I am reading this paper by E. Weinberg - Fundamental monopoles and multimonopole solutions for arbitrary simple gauge groups.
I am having a problem with a calculation. I don't have much experience with such calculations, and have never come across the gamma matrices mentioned here. So, it would be great if someone could help. M is an position independent quantity.
The author claims that the divergence of the following quantity $$J_i=\frac{1}{2}tr
tr is the trace over group and spinor indices.
$\gamma$ are the gamma matrices forming the Clifford algebra. $D_4=\Phi$ where $\Phi^{ac}=f_{abc}\phi^b$, $\phi$ is the scalar higgs field. and $D$ is the usual covariant derivative in Yang-Mills theory given by $(D_{\mu}\phi)^a=\partial_{\mu}\phi^a + f_{abc}A_{\mu}^b\phi^c$
Attempt at a solution
$$\partial_iJ_i=\frac{1}{2}tr
Divergence = $$\frac{1}{2}tr \ \gamma_5 \gamma_i \partial_i(\gamma \cdot D)
I don't know how to proceed after this. The problem is the denominator is squared in the second term, where is should have power 1. Help will be very much appreciated.
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