On the class \( \mathcal{S} \) origin of spindle solutions

We analyse the backreacted geometry corresponding to a stack of M5-branes wrapped on a spindle, with a view towards precision tests of the dual $\mathcal{N}$ \( \mathcal{N} \) = 1 superconformal field theory. We carefully study the singular loci of the uplifted geometry and show that these correspond to C³/Z_n conical singularities. Therefore, these solutions present one of the first explicit realisations of honest locally $\mathcal{N}$ \( \mathcal{N} \) = 1 preserving punctures in class $\mathcal{S}$ \( \mathcal{S} \) . Additionally we study the symmetries and anomalies of the dual field theory through anomaly inflow and compute a variety of holographic observables including dimensions of BPS operators. This work paves the way for advancements in the study and identification of the precise dual field theories.