Distinction or rule about $d$ and $\delta$ in differential forms

Question

I have in trouble understanding the differential forms.

For $k$ form $\alpha$ and $l$ form $\beta$ we have \begin{align} \alpha \wedge \beta = (-1)^{kl} \beta \wedge \alpha \end{align} And for differentiation, we have \begin{align} d(\alpha \wedge \beta) = d \alpha \wedge \beta + (-1)^k \alpha \wedge d \beta \end{align}

Apply this for usual Riemannian case, for 1-form spin connection $w$, and two form curvature $R= dw+ w\wedge w$ \begin{align} d R = d^2 w + dw \wedge w - w \wedge dw = d w \wedge w - w \wedge dw \end{align}

Now i want to variation of differential forms. How about variation? is the same rule holds?

In page 10 of lecture note and some computation in https://physics.stackexchange.com/questions/222100/variations-of-actions-of-lie-algebra-valued-differential-forms, it seems they treat $\delta$, following usual Lebiniz rule. $i.e$, \begin{align} \delta R = \delta d w+ \delta w \wedge w + w \wedge \delta w \end{align}

Is their procedure right?

In usual differentiation or variation case, this does not be a big problem, (As far as i known, the role of differentiation and variation are similar whether they treat function or functional) but in terms of differential form. I got confused.

Can you give me some formula for variation $\delta$ acting on $(\alpha \wedge \beta)$?

How about Lie derivatives?

I tried to find some reference related with variation on differential form, but they only treat differentiation. Recommendation of any kinds of references are welcomed

Answer 1

I think what needs to be emphasised is that $d$ and this variational $\delta$ (as opposed to the codifferential $\delta$ that is the adjoint of $d$) are two very different operations, that both happen to be a bit like derivatives:

• Imprecisely, $d$ talks about the variation in the form $\alpha$ near $x$ as we move around on the manifold. A simple, though coarse, analogue is the derivative $d/dx$.
• Whereas $\delta$ is talking about what happens to a function of $\alpha$ if we change $\alpha$ by a small amount; I find the physicists' notation is rather lacking here. The (more precise) analogue is the functional derivative, $DF[\alpha](\phi) = \lim_{h \to 0} (F[\alpha+h\phi]-F[\alpha])/h $.

In particular, there is antisymmetry built into the definition of $d$, but $\delta$, while superficially looking the same, is an operation talking about different sorts of variations in a different place. A more mathematical way to write the variation is to expand $F[\alpha+h\phi]$ to first order in $h$, so, for example, $$ (\alpha+h\phi) \wedge (\alpha+h\phi) = \alpha \wedge \alpha + h(\phi \wedge \alpha + \alpha \wedge \phi) + o(h), $$ and subtracting and taking $h \to 0$ gives Leibniz for the variational derivative.