Hey folks, I'm trying to understand a couple of calculations in Chern-Simons theory and I'm stuck.(adsbygoogle = window.adsbygoogle || []).push({});

I want to prove the following in four spacetime dimensions:

Let [tex]G[/tex] be a Lie Group with generators [tex]T_{a}[/tex]. Further let

[tex]A=T_{a} A^{a}_{\mu} dx^{\mu}[/tex]

be a connection and

[tex]F=dA+A \wedge A[/tex]

its field strength. Now I want to prove

[tex]Tr F \wedge F= d Tr(A \wedge dA+\frac{2}{3}A\wedge A\wedge A)[/tex]

First I tried the following

[tex]F\wedge F=dA\wedge dA+dA \wedge A\wedge A+A\wedge A\wedge dA+A\wedge A\wedge A\wedge A[/tex]

[tex]d (A \wedge dA+\frac{2}{3}A\wedge A\wedge A)= dA\wedge dA+\frac{2}{3}dA\wedge A\wedge A +\frac{2}{3} A\wedge dA\wedge A+\frac{2}{3} A\wedge A \wedge dA [/tex]

When I take the trace, I can make a [tex]A\wedge A\wedge dA[/tex] into [tex]dA \wedge A\wedge A[/tex], but that leaves the questions of why the trace of [tex]A^4[/tex] has to vanish and where the factors of 2/3 come from.

Alternatively, I tried the following:

Let [tex]A_t=tA[/tex] and

[tex]F_t=dA_t+A_t\wedge A_t=dt\wedgeA+t dA+t^2 A\wedge A[/tex]

Now one can evaluate [tex]Tr F_t \wedge F_t [/tex] and afterwards set t=1. This works out fine, but I still don't understand why the trace of [tex]A^4[/tex] has to vanish.

So for the moment, it all boils down to: why is [tex]Tr A^4=0[/tex]?

Thank you for your help!

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# Chern Simons form: calculation

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