# Chern Simons form: calculation

1. Jul 31, 2010

### SergejVictorov

Hey folks, I'm trying to understand a couple of calculations in Chern-Simons theory and I'm stuck.

I want to prove the following in four spacetime dimensions:

Let $$G$$ be a Lie Group with generators $$T_{a}$$. Further let
$$A=T_{a} A^{a}_{\mu} dx^{\mu}$$
be a connection and
$$F=dA+A \wedge A$$
its field strength. Now I want to prove
$$Tr F \wedge F= d Tr(A \wedge dA+\frac{2}{3}A\wedge A\wedge A)$$

First I tried the following
$$F\wedge F=dA\wedge dA+dA \wedge A\wedge A+A\wedge A\wedge dA+A\wedge A\wedge A\wedge A$$

$$d (A \wedge dA+\frac{2}{3}A\wedge A\wedge A)= dA\wedge dA+\frac{2}{3}dA\wedge A\wedge A +\frac{2}{3} A\wedge dA\wedge A+\frac{2}{3} A\wedge A \wedge dA$$

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

Alternatively, I tried the following:

Let $$A_t=tA$$ and
$$F_t=dA_t+A_t\wedge A_t=dt\wedgeA+t dA+t^2 A\wedge A$$

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

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

$$Tr\left[dA\wedge A\wedge A \right]= \epsilon^{ijkl}Tr\left[(\partial_{i}A_{j})A_{k}A_{l} \right]= \epsilon^{ijkl}Tr\left[A_{l}(\partial_{i}A_{j})A_{k} \right] = -\epsilon^{ijkl}Tr\left[A_{i}(\partial_{j}A_{k})A_{l} \right]=-Tr\left[A\wedge dA\wedge A \right]$$,
in which the cyclic property of trace and the antisymmetry of the $$\displaystyle{\epsilon}$$ tensor is used.
$$Tr\left[A^{4}\right]=0$$ can be proved similarly.