# Chern-Simons theory

The Chern-Simons 3-form is given by

$\omega_3={\rm Tr} \left[ A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right]$

where $A$ is a connection one-form in the adjoint representation of a non-Abelian gauge group.

My differential geometry is rather rusty (and this is new to me too) hence my questions;

$A$ is a 1-form. By definition of the wedge product between a $p$ form $\alpha$ and $q$ form $\beta$ we have $\alpha\wedge\beta=(-1)^{pq}\beta\wedge\alpha$. So we should have $A\wedge A=-A\wedge A=0$.

Why is this not the case?

Next question; I want to calculate $d\omega_3$ Does the fact that everything is inside the trace effect my calculation? In other words does the differential operator pass through the trace and only act on the forms?

That's because you are forgetting that $A$ has a Yang-Mills index. You better write this in components, which reads

$\epsilon^{\mu\nu\rho} g_{IJ} \Big( A^I_\mu \partial_\nu A_\rho^J + \frac{1}{3} f^J{}_{KL} A^I_\mu A^K_\nu A^L_\rho \Big)$

• Right so each $A^I$ is a one-form in the sense of my original post which satisfies the wedge product property I listed? i.e $A^I\wedge A^J=-A^J\wedge A^I$ Jan 23, 2016 at 20:13
• Yes. That's why "$A^I$ is a connection one-form in the adjoint representation of a non-Abelian gauge group". In the case of Abelian, that term just vanishes. Jan 23, 2016 at 20:18
• Is something like $A\wedge A$ written in component form as $g_{IJ}A^I\wedge A^J$ or do I need to include the Lorentz indicies too? Jan 23, 2016 at 21:20
• $A \wedge A = A_\mu^I A_\nu^J d x^\mu \wedge dx^\nu T_I T_J = A_\mu^I A_\nu^J d x^\mu \wedge dx^\nu [T_I, T_J] = A_\mu^I A_\nu^J ( d x^\mu \wedge dx^\nu ) f^K{}_{IJ} T_K$. Note that I was not careful with the numerical factors. Jan 23, 2016 at 22:37
This is an example of lie-algebra valued 1-forms. Actually you may write explicitly, $A = A_{\mu} ^a T^a dx^{\mu}$. Since the generators also anti-commute so we get the result. And for the same reason sometimes you will find expressions like $[A,A]$ in literature for your term.