Vector valued differential forms

exercise from Chern-Simons theory

• Chern-Simons invariant

\[S=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A)=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{1}{3}A\wedge [A,A])\]

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 [Notes on Differential Geometry and Lie Groups http://www.cis.upenn.edu/%7Ecis610/diffgeom-n.pdf] by Jean Gallier
• 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

related items

• Knizhnik–Zamolodchikov equations(KZ equations)

encyclopedia

• http://en.wikipedia.org/wiki/Vector-valued_form

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