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<title>Chern–Simons form</title>

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

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>Chern–Simons forms</b> are certain secondary <a href="Characteristic_class" title="Characteristic class">characteristic classes</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The theory is named for <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>, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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

<p>Given a <a href="Manifold" title="Manifold">manifold</a> and a <a href="Lie_algebra" title="Lie algebra">Lie algebra</a> valued <a href="Multilinear_form" title="Multilinear form">1-form</a> <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 \mathbf {A} }">

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

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</math></span><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"></span> over it, we can define a family of <a href="Multilinear_form" title="Multilinear form"><i>p</i>-forms</a>:<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>In one dimension, the <b>Chern–Simons</b> <a href="Multilinear_form" title="Multilinear form">1-form</a> is given by

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} [\mathbf {A} ].}</annotation>

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</math></span><img src="./092c8e445b62d5b43e5f856352a1b4d4d00d6550.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.55ex; height:2.843ex;" alt="{\displaystyle \operatorname {Tr} [\mathbf {A} ].}" loading="lazy"></span></dd></dl>

<p>In three dimensions, the <b>Chern–Simons 3-form</b> is given by

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} \left[\mathbf {F} \wedge \mathbf {A} -{\frac {1}{3}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right]=\operatorname {Tr} \left[d\mathbf {A} \wedge \mathbf {A} +{\frac {2}{3}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right].}</annotation>

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</math></span><img src="./2cea87fcbf697dafba00a19e0d9eeddbc49940cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:60.082ex; height:6.176ex;" alt="{\displaystyle \operatorname {Tr} \left[\mathbf {F} \wedge \mathbf {A} -{\frac {1}{3}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right]=\operatorname {Tr} \left[d\mathbf {A} \wedge \mathbf {A} +{\frac {2}{3}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right].}" loading="lazy"></span></dd></dl>

<p>In five dimensions, the <b>Chern–Simons 5-form</b> is given by

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\operatorname {Tr} \left[\mathbf {F} \wedge \mathbf {F} \wedge \mathbf {A} -{\frac {1}{2}}\mathbf {F} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} +{\frac {1}{10}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right]\\[6pt]={}&\operatorname {Tr} \left[d\mathbf {A} \wedge d\mathbf {A} \wedge \mathbf {A} +{\frac {3}{2}}d\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} +{\frac {3}{5}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right]\end{aligned}}}</annotation>

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</math></span><img src="./c8710899b798ec98637895e5c5a11df81674e471.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:69.052ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}&\operatorname {Tr} \left[\mathbf {F} \wedge \mathbf {F} \wedge \mathbf {A} -{\frac {1}{2}}\mathbf {F} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} +{\frac {1}{10}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right]\\[6pt]={}&\operatorname {Tr} \left[d\mathbf {A} \wedge d\mathbf {A} \wedge \mathbf {A} +{\frac {3}{2}}d\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} +{\frac {3}{5}}\mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \wedge \mathbf {A} \right]\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>where the curvature <b>F</b> is defined as

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} .}</annotation>

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</math></span><img src="./14a0a39307e25aca9a9f8d7e168b86e04a733363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.125ex; height:2.343ex;" alt="{\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} .}" loading="lazy"></span></dd></dl>

<p>The general Chern–Simons form <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 _{2k-1}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{2k-1}}</annotation>

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</math></span><img src="./04dabca606e414d644d84bffd67d4ce6edd0d6e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.457ex; height:2.009ex;" alt="{\displaystyle \omega _{2k-1}}" loading="lazy"></span> is defined in such a way that

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<annotation encoding="application/x-tex">{\displaystyle d\omega _{2k-1}=\operatorname {Tr} (F^{k}),}</annotation>

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</math></span><img src="./82cba627296c354495969fe5f54de479df4eb7b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.721ex; height:3.176ex;" alt="{\displaystyle d\omega _{2k-1}=\operatorname {Tr} (F^{k}),}" loading="lazy"></span></dd></dl>

<p>where the <a href="Wedge_product" class="mw-redirect" title="Wedge product">wedge product</a> is used to define <i>F<sup>k</sup></i>. The right-hand side of this equation is proportional to the <i>k</i>-th <a href="Chern_character" class="mw-redirect" title="Chern character">Chern character</a> of the connection <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 \mathbf {A} }">

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</math></span><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"></span>.

</p><p>In general, the Chern–Simons <a href="Multilinear_form" title="Multilinear form"><i>p</i>-form</a> is defined for any odd <i>p</i>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Application_to_physics">Application to physics</h2></div>

<p>In 1978, <a href="Albert_Schwarz" title="Albert Schwarz">Albert Schwarz</a> formulated <a href="Chern%E2%80%93Simons_theory" title="Chern–Simons theory">Chern–Simons theory</a>, early <a href="Topological_quantum_field_theory" title="Topological quantum field theory">topological quantum field theory</a>, using Chern-Simons forms.<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>

</p><p>In the <a href="Gauge_theory" title="Gauge theory">gauge theory</a>, the <a href="Differential_form" title="Differential form">integral</a> of Chern-Simons form is a global geometric invariant, and is typically <a href="Gauge_invariant" class="mw-redirect" title="Gauge invariant">gauge invariant</a> modulo addition of an integer.

</p>

<ul><li><a href="Chern%E2%80%93Weil_homomorphism" title="Chern–Weil homomorphism">Chern–Weil homomorphism</a></li>

<li><a href="Chiral_anomaly" title="Chiral anomaly">Chiral anomaly</a></li>

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

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

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

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</style><cite id="CITEREFFreed2009" class="citation web cs1">Freed, Daniel (January 15, 2009). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01243-9/S0273-0979-09-01243-9.pdf">"Remarks on Chern–Simons theory"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">April 1,</span> 2020</span>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFChernTianLi1996" class="citation book cs1">Chern, Shiing-Shen; Tian, G.; Li, Peter (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uOfSa0sfJr0C&q=Characteristic+Forms+and+Geometric+Invariants&pg=PA363"><i>A Mathematician and His Mathematical Work: Selected Papers of S.S. Chern</i></a>. World Scientific. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-02-2385-4</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/Chern-Simons+form">"Chern-Simons form in nLab"</a>. <i>ncatlab.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">May 1,</span> 2020</span>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFMoore2019" class="citation web cs1">Moore, Greg (June 7, 2019). <a rel="nofollow" class="external text" href="http://www.physics.rutgers.edu/~gmoore/TASI-ChernSimons-StudentNotes.pdf">"Introduction To Chern-Simons Theories"</a> <span class="cs1-format">(PDF)</span>. <i>University of Texas</i><span class="reference-accessdate">. Retrieved <span class="nowrap">June 7,</span> 2019</span>.</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="CITEREFSchwartz1978" class="citation journal cs1">Schwartz, A. S. (1978). "The partition function of degenerate quadratic functional and Ray-Singer invariants". <i>Letters in Mathematical Physics</i>. <b>2</b> (3): <span class="nowrap">247–</span>252. <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/1978LMaPh...2..247S">1978LMaPh...2..247S</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%2FBF00406412">10.1007/BF00406412</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:123231019">123231019</a>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFChernSimons1974" class="citation journal cs1"><a href="Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern, S.-S.</a>; <a href="James_Harris_Simons" class="mw-redirect" title="James Harris Simons">Simons, J.</a> (1974). "Characteristic forms and geometric invariants". <i><a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>. Second Series. <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="CITEREFBertlmann2001" class="citation book cs1">Bertlmann, Reinhold A. (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=FC_DRRUHFXEC&pg=PA321">"Chern–Simons form, homotopy operator and anomaly"</a>. <i>Anomalies in Quantum Field Theory</i> (Revised ed.). <a href="Clarendon_Press" class="mw-redirect" title="Clarendon Press">Clarendon Press</a>. pp. <span class="nowrap">321–</span>341. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-850762-3</bdi>.</cite></li></ul>

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</style><div id="String_theory190" style="font-size:114%;margin:0 4em"><a href="String_theory" title="String theory">String theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Background</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="String_(physics)" title="String (physics)">Strings</a></li>

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

<li><a href="History_of_string_theory" title="History of string theory">History of string theory</a>

<ul><li><a href="First_superstring_revolution" class="mw-redirect" title="First superstring revolution">First superstring revolution</a></li>

<li><a href="Second_superstring_revolution" class="mw-redirect" title="Second superstring revolution">Second superstring revolution</a></li></ul></li>

<li><a href="String_theory_landscape" title="String theory landscape">String theory landscape</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Theory</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="Nambu%E2%80%93Goto_action" title="Nambu–Goto action">Nambu–Goto action</a></li>

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

<li><a href="Bosonic_string_theory" title="Bosonic string theory">Bosonic string theory</a></li>

<li><a href="Superstring_theory" title="Superstring theory">Superstring theory</a>

<ul><li><a href="Type_I_string_theory" title="Type I string theory">Type I string</a></li>

<li><a href="Type_II_string_theory" title="Type II string theory">Type II string</a>

<ul><li><a href="Type_II_string_theory" title="Type II string theory">Type IIA string</a></li>

<li><a href="Type_II_string_theory" title="Type II string theory">Type IIB string</a></li></ul></li>

<li><a href="Heterotic_string_theory" title="Heterotic string theory">Heterotic string</a></li></ul></li>

<li><a href="N%3D2_superstring" class="mw-redirect" title="N=2 superstring">N=2 superstring</a></li>

<li><a href="F-theory" title="F-theory">F-theory</a></li>

<li><a href="String_field_theory" title="String field theory">String field theory</a></li>

<li><a href="Matrix_string_theory" title="Matrix string theory">Matrix string theory</a></li>

<li><a href="Non-critical_string_theory" title="Non-critical string theory">Non-critical string theory</a></li>

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

<li><a href="Tachyon_condensation" title="Tachyon condensation">Tachyon condensation</a></li>

<li><a href="RNS_formalism" title="RNS formalism">RNS formalism</a></li>

<li><a href="GS_formalism" title="GS formalism">GS formalism</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="String_duality" title="String duality">String duality</a></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="T-duality" title="T-duality">T-duality</a></li>

<li><a href="S-duality" title="S-duality">S-duality</a></li>

<li><a href="U-duality" title="U-duality">U-duality</a></li>

<li><a href="Montonen%E2%80%93Olive_duality" title="Montonen–Olive duality">Montonen–Olive duality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Particles and fields</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="Graviton" title="Graviton">Graviton</a></li>

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

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

<li><a href="Ramond%E2%80%93Ramond_field" title="Ramond–Ramond field">Ramond–Ramond field</a></li>

<li><a href="Kalb%E2%80%93Ramond_field" title="Kalb–Ramond field">Kalb–Ramond field</a></li>

<li><a href="Magnetic_monopole" title="Magnetic monopole">Magnetic monopole</a></li>

<li><a href="Dual_graviton" title="Dual graviton">Dual graviton</a></li>

<li><a href="Dual_photon" title="Dual photon">Dual photon</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="Brane" title="Brane">Branes</a></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="D-brane" title="D-brane">D-brane</a></li>

<li><a href="NS5-brane" title="NS5-brane">NS5-brane</a></li>

<li><a href="M2-brane" title="M2-brane">M2-brane</a></li>

<li><a href="M5-brane" title="M5-brane">M5-brane</a></li>

<li><a href="S-brane" title="S-brane">S-brane</a></li>

<li><a href="Black_brane" title="Black brane">Black brane</a></li>

<li><a href="Black_hole" title="Black hole">Black holes</a></li>

<li><a href="Black_string" class="mw-redirect" title="Black string">Black string</a></li>

<li><a href="Brane_cosmology" title="Brane cosmology">Brane cosmology</a></li>

<li><a href="Quiver_diagram" title="Quiver diagram">Quiver diagram</a></li>

<li><a href="Hanany%E2%80%93Witten_transition" title="Hanany–Witten transition">Hanany–Witten transition</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></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="Virasoro_algebra" title="Virasoro algebra">Virasoro algebra</a></li>

<li><a href="Mirror_symmetry_(string_theory)" title="Mirror symmetry (string theory)">Mirror symmetry</a></li>

<li><a href="Conformal_anomaly" title="Conformal anomaly">Conformal anomaly</a></li>

<li><a href="Conformal_symmetry" title="Conformal symmetry">Conformal algebra</a></li>

<li><a href="Superconformal_algebra" title="Superconformal algebra">Superconformal algebra</a></li>

<li><a href="Vertex_operator_algebra" title="Vertex operator algebra">Vertex operator algebra</a></li>

<li><a href="Loop_algebra" title="Loop algebra">Loop algebra</a></li>

<li><a href="Kac%E2%80%93Moody_algebra" title="Kac–Moody algebra">Kac–Moody algebra</a></li>

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

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="Gauge_theory" title="Gauge theory">Gauge theory</a></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="Anomaly_(physics)" title="Anomaly (physics)">Anomalies</a></li>

<li><a href="Instanton" title="Instanton">Instantons</a></li>

<li><a href="Bogomol'nyi%E2%80%93Prasad%E2%80%93Sommerfield_bound" title="Bogomol'nyi–Prasad–Sommerfield bound">Bogomol'nyi–Prasad–Sommerfield bound</a></li>

<li><a href="Exceptional_Lie_group" class="mw-redirect" title="Exceptional Lie group">Exceptional Lie groups</a> (<a href="G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a>, <a href="F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a>, <a href="E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a>, <a href="E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a>, <a href="E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a>)</li>

<li><a href="ADE_classification" title="ADE classification">ADE classification</a></li>

<li><a href="Dirac_string" title="Dirac string">Dirac string</a></li>

<li><a href="P-form_electrodynamics" title="P-form electrodynamics"><i>p</i>-form electrodynamics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Geometry</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="Worldsheet" title="Worldsheet">Worldsheet</a></li>

<li><a href="Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li>

<li><a href="Compactification_(physics)" title="Compactification (physics)">Compactification</a></li>

<li><a href="Why_10_dimensions" class="mw-redirect" title="Why 10 dimensions">Why 10 dimensions</a>?</li>

<li><a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifold</a></li>

<li><a href="Ricci-flat_manifold" title="Ricci-flat manifold">Ricci-flat manifold</a>

<ul><li><a href="Calabi%E2%80%93Yau_manifold" title="Calabi–Yau manifold">Calabi–Yau manifold</a></li>

<li><a href="Hyperk%C3%A4hler_manifold" title="Hyperkähler manifold">Hyperkähler manifold</a>

<ul><li><a href="K3_surface" title="K3 surface">K3 surface</a></li></ul></li>

<li><a href="G2_manifold" title="G2 manifold">G<sub>2</sub> manifold</a></li>

<li><a href="Spin(7)-manifold" title="Spin(7)-manifold">Spin(7)-manifold</a></li></ul></li>

<li><a href="Generalized_complex_structure" title="Generalized complex structure">Generalized complex manifold</a></li>

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

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

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

<li><a href="Moduli_space" title="Moduli space">Moduli space</a></li>

<li><a href="Ho%C5%99ava%E2%80%93Witten_theory" title="Hořava–Witten theory">Hořava–Witten theory</a></li>

<li><a href="K-theory_(physics)" title="K-theory (physics)">K-theory (physics)</a></li>

<li><a href="Twisted_K-theory" title="Twisted K-theory">Twisted K-theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="Supersymmetry" title="Supersymmetry">Supersymmetry</a></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="Supergravity" title="Supergravity">Supergravity</a></li>

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

<li><a href="Type_I_supergravity" title="Type I supergravity">Type I supergravity</a></li>

<li><a href="Type_IIA_supergravity" title="Type IIA supergravity">Type IIA supergravity</a></li>

<li><a href="Type_IIB_supergravity" title="Type IIB supergravity">Type IIB supergravity</a></li>

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

<li><a href="Lie_superalgebra" title="Lie superalgebra">Lie superalgebra</a></li>

<li><a href="Lie_supergroup" class="mw-redirect" title="Lie supergroup">Lie supergroup</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="Holography" title="Holography">Holography</a></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="Holographic_principle" title="Holographic principle">Holographic principle</a></li>

<li><a href="AdS/CFT_correspondence" title="AdS/CFT correspondence">AdS/CFT correspondence</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="M-theory" title="M-theory">M-theory</a></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="Matrix_theory_(physics)" title="Matrix theory (physics)">Matrix theory</a></li>

<li><a href="Introduction_to_M-theory" title="Introduction to M-theory">Introduction to M-theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">String theorists</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="Mina_Aganagi%C4%87" title="Mina Aganagić">Aganagić</a></li>

<li><a href="Nima_Arkani-Hamed" title="Nima Arkani-Hamed">Arkani-Hamed</a></li>

<li><a href="Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li>

<li><a href="Tom_Banks_(physicist)" title="Tom Banks (physicist)">Banks</a></li>

<li><a href="David_Berenstein" title="David Berenstein">Berenstein</a></li>

<li><a href="Raphael_Bousso" title="Raphael Bousso">Bousso</a></li>

<li><a href="Thomas_Curtright" title="Thomas Curtright">Curtright</a></li>

<li><a href="Robbert_Dijkgraaf" title="Robbert Dijkgraaf">Dijkgraaf</a></li>

<li><a href="Jacques_Distler" title="Jacques Distler">Distler</a></li>

<li><a href="Michael_R._Douglas" title="Michael R. Douglas">Douglas</a></li>

<li><a href="Gia_Dvali" class="mw-redirect" title="Gia Dvali">Dvali</a></li>

<li><a href="Sergio_Ferrara" title="Sergio Ferrara">Ferrara</a></li>

<li><a href="Willy_Fischler" title="Willy Fischler">Fischler</a></li>

<li><a href="Daniel_Friedan" title="Daniel Friedan">Friedan</a></li>

<li><a href="Sylvester_James_Gates" title="Sylvester James Gates">Gates</a></li>

<li><a href="Ferdinando_Gliozzi" title="Ferdinando Gliozzi">Gliozzi</a></li>

<li><a href="Rajesh_Gopakumar" title="Rajesh Gopakumar">Gopakumar</a></li>

<li><a href="Michael_Green_(physicist)" title="Michael Green (physicist)">Green</a></li>

<li><a href="Brian_Greene" title="Brian Greene">Greene</a></li>

<li><a href="David_Gross" title="David Gross">Gross</a></li>

<li><a href="Steven_Gubser" title="Steven Gubser">Gubser</a></li>

<li><a href="Sergei_Gukov" title="Sergei Gukov">Gukov</a></li>

<li><a href="Andrew_J._Hanson" title="Andrew J. Hanson">Hanson</a></li>

<li><a href="Jeffrey_A._Harvey" title="Jeffrey A. Harvey">Harvey</a></li>

<li><a href="Gerard_'t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li>

<li><a href="Petr_Ho%C5%99ava_(theorist)" class="mw-redirect" title="Petr Hořava (theorist)">Hořava</a></li>

<li><a href="Gary_Gibbons" title="Gary Gibbons">Gibbons</a></li>

<li><a href="Shamit_Kachru" title="Shamit Kachru">Kachru</a></li>

<li><a href="Renata_Kallosh" title="Renata Kallosh">Kallosh</a></li>

<li><a href="Theodor_Kaluza" title="Theodor Kaluza">Kaluza</a></li>

<li><a href="Anton_Kapustin" title="Anton Kapustin">Kapustin</a></li>

<li><a href="Igor_Klebanov" title="Igor Klebanov">Klebanov</a></li>

<li><a href="Vadim_Knizhnik" title="Vadim Knizhnik">Knizhnik</a></li>

<li><a href="Maxim_Kontsevich" title="Maxim Kontsevich">Kontsevich</a></li>

<li><a href="Oskar_Klein" title="Oskar Klein">Klein</a></li>

<li><a href="Andrei_Linde" title="Andrei Linde">Linde</a></li>

<li><a href="Juan_Mart%C3%ADn_Maldacena" class="mw-redirect" title="Juan Martín Maldacena">Maldacena</a></li>

<li><a href="Stanley_Mandelstam" title="Stanley Mandelstam">Mandelstam</a></li>

<li><a href="Donald_Marolf" title="Donald Marolf">Marolf</a></li>

<li><a href="Emil_Martinec" title="Emil Martinec">Martinec</a></li>

<li><a href="Shiraz_Minwalla" title="Shiraz Minwalla">Minwalla</a></li>

<li><a href="Greg_Moore_(physicist)" title="Greg Moore (physicist)">Moore</a></li>

<li><a href="Sunil_Mukhi" title="Sunil Mukhi">Mukhi</a></li>

<li><a href="Robert_Myers_(physicist)" title="Robert Myers (physicist)">Myers</a></li>

<li><a href="Dimitri_Nanopoulos" title="Dimitri Nanopoulos">Nanopoulos</a></li>

<li><a href="Hora%C8%9Biu_N%C4%83stase" title="Horațiu Năstase">Năstase</a></li>

<li><a href="Nikita_Nekrasov" title="Nikita Nekrasov">Nekrasov</a></li>

<li><a href="Andr%C3%A9_Neveu" title="André Neveu">Neveu</a></li>

<li><a href="Holger_Bech_Nielsen" title="Holger Bech Nielsen">Nielsen</a></li>

<li><a href="Peter_van_Nieuwenhuizen" title="Peter van Nieuwenhuizen">van Nieuwenhuizen</a></li>

<li><a href="Sergei_Novikov_(mathematician)" title="Sergei Novikov (mathematician)">Novikov</a></li>

<li><a href="David_Olive" title="David Olive">Olive</a></li>

<li><a href="Hirosi_Ooguri" title="Hirosi Ooguri">Ooguri</a></li>

<li><a href="Burt_Ovrut" title="Burt Ovrut">Ovrut</a></li>

<li><a href="Joseph_Polchinski" title="Joseph Polchinski">Polchinski</a></li>

<li><a href="Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Polyakov</a></li>

<li><a href="Arvind_Rajaraman" title="Arvind Rajaraman">Rajaraman</a></li>

<li><a href="Pierre_Ramond" title="Pierre Ramond">Ramond</a></li>

<li><a href="Lisa_Randall" title="Lisa Randall">Randall</a></li>

<li><a href="Seifallah_Randjbar-Daemi" title="Seifallah Randjbar-Daemi">Randjbar-Daemi</a></li>

<li><a href="Martin_Ro%C4%8Dek" title="Martin Roček">Roček</a></li>

<li><a href="Augusto_Sagnotti" title="Augusto Sagnotti">Sagnotti</a></li>

<li><a href="Jo%C3%ABl_Scherk" title="Joël Scherk">Scherk</a></li>

<li><a href="John_Henry_Schwarz" title="John Henry Schwarz">Schwarz</a></li>

<li><a href="Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li>

<li><a href="Stephen_Shenker" title="Stephen Shenker">Shenker</a></li>

<li><a href="Warren_Siegel" title="Warren Siegel">Siegel</a></li>

<li><a href="Eva_Silverstein" title="Eva Silverstein">Silverstein</a></li>

<li><a href="Matthias_Staudacher" title="Matthias Staudacher">Staudacher</a></li>

<li><a href="Paul_Steinhardt" title="Paul Steinhardt">Steinhardt</a></li>

<li><a href="Andrew_Strominger" title="Andrew Strominger">Strominger</a></li>

<li><a href="Raman_Sundrum" title="Raman Sundrum">Sundrum</a></li>

<li><a href="Leonard_Susskind" title="Leonard Susskind">Susskind</a></li>

<li><a href="Paul_Townsend" title="Paul Townsend">Townsend</a></li>

<li><a href="Sandip_Trivedi" title="Sandip Trivedi">Trivedi</a></li>

<li><a href="Neil_Turok" title="Neil Turok">Turok</a></li>

<li><a href="Gabriele_Veneziano" title="Gabriele Veneziano">Veneziano</a></li>

<li><a href="Erik_Verlinde" title="Erik Verlinde">Verlinde</a></li>

<li><a href="Herman_Verlinde" title="Herman Verlinde">Verlinde</a></li>

<li><a href="Edward_Witten" title="Edward Witten">Witten</a></li>

<li><a href="Tamiaki_Yoneya" title="Tamiaki Yoneya">Yoneya</a></li>

<li><a href="Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Zamolodchikov</a></li>

<li><a href="Alexei_Zamolodchikov" title="Alexei Zamolodchikov">Zamolodchikov</a></li>

<li><a href="Eric_Zaslow" title="Eric Zaslow">Zaslow</a></li>

<li><a href="Bruno_Zumino" title="Bruno Zumino">Zumino</a></li>

<li><a href="Barton_Zwiebach" title="Barton Zwiebach">Zwiebach</a></li></ul>

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This article is issued from <a class="external text" title="Last edited on 2023-12-30" href="https://en.wikipedia.org/wiki/?title=Chern%E2%80%93Simons_form&oldid=1192688966">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.

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