# Winding number in 4D & $SU(2)$ group

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 the winding number $$n$$ associated. $$tr$$ 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.

• I asked this in Math's site but I couldn't find any answer, so taking into account that there is Physics related to Winding number and my interest on it comes from theta vacua and instantons, I have cross-posted it May 30, 2019 at 5:54
• Crossposted from math.stackexchange.com/q/3179213/11127 May 30, 2019 at 5:59

I don't have Sredniki to hand, but it seems to be that your "four dimensional" space of the $$x^\mu$$ at the beginning of the question is really the three dimensional space forming the surface of a three sphere and the integral is over this three-dimensional space. Your $$U\in {\rm SU}(2)$$ is also parametrized by a three-sphere, so you have a winding number of 3-sphere about 3-sphere.
In the second equation, however the $$x^\mu$$ are clearly intended to be an actual four dimensional space and one is choosing some general (no longer necessarily a sphere) three dimensional surface embedded in that space. If you choose a surface $$x^0=0$$ for example then $$\epsilon^{0abc}= \epsilon^{abc}$$ and you are back to the original integral. Even if the new surface is not compact, you will still get an integer value for $$n$$ if $$U\to {\rm identity}$$ at infinity.