Dirac string describes the string connecting the U(1) magnetic monopole to the U(1) anti-magnetic monopole in the U(1) gauge theory.

Since U(1) is a subgroup of SU(2) and SO(3), we may embed the U(1) gauge theory to SU(2) gauge theory or SO(3) gauge theory. Of course, the outcome phases of the gauge theories and their operator spectra are very different from each other (U(1) v.s. SO(3) v.s. SU(2).)

But one can still ask, what is the fate of U(1) Dirac string in the SU(2) gauge theory or SO(3) gauge theory?


1 Answer 1


There are a couple aspects here.

First, Monopoles usually fit into some kind of symmetry breaking scheme.

The idea is, when all you have is the monopole, a source of magnetic flux, then it can't spontaneously go away. But if you had some additional symmetry available, then you could continuously deform the field to get rid of the magnetic charge. It's pretty hard to show this in 3-D, because topological charge is very difficult to see in 3-D. The "hairy ball theorem" has a cool name but it's tough to show. But in 2-D it's a lot easier. Well... Less difficult anyway.

I need my visual aid here, a thing that hung above my desk for 3 years of my PhD. Imagine a torus. Now imagine that in this torus, you inscribe a Mobius strip, such that the strip goes all the way the long-way around the torus, with the edges of the Mobius on the outside of the torus. The cross-section through the torus, a disk, represents the full symmetry. Rotation around the outside of the small circle, and so twisting the surface of the Mobius, of the torus represents the remainder after symmetry breaking.

So, under the full symmetry, we can continuously deform the Mobius strip to get rid of the change in orientation. We can do that because we can change the strip so that it goes to zero width, and twists. But if we can only rotate each location, and we have to keep things continuous, then we can't get rid of the kink in the strip.

So, if the full symmetry is restored at high enough energy, then we can create a monopole at those high energies, cool it, and have it become topologically frozen in.

The other aspect of the monopole is, it may act very weirdly in relation to non-Abelian fields. That definitely includes SU(2), etc.



The basic idea is this. You build a monopole in a principal fiber bundle by constructing it in gauge patches. For example, you build a north and south hemisphere, and you make structure transformations between them. You build it so that in the north hemisphere, the pole is in the south, and in the south, the pole is in the north. So you never hit the string.

What you find with monopoles that have non-Abelian charge, there is a problem. A gauge transform can move the pole from one hemisphere to the other. That is, for non-Abelian monopole charge, you can not properly build the hemispheres so as to hide the string.


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