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In QCD you may add the term $\mathcal{L}_{\theta} = \theta\dfrac{g^2}{16\pi^2} \text{Tr}F\tilde{F}$, which turns out to be a total derivative. Now, it can be proven that the action of this lagrangian is $S_\theta = \theta\nu$, where $\nu\in\mathbb{N}$ is the winding number of the field configuration. Now, I know what the winding number is for a closed 1D curve. But I am unsure about how to visualize the winding number of a 4D field (or even a 2 or 3D one!).

So, my question is, how should I qualitatively visualize what the gauge field looks like when it has a non-null winding number?

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    $\begingroup$ Not sure if you saw this post, but it may help: physics.stackexchange.com/questions/483258/… $\endgroup$
    – ad2004
    Commented May 18 at 19:02
  • $\begingroup$ why do you think you can "qualitatively visualize" something in 4d other than just visualizing the lower-dimensional analogue, given that most human imagination is probably not more than 3d? $\endgroup$
    – ACuriousMind
    Commented May 18 at 22:40
  • $\begingroup$ @ACuriousMind Visualizing the 3D - or even the 2D - analogue would help. My knowledge of algebraic topology is sadly quite limited, so I cannot use that kind of intuition... $\endgroup$ Commented May 18 at 22:46
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    $\begingroup$ The topological defect associated with this winding number is called an instanton (en.wikipedia.org/wiki/Instanton). It is analogues to a phase vortex (en.wikipedia.org/wiki/Optical_vortex). Perhaps the visualization of a phase vortex can give one some idea of it. $\endgroup$ Commented May 19 at 3:25

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The proper answer is that fully visualizing this winding would involve imagining the winding on a 3-sphere. Just in case someone reading this doesn't know why that is, it's because the boundary conditions demanding the fields vanish at infinity mean we can "add a point" there to turn $\mathbb{R}^3 \to S^3$. That implies that at a fixed moment in time, we can view the connection as a map from $S^3 \to G$, because each component takes in a point of space and out puts a Lie algebra element (which can be exponentiated to a group element). The equivalence classes of all maps of $S^3 \hookrightarrow G$ under continuous deformations is given the name $\pi_3(G)$, the third fundamental group of G. It turns out that for every Lie group, $\pi_3(G) = \mathbb{Z}$, which is pretty remarkable since the same universal behavior is not true for, say $\pi_1(G)$. So it's pretty special to four spacetime dimensions that the winding number has such universal behavior regardless of gauge group. Also, that winding number $\nu$ is exactly the element of $\mathbb{Z}$ it belongs to.

Anyways, given the usual caveats of the fact you can't actually visualize four dimensions (or even three if it's curved), there are some analogies in lower dimensions that can help give intuition. The obvious example in 1d is winding around a circle: for instance, a tangent vector field has non-trivial winding. But a nice example in 2d is a skyrmion, and in particular the magnetic skyrmion page has some nice pictures. Again, it's not exactly the same as the case you mentioned because this example is of non-trivial winding on a 2-sphere. But hopefully, comparing and contrasting to the 1d case will help you get intuition for the higher dimensional case.

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  • $\begingroup$ Thank you, this was very helpful. Especially the last paragraph! $\endgroup$ Commented May 19 at 11:52

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