Generalized Geometry in AdS/CFT and Volume Minimization

TL;DR

This paper introduces generalized Sasakian geometry within AdS_5/CFT_4, linking supergravity solutions to volume minimization and central charge calculations in dual superconformal field theories.

Contribution

It defines generalized Sasakian structures and shows their role in determining supersymmetric AdS_5 solutions via volume minimization, connecting geometry with field theory data.

Findings

Contact volume minimization determines Reeb vector fields.

Contact volume inversely relates to trial central charge.

Computed volumes match field theory central charges and R-charges.

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.