Fibred GK geometry and supersymmetric AdS solutions

Abstract

Abstract We continue our study of a general class of $$ \mathcal{N} $$ N = 2 supersymmetric AdS 3 x Y 7 and AdS 2 x Y 9 solutions of type IIB and D = 11 supergravity, respectively. The geometry of the internal spaces is part of a general family of “GK geometries”, Y 2n+l, n ≥ 3, and here we study examples in which Y 2n+l fibres over a Kahler base manifold B 2k , 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 Y 7 or Y 9 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.