# Visualization of a gauge field with non-null winding number

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?

• Not sure if you saw this post, but it may help: physics.stackexchange.com/questions/483258/… Commented May 18 at 19:02
• 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? Commented May 18 at 22:40
• @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... Commented May 18 at 22:46
• 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. Commented May 19 at 3:25

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.