Covariant derivative on $n$-forms

Question

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.

Answer 1

Gauge theory, in mathematical terms, is the mathematics of fibre bundles and in particular principal and vector bundles. It can be thought through in two different languages, the language of bundles and that of sheafs.

Probably the best book, given your background, is Baez & Munion's book, Gauge Theory, Gravity & Knots. They take a fairly intuitive physical approach but do describe all the major constructions in differential geometry whilst tying it to the physics of gauge theory. They do discuss how to induce a connection on a vector bundle from a principal bundle via a representation of the gauge structure group. Since connections induce covariant derivatives on vector bundles this should give a derivation of the formula you are asking about.

If you are interested in a fairly elementary book on the mathematics of differential geometry, a very good one is La Fontaine's An Introduction to Differential Manifolds. It's elementary in that it doesn't tackle either the language of bundles or sheafs. A good choice after this which does discuss these constructions is Conlon's Differential Geometry which tackle bundles and implicitly, sheaves.