# Toric geometry and the dual of $c$-extremization

## TL;DR

This paper develops a geometric method to compute central charges and R-charges for a broad class of 2D theories derived from D3-branes on toric Calabi-Yau singularities, matching field theory results via $c$-extremization.

## Contribution

It introduces a toric geometric approach to calculate supergravity central charges and R-charges, confirming their agreement with field theory $c$-extremization for various singularities.

## Key findings

- Supergravity formulas match field theory $c$-extremization results.
- The trial central charge $ ext{Z}$ in gravity matches the field theory trial $c$-function.
- Validated geometric formulas against known supergravity solutions.

## Abstract

We consider D3-brane gauge theories at an arbitrary toric Calabi-Yau 3-fold cone singularity that are then further compactified on a Riemann surface $\Sigma_g$, with an arbitrary partial topological twist for the global $U(1)$ symmetries. This constitutes a rich, infinite class of two-dimensional $(0,2)$ theories. Under the assumption that such a theory flows to a SCFT, we show that the supergravity formulas for the central charge and $R$-charges of BPS baryonic operators of the dual AdS$_3$ solution may be computed using only the toric data of the Calabi-Yau 3-fold and the topological twist parameters. We exemplify the procedure for both the $Y^{p,q}$ and $X^{p,q}$ 3-fold singularities, along with their associated dual quiver gauge theories, showing that the new supergravity results perfectly match the field theory results obtained using $c$-extremization, for arbitrary twist over $\Sigma_g$. We furthermore conjecture that the trial central charge $\mathscr{Z}$, which we define in gravity, matches the field theory trial $c$-function off-shell, and show this holds in non-trivial examples. Finally, we check our general geometric formulae against a number of explicitly known supergravity solutions.