The role of the Hopf Fibration in Kaluza-Klein theory

Question

I have been learning recently about the Hopf Fibration and its relation to physics. My professor has told me that it is one of the simplest methods of dimensional reduction in Kaluza-Klein theory. After looking through various sources on KK theory, I see similarities and can imagine where the Hopf Fibration might fit, for example, the Hopf Fibration maps from $S^3$ to $S^2$, and this corresponds to the 3 spatial dimensions we observe, if we take time to be independent of the fibration, and of course the $U(1)$ fibres are the extra dimension compactified.

My question is, in practical terms, how does the Hopf Fibration actually come in to the theory? The line element in the KK metric has the form \begin{align*} ds^2 = g_{\mu \nu}dx^\mu dx^\nu + \left( dx^4 - A_\mu dx^\mu \right)^2 \end{align*} and this is similar in form to a metric I derived on the total space of the Hopf Fibration (particularly in the term in brackets, which looks just like the connection in the metric of the HF). When dimensionally reducing the theory using the Hopf Fibration, what would you do calculationally? I feel lost here - my first thought would be to just swap out the connection above with the one from the HF but I feel like i'm almost definitely wrong.

Also wouldn't the HF only be applicable if space had the topological structure of $S^2$ ?

Any help would be greatly appreciated - thanks in advance!

Answer 1

Hi I am also interested in this question. I have done some work on the Hopf-Fibration but do not have much knowledge of the KK theory - I would certainly like to know more about it though.

The Hopf-Fibration is an aspect of quaternion geometry. The quaternion is a 4-tuple which describes the 3-sphere $\mathbb{S}^3\subset\mathbb{R}^4$.

It may be projected down to 3-dimensional space, and then the 4th dimension is compactified as you said. As I understand it, the hopf-fibration is a broad area of research with a lot of unknowns.

For the sterographic projection of the hopf-fibration there are a bunch of good resources on Niels Johnson's website.

https://nilesjohnson.net/hopf.html

The stereographic projection is a mapping of the form $\mathbb{S}^2\mapsto_\mathbb{S^1}\mathbb{R}^3$ which is different than the standard projection $$\mathbb{S}^3\mapsto_\mathbb{S^1}\mathbb{S}^2$$

which seems to be what you are interested in. There is a good introductory article on this type of projection "Unit quaternions and the Bloch Sphere" found on the arXiv at

https://arxiv.org/abs/1411.4999

I hope that helps!

I also have a question on the KK theory - is the general idea of KK that the structure is a 4+1 spacetime; In the sense that there are 4 spatial directions and 1 time parameter?