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.
Recall the following results on homotopy groups;
\pi_{n} (S^{n}) = Z,
\pi_{n} (S^{1}) = 0, \ \ \mbox{for} \ n > 1
where \pi_{n} (S^{m}) refers to the homotopy group for the mapping S^{n}\rightarrow S^{m} and Z is the group of integers.
If the gauge group is U(1), every mapping of S^3 (the boundary of the (3+1)-dimensional Euclidean domain) into S^1 (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
