Proving the equivalence of $c$-extremization and its gravitational dual for all toric quivers

The gravitational dual of $c$-extremization for a class of $(0,2)$ two-dimensional theories obtained by twisted compactifications of D3-brane gauge theories living at a toric Calabi-Yau three-fold has been recently proposed. The equivalence of this construction with $c$-extremization has been checked in various examples and holds also off-shell. In this note we prove that such equivalence holds for an arbitrary toric Calabi-Yau. We do it by generalizing the proof of the equivalence between $a$-maximization and volume minimization for four-dimensional toric quivers. By an explicit parameterization of the R-charges we map the trial right-moving central charge $c_r$ into the off-shell functional to be extremized in gravity. We also observe that the similar construction for M2-branes on ${\mathbb{C}}^4$ is equivalent to the $\mathcal{I}$-extremization principle that leads to the microscopic counting for the entropy of magnetically charged black holes in AdS${}_4\times S^7$. Also this equivalence holds off-shell.