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<title>Chern-Simons-Form</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Chern-Simons-Form</h1>

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Die Chern-Simons-Formen sind bei der Definition von sekundären <a href="Charakteristische_Klasse" title="Charakteristische Klasse">charakteristischen Klassen</a> verwendete <a href="Differentialform" title="Differentialform">Differentialformen</a>, die in der <a href="Mathematik" title="Mathematik">Mathematik</a> in <a href="Differentialgeometrie" title="Differentialgeometrie">Differentialgeometrie</a> und <a href="Differentialtopologie" title="Differentialtopologie">Differentialtopologie</a> in verschiedenen Zusammenhängen vorkommen, insbesondere in <a href="Eichtheorie" title="Eichtheorie">Eichtheorien</a>. Die Chern-Simons-3-Form definiert das <a href="Chern-Simons-Funktional" title="Chern-Simons-Funktional">Wirkungsfunktional der Chern-Simons-Theorie</a>. Sie sind benannt nach <a href="Shiing-Shen_Chern" title="Shiing-Shen Chern">Shiing-Shen Chern</a> und <a href="James_Harris_Simons" class="mw-redirect" title="James Harris Simons">James Harris Simons</a>, den Autoren der 1974 veröffentlichten Arbeit Characteristic Forms and Geometric Invariants.

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

Sei M eine <a href="Riemannsche_Mannigfaltigkeit" title="Riemannsche Mannigfaltigkeit">Riemannsche Mannigfaltigkeit</a>. Der <a href="Zusammenhang_(Prinzipalb%C3%BCndel)#Riemannscher_Zusammenhang" title="Zusammenhang (Prinzipalbündel)">Riemannsche Zusammenhang</a>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\in \Omega ^{1}(P(M),{\mathfrak {gl}}(n))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba0438b92810c98af0518b60521fafc13c3f9177.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.183ex; height:3.176ex;" alt="{\displaystyle A\in \Omega ^{1}(P(M),{\mathfrak {gl}}(n))}" loading="lazy"></dd></dl>

ist eine <a href="Lie-Algebra" title="Lie-Algebra">Lie-Algebra</a>-wertige <a href="Differentialform" title="Differentialform">1-Form</a> auf dem <a href="Rahmenb%C3%BCndel" class="mw-redirect" title="Rahmenbündel">Rahmenbündel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(M)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afa81e21b1d085edd94f267faa1546609999affd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.997ex; height:2.843ex;" alt="{\displaystyle P(M)}" loading="lazy">.

Die Chern-Simons-1-Form wird definiert durch

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02a6342070f1d6997a6478bfe9ef03d6b554d2b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.903ex; height:2.843ex;" alt="{\displaystyle \operatorname {Tr} [\mathbf {A} ]}" loading="lazy">,</dd></dl>

wobei Tr die <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> von Matrizen bezeichnet.

Die Chern-Simons-3-Form wird definiert durch

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<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].}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a477415134e34626488680f21494bf87f7dc6720.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.039ex; 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].}" loading="lazy"></dd></dl>

Die Chern-Simons-5-Form wird definiert durch

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/115ca07c893a9ba560e0791d6d78096e8d7eb346.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.352ex; height:6.176ex;" alt="{\displaystyle \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]}" loading="lazy"></dd></dl>

wobei die Krümmung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da18bef8c979f3548bb0d8976f5844012d7b8256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }" loading="lazy"> definiert ist durch

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14a0a39307e25aca9a9f8d7e168b86e04a733363.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"></dd></dl>

Die allgemeine Chern-Simons-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{2k-1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04dabca606e414d644d84bffd67d4ce6edd0d6e7.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"> ist definiert, so dass

<msub>

</mrow>

</msub>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b557699b4336b523aba875717dcd8d785515d9a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.909ex; height:3.343ex;" alt="{\displaystyle d\omega _{2k-1}=\operatorname {Tr} \left(\mathbf {F} ^{k}\right),}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} ^{k}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04da87c00d4b100ee159ffe80257b0ee344ca485.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.771ex; height:2.676ex;" alt="{\displaystyle \mathbf {F} ^{k}}" loading="lazy"> durch das <a href="%C3%84u%C3%9Feres_Produkt" class="mw-disambig" title="Äußeres Produkt">äußere Produkt</a> von Differentialformen definiert wird.

Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> eine <a href="Parallelisierbare_Mannigfaltigkeit" class="mw-redirect" title="Parallelisierbare Mannigfaltigkeit">parallelisierbare</a> 2k-1-dimensionale <a href="Mannigfaltigkeit" title="Mannigfaltigkeit">Mannigfaltigkeit</a> ist (zum Beispiele eine orientierbare <a href="3-Mannigfaltigkeit" title="3-Mannigfaltigkeit">3-Mannigfaltigkeit</a>), dann gibt es einen Schnitt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s:M\rightarrow P(M)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s:M\rightarrow P(M)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd4150322a316bc843dd1c5bc8932ffd4b7471cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.081ex; height:2.843ex;" alt="{\displaystyle s:M\rightarrow P(M)}" loading="lazy"> und das Integral von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{*}\omega _{2k-1}}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/267aadbbf09b2cfae449503f3a06b7628501634d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.602ex; height:2.676ex;" alt="{\displaystyle s^{*}\omega _{2k-1}}" loading="lazy"> über die Mannigfaltigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> ist eine globale Invariante, die modulo der Addition ganzer Zahlen wohldefiniert ist. (Für verschiedene Schnitte unterscheiden sich die Integrale nur um ganze Zahlen.)

Die so definierte Invariante ist die Chern-Simons-Invariante

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {cs} (M)\in \mathbb {R} /\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c5e85cd44c595c834c256440c51e9ba2594b8e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.432ex; height:2.843ex;" alt="{\displaystyle \operatorname {cs} (M)\in \mathbb {R} /\mathbb {Z} }" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Allgemeine_Definition_für_Prinzipalbündel_und_invariante_Polynome">Allgemeine Definition für Prinzipalbündel und invariante Polynome</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> eine <a href="Lie-Gruppe" title="Lie-Gruppe">Lie-Gruppe</a> mit <a href="Lie-Gruppe#Lie-Algebra_der_Lie-Gruppe" title="Lie-Gruppe">Lie-Algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in I^{k}({\mathfrak {g}})}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in I^{k}({\mathfrak {g}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d5b85dfbebd37a2247915ef59a2f4437d8abb14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.405ex; height:3.176ex;" alt="{\displaystyle f\in I^{k}({\mathfrak {g}})}" loading="lazy"> ein <a href="Invariantes_Polynom#Invariante_Polynome_in_der_Theorie_der_Lie-Gruppen" title="Invariantes Polynom">invariantes Polynom</a>.

Jedem invarianten Polynom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> entspricht eine Chern-Simons-Form von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-Prinzipalbündeln wie folgt.

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :P\rightarrow M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi :P\rightarrow M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01278d9db7a95e3636ebc383271ffd95af73f9b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.071ex; height:2.176ex;" alt="{\displaystyle \pi :P\rightarrow M}" loading="lazy"> ein <a href="Prinzipalb%C3%BCndel" class="mw-redirect" title="Prinzipalbündel">Prinzipalbündel</a> mit Strukturgruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">. Man wähle eine <a href="Zusammenhang_(Prinzipalb%C3%BCndel)" title="Zusammenhang (Prinzipalbündel)">Zusammenhangsform</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Omega ^{1}(P,{\mathfrak {g}})}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \in \Omega ^{1}(P,{\mathfrak {g}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6037692d66c7c173b726a658ad0fd3af2ff8f0ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.779ex; height:3.176ex;" alt="{\displaystyle \omega \in \Omega ^{1}(P,{\mathfrak {g}})}" loading="lazy"> und bezeichne mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \in \Omega ^{2}(P,{\mathfrak {g}})}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega \in \Omega ^{2}(P,{\mathfrak {g}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fef3d4b1868bb36a69c84f9f48c83a23966447a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.011ex; height:3.176ex;" alt="{\displaystyle \Omega \in \Omega ^{2}(P,{\mathfrak {g}})}" loading="lazy"> ihre <a href="Zusammenhang_(Prinzipalb%C3%BCndel)#Krümmung" title="Zusammenhang (Prinzipalbündel)">Krümmungsform</a>. Dann ist die Chern-Simons-Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Tf\in \Omega ^{2k-1}(P,\mathbb {R} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Tf\in \Omega ^{2k-1}(P,\mathbb {R} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/128ec7064d4c8244142b780dba2e9d6f86984203.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.712ex; height:3.176ex;" alt="{\displaystyle Tf\in \Omega ^{2k-1}(P,\mathbb {R} )}" loading="lazy"> definiert durch

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Tf=\sum _{i=0}^{k-1}A_{i}f\left(\omega \wedge [\omega ,\omega ]^{i}\wedge \Omega ^{k-i-1}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2b12cea9d44c90d474b55b4c335c1da048a96ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.437ex; height:7.343ex;" alt="{\displaystyle Tf=\sum _{i=0}^{k-1}A_{i}f\left(\omega \wedge [\omega ,\omega ]^{i}\wedge \Omega ^{k-i-1}\right)}" loading="lazy"></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}:=(-1)^{i}{\frac {k!(k-1)!}{2^{i}(k+i)!(k-1-i)!}}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{i}:=(-1)^{i}{\frac {k!(k-1)!}{2^{i}(k+i)!(k-1-i)!}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6620c3894269d8b5be20f1e8fdc57dd0b4808ce3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:33.289ex; height:6.676ex;" alt="{\displaystyle A_{i}:=(-1)^{i}{\frac {k!(k-1)!}{2^{i}(k+i)!(k-1-i)!}}}" loading="lazy">.

Im Fall flacher Bündel vereinfacht sich diese Formel zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{k-1}{\frac {k!(k-1)!}{2^{k-1}(2k-1)!}}f\left(\omega \wedge [\omega ,\omega ]^{k-1}\right)}">

<msup>

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<mfrac>

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<mrow>

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</msup>

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<mrow>

<mrow>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle (-1)^{k-1}{\frac {k!(k-1)!}{2^{k-1}(2k-1)!}}f\left(\omega \wedge [\omega ,\omega ]^{k-1}\right)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/340d5ad67330b06221e1b51d52751831e91a7a93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.222ex; height:6.843ex;" alt="{\displaystyle (-1)^{k-1}{\frac {k!(k-1)!}{2^{k-1}(2k-1)!}}f\left(\omega \wedge [\omega ,\omega ]^{k-1}\right)}" loading="lazy">.

Es gilt die Gleichung

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/453bc3b8cee7cdd179f627bf088fd7b1f080dab8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.851ex; height:2.843ex;" alt="{\displaystyle dTf=f(\Omega ,\ldots ,\Omega )}" loading="lazy">,</dd></dl>

im Fall flacher Bündel also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dTf=0}">

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<annotation encoding="application/x-tex">{\displaystyle dTf=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f340c2e9e6a2008d327dbc71df324934a22d44e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.392ex; height:2.509ex;" alt="{\displaystyle dTf=0}" loading="lazy">.

Bekanntlich entspricht jede charakteristische Klasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in H^{*}(BG,\mathbb {Z} )}">

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<annotation encoding="application/x-tex">{\displaystyle u\in H^{*}(BG,\mathbb {Z} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc0318dd9c6acda0ac7a998c96778098fd55de1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.312ex; height:2.843ex;" alt="{\displaystyle u\in H^{*}(BG,\mathbb {Z} )}" loading="lazy"> einem invarianten Polynom, siehe <a href="Chern-Weil-Theorie" title="Chern-Weil-Theorie">Chern-Weil-Theorie</a>. Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\Omega ,\ldots ,\Omega )=0}">

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<annotation encoding="application/x-tex">{\displaystyle f(\Omega ,\ldots ,\Omega )=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dcf59ae84566f5d5478840a464e0ce1fec230e86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.883ex; height:2.843ex;" alt="{\displaystyle f(\Omega ,\ldots ,\Omega )=0}" loading="lazy">, dann verschwindet nach Chern-Weil-Theorie die entsprechende charakteristische Klasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{f}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0f1dd81abca1620bc42ed439b198dcfe95e350e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.466ex; height:2.343ex;" alt="{\displaystyle u_{f}}" loading="lazy"> in reeller Kohomologie. Die Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Tf}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0174e6271729d41db5196ca3504121d11ea831e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.915ex; height:2.509ex;" alt="{\displaystyle Tf}" loading="lazy"> ist in diesem Fall geschlossen und definiert zunächst eine Klasse in der Kohomologie von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy">. Zurückziehen mittels eines Schnittes definiert eine Kohomologieklasse von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, welche modulo ganzer Zahlen wohldefiniert ist. Die so definierte Kohomologieklasse in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{*}(M,\mathbb {R} /\mathbb {Z} )}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60f7b7b20b68bd5c916a4261ceffa4a2b06df5a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.834ex; height:2.843ex;" alt="{\displaystyle H^{*}(M,\mathbb {R} /\mathbb {Z} )}" loading="lazy"> passt in die <a href="Bockstein-Folge" title="Bockstein-Folge">Bockstein-Folge</a>

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<annotation encoding="application/x-tex">{\displaystyle H^{2k-1}(M,\mathbb {R} /\mathbb {Z} )\rightarrow H^{2k}(M,\mathbb {Z} )\rightarrow H^{2k}(M,\mathbb {R} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fee1e4837a2fc15da9620da8a808f4dec76b8506.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.847ex; height:3.176ex;" alt="{\displaystyle H^{2k-1}(M,\mathbb {R} /\mathbb {Z} )\rightarrow H^{2k}(M,\mathbb {Z} )\rightarrow H^{2k}(M,\mathbb {R} )}" loading="lazy">,</dd></dl>

wo sie auf die charakteristische Klasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{f}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0f1dd81abca1620bc42ed439b198dcfe95e350e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.466ex; height:2.343ex;" alt="{\displaystyle u_{f}}" loading="lazy"> abgebildet wird, deren Bild in reeller Kohomologie verschwindet.

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Vierdimensionale_Chern-Simons-Theorie" title="Vierdimensionale Chern-Simons-Theorie">Vierdimensionale Chern-Simons-Theorie</a></li>

<li><a href="Sechsdimensionale_holomorphe_Chern-Simons-Theorie" title="Sechsdimensionale holomorphe Chern-Simons-Theorie">Sechsdimensionale holomorphe Chern-Simons-Theorie</a></li>

<li><a href="Chirale_Anomalie" class="mw-redirect" title="Chirale Anomalie">Chirale Anomalie</a></li>

<li><a href="Jones-Polynom" title="Jones-Polynom">Jones-Polynom</a></li></ul>

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

<ul><li>Chern, S.-S.; Simons, J.: Characteristic forms and geometric invariants. The <a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>, Second Series 99, 1974, S. 48–69.</li></ul></div><div><div class="zim-footer">

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