Toric geometry and the dual of ℐ-extremization

Abstract

We consider d = 3, $\mathcal{N}=2$ gauge theories arising on membranes sitting at the apex of an arbitrary toric Calabi-Yau 4-fold cone singularity that are then further compactified on a Riemann surface, Σ_g , with a topological twist that preserves two supersymmetries. If the theories flow to a superconformal quantum mechanics in the infrared, then they have a D = 11 supergravity dual of the form AdS₂ × Y₉, with electric four-form flux and where Y₉ is topologically a fibration of a Sasakian Y₇ over Σ_g . These D = 11 solutions are also expected to arise as the near horizon limit of magnetically charged black holes in AdS₄ × Y₇, with a Sasaki-Einstein metric on Y₇. We show that an off-shell entropy function for the dual AdS₂ solutions may be computed using the toric data and Kähler class parameters of the Calabi-Yau 4-fold, that are encoded in a master volume, as well as a set of integers that determine the fibration of Y₇ over Σ_g and a Kähler class parameter for Σ_g . We also discuss the class of supersymmetric AdS₃ × Y₇ solutions of type IIB supergravity with five-form flux only in the case that Y₇ is toric, and show how the off-shell central charge of the dual field theory can be obtained from the toric data. We illustrate with several examples, finding agreement both with explicit supergravity solutions as well as with some known field theory results concerning ℐ-extremization.