In the book 'Quantum field theory' by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such that

$$\hat{x} = (\sin\chi \sin\psi \cos\phi, \sin\chi \sin\psi \sin\phi, \sin\chi \cos\psi, \cos\chi), \quad \sum_\mu \hat{x}_\mu \hat{x}_\mu = 1$$

$n$ is given by

$$ n = -\frac{1}{24\pi^2}\int_0^\pi d\chi\int_0^\pi d\psi \int_0^{2\pi} d\phi\ \epsilon^{\alpha\beta\gamma}tr\{(U\partial_\alpha U^\dagger) (U\partial_\beta U^\dagger) (U\partial_\gamma U^\dagger)\}, \quad \epsilon^{\chi\psi\phi} = +1 $$

Where $U$ is only dependent on $\hat{x}$, belongs to $SU(2)$ and has associated the winding number $n$. $tr$ represents the trace.

But suddenly Srednicki says that you can write $n$ as an integral over the surface of this 4-dimensional space of the form

$$ n = \frac{1}{24\pi^2}\int dS_\mu\ \epsilon^{\mu\nu\sigma\tau}tr\{(U\partial_\nu U^\dagger) (U\partial_\sigma U^\dagger) (U\partial_\tau U^\dagger)\}, \quad \partial_\nu = \partial/\partial x^\nu\ {\rm and\ so\ on} $$

I don't understand how you can go from one expression of $n$ to the other.


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