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In Yang-Mills theory, the covariant differential is defined as

$d_A \phi = d\phi+[A,\phi].$

The curvature is defined as

$F = dA + \frac{1}{2}[A,A],$

note the factor of $\frac{1}{2}$. Yet, I often see authors define the curvature as

$F = d_A A.$

How does this definition make sense, given the factor of $\frac{1}{2}$ difference?

Notice that this factor is mostly a matter of convention. By simply rescaling the structure constants in the Lie algebra (which may always be done) it may be changed at will

This corresponds to the case that the G-bundle over which you consider the connection 1-form $A$ and hence the curvature $F$ is non-abelian. That is, in the case you consider, say a $U(1)$-bundle the curvature on the associated bundle will be just

$$ F = dA$$

The symbol $d_A$ means you are dealing with the non-abelian case and hence you get the extra commutator. The factor of $1/2$ is there due to antisymmetry. Try to expand the brackets by picking a trivilization over frame and you will see that the factor is needed so that all is good.

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