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On my QFT lecture note there is a comment that says 'Non-abelian gauge theory is extremely unique in 4-dimensional spacetime'. However, I didn't really catch what that means. Why is it extremely unique? Could someone give me some examples for that?

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  • $\begingroup$ There’s nothing unique about the choice of the gauge group. But that choice is presumably not what the comment was about. $\endgroup$
    – Ghoster
    Apr 25, 2023 at 0:20

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4 spacetime dimensions are the only dimensions in which the (anti-)self duality equation $$ F = \pm{\star} F$$ (for $\star$ the Hodge star) makes sense since both $F$ and ${\star}F$ are 2-forms in 4d, which is more or less directly related to the topological term $F\wedge F$ being a possible term in the Lagrangian in addition to the standard Yang-Mills "kinetic" term $F\wedge{\star}F$.

The classical minima of an action $\int\mathrm{tr}(F\wedge{\star}F) + \mathrm{tr}(F\wedge F)$ like are precisely the (anti-)self dual field configurations, i.e. instantons are the classical solutions to the equations of motion.

Instantons have a lot of special features that make them of interest both physically (for instance as vacua/mediators between vacua, see e.g. this answer or this answer of mine) and mathematically (for instance as Donaldson invariance, see this answer of mine). While there are topological features of Yang-Mills theories in all dimensions, the anti-self duality condition makes the 4d theory uniquely rich.

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One of the features of the non-Abelian gauge theory is that it is renormalizable and asymptotically free—meaning the quantum corrections are systematically calculable in the same way as in quantum electrodynamics, but (unlike in QED), the quantum corrections lead to the interaction getting weaker (rather than stronger) at very large momentum (or short distance) scales. This has all sorts of phenomenalistic consequences in QCD, and it also has formal consequences for the structure of the theory. (There reasons to think that only asymptotically free theories may have well-defined nonperturbative continuum formulations.) Theories with non-Abelian gauge bosons are the only such asymptotically free theories in 3 + 1 dimensions, and in my experience, when people talk about the uniqueness of non-Abelian gauge theories, this is the property that they are primarily referring to.

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