# Instantons in non-abelian theories

#### Bobhawke

Im trying to learn about instantons at the moments from Cheng and Li's book. It seems to suggest that instantons only occur in theories that are non-abelian. Why is this so?

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#### samalkhaiat

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;

$$\pi_{n} (S^{n}) = Z,$$
$$\pi_{n} (S^{1}) = 0, \ \ \mbox{for} \ n > 1$$

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

Last edited:

Thank you Samal!

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