Gravitational Blocks, Spindles and GK Geometry
Andrea Boido, Jerome P. Gauntlett, Dario Martelli, James Sparks
Abstract
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 $$_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow/> <mml:mn>3</mml:mn> </mml:msub> </mml:math> solutions of type IIB supergravity, sourced by D3-branes, and supersymmetric AdS $$_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow/> <mml:mn>2</mml:mn> </mml:msub> </mml:math> solutions of $$D=11$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>11</mml:mn> </mml:mrow> </mml:math> 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 $$_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow/> <mml:mn>3</mml:mn> </mml:msub> </mml:math> 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 $$_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow/> <mml:mn>2</mml:mn> </mml:msub> </mml:math> solutions our gravitational block formula allows us to obtain the entropy for supersymmetric, magnetically charged and accelerating black holes in AdS $$_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow/> <mml:mn>4</mml:mn> </mml:msub> </mml:math> .