Global gauge anomalies for simple Lie algebras

Abstract

We generalize the formula by Elitzur and Nair on the global-anomaly coefficients in even (D =2n)-dimensional space and analyze global anomalies for Sp(2M, SO(M, and SU(M groups. In particular, we show that any irreducible representation of any Sp(M and SU(2) group has no global anomalies in D = 8k dimensions. In D = 8k +4 dimensions, SU(2) has 2,-type global anomalies only if the spin J of an irreducible representation has the form J = f ( 1 +41) = i, $, P,. . . . For any SU(M group in D =2n, the global-anomaly coefficients can be expressed in terms of so-called unstable James numbers of Stiefel manifold SU(n +l)/SU(n -k) and generalized Dynkin indices Q. +,(a) for SU(n + 1).

Some Global Aspects of Gauge Anomalies of Semisimple Gauge Groups and Fermion Generations in Gut and Superstring Theories

“…A simple way to see a j * for our purpose is to note that the top row is exact for any subgroup SU(2) of SU (8). We can choose the SU(2) in the top row as identical as one of the SU(2) subgroups from the 2nd row, then j is an isomorphism into a SU(2) subgroup in the 2nd row or j * is an isomorphism from Z 2 into one of the Z 2 's in the 2nd row.…”

“…This formula may need to be slightly modified for G = SO(2n + 2) with n = odd for reasons explained in ref. 8, but it will not so relevent to our discussions here. We note here that Eq.…”

“…Therefore, with Q 2 ( ) = 1 for SU (8) and the assumption of the topological result which will be proven in details in the next section, the anomaly coefficient in our consideration of H = (SU(2)) 3 in the rep ω = (2, 2, 2) with 2 denoting the corresponding dimensions is A(ω) = −1, so that it has a Z 2 global gauge anomaly.…”

“…Furthermore, the local anomaly-free condition Eq. (5) is automatically obeyed for the H. Therefore, the possible global gauge anomaly for the gauge group H = (SU(2)) 3 may be studied as the local gauge anomalies in the G = SU (8).…”

More anomaly free models of six-dimensional gauged supergravity

“…The conditions for the cancellation of global gauge anomaly have been investigated through the works of [27,28,29,30,31] for the case of Weyl spinors. The conditions with symplectic Majorana Weyl spinors in six dimensions seem to be absent in the literature, so we will give the derivation in appendix B.…”