High-energy theory
What the spectrum has to contain
About 6 min if you have seen this material before, 25 min or so if it is new to you.
Background
Assumed background: left-handed Weyl fermions, representations, and that a chiral gauge theory with a nonzero cubic anomaly has no consistent quantisation.
Model building runs backwards. You do not choose a spectrum and then test it; you write down the consistency conditions and see which spectra survive, and the conditions are restrictive enough that very little does.
For with left-handed Weyl fermions, the The failure of a classical gauge symmetry to survive quantisation, which makes the theory inconsistent unless the contributions cancel. adds over the spectrum and the total must vanish. Normalise ; then , and the two-index antisymmetric representation has .
One more constraint. A representation together with its conjugate admits a gauge-invariant mass term, so those fields do not stay in the massless spectrum. A theory built to be chiral cannot use them.
Consider with one Weyl fermion in the two-index antisymmetric representation (the ) and Weyl fermions in the antifundamental, and nothing else.
What is ?
Past 45 min on this? That usually means a missing prerequisite rather than a lack of effort — worth reading up before pushing further, starting with Chiral anomaly (Wikipedia).