Populating the swampland : the case of U ( 1 ) 496 and E 8 × U ( 1 ) 248

For d = 10 N = 1 SUGRA coupled to d = 10 N = 1 SYM, anomaly cancellation places severe constraints on the allowed gauge groups. Besides the ones known to appear in string theory, only U(1) and E8 × U(1) are allowed. There are no known theories of quantum gravity that reduce in some limit to these two last supergravity theories, and in this note I present some evidence that those quantum theories might not exist. The first observation is that, upon compactification, requring that the quantum theory possesses a moduli space with finite volume typically implies the existence of singularities where the 4d gauge group is enhanced, but for these two theories that gauge enhancement is problematic from the 10d point of view. I also point out that while these four supergravity theories present repulson-type singularities, the known mechanism that repairs those singularities for the first two the non-Abelian enhançon is not available for the last two theories. In short, these two supergravity theories might be too Abelian for their own good.