A geometric dual of c-extremization

Abstract

A bstract We consider supersymmetric AdS 3 × Y 7 and AdS 2 × Y 9 solutions of type IIB and D = 11 supergravity, respectively, that are holographically dual to SCFTs with (0 , 2) supersymmetry in two dimensions and $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 supersymmetry in one dimension. The geometry of Y 2 n +1 , which can be defined for n ≥ 3, shares many similarities with Sasaki-Einstein geometry, including the existence of a canonical R-symmetry Killing vector, but there are also some crucial differences. We show that the R-symmetry Killing vector may be determined by extremizing a function that depends only on certain global, topological data. In particular, assuming it exists, for n = 3 one can compute the central charge of an AdS 3 × Y 7 solution without knowing its explicit form. We interpret this as a geometric dual of c -extremization in (0 , 2) SCFTs. For the case of AdS 2 × Y 9 solutions we show that the extremal problem can be used to obtain properties of the dual quantum mechanics, including obtaining the entropy of a class of supersymmetric black holes in AdS 4 . We also study many specific examples of the type AdS 3 × T 2 × Y 5 , including a new family of explicit supergravity solutions. In addition we discuss the possibility that the (0 , 2) SCFTs dual to these solutions can arise from the compactification on T 2 of certain d = 4 quiver gauge theories associated with five-dimensional Sasaki-Einstein metrics and, surprisingly, come to a negative conclusion.