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Chern–Simons form
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<h2 id="definition">Definition</h2> <p>Given a <a href="/facts/Manifold/rXPERJtu">manifold</a> and a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> valued <a href="/facts/Multilinear_form/Rk420AcK">1-form</a> A {\displaystyle \mathbf {A} } over it, we can define a family of <a href="/facts/Multilinear_form/Rk420AcK"><i>p</i>-forms</a>:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>In one dimension, the Chern–Simons <a href="/facts/Multilinear_form/Rk420AcK">1-form</a> is given by </p> Tr [ A ] . {\displaystyle \operatorname {Tr} [\mathbf {A} ].} <p>In three dimensions, the Chern–Simons 3-form is given by </p> Tr [ F ∧ A − 1 3 A ∧ A ∧ A ] = Tr [ d A ∧ A + 2 3 A ∧ A ∧ A ] . {\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].} <p>In five dimensions, the Chern–Simons 5-form is given by </p> Tr [ F ∧ F ∧ A − 1 2 F ∧ A ∧ A ∧ A + 1 10 A ∧ A ∧ A ∧ A ∧ A ] = Tr [ d A ∧ d A ∧ A + 3 2 d A ∧ A ∧ A ∧ A + 3 5 A ∧ A ∧ A ∧ A ∧ A ] {\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}}} <p>where the curvature F is defined as </p> F = d A + A ∧ A . {\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} .} <p>The general Chern–Simons form ω 2 k − 1 {\displaystyle \omega _{2k-1}} is defined in such a way that </p> d ω 2 k − 1 = Tr ( F k ) , {\displaystyle d\omega _{2k-1}=\operatorname {Tr} (F^{k}),} <p>where the <a href="/facts/Wedge_product/rdN4TvpR">wedge product</a> is used to define <i>Fk</i>. The right-hand side of this equation is proportional to the <i>k</i>-th <a href="/facts/Chern_character/Tw2WD5hM">Chern character</a> of the connection A {\displaystyle \mathbf {A} } . </p><p>In general, the Chern–Simons <a href="/facts/Multilinear_form/Rk420AcK"><i>p</i>-form</a> is defined for any odd <i>p</i>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="application-to-physics">Application to physics</h2> <p>In 1978, <a href="/facts/Albert_Schwarz/aBhgfdYC">Albert Schwarz</a> formulated <a href="/facts/Chern%25E2%2580%2593Simons_theory/AQ3Kayn3">Chern–Simons theory</a>, early <a href="/facts/Topological_quantum_field_theory/zHjrp3b8">topological quantum field theory</a>, using Chern-Simons forms.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>In the <a href="/facts/Gauge_theory/fSW3gggv">gauge theory</a>, the <a href="/facts/Differential_form/T9gyHtMv">integral</a> of Chern-Simons form is a global geometric invariant, and is typically <a href="/facts/Gauge_invariant/fSW3gggv">gauge invariant</a> modulo addition of an integer. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Chern%25E2%2580%2593Weil_homomorphism/zcBkde4R">Chern–Weil homomorphism</a></li> <li><a href="/facts/Chiral_anomaly/xs7QFslR">Chiral anomaly</a></li> <li><a href="/facts/Topological_quantum_field_theory/zHjrp3b8">Topological quantum field theory</a></li> <li><a href="/facts/Jones_polynomial/c73vElt9">Jones polynomial</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li><a href="/facts/Shiing-Shen_Chern/FpfIQsEH">Chern, S.-S.</a>; <a href="/facts/James_Harris_Simons/5bsjBRyz">Simons, J.</a> (1974). "Characteristic forms and geometric invariants". <i><a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a></i>. Second Series. 99 (1): 48–69. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1971013">10.2307/1971013</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1971013">1971013</a>.</li> <li>Bertlmann, Reinhold A. (2001). <a 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="/facts/Clarendon_Press/CYyTzzWG">Clarendon Press</a>. pp. 321–341. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-850762-3.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Freed, Daniel (January 15, 2009). "Remarks on Chern–Simons theory" (PDF). Retrieved April 1, 2020. <a href="https://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01243-9/S0273-0979-09-01243-9.pdf" target="_blank">https://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01243-9/S0273-0979-09-01243-9.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Chern, Shiing-Shen; Tian, G.; Li, Peter (1996). A Mathematician and His Mathematical Work: Selected Papers of S.S. Chern. World Scientific. ISBN 978-981-02-2385-4. <a href="978-981-02-2385-4" target="_blank">978-981-02-2385-4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"Chern-Simons form in nLab". ncatlab.org. Retrieved May 1, 2020. <a href="https://ncatlab.org/nlab/show/Chern-Simons+form" target="_blank">https://ncatlab.org/nlab/show/Chern-Simons+form</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Moore, Greg (June 7, 2019). "Introduction To Chern-Simons Theories" (PDF). University of Texas. Retrieved June 7, 2019. <a href="http://www.physics.rutgers.edu/~gmoore/TASI-ChernSimons-StudentNotes.pdf" target="_blank">http://www.physics.rutgers.edu/~gmoore/TASI-ChernSimons-StudentNotes.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Schwartz, A. S. (1978). "The partition function of degenerate quadratic functional and Ray-Singer invariants". Letters in Mathematical Physics. 2 (3): 247–252. Bibcode:1978LMaPh...2..247S. doi:10.1007/BF00406412. S2CID 123231019. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>