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, Σg , 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 Σg . 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 Σg and a Kähler class parameter for Σg . 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.
Highlights
It can be obtained via c-extremization [3]
We 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
The best understood examples are those associated with Sasaki-Einstein (SE) geometry, the class of AdS5 × SE5 solutions of type IIB and the AdS4 × SE7 solutions of D = 11 supergravity that are dual to N = 1 SCFTs in d = 4 and N = 2 SCFTs in d = 3, respectively
Summary
The complex structure pairs the radial vector r∂r with a canonically defined vector ξ. The vector ξ has unit norm and defines a foliation Fξ of Y7. For the class of geometries of interest [5], we require the vector ξ to be a Killing vector for the metric on Y7, with ds27 = η2 + ds26(ω) ,. Where the metric ds26(ω) transverse to the foliation Fξ is conformally Kahler, with Kahler two-form ω. We choose a basis so that the holomorphic (4, 0)-form has unit charge under ∂φ1 and is uncharged under ∂φi, i = 2, 3, 4. Where ρ denotes the Ricci two-form of the transverse Kahler metric, and [ρ] = 2πcB1 , where cB1 is the basic first Chern class of the foliation
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