I apologize for this simple question, but I lost a factor of 2 and can't find it anymore, so now I'm looking on the internet, perhaps one of you has some information about its whereabouts. :-)
Consider the electromagnetic field tensor $F_{\mu\nu}$ which corresponds to a differential form $F =\frac12 F_{\mu\nu} dx^\mu \wedge dx^\nu$. The wedge product of $F$ with itself can be expressed as
$$ F \wedge F = \frac14 F_{\mu\nu}\mathcal F^{\mu\nu} $$
where $\mathcal F^{\mu\nu} = \frac12 \epsilon^{\mu\nu\sigma\tau}F_{\sigma\tau}$ is the dual field tensor.
Writing $A = A_\mu dx^\mu$ for the vector potential, we have
$$ \int_V F \wedge F = \int_V (d A) \wedge F = \int_V d(A \wedge F) = \int_{\partial V} A \wedge F$$
But it appears to me that the latter integral is equal to
$$ \dots = \int_{\partial V} dx^\mu\ A_\nu \mathcal F^{\mu\nu} .$$
Yet, at the same time, we can write
$$ \int_V dx^\alpha \frac12 F_{\mu\nu}\mathcal F^{\mu\nu} = \int_V dx^\alpha (\partial_\mu A_\nu)\mathcal F^{\mu\nu} = \int_V dx^\alpha \partial_\mu (A_\nu\mathcal F^{\mu\nu}) = \int_{\partial V} dx^\mu\ A_\nu \mathcal F^{\mu\nu} .$$
Clearly, that cannot be.
I have lost a factor of two. Where is it?
