Note on gauge and gravitational anomalies of discrete $Z_N$ symmetries

Abstract

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 $Z_N$ $\mathcal{R}$-symmetries as well as non-$\mathcal{R}$ $Z_N$ 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 $Z_3$ $\mathcal{R}$-symmetry in the MSSM. Furthermore, we point out that in a theory with one Majorana spin 3/2 gravitino, a large class of $Z_4$ $\mathcal{R}$-symmetries are violated in the presence of Eguchi-Hanson (EH) gravitational instanton. This is also in general true for higher $Z_N$ $\mathcal{R}$-symmetries. We also notice that in 4 dimensional $\mathcal{N}=1$ supergravity, the global $U(1)$ $\mathcal{R}$-symmetry is always violated by the EH instanton irrespective of the matter content of the theory.

Comments: Version to appear in JHEP