the condition that the decay constant f of an axion cannot exceed $M_{\text{p}}$ [@swampland; @WGC; @WGC1]. In the present case this comes about because otherwise the scalar interactions would be weaker than gravitation.

 The extremal equation: towers of states and an emerging dimension {#clave}
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We can consider the [*extremal*]{} case for a single scalar in which the scalar interactions equal the gravitational one. Then the constraint may be written as a differential equation on the field dependent mass $m^2(\phi)$: 2((m\^[2]{})\^[ ]{})\^2  -  m\^2( (m\^[2]{})\^[ ]{})-  = 0  . \[guay\] One obtains the extremal solutions for $m^2$ (with $\phi$ in $M_{\text{p}}$ units): m\^2() =    . For this to be a solution one must have $B\geq0$. Concerning $A$, we chose it positive (otherwise $m^2$ would always be negative for all $\phi$). Defining a field $R=e^{\phi/2}$, with kinetic metric $2(dR/R)^2$ one can rewrite the above expression in the more suggestive way m\^2 =   . For $N,M\not=0$ one can also write m\^2 =  m\_0\^2 ,  [M]{}\_[N,M]{}\^2=N\^2R\^2 +  . \[masilla\] Here ${\cal M}_{N,M}$ looks like the spectrum of a string compactification in a circle, with the duality invariance R 1/R  ;  M  N  . Note also that for large(small) $R$ one gets the limits: 