Mirror fermions / mirror families. How does it work?

[arivero]

From time to time we have some minor threads mentioning real vs complex representations of fermions, chiral theories, etc and how a loophole is to use mirror generations, but I do not remember some detailed discussion of how does it work.

For starters, do we need an even number of parity-invariant generations, and then half go to low energy, half remain high? Or can we do it with odd numbers?

 

[mitchell porter]

I think you have it backwards? The significance of a mirror generation or anti-generation is that, even if you have a generation of chiral fermions, if you also have an anti-generation, they will pair off into non-chiral vector-like fermions. Naturalness arguments then imply that these vector fermions will be heavy unless there is a special "mirror parity" symmetry. So each anti-generation removes a generation from low-energy phenomenology.

For example, in heterotic phenomenology, the net number of generations is half the Euler character of the Calabi-Yau, which is the difference between two Hodge numbers, one of which gives the number of generations and the other the number of anti-generations. (I believe the idea is that you start with a 27 superfield of E6 coming from the string, then you get a copy of that for each fermionic zero mode of that superfield on the Calabi-Yau, and the number of those zero modes equals the number of harmonic forms which is given by the relevant Hodge number; and then something analogous happens for 27bar.)

So you might have four generations and one anti-generation, but the anti-generation will pair off with a generation and become heavy, leading to a net total of three light generations.

If you go the other way, and start with vector-like fermions but try to get light chiral fermions... Nir Polonsky's investigations in N=2 phenomenology would be relevant. But I think that implies a lot of BSM effects that aren't seen.