Geometric Extremization for Supersymmetric $AdS_3$ and $AdS_2$ Solutions

We consider supersymmetric $AdS_3\times Y_7$ solutions of type IIB supergravity dual to N=(0,2) SCFTs in d=2, as well as $AdS_2\times Y_9$ solutions of D=11 supergravity dual to N=2 supersymmetric quantum mechanics, some of which arise as the near horizon limit of supersymmetric, 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 SCFTs.

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) SCFTs analogous to the well known geometric duals of a-maximization of d=4 N=1 SCFTs and F-extremization of d=3 N=2 SCFTs in the context of Sasaki-Einstein geometry. 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.