I am trying to evaluate $A \land A \land A$, but I am a bit confused on how exactly to do it and produce the usual notation used in physics. I am trying to use the definition of the wedge product of Lie algebra valued forms given here Lie algebra-valued differential form, but I am not sure how to proceed. Any input is very much appreciated.
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1$\begingroup$ The CS action has ${\rm tr}(A\wedge A \wedge A)$ and the trace simplifies things greatly. You only need the structure funtions and the quadratic Casimir in the representation in which the $A$'s live. $\endgroup$– mike stoneCommented Jun 29, 2021 at 16:08
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$\begingroup$ @mike stone The trace does simplify things, but I would like to see how would one go from $tr(A \wedge A \wedge A)$ to $\epsilon ^{\mu \nu \rho} tr(A_{\mu} A_{\nu} A_{\rho})$ which is how it is usually introduced in physics. $\endgroup$– JoelCommented Jun 29, 2021 at 16:18
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Does this help:
Consider a lie-lagebra valued form $A=A_\mu^a \lambda_a dx^\mu$ then $$ {\rm tr}(A^3)= {\rm tr} \{\lambda_a\lambda_b \lambda_c\} A^a_\alpha A^b_\beta A^c_\gamma dx^\alpha \wedge dx^\beta \wedge dx^\gamma\\ {\rm tr} \{\lambda_a\lambda_b \lambda_c\} A^a_\alpha A^b_\beta A^c_\gamma \epsilon^{\alpha\beta\gamma} dx^1\wedge dx^2 \wedge dx^3\\ \frac 12 {\rm tr} \{\lambda_a[\lambda_b, \lambda_c]\} A^a_\alpha A^b_\beta A^c_\gamma \epsilon^{\alpha\beta\gamma} dx^1\wedge dx^2 \wedge dx^3. $$
answered Jun 29, 2021 at 16:26