"Vector valued differential forms"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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==introduction== | ==introduction== | ||
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==exercise from Chern-Simons theory== | ==exercise from Chern-Simons theory== | ||
− | * [[Chern-Simons gauge theory and invariant|Chern-Simons invariant]] | + | * [[Chern-Simons gauge theory and invariant|Chern-Simons invariant]] |
+ | :<math>S=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A)=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{1}{3}A\wedge [A,A])</math> | ||
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Let <math>A=(a_1dx+A_1 dt)\otimes X_1 +(a_2dx+A_2dt)\otimes X_2</math> be a vector valued form | Let <math>A=(a_1dx+A_1 dt)\otimes X_1 +(a_2dx+A_2dt)\otimes X_2</math> be a vector valued form | ||
+ | :<math>A\wedge A =(a_1A_2-a_2A_1)dx\wedge dt \otimes [X_1, X_2]</math> | ||
+ | :<math>[A,A]=2(a_1A_2-a_2A_1)dx\wedge dt \otimes [X_1, X_2]</math> | ||
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− | + | * http://library.wolfram.com/infocenter/MathSource/683/ | |
− | + | * http://en.wikipedia.org/wiki/Vector-valued_form | |
− | + | * chapter 8 of [http://www.cis.upenn.edu/%7Ecis610/diffgeom-n.pdf http://www.cis.upenn.edu/~cis610/diffgeom-n.pdf] | |
− | + | * Integrable Hamiltonian hierarchies: spectral and geometric methods http://books.google.com/books?id=uoWMwhA4UNkC&pg=PA466&lpg=PA466&dq=commutator+vector+valued+forms&source=bl&ots=BAl_WcubCU&sig=Hn05qnd8ahVchDY3U3uOerMQ4kY&hl=ko&ei=ZYefTdWnN4fbiALArfyCAw&sa=X&oi=book_result&ct=result&resnum=5&ved=0CDsQ6AEwBA#v=onepage&q=commutator%20vector%20valued%20forms&f=false | |
− | http://library.wolfram.com/infocenter/MathSource/683/ | + | * http://math.stackexchange.com/questions/3192/gauge-transformations-in-differential-forms |
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− | chapter 8 of [http://www.cis.upenn.edu/%7Ecis610/diffgeom-n.pdf http://www.cis.upenn.edu/~cis610/diffgeom-n.pdf] | ||
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− | Integrable Hamiltonian hierarchies: spectral and geometric methods http://books.google.com/books?id=uoWMwhA4UNkC&pg=PA466&lpg=PA466&dq=commutator+vector+valued+forms&source=bl&ots=BAl_WcubCU&sig=Hn05qnd8ahVchDY3U3uOerMQ4kY&hl=ko&ei=ZYefTdWnN4fbiALArfyCAw&sa=X&oi=book_result&ct=result&resnum=5&ved=0CDsQ6AEwBA#v=onepage&q=commutator%20vector%20valued%20forms&f=false | ||
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− | http://math.stackexchange.com/questions/3192/gauge-transformations-in-differential-forms | ||
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==history== | ==history== | ||
43번째 줄: | 29번째 줄: | ||
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
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==related items== | ==related items== | ||
51번째 줄: | 37번째 줄: | ||
* [[Knizhnik–Zamolodchikov equations(KZ equations)]] | * [[Knizhnik–Zamolodchikov equations(KZ equations)]] | ||
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==encyclopedia== | ==encyclopedia== | ||
* http://en.wikipedia.org/wiki/Vector-valued_form | * http://en.wikipedia.org/wiki/Vector-valued_form | ||
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==question and answers(Math Overflow)== | ==question and answers(Math Overflow)== | ||
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* http://math.stackexchange.com/questions/5962/some-questions-about-differential-forms | * http://math.stackexchange.com/questions/5962/some-questions-about-differential-forms | ||
* http://math.stackexchange.com/questions/3194/taking-trace-of-vector-valued-differential-forms | * http://math.stackexchange.com/questions/3194/taking-trace-of-vector-valued-differential-forms | ||
* http://mathoverflow.net/questions/31905/some-basic-questions-about-chern-simons-theory | * http://mathoverflow.net/questions/31905/some-basic-questions-about-chern-simons-theory | ||
* http://mathoverflow.net/questions/12248/gauge-connections-and-lie-algebras | * http://mathoverflow.net/questions/12248/gauge-connections-and-lie-algebras | ||
− | * http://mathoverflow.net/questions/36178/what-is-the-trace-in-the-chern-simons-action | + | * http://mathoverflow.net/questions/36178/what-is-the-trace-in-the-chern-simons-action |
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[[분류:개인노트]] | [[분류:개인노트]] | ||
[[분류:math and physics]] | [[분류:math and physics]] |
2013년 4월 29일 (월) 07:07 판
introduction
exercise from Chern-Simons theory
\[S=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A)=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{1}{3}A\wedge [A,A])\]
Let \(A=(a_1dx+A_1 dt)\otimes X_1 +(a_2dx+A_2dt)\otimes X_2\) be a vector valued form \[A\wedge A =(a_1A_2-a_2A_1)dx\wedge dt \otimes [X_1, X_2]\] \[[A,A]=2(a_1A_2-a_2A_1)dx\wedge dt \otimes [X_1, X_2]\]
- http://library.wolfram.com/infocenter/MathSource/683/
- http://en.wikipedia.org/wiki/Vector-valued_form
- chapter 8 of http://www.cis.upenn.edu/~cis610/diffgeom-n.pdf
- Integrable Hamiltonian hierarchies: spectral and geometric methods http://books.google.com/books?id=uoWMwhA4UNkC&pg=PA466&lpg=PA466&dq=commutator+vector+valued+forms&source=bl&ots=BAl_WcubCU&sig=Hn05qnd8ahVchDY3U3uOerMQ4kY&hl=ko&ei=ZYefTdWnN4fbiALArfyCAw&sa=X&oi=book_result&ct=result&resnum=5&ved=0CDsQ6AEwBA#v=onepage&q=commutator%20vector%20valued%20forms&f=false
- http://math.stackexchange.com/questions/3192/gauge-transformations-in-differential-forms
history
encyclopedia
question and answers(Math Overflow)
- http://math.stackexchange.com/questions/5962/some-questions-about-differential-forms
- http://math.stackexchange.com/questions/3194/taking-trace-of-vector-valued-differential-forms
- http://mathoverflow.net/questions/31905/some-basic-questions-about-chern-simons-theory
- http://mathoverflow.net/questions/12248/gauge-connections-and-lie-algebras
- http://mathoverflow.net/questions/36178/what-is-the-trace-in-the-chern-simons-action