Toric geometry and the dual of ${\cal I}$-extremization

Understand

We consider $d=3$, $\mathcal{N}=2$ gauge theories arising on membranes sitting at the apex of an arbitrary toric Calabi-Yau 4-fold cone singularity that are then further compactified on a Riemann surface, $\Sigma_g$, with a topological twist that preserves two supersymmetries.

• If the theories flow to a superconformal quantum mechanics in the infrared, then they have a $D=11$ supergravity dual of the form AdS$_2\times Y_9$, with electric four-form flux and where $Y_9$ is topologically a fibration of a Sasakian $Y_7$ over $\Sigma_g$.
• These $D=11$ solutions are also expected to arise as the near horizon limit of magnetically charged black holes in AdS$_4\times Y_7$, with a Sasaki-Einstein metric on $Y_7$.
• We show that an off-shell entropy function for the dual AdS$_2$ solutions may be computed using the toric data and K\"ahler class parameters of the Calabi-Yau 4-fold, that are encoded in a master volume, as well as a set of integers that determine the fibration of $Y_7$ over $\Sigma_g$ and a K\"ahler class parameter for $\Sigma_g$.