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1. • CommentRowNumber1.
   • CommentAuthorcfgauss
   • CommentTimeAug 2nd 2010
   • (edited Aug 2nd 2010)
   • PermaLink
   Author: cfgauss
   Format: MarkdownItexI'm a physics grad student and I've been looking through Daniel Freed's lectures in QFT and strings: a course for mathematicians, and had some notation questions. Trying to google to find help seems to only lead me to an n-lab page (http://ncatlab.org/nlab/show/curvature+characteristic+form), and Daniel Freeds web page and papers, which didn't really help me, so I thought this would be a good place to ask! In his lectures on classical field theories, he gives the example of a lagrangian of classical mechanics, and writes the symplectic 2-form we get out of the action as $\omega = \delta \gamma = \langle \delta x' \wedge \delta x \rangle$ where $\gamma = \langlex' , \delta x \rangle$ and $\langle .,. \rangle$ is the inner product. But I do not understand what the notation $\langle A \wedge A \rangle$ could mean? If I vary gamma I would think I should get: $\langle \delta x', \delta x \rangle + \langle x', \delta^2 x \rangle = \langle \delta x', \delta x \rangle$ which looks almost like the \omega above except for a wedge instead of a comma! But this notation is used again later several times, e.g., in defining the chern-simons action as $S = \int \langle A \wedge F \rangle - \frac{1}{6} \langle A \wedge [A \wedge A] \rangle$ when the usual form looks similar but with a $+\frac{2}{3}$ and a trace instead of the brackets. It also seems that $1/2 [A\wedge A] = [A,A]$? Where does this come from? I don't see any explanation of either of these notations and don't recall seeing it in any of my differential geometry books. The link above claims <.> is used for the "curvature characteristic form" but that's not something I've ever heard of, and it doesn't seem to really define it... So does anyone know of any texts / resources where this is defined? Or is there a simple definition in terms of the inner product as the notation would indicate?

   I’m a physics grad student and I’ve been looking through Daniel Freed’s lectures in QFT and strings: a course for mathematicians, and had some notation questions. Trying to google to find help seems to only lead me to an n-lab page (http://ncatlab.org/nlab/show/curvature+characteristic+form), and Daniel Freeds web page and papers, which didn’t really help me, so I thought this would be a good place to ask!

   In his lectures on classical field theories, he gives the example of a lagrangian of classical mechanics, and writes the symplectic 2-form we get out of the action as $\omega = \delta \gamma = \langle \delta x' \wedge \delta x \rangle$ where $\gamma = \langlex' , \delta x \rangle$ and $\langle .,. \rangle$ is the inner product.

   But I do not understand what the notation $\langle A \wedge A \rangle$ could mean? If I vary gamma I would think I should get: $\langle \delta x', \delta x \rangle + \langle x', \delta^2 x \rangle = \langle \delta x', \delta x \rangle$ which looks almost like the \omega above except for a wedge instead of a comma!

   But this notation is used again later several times, e.g., in defining the chern-simons action as $S = \int \langle A \wedge F \rangle - \frac{1}{6} \langle A \wedge [A \wedge A] \rangle$ when the usual form looks similar but with a $+\frac{2}{3}$ and a trace instead of the brackets. It also seems that $1/2 [A\wedge A] = [A,A]$? Where does this come from?

   I don’t see any explanation of either of these notations and don’t recall seeing it in any of my differential geometry books.

   The link above claims <.> is used for the “curvature characteristic form” but that’s not something I’ve ever heard of, and it doesn’t seem to really define it… So does anyone know of any texts / resources where this is defined? Or is there a simple definition in terms of the inner product as the notation would indicate?

2. • CommentRowNumber2.
   • CommentAuthorEric
   • CommentTimeAug 2nd 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexHi cfgauss, I don't have an immediate answer to your question, just a note about formatting. If you edit your comment, throw some dollar signs around your tex commands and click "Markdown+Itex", it should render as math. Gotta run for now!

   Hi cfgauss,

   I don’t have an immediate answer to your question, just a note about formatting.

   If you edit your comment, throw some dollar signs around your tex commands and click “Markdown+Itex”, it should render as math.

   Gotta run for now!