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**Toric geometry and the dual of \(\mathcal{I}\)-extremization.**
*(English)*
Zbl 1416.81155

Summary: 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, \({\Sigma}_g\), 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 \(\mathrm{AdS}_2 \times Y_9\), with electric four-form flux and where \(Y_9\) is topologically a fibration of a Sasakian \(Y_7\) over \({\Sigma}_g\). These \(D=11\) solutions are also expected to arise as the near horizon limit of magnetically charged black holes in \(\mathrm{AdS}_4 \times Y_7\), with a Sasaki-Einstein metric on \(Y_7\). We show that an off-shell entropy function for the dual \(\mathrm{AdS}_2\) 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 \(Y_7\) over \({\Sigma}_g\) and a Kähler class parameter for \({\Sigma}_g\). We also discuss the class of supersymmetric \(\mathrm{AdS}_3 \times Y_7\) solutions of type IIB supergravity 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 \(\mathcal{I}\)-extremization.

### MSC:

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

83C57 | Black holes |

83E30 | String and superstring theories in gravitational theory |

81T60 | Supersymmetric field theories in quantum mechanics |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

83E50 | Supergravity |

### Keywords:

AdS-CFT correspondence; gauge-gravity correspondence; black holes in string theory; supersymmetric gauge theory
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\textit{J. P. Gauntlett} et al., J. High Energy Phys. 2019, No. 6, Paper No. 140, 65 p. (2019; Zbl 1416.81155)

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