Exotic supergravities and the Swampland

In six dimensions, there is an exotic $\mathcal{N}=(4,0)$ supermultiplet that contains only fields of spin $\leq 2$, but no graviton, and that on a circle reduces to 5D $\mathcal{N}=4$ supergravity. It has been proposed that, if suitable interactions exist, the $(4,0)$ theory might provide a consistent alternative UV completion for $\mathcal{N}=4$ 5D supergravity, realizing a supersymmetric version of asymptotic safety. In this note we argue that any Lorentz-invariant $(4,0)$ theory (interacting or not) carries an exact global symmetry when compactified on $S^1$, and is therefore incompatible with the Swampland no global symmetries conjecture. Another example of exotic supergravity, the 6D $(3,1)$ theory, does not have this problem. We study the general case and find that the only exotic spin-2 field that reduces to Einsteinian gravity and has no global symmetries when compactified on a high-dimensional torus is that of the $(3,1)$ theory. All other possibilities either yield several gravitons or have global symmetries.