Consistent Truncations from Duality Symmetries and Desingularization of Orbifold Uplifts

This paper is an extension of the results presented in \cite{Guarino:2024gke}. We study $G_S$-invariant subsectors of maximal gauged supergravities and show that such models can provide consistent truncations even when $G_S$ is not a symmetry of the original supergravity. We show that this construction is key to building pure supergravities around a supersymmetric AdS$_D$ solution. We illustrate this construction by building a consistent $\mathcal{N}=4$ subsector of the $D=4$ $\mathcal{N}=8$ $[\mathrm{SO}(6)\times \mathrm{SO}(1,1)]\ltimes \mathbb{R}^{12}$ gauged supergravity. We use this result to build the uplift of the multicharge spindle solutions in type IIB and we define a simple criterion for assessing the regularity of the uplift. We show that the type IIB uplift of the spindle is always non-regular, admitting eight codimension-six orbifold singularities. We apply the same criterion to other spindle uplifts, recovering known results and making predictions on the regularity of spindles on (quasi-)regular SE$_7$ manifolds.