Abstract
We continue our study of a general class of mathcal{N} = 2 supersymmetric AdS3 x Y7 and AdS2 x Y9 solutions of type IIB and D = 11 supergravity, respectively. The geometry of the internal spaces is part of a general family of “GK geometries”, Y2n+l, n ≥ 3, and here we study examples in which Y2n+l fibres over a Kahler base manifold B2k, with toric fibres. We show that the flux quantization conditions, and an action function that determines the supersymmetric R-symmetry Killing vector of a geometry, may all be written in terms of the “master volume” of the fibre, together with certain global data associated with the Kähler base. In particular, this allows one to compute the central charge and entropy of the holographically dual (0, 2) SCFT and dual superconformal quantum mechanics, respectively, without knowing the explicit form of the Y7 or Y9 geometry. We illustrate with a number of examples, finding agreement with explicit supergravity solutions in cases where these are known, and we also obtain new results. In addition we present, en passant, new formulae for calculating the volumes of Sasaki-Einstein manifolds.
Highlights
An interesting arena for exploring the AdS/CFT correspondence, both from the geometric and the field theory points of view, is the class of supersymmetric AdS3 × Y7 solutions of type IIB supergravity of [1] and AdS2 × Y9 solutions of D = 11 supergravity of [2]
The geometry of the internal spaces is part of a general family of “GK geometries”, Y2n+1, n ≥ 3, and here we study examples in which Y2n+1 fibres over a Kahler base manifold B2k, with toric fibres
We show that the flux quantization conditions, and an action function that determines the supersymmetric R-symmetry Killing vector of a geometry, may all be written in terms of the “master volume” of the fibre, together with certain global data associated with the Kahler base
Summary
In the case of n = 3, i.e. Y7, this variational problem is a geometric realisation of the c-extremization principle for the dual (0, 2) d = 2 SCFTs proposed in [7] and allows one to obtain, for example, the central charge of the dual field theory without knowing the explicit AdS3 × Y7 solution. Show that the extremal problem of [4] can again be implemented using the master volume formula for the toric fibres, as in the cases studied in [9, 16]. We have included an appendix E, which explains how the formalism developed in [9, 16] and the present paper allows one to efficiently compute the Sasakian volume function of [5, 6]
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