Sasaki-Einstein Geometry, GK Geometry and the AdS/CFT correspondence

TL;DR

The paper surveys Sasaki-Einstein (SE) and GK geometry in the context of AdS/CFT, highlighting how physical principles induce geometric extremization: volume minimization for SE metrics realizes a- and F-extremization in dual SCFTs, while a parallel GK extremization yields central charges and black-hole entropies across AdS$_3$, AdS$_2$, and spindle fibrations. It details explicit SE examples (e.g., $S^5$, $T^{1,1}$, $Y^{p,q}$, $L^{a,b,c}$) and toric constructions that render the extremization problem tractable from toric data, and it extends these ideas to GK geometry with a corresponding off-shell action $S_{SUSY}$ and master volumes that encode physical observables without explicit solutions. The work connects geometric data to field theory via quiver gauge theories, $a$-maximization, $F$- and $c$-extremization, and, in GK settings, $c$- and $I$-extremization, as well as to microstate counts and black-hole entropy through gravitational blocks. It also discusses obstructions (Bishop, Lichnerowicz) and stability notions (K-stability) that govern existence and uniqueness, providing a broad, unifying framework for AdS/CFT across diverse dimensions and fibrations.

Abstract

We review various aspects of Sasaki-Einstein and GK geometry, emphasising their similarities, interconnections and significance for the AdS/CFT correspondence. In particular, we highlight the key role that physical considerations have played in formulating geometric extremization principles, which have been instrumental in both understanding the geometry and identifying the corresponding dual field theories.