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) $

This component notation also answers your second question.

  • $\begingroup$ 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 $A^I\wedge A^J=-A^J\wedge A^I$ $\endgroup$ – Okazaki Jan 23 '16 at 20:13
  • $\begingroup$ 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. $\endgroup$ – John Doe Jan 23 '16 at 20:18
  • $\begingroup$ 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? $\endgroup$ – Okazaki Jan 23 '16 at 21:20
  • $\begingroup$ $A \wedge A = A_\mu^I A_\nu^J d x^\mu \wedge dx^\nu T_I T_J = A_\mu^I A_\nu^J d x^\mu \wedge dx^\nu [T_I, T_J] = A_\mu^I A_\nu^J ( d x^\mu \wedge dx^\nu ) f^K{}_{IJ} T_K$. Note that I was not careful with the numerical factors. $\endgroup$ – John Doe Jan 23 '16 at 22:37
  • $\begingroup$ You can read the (unofficial) notes by Malcolm Perry on Applications of Differential Geometry to Physics, link: aei.mpg.de/~gielen/diffgeo.pdf $\endgroup$ – John Doe Jan 23 '16 at 22:39

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.


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