Generalized Geometry in AdS/CFT and Volume Minimization

Abstract

We study the general structure of the AdS_5/CFT_4 correspondence in type IIB string theory from the perspective of generalized geometry. We begin by defining a notion of "generalized Sasakian geometry," which consists of a contact structure together with a differential system for three symplectic forms on the four-dimensional transverse space to the Reeb vector field. A generalized Sasakian manifold which satisfies an additional "Einstein" condition provides a general supersymmetric AdS_5 solution of type IIB supergravity with fluxes. We then show that the supergravity action restricted to a space of generalized Sasakian structures is simply the contact volume, and that its minimization determines the Reeb vector field for such a solution. We conjecture that this contact volume is equal to the inverse of the trial central charge whose maximization determines the R-symmetry of any four-dimensional N=1 superconformal field theory. This variational procedure allows us to compute the contact volumes for a predicted infinite family of solutions, and we find perfect agreement with the central charges and R-charges of BPS operators in the dual mass-deformed generalized conifold theories.

70 pages, 4 figures

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• Sasaki-Einstein Manifolds and Volume Minimisation
• Lectures on Generalized Complex Geometry for Physicists
• Generalized non-supersymmetric flux vacua
• Reformulating Supersymmetry with a Generalized Dolbeault Operator
• Topological sigma-models with H-flux and twisted generalized complex manifolds
• Gravity duals to deformed SYM theories and Generalized Complex Geometry
• Generalized Kaehler Potentials from Supergravity
• Conformal blocks for AdS5 singletons
• The central charge of supersymmetric AdS_5 solutions of type IIB supergravity

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• The exceptional generalised geometry of supersymmetric AdS flux backgrounds
• Free Energy vs Sasaki-Einstein Volume for Infinite Families of M2-Brane Theories
• On AdS${}_3$ solutions of Type IIB
• The free energy of N=2 supersymmetric AdS_4 solutions of M-theory
• Generalised Structures for $\mathcal{N}=1$ AdS Backgrounds
• Wilson loops and the geometry of matrix models in AdS_4/CFT_3
• A Deformation of Sasakian Structure in the Presence of Torsion and Supergravity Solutions
• Supersymmetric AdS backgrounds and weak generalised holonomy
• $N=(2,0)$ AdS${}_3$ Solutions of M-theory
• Integrable Hopf twists, marginal deformations and generalised geometry
• Can you hear the shape of dual geometries?