JHEP06(2019)140 Published for SISSA by Springer Received: May 4, 2019 Accepted: June 20, 2019 Published: June 28, 2019 Toric geometry and the dual of I -extremization Jerome P. Gauntlett, a Dario Martelli b,1 and James Sparks c a Blackett Laboratory, Imperial College, Prince Consort Rd., London, SW7 2AZ, U.K. b Department of Mathematics, King’s College London, The Strand, London, WC2R 2LS, U.K. c Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, U.K. E-mail: j.gauntlett@imperial.ac.uk, dario.martelli@kcl.ac.uk, sparks@maths.ox.ac.uk Abstract: We consider d = 3, 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 super- symmetries. If the theories flow to a superconformal quantum mechanics in the infrared, then they have a D = 11 supergravity dual of the form AdS 2 × Y 9 , with electric four-form flux and where Y 9 is topologically a fibration of a Sasakian Y 7 over Σ g . These D = 11 solutions are also expected to arise as the near horizon limit of magnetically charged black holes in AdS 4 × Y 7 , with a Sasaki-Einstein metric on Y 7 . We show that an off-shell entropy function for the dual AdS 2 solutions may be computed using the toric data and K¨ ahler 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 Y 7 over Σ g and a K¨ ahler class parameter for Σ g . We also discuss the class of supersymmetric AdS 3 × Y 7 solutions of type IIB su- pergravity with five-form flux only in the case that Y 7 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. Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence, Black Holes in String Theory, Supersymmetric Gauge Theory ArXiv ePrint: 1904.04282 1 On leave at the Galileo Galilei Institute, Largo Enrico Fermi, 2, 50125 Firenze, Italy. Open Access,c  The Authors. Article funded by SCOAP 3 . https://doi.org/10.1007/JHEP06(2019)140