I Covariant derivative of field strength tensor

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1. Jan 5, 2017

d8586

Hi,

I am struggling to derive the relations on the right hand column of eq.(4) in https://arxiv.org/pdf/1008.4884.pdf

Even the easy abelian one (third row)

which is

$$D_\rho B_{\mu\nu}=\partial_\rho B_{\mu\nu}$$

doesn't match my calculation

Since

$$D_\rho B_{\mu\nu}=(\partial_\rho+i g B_\rho)(\partial_\mu B_\nu-\partial_\nu B_\mu)$$

the equation seems to imply that

$$B_\rho (\partial_\mu B_\nu-\partial_\nu B_\mu)=0$$

What am I missing?

2. Jan 5, 2017

Orodruin

Staff Emeritus
The field tensor transforms according to the adjoint representation, not the fundamental one. The structure constants for an Abelian group are zero.

3. Jan 5, 2017

d8586

Thank you very much!