# Covariant derivative on $n$-forms

I am new to form notation. I have recently read that one can write (non-abelian) gauge theory in terms of forms. I am stuck and can't derive this equation below:

$$\nabla_{A} \alpha_p = d \alpha_p + A \wedge \alpha_p - (-1)^p \alpha_p \wedge A$$

Gauge covariant derivative on form

For example, applying this formula for a 1-form A to calculate curvature and Bianchi's identity:

$$F = \nabla_A A = d A + A \wedge A + A \wedge A$$ This formula seems incorrect for 1 forms since textbooks say this should be: $$F = dA + A \wedge A$$

For 2 forms (Bianchi's identity) $$\nabla_A F = d A + A \wedge F - F \wedge A = 0$$ which seems correct.

I'm familiar with wedge product and basics of exterior differentiation, but not at a formal mathematical level (typical QFT textbook level). Something intuitive will help.

The terminology is:

• $$\nabla_A$$ is the covariant derivative which uses the matrix valued 1-form $$A = A^a_\mu T^a dx^\mu$$ written in terms of the basis of the lie algebra $$T^a$$. The functions $$A^a_\mu$$ are functions of spacetime.

• $$\alpha_p$$ is a $$p$$-form on spacetime.