Gravitational Blocks, Spindles and GK Geometry

TL;DR

This paper develops a gravitational block formula for supersymmetric AdS solutions in supergravity, enabling the computation of actions, entropies, and holographic quantities for various wrapped brane configurations and generalizing previous results.

Contribution

It introduces a new gravitational block formula for GK geometry that unifies and extends calculations of supersymmetric AdS solutions and black hole entropies.

Findings

Derived a gravitational block formula for AdS$_3$ and AdS$_2$ solutions.

Proved the off-shell supersymmetric action matches the dual field theory c-function.

Applied the formula to compute black hole entropy in AdS$_4$.

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 two-sphere. 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$_4$.