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Descrizione
In the past two years there has been a surge in the interest towards low dimensional gauged supergravity solutions where the spacetime metric includes a $2\text{d}$ orbifold known as spindle. Topologically a spindle is a 2-sphere, but it has conical singularities at the two poles. Remarkably, uplifting such solutions to type IIB/$11\text{d}$ supergravity on Sasaki-Einstein manifolds leads to perfectly smooth geometries. In this talk, I will introduce some recent developments stemming from applying the geometric extremization procedure on generic $AdS_2 \times Y_9$ and $AdS_3 \times Y_7$ geometries, where $Y_9$ and $Y_7$ are fibrations of respectively $7\text{d}$ and $5\text{d}$ Sasaki-Einstein manifolds $X_7$ and $X_5$ over the spindle $\Sigma$. When put on-shell, such geometries are solutions of M-theory and type IIB supergravity respectively, and they are expected to arise as near horizon limit of supersymmetric magnetically charged accelerating $AdS_4$ black holes uplifted on $X_7$ and supersymmetric accelerating $AdS_5$ black strings uplifted on $X_5$. The result is a gravitational block formula for respectively the entropy function of the $AdS_4$ black holes and the trial central charge of the $2\text{d}$ $\mathcal {N}=(2,0)$ SCFTs dual to the $AdS_3$ solutions. This formula looks like a sum of two contributions ("blocks") localized over the two poles of the spindle that depend only on geometric data of the the fibers $X_7$ and $X_5$ as well as on how these are twisted over $\Sigma$. Remarkably, by algebraically extremizing this quantity over the possible R-symmetry vectors one can obtain the on-shell entropy/central charge without ever having to solve the supergravity equations of motion.
