Gravitational Blocks, Spindles and GK Geometry

Abstract

We derive a gravitational block formula for the supersymmetric action for a general class of supersymmetric AdS solutions, described by GK geometry. Extremal points of this action describe supersymmetric AdS 3 solutions of type IIB supergravity, sourced by D3-branes, and supersymmetric AdS 2 solutions of D = 11 supergravity, sourced by M2-branes. In both cases, the branes are also wrapped over a two-dimensional orbifold known as a spindle, or a twosphere. We develop various geometric methods for computing the gravitational block contributions, allowing us to recover previously known results for various explicit supergravity solutions, and to significantly generalize these results to other compactifications. For the AdS 3 solutions we give a general proof that our off-shell supersymmetric action agrees with an appropriate off-shell c-function in the dual field theory, establishing a very general exact result in holography. For the AdS 2 solutions our gravitational block formula allows us to obtain the entropy for supersymmetric, magnetically charged and accelerating black holes in AdS 4 .

Maximally symmetric nuts in 4d 𝒩 = 2 higher derivative supergravity

“…The interest in allowing conical defects to asymptotically AdS solutions in various dimensions has recently spiked due to the original work of [73,74] on the so-called spindle geometry, which was also shown to make sense holographically. The gluing rules applicable to black holes with spindle horizons were later proposed in [75] and generalized in various directions in [65] and [66]. From the point of view of the present work, focusing on the 4d perspective, the black spindles are backgrounds exhibiting two different fixed points, near which the metric is locally C × C/Z n 1 and C × C/Z n 2 , respectively.…”

“…In turn one can also investigate the quantum corrections coming on top of the higher derivative terms, see again [1,50], such that we complete the quantum gravitational block (conjectured in [2] via holography to be the complete Airy function for the AdS 4 ×S 7 /Z k example). It would likewise be very interesting to fully understand the analogous gravitational building blocks in different dimensions (see [18,65]) and even directly inside string compactifications via the internal geometry (see [66]). We hope to come back to all of these problems.…”