Note on gauge and gravitational anomalies of discrete ZN symmetries

In this note, we discuss the consistency conditions which a discrete Z(N) symmetry should satisfy in order that it is not violated by gauge and gravitational instantons. As examples, we enlist all the Z(N) R-symmetries as well as non-R Z(N) symmetries (N=2,3,4) in the minimally supersymmetric standard model (MSSM) that are free from gauge and gravitational anomalies. We show that there exists non-anomalous discrete symmetries that forbid Baryon number violation up to dimension 6 level (in superspace). We also observe that there exists no non-anomalous Z(3) R-symmetry in the MSSM. Furthermore, we point out that in a theory with one Majorana spin 3/2 gravitino, a large class of Z(4) R-symmetries are violated in the presence of Eguchi-Hanson (EH) gravitational instanton. This is also in general true for higher Z(N) R-symmetries. We also notice that in 4 dimensional N = 1 supergravity, the global U(1) R-symmetry is always violated by the EH instanton irrespective of the matter content of the theory.