Topological number integral in the Yang-Mills theory in boundary and volume forms

Question

Suppose non-trivial vacuum configuration of the Yang-Mills theory with the winding number $n$:
$$
\tag 1 A_{\mu}(x) = g_{(n)}(x)\partial_{\mu}g^{-1}_{(n)}(x)
$$

The winding number is given by the surface integral of topological density over the 3-sphere:
$$
\tag 2 n = \frac{1}{24\pi^{2}}\int_{S^{3}} d\sigma_{\mu}\epsilon^{\mu\nu\alpha\beta}\text{tr}\big[(g_{(n)}\partial_{\nu}g_{(n)}^{-1})(g_{(n)}\partial_{\alpha}g_{(n)}^{-1} )(g_{(n)}\partial_{\beta}g_{(n)}^{-1} )\big] \
$$

In various literature sources (for example, in Rubakov's "Classical gauge fields. Bosons") people often rewrite the surface integral $(2)$ in terms of volume integral:
$$
\tag 3 n = -\frac{1}{16\pi^{2}}\int d^{4}x\text{tr}\big[F_{\mu\nu}\tilde{F}^{\mu\nu}\big],
$$

where $F$ is the gauge field strength and $\tilde{F} = *F$ is its dual. They claim that $(3)$ and $(2)$ are equivalent. But in fact $(2)$ gives non-zero integer result for the pure gauge $(1)$, while $(3)$ vanishes! This can be seen by choosing the 4-dimensional euclidean manifold to be the "cylinder", with the planes being defined by $\tau = \pm \infty$. Then in the gauge $A_{0} = 0$ we obtain from $(3)$
$$
\tag 4 n = n(\tau = \infty) - n(\tau = -\infty)
$$

The precise reason is that we include $\epsilon^{\mu\nu\alpha\beta}\text{tr}\big[\partial_{\mu}F_{\nu\alpha}A_{\beta}\big]$ term in the action when converting the surface integral into the volume integral.

So why do people say that $(2)$ and $(3)$ are equivalent?

Comments

Too bad this got closed. Many readers of the standard physics textbooks share this confusion, or should share it if they really try to check the text. Here is an expository explanation of what is really going on: SU(2)-instantons from the correct maths to the traditional physics story

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@UrsSchreiber I reopend it (you could have done this too), as I dont know why the author closed his nice question. Maybe it was a misclick (?) ...

Answer 1

I think by "equivalent" they probably meant your equation (4). The two n's you defined in (2) and (3) are not the same: the first is a Chern-Simons number $n_{CS}$ of some vacuum, the second is the instanton number $n_{ins}$. If we consider a vacuum to vacuum transition due to instantons, each vacuum with its own $n_{CS}$, then the $n_{CS}$ and $n_{ins}$ are precisely related by your equation (4), this is a consequence of Stokes theorem.

Comments

I want to agree with You, but still have some doubts.

Answer 2

Many readers of the standard physics textbooks share this confusion, or should share it if they really try to check the text. Here is an expository explanation of what is really going on: SU(2)-instantons from the correct maths to the traditional physics story

Comments

Thank You, it is important to be familiar with correct description of instantons, I'll read. However, does it directly related to my question about different forms of Maurer-Cartan invariant?

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As far as I understand, the stories about instantons and about vacuum configurations are different. The reason is that instantons live in stereographically projected space $R^{4} \to S^{4}$ (the latter follows from conformal invariance of Yang-Mills theory and finiteness of the action), while vacuum configurations belonging to non-trivial homotopic class are initially defined as mapping $S^{3} \to G$.

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I meant to explain why your (3) does not vanish. Namely the 4-form being integrated is not in fact globally exact on its correct domain of definition, which is \(S^4\). I also meant to explain why the winding number of \(S^3 \to SU(2)\) is just another incarnation of the instanton number.

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@UrsSchreiber : Thank You. But how exactly this prevents me to ensure that the expression $(3)$ vanishes after substituting the vacuum solution $(1)$? The fact that it is defined only on $S^{3}$?

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In the entry that I pointed to there is meant to be explanation of how you may choose this "vacuum configuration" locally, but not globally on the 4-sphere.

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But what to do with the expression $(4)$? How to interpret it as the winding number of the vacuum gauge configuration?

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As I meant to have explained, you can not assume that the 4d space is a cyclinder. Instead you need to take it to be the 4-sphere.

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Not sure what Blalaton is meaning to make fun of, but indeed instantons are seen in computer lattice models, see for instance Gruber 13, especially sections 5 and 7