Abstract

We consider supersymmetric AdS_3 x Y_7 solutions of type IIB supergravity dual to N=(0,2) SCF Ts in d=2, as well as AdS_2 x Y_9 solutions of D=11 supergravity dual to N=2 supersymmetric quantum mechanics, some of which arise as the near horizon limit of supersymmetric, magnetically charged black hole solutions in AdS_4. The geometry underlying these solutions was first identified in 2005-2007. Around that time infinite classes of explicit supergravity solutions were also found but, surprisingly, there was little progress in identifying the dual SCF Ts.

We will discuss new results concerning the Y_{2n+1} geometries that provide significant new insights. For the case of Y_7, there is a novel variation principle that allows one to calculate the central charge of the dual SCFT without knowing the explicit metric. This provides a geometric dual of c-extremization for d=2 N=(0,2) SCF Ts analogous to the well known geometric duals of a-maximization of d=4 N=1 SCF Ts and F-extremization of d=3 N=2 SCF Ts. In the case of Y_9 the variational principle can also be used to obtain properties of the dual N=2 quantum mechanics as well as the entropy of a class of supersymmetric black holes in AdS_4 thus providing a geometric dual of I-extremization. We have also developed some powerful new tools based on a novel kind of toric geometry, which lead to additional insights as well as the prospect of making further significant progress in this area.