There are two striking aspects of non-Abelian gauge groups (compared to their Abelian cousins):

(1) The pure gauge parts of non-Abelian Lagrangians contain self-interaction terms that are trilinear and quadrilinear in the gauge fields.

(2) Non-Abelian gauge groups have topologically non-trivial field configurations (e.g. instantons).

Are these two aspects of non-Abelian gauge groups completely unrelated or is there an intuitive connection between (1) and (2)?

For example, 't Hooft showed that fermionic zero modes in the instanton background lead to a non-perturbative multi-fermion interaction in the low energy effective action - can that somehow be related to the self-interaction in the pure gauge parts?

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    $\begingroup$ Your 't Hooft reference is only an erratum. Can you give the full reference? $\endgroup$ – Bob Bee Jun 6 '17 at 3:12

I think it's fair to say that they're unrelated.

Abelian gauge groups can exhibit instantons and other non-trivial topological behavior. Likewise there's Abelian chern-Simons theory, where the natural Lagrangian has a cubic interaction.


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