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1063%2F1.1471365&rft_id=info%3Abibcode%2F2002JMP....43.2355K&rft.aulast=Kim&rft.aufirst=Seongtag&rft.au=Kim%2C+Yoonbai&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChern%E2%80%93Simons+theory" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><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="/wiki/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="/wiki/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="/wiki/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="/wiki/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 title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+D&rft.atitle=Effect+of+Chern-Simons+dynamics+on+the+energy+of+electrically+charged+and+spinning+vortices&rft.volume=95&rft.issue=8&rft.pages=085016&rft.date=2017&rft_id=info%3Aarxiv%2F1612.05835&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A62882649%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.95.085016&rft_id=info%3Abibcode%2F2017PhRvD..95h5016N&rft.aulast=Navarro-L%C3%A9rida&rft.aufirst=Francisco&rft.au=Radu%2C+Eugen&rft.au=Tchrakian%2C+D.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AChern%E2%80%93Simons+theory" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a 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class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_field_theories" title="Special:EditPage/Template:Quantum field theories"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_field_theories" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/Algebraic_quantum_field_theory" title="Algebraic quantum field theory">Algebraic QFT</a></li> <li><a href="/wiki/Axiomatic_quantum_field_theory" title="Axiomatic quantum field theory">Axiomatic QFT</a></li> <li><a href="/wiki/Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></li> <li><a href="/wiki/Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li> <li><a href="/wiki/Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative QFT</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></li> <li><a href="/wiki/Quantum_field_theory_in_curved_spacetime" title="Quantum field theory in curved spacetime">QFT in curved spacetime</a></li> <li><a href="/wiki/String_theory" title="String theory">String theory</a></li> <li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li> <li><a href="/wiki/Thermal_quantum_field_theory" title="Thermal quantum field theory">Thermal QFT</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological QFT</a></li> <li><a href="/wiki/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="/wiki/Born%E2%80%93Infeld_model" title="Born–Infeld model">Born–Infeld</a></li> <li><a href="/wiki/Euler%E2%80%93Heisenberg_Lagrangian" title="Euler–Heisenberg Lagrangian">Euler–Heisenberg</a></li> <li><a href="/wiki/Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau</a></li> <li><a href="/wiki/Non-linear_sigma_model" title="Non-linear sigma model">Non-linear sigma</a></li> <li><a href="/wiki/Proca_action" title="Proca action">Proca</a></li> <li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li><a href="/wiki/Quartic_interaction" title="Quartic interaction">Quartic interaction</a></li> <li><a href="/wiki/Scalar_electrodynamics" title="Scalar electrodynamics">Scalar electrodynamics</a></li> <li><a href="/wiki/Scalar_chromodynamics" title="Scalar chromodynamics">Scalar chromodynamics</a></li> <li><a href="/wiki/Soler_model" title="Soler model">Soler</a></li> <li><a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills</a></li> <li><a href="/wiki/Yang%E2%80%93Mills%E2%80%93Higgs_equations" title="Yang–Mills–Higgs equations">Yang–Mills–Higgs</a></li> <li><a href="/wiki/Yukawa_interaction" 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="/wiki/Two-dimensional_Yang%E2%80%93Mills_theory" title="Two-dimensional Yang–Mills theory">2D Yang–Mills</a></li> <li><a href="/wiki/Bullough%E2%80%93Dodd_model" title="Bullough–Dodd model">Bullough–Dodd</a></li> <li><a href="/wiki/Gross%E2%80%93Neveu_model" title="Gross–Neveu model">Gross–Neveu</a></li> <li><a href="/wiki/Schwinger_model" title="Schwinger model">Schwinger</a></li> <li><a href="/wiki/Sine-Gordon_equation" title="Sine-Gordon equation">Sine-Gordon</a></li> <li><a href="/wiki/Thirring_model" title="Thirring model">Thirring</a></li> <li><a href="/wiki/Thirring%E2%80%93Wess_model" title="Thirring–Wess model">Thirring–Wess</a></li> <li><a href="/wiki/Toda_field_theory" title="Toda field theory">Toda</a></li></ul> </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="/wiki/Massless_free_scalar_bosons_in_two_dimensions" title="Massless free scalar bosons in two dimensions">2D free massless scalar</a></li> <li><a href="/wiki/Liouville_field_theory" title="Liouville field theory">Liouville</a></li> <li><a href="/wiki/Minimal_model_(physics)" title="Minimal model (physics)">Minimal</a></li> <li><a href="/wiki/Polyakov_action" title="Polyakov action">Polyakov</a></li> <li><a href="/wiki/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"> <ul><li><a href="/wiki/4D_N_%3D_1_global_supersymmetry" title="4D N = 1 global supersymmetry">4D N = 1</a></li> <li><a href="/wiki/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="/wiki/Seiberg%E2%80%93Witten_theory" title="Seiberg–Witten theory">Seiberg–Witten</a></li> <li><a href="/wiki/Super_QCD" title="Super QCD">Super QCD</a></li> <li><a href="/wiki/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"> <ul><li><a href="/wiki/6D_(2,0)_superconformal_field_theory" title="6D (2,0) superconformal field theory">6D (2,0)</a></li> <li><a href="/wiki/ABJM_superconformal_field_theory" title="ABJM superconformal field theory">ABJM</a></li> <li><a href="/wiki/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"> <ul><li><a href="/wiki/Pure_4D_N_%3D_1_supergravity" title="Pure 4D N = 1 supergravity">Pure 4D N = 1</a></li> <li><a href="/wiki/4D_N_%3D_1_supergravity" title="4D N = 1 supergravity">4D N = 1</a></li> <li><a href="/wiki/N_%3D_8_supergravity" title="N = 8 supergravity">4D N = 8</a></li> <li><a href="/wiki/Higher-dimensional_supergravity" title="Higher-dimensional supergravity">Higher dimensional</a></li> <li><a href="/wiki/Type_I_supergravity" title="Type I supergravity">Type I</a></li> <li><a href="/wiki/Type_IIA_supergravity" title="Type IIA supergravity">Type IIA</a></li> <li><a href="/wiki/Type_IIB_supergravity" title="Type IIB supergravity">Type IIB</a></li> <li><a href="/wiki/Eleven-dimensional_supergravity" title="Eleven-dimensional supergravity">11D</a></li></ul> </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><li><a href="/wiki/BF_model" title="BF model">BF</a></li> <li><a class="mw-selflink selflink">Chern–Simons</a></li></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="/wiki/Chiral_model" title="Chiral model">Chiral</a></li> <li><a href="/wiki/Fermi%27s_interaction" title="Fermi's interaction">Fermi</a></li> <li><a href="/wiki/Minimal_Supersymmetric_Standard_Model" title="Minimal Supersymmetric Standard Model">MSSM</a></li> <li><a href="/wiki/Nambu%E2%80%93Jona-Lasinio_model" title="Nambu–Jona-Lasinio model">Nambu–Jona-Lasinio</a></li> <li><a href="/wiki/Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li> <li><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a></li> <li><a href="/wiki/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="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Cosmic_string" title="Cosmic string">Cosmic string</a></li> <li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li> <li><a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li> <li><a href="/wiki/Loop_quantum_cosmology" title="Loop quantum cosmology">Loop quantum cosmology</a></li> <li><a href="/wiki/On_shell_and_off_shell" title="On shell and off shell">On shell and off shell</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_foam" title="Quantum foam">Quantum foam</a></li> <li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuations</a> <ul><li><a href="/wiki/Template:Quantum_electrodynamics" title="Template:Quantum electrodynamics">links</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a> <ul><li><a href="/wiki/Template:Quantum_gravity" title="Template:Quantum gravity">links</a></li></ul></li> <li><a href="/wiki/Quantum_hadrodynamics" title="Quantum hadrodynamics">Quantum hadrodynamics</a></li> <li><a href="/wiki/Quantum_hydrodynamics" title="Quantum hydrodynamics">Quantum hydrodynamics</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a> <ul><li><a href="/wiki/Template:Quantum_information" title="Template:Quantum information">links</a></li></ul></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/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"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/16px-Symbol_template_class_pink.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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