c-extremization from toric geometry

Understand

We derive a geometric formulation of the 2d central charge $c_r$ from infinite families of 4d $\mathcal{N}=1$ superconformal field theories topologically twisted on constant curvature Riemann surfaces.

• They correspond to toric quiver gauge theories and are associated to D3 branes probing five dimensional Sasaki-Einstein geometries in the AdS/CFT correspondence.
• We show that $c_r$ can be expressed in terms of the areas of the toric diagram describing the moduli space of the 4d theory, both for toric geometries with smooth and singular horizons.
• We also study the relation between a-maximization in 4d and c-extremization in 2d, giving further evidences of the mixing of the baryonic symmetries with the exact R-current in two dimensions.