Are tiny gauge couplings out of the Swampland?

There is significant evidence suggesting that continuous global symmetries are always gauged in quantum gravity. However, very weakly gauged symmetries seem global to an effective field theory expansion in powers of Newton's constant. We show that, at least for Einsteinian quantum gravity on AdS, such extremely weak gaugings are indeed in the Swampland: Consistency with AdS black hole thermodynamics requires the bulk gauge coupling $g^2$ not to vanish faster than $\sim\exp(\ell^{d-1}/G)$, where $\ell$ is the $AdS_{d+1}$ radius and $G$ is Newton's constant as we take the $G\rightarrow0$ limit. This translates to a constraint in the dual large $N$ CFT, namely, that the two-point function coefficient of the current $C_J$ cannot grow faster than $\exp(N^2)$ in the large $N$ limit. We also recover a previously known logarithmic relationship between the cutoff of the effective field theory in AdS, Planck's mass, and the AdS radius