Kaluza-Klein tower and bubble nucleation in six dimensional Einstein-Maxwell theory

We study the implication of the distance and the cobordism conjecture on the 6-dimensional Einstein-Maxwell theory compactified on $S^2$. In this toy model, the radion potential is stabilized by the conspiracy of the curvature of $S^2$ and the flux through $S^2$ parametrized by $f$, and uplifted by the positive 6-dimensional cosmological constant parametrized by $\lambda$. When $\lambda=0$, the radion is stabilized at the anti-de Sitter (AdS) vacuum, which cannot be interpolated to the Minkowski vacuum since the Kaluza-Klein (KK) tower descends from UV in the vanishing limit of the 4-dimensional cosmological constant. For nonzero $\lambda$ which realizes the metastable de Sitter (dS) vacuum, as well as the AdS and the Minkowski vacuum, such an obstruction can be found provided the combination $f^2\lambda$ is fixed and the limit $\lambda\to 0$ is taken. Moreover, the 6-dimensional Einstein-Maxwell theory allows the transition between vacua through the nucleation of the bubble. In this case, the values of the 4-dimensional cosmological constant inside and outside the bubble are different as $f$ is changed at the bubble wall, while $\lambda$ remains unchanged. Regarding the AdS vacuum with the vanishing curvature radius as the `nothing', we find that the transition from the metastable dS vacuum to the nothing is not prevented by the descent of the KK tower since $f^2\lambda$ is not fixed.