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 C-4 is equivalent to the I-extremization principle that leads to the microscopic counting for the entropy of magnetically charged black holes in AdS(4) x S-7. Also this equivalence holds off-shell.