than the gravitational force for all values of the scalar field. This must be true for any scalar in the theory, and not only for a particular set of WGC states. The extremal version of the equation yields a surprise: the solutions are compatible with towers of momenta and winding of an emerging dimension. Those towers become massless for $|\phi|\rightarrow \infty$, in agreement, with expectations from the swampland distance conjecture [@distance; @review; @vafafederico]. We interpret these towers as massive solitonic states which appear playing the role of WGC states. This structure is analogous to the behaviour in string theory in which towers of solitonic states (D-branes) become massless or tensionless for large fields. Thus the simplest theory you can think of, a scalar coupled to quantum gravity, secretly contains several features of string theory: emerging extra dimensions, winding strings and duality.

The scalar weak gravity conjecture
----------------------------------

The WGC case for purely scalar interactions is not obvious, since clear arguments based on blackhole physics are lacking. Still it has been argued that a variant of the WGC applies to axions with masses replaced by instanton actions and gauge couplings replaced by $1/f$, with $f$ the axion decay constant. For axions 