axial anomaly for adjoint fermion v.s. fundamental fermion

Question

It is known that the axial anomaly (chiral anomaly, the left L- right R) shows that $U(1)_A$-axial symmetry is not a global symmetry at quantum level.

In particular, one can consider the (1) fundamental fermion and (2) adjoit fermion for QCD. The $U(1)_A$-axial symmetry is not a symmetry for both cases, but there are discrete sectors that survive as a symmetry within $U(1)_A$.

What are the remained discrete axial symmetries for (1) fundamental fermion and (2) adjoint fermion, say for a Lie group G (like SU(N))?

This post imported from StackExchange Physics at 2020-11-05 13:28 (UTC), posted by SE-user annie marie heart

Answer 1

The discrete symmetry is the symmetry of the corresponding 't Hooft vertex. For an $SU(N_c)$ gauge group this symmetry is $Z_{N_f}$ in the case of fundamental fermions, and $Z_{N_c}$ in the case of adjoint fermions.

This post imported from StackExchange Physics at 2020-11-05 13:28 (UTC), posted by SE-user Thomas