Instantons are phenomena that exclusively occur in non-abelian gauge theories, as established in Cheng and Li's book. In (3+1) dimensions, Abelian gauge theories, such as those with a U(1) gauge group, lack non-trivial instanton solutions due to their homotopy groups allowing only trivial mappings. However, in (1+1) dimensions, non-trivial instantons can exist in Abelian models, exemplified by the CP(N) model. This distinction highlights the necessity of non-abelian frameworks for discussing instantons in four-dimensional spacetime.
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Bobhawke said:instantons only occur in theories that are non-abelian.
This is not a general rule, in (1+1) dimensions,there are many Abelian models with non-trivial instanton solutions.
Why is this so?
Recall the following results on homotopy groups;
[tex]\pi_{n} (S^{n}) = Z,[/tex]
[tex]\pi_{n} (S^{1}) = 0, \ \ \mbox{for} \ n > 1[/tex]
where [itex]\pi_{n} (S^{m})[/itex] refers to the homotopy group for the mapping [itex]S^{n}\rightarrow S^{m}[/itex] and Z is the group of integers.
If the gauge group is U(1), every mapping of [itex]S^3[/itex] (the boundary of the (3+1)-dimensional Euclidean domain) into [itex]S^1[/itex] (the range of boundary values = U(1) manifold) is continueosly deformable into a single point (the trivial mapping). Thus, in (3+1)-dimensional Euclidean spacetime, Abelian gauge theories have no analog of the winding number, i.e., no non-trivial instanton sectors. This is why peopel choose non-abelian systems to discuss instantons in 4 dimensions.
It is only in (1+1) dimensions that the Abelian instantons exist with integral homotopy indices. Look up the very important Abelian model CP(N).
regards
sam