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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.

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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.

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  • $\begingroup$ Hi thank you for the reference. I am particularly interested to know if the formula for covariant differentiation is correct. Most books have very few computational examples making it hard to practice. I will check out Baez's book. $\endgroup$ Dec 12, 2021 at 7:17
  • $\begingroup$ @Physicsdude: You're welcome. I've had similar problems with calculations. Baez's book is great, it's the one that made me want to learn differential geometry. Good luck. $\endgroup$ Dec 12, 2021 at 7:27
  • $\begingroup$ @physicsdude: By the way, differential forms on a manifold are exactly the same as the anti-symmetric tensor fields used in GR. There's a covariant derivative thats already used there for tensor fields and this should match the covariant derivative for differential forms. The covariant derivative that's used in gauge theory is not used over a manifold but over a bundle. In maths, its called the exterior covariant derivative and this is the gauge covariant derivative you mention above. $\endgroup$ Dec 12, 2021 at 7:59

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