Why complex reps of gauge group for chiral theory?

[Lapidus]

Why must the gauge group be in a complex representation so that chirality of the fermions is respected?

thanks

 

[Roy_1981]

Why must the gauge group be in a complex representation so that chirality of the fermions is respected?

thanks

I think your confusion is a matter of semantics or definition/convention. By convention all standard model fermions are taken to be left handed weyl spinors i.e. massless. Then property that these massless Weyl fermions are in a complex representation of the standard model group is called or defined to be "chirality". Not to be confused with chirality matrix γ₅ which determines handedness chirality of fermions. Chiral (handedness eigenstates) fermions when appear in complex representations of the gauge group makes the THEORY chiral.
Recall in QCD, we also have complex representations, e.g. fundamental of color gauge group SU(3). But this does NOT make QCD a chiral theory because the quarks/fermions being massive are not chirality eigenstates i.e. constituents are not chiral fermions.

So we need 2 things for a theory to be chiral, a) fermions must be massless, ie. chiral fermions b) they should come in complex representations of the gauge group.

 

[arivero]

al.
Recall in QCD, we also have complex representations, e.g. fundamental of color gauge group SU(3). But this does NOT make QCD a chiral theory

In fact there is a bit of Doctrine here, as we could equally define that QCD is non-chiral SU(3), similar to electromagnetism in this aspect.