Why semi-simple and compact Gauge Group in YM Theory? [duplicate]

Question

I'm studying the Yang-Mills theory, with the Action: $$ S=-\frac{1}{2}\int\mathrm{tr}_{\rho}(\mathcal{F}\wedge\star\mathcal{F}) $$ where $\mathcal{F}:=\mathrm{d} \mathcal{A}+\frac{1}{2}[\mathcal{A},\mathcal{A}]$ is the field strength ($\mathcal{A}$ is the Gauge Field), and the trace $\mathrm{tr}_\rho$ is taken over some Represenation $\rho$ of the Gauge Group $G$, and $\star$ is the Hodge star operator. (I'm following the notations in Nakahara's Geometry, topology and physics. )

Then my question is, it seems that mathematically speaking the Gauge Group $G$ and the Representation $\rho$ can be both taken to be arbitary (Nakahara's chapter 10), but why in physics (Peskin's QFT book), we are only considering Semisimple and Compact Gauge Group and unitary represenation? I know they have nice classifying properties, but are there some physical principles that can restrict us to only consider them?

Answer 1

The compact criterion comes from the fact that the kinetic term showing up in the lagrangian is actually, strictly defined via a associative bilinear form (often the killing form). But for the theory to make sense physically, it needs to be (semi)definite. This is only the case for compact Lie algebras. Regarding semi-simplicity, this does not need to be the case (see U(1) YM theory).

In case one chooses the killing form, which is mostly the case, one has the Cartan criterium and Weyl theorem. They tell us that for the killing form to be non-degenerate and definite, one needs semisimple and compact lie algebras.

Regarding unitary representations, once you have a compact lie group you can unitarize and thus assume this w.l.o.g.