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

## TL;DR

This paper proves the general equivalence between $c$-extremization and its gravitational dual for all toric Calabi-Yau three-folds, extending previous specific examples and connecting to M2-brane theories and black hole entropy counting.

## Contribution

It generalizes the proof of the equivalence between $c$-extremization and gravity duals to all toric Calabi-Yau three-folds, unifying previous case-by-case checks.

## Key findings

- Proves off-shell equivalence for all toric Calabi-Yau cases.
- Maps trial R-charges to gravitational extremization functional.
- Connects $c$-extremization with $ ext{I}$-extremization for M2-branes.

## Abstract

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.