Global Properties of the Conformal Manifold for S-Fold Backgrounds

We study a one-parameter family of $\mathcal{N}=2$ anti-de Sitter vacua with $\mathrm{U}(1{)}^2$ symmetry of gauged four-dimensional maximal supergravity, with dyonic gauge group $[\mathrm{S}\mathrm{O}(6)\times \mathrm{S}\mathrm{O}(1,1)]⋉{\mathbb{R}}^{12}$. These backgrounds are known to correspond to Type IIB S-fold solutions with internal manifold of topology $S^1\times S^5$. The family of AdS${}_4$ vacua is parametrized by a modulus $\chi$. Although $\chi$ appears non-compact in the four-dimensional supergravity, we show that this is just an artefact of the four-dimensional description. We give the 10-dimensional geometric interpretation of the modulus and show that it actually has periodicity of $\frac{2\pi }{T}$, which is the inverse radius of $S^1$. We deduce this by providing the explicit $D=10$ uplift of the family of vacua as well as computing the entire modulus-dependent Kaluza-Klein spectrum as a function of $\chi$. At the special values $\chi =0$ and $\chi =\frac{\pi }{T}$, the symmetry enhances according to $\mathrm{U}(1{)}^2\to \mathrm{U}(2)$, giving rise however to inequivalent Kaluza-Klein spectra. At $\chi =\frac{\pi }{T}$, this realizes a bosonic version of the “space invaders” scenario with additional massless vector fields arising from formerly massive fields at higher Kaluza-Klein levels.