ABSTRACT |
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 relevant geometry on $Y_{2n+1}$, $n\ge 3$, shares some similarities with Sasaki-Einstein geometry but also some key differences. This geometry was first identified in 2005-2007 and around that time infinite classes of explicit examples 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, analogous to volume minimisation for Sasaki-Einstein metrics, 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. |