In Wikipedia, it states about the magnetic monopole of the gauge theory is determined by the fact:
This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It generalizes to $d + 1$ spacetime dimensions with $d \geq 2$ in several ways. One way is to extend everything into the extra dimensions, so that U(1) monopoles become sheets of dimension $d − 3$. Another way is to examine the type of topological singularity at a point with the homotopy group $$ \pi_{d −2}(G). $$
My goal and my questions are trying to get generic definitions and classifications with reasonable assumptions. Do you have one?
Can you explain the Wik quote above?
Can I understand the above statement from Wikipedia as follows:
My attempt: The magnetic monopole's wordline is $d+1-3=d-2$ dimensions (which generally has a worldvolume of $d-2$ dimensions). Thus the magnetic monopole itself is $d-3$ dimension (as a constant time slice of $d-2$ dimensions).
- magnetic monopole's wordline is the so-called 't Hooft operator.
Consider the full spacetime as $S^{d+1}$. Cutting out the ${d-2}$ dimensional worldvolume as the time trajectory of the magnetic monopole. We can consider the cut with a tubular neighborhood around this, thus we have $$D^{d-3} \times S^{1},$$ where the closed time trajectory is $S^{1}$. I think what I should really do about the surgery is $$ S^{d+1} \smallsetminus (D^{2} \times D^{d-3} \times S^{1}) = ? $$
But all I know so far is that $$ S^{d+1} \smallsetminus D^{d} \times S^{1} = S^{d-1} \times D^{2}. $$
So I can consider the covering charts over $S^{d-1}$. I suppose that minimal overlapping of the charts form a trajectory of $S^{d-2}$ (True of false?).
Thus consider the homotopy group (for the map from $S^{d-2}$ to the gauge group $G$) $$ \pi_{d −2}(G) $$ seems reasonable. Is my argument above correct, or is it incorrect?
