Let's begin with a simple model of a field theory:

$$ \mathcal{H} = \int ( \nabla \phi ) ^2 $$

where $\phi$ is an angle valued field defined on some space. We suppose for the moment to freeze out the temporal dimention.

We will now see the usual argument that is employed to introduce topological defects. Start by considering a configuration of the field:

$$\phi: D^d \rightarrow U(1)$$

where $D^d$ is the $d$-dimentional disc. The boundary conditions are given by a smooth function:

$$\phi_0: S^{d-1} \rightarrow U(1)$$

The functions $\phi_0$ (up to smooth deformations) can be classified by the elements of the homotopy group $\pi_{d-1} \left( U(1) \right)$. Generally speaking for a $M$-valued field we can classify the boundary conditions with the homotopy groups $\pi_{d-1} \left( M \right)$. If for some dimentions it is non trivial then we can have topologicaly non trivial boundary conditions and we say that the bulk contains a topological defect. Indeed the whole point of having topological defect is to have boundary conditions that are not smoothly deformable to the ones of a gound state solution (i.e. a constant angle).

By this argument one can see that for this field theory there are no domain walls (which are associated to the fact that $U(1)$ is connected) but there can be vortex (and antivortex), no monopoles, etc., ...

Vortex solutions are possible in 2D. I know that this theory can support vortex also in 3D in the form of an extended 1D singularity. The argument above doesn't seem to contain this case because a 1D vortex in a 3D fluid will produce boundary conditions on $S^2$ which are outside the hypothesis of smoothness. Indeed one would have two singularities at the two extremum of the 1D string. Unless one has a vortex which closes onto itself by gluing the two ends into a cicle. This way one would have a smooth boundary contition I think.

My question now is how to properly take in account the existence of those defects in dimensions different from the one they give smooth boundary conditions. As an example: how can I theoretically justify the presence of 1D vortex in a 3D fluid when the precedent justification (by definition basically) only allows vortex in 2D?



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