We propose two models where a U(1) Peccei-Quinn (PQ) global symmetry arises accidentally
and is respected up to high-dimensional operators, so that the axion solution to the strong
CP problem is successful even in the presence of Planck-suppressed operators. One model is
SU(N) gauge interactions with fermions in the fundamental and a scalar in the symmetric.
The axion arises from spontaneous symmetry breaking to SO(N), that confines at a lower
energy scale. Axion quality in the model needs N=>10. SO bound states and possibly
monopoles provide extra Dark Matter (DM) candidates beyond the axion. In the second
model, the scalar is in the anti-symmetric: SU(N) broken to Sp(N) needs even N=>20.
The cosmological DM abundance, consisting of axions and/or super-heavy relics, can be
reproduced if the PQ symmetry is broken before inflation (Boltzmann-suppressed production
of super-heavy relics) or after (super-heavy relics in thermal equilibrium get partially diluted
by dark glue-ball decays)
