James Sparks

We explain how equivariant localization may be applied to AdS/CFT to compute various BPS observables in gravity, such as central charges and conformal dimensions of chiral primary operators, without solving the supergravity equations. The key ingredient is that supersymmetric AdS solutions with an R-symmetry are equipped with a set of equivariantly closed forms. These may in turn be used to impose flux quantization and compute observables for supergravity solutions, using only topological information and the Berline-Vergne-Atiyah-Bott fixed point formula. We illustrate the formalism by considering AdS 5× M 6 and AdS 3× M 8 solutions of D= 11 supergravity. As well as recovering results for many classes of well-known supergravity solutions, without using any knowledge of their explicit form, we also compute central charges for which explicit supergravity solutions have not been constructed.

We use equivariant localization to construct off-shell entropy functions for supersymmetric black holes in , gauged supergravity coupled to matter. This allows one to compute the black hole entropy without solving the supergravity equations of motion and provides a novel generalization of the attractor mechanism. We consider magnetically charged black holes in which have an near horizon geometry, where is a sphere or a spindle, and we also obtain entropy functions for ungauged supergravity as a simple corollary. We derive analogous results for black strings and rings in supergravity which have an near horizon geometry, and in this setting we derive an off-shell expression for the central charge of the dual , ~SCFT.

We show that supersymmetric supergravity solutions with an R-symmetry Killing vector are equipped with a set of equivariantly closed forms. Various physical observables may be expressed as integrals of these forms, and then evaluated using the Berline-Vergne-Atiyah-Bott fixed point theorem. We illustrate with a variety of holographic examples, including on-shell actions, black hole entropies, central charges, and scaling dimensions of operators. The resulting expressions depend only on topological data and the R-symmetry vector, and hence may be evaluated without solving the supergravity equations.

We introduce an entropy function for supersymmetric accelerating black holes in four-dimensional anti–de Sitter space that uplift on general Sasaki-Einstein manifolds X 7 to solutions of M theory. This allows one to compute the black hole entropy without knowing the explicit solutions. A dual holographic microstate counting would follow from computing certain supersymmetric partition functions of Chern-Simons-matter theories compactified on a spindle. We make a general prediction for a class of such partition functions in terms of “blocks,” with each block being constructed from the partition function on a three-sphere.

Supersymmetric, magnetically charged (and possibly accelerating) black holes in AdS that uplift on Sasaki-Einstein manifolds to M-theory have a dual matrix model description. The matrix model in turn arises by localization of the 3d SCFTs, dual to the AdS vacuum, on the black hole horizon geometry, which is a Riemann surface (or a spindle ). We identify the imaginary part of the continuously distributed eigenvalues in the matrix model, and their density function , with natural geometric quantities associated with the M-theory circle action on the near-horizon geometry AdS, the internal space being a fibration over (or ). Moreover, we argue that the points where is discontinuous match with the classical action of BPS probe M2-branes wrapping AdS and the M-theory circle. We illustrate our findings with the ABJM and ADHM theories, whose duals have , and some of their flavoured variants corresponding to other toric .

We consider d= 2, N=(0, 2) SCFTs that can arise from M5-branes wrapping four-dimensional, complex, toric manifolds and orbifolds. We use equivariant localization to compute the off-shell central charge of the dual supergravity solutions, obtaining a result that can be written as a sum of gravitational blocks and precisely agrees with a field theory computation using anomaly polynomials and c-extremization.

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 solutions of type IIB supergravity, sourced by D3-branes, and supersymmetric AdS solutions of 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 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 …

We construct supersymmetric AdS 3 solutions of D= 11 supergravity, dual to d= 2,=(0, 2) SCFTs, that are associated with M5-branes wrapping two different four-dimensional orbifolds. In one case the orbifold is a spindle fibred over another spindle, while in the other it is a spindle fibred over a Riemann surface with genus g> 1. We show that the central charges of the d= 2 SCFTs calculated from the gravity solutions agree with field theory computations using anomaly polynomials. The new D= 11 solutions are obtained after constructing a new consistent Kaluza-Klein truncation of maximal D= 7 gauged supergravity reduced on a spindle down to D= 5 minimal gauged supergravity.

We construct a family of multidyonically charged and rotating supersymmetric AdS 2× Σ solutions of D= 4, N= 4 gauged supergravity, where Σ is a sphere with two conical singularities known as a spindle. We argue that these arise as near horizon limits of extremal dyonically charged rotating and accelerating supersymmetric black holes in AdS 4 that we conjecture to exist. We demonstrate this in the nonrotating limit, constructing the accelerating black hole solutions and showing that the nonspinning spindle solutions arise as the near horizon limit of the supersymmetric and extremal subclass of these black holes. From the near horizon solutions we compute the Bekenstein-Hawking entropy of the black holes as a function of the conserved charges, and show that this may equivalently be obtained by extremizing a simple entropy function. For appropriately quantized magnetic fluxes, the solutions uplift on S 7, or its N …

In the context of holography, we analyse aspects of supersymmetric geometries based on two-dimensional orbifolds known as spindles. By analysing spin c spinors on a spindle with an azimuthal rotation symmetry we show that under rather general conditions there are just two possibilities, called the ‘twist’and the ‘anti-twist’, which are determined by the quantized magnetic flux through the spindle. A special case of the twist is the standard topological twist which is associated with constant and chiral spinors. We construct solutions of D= 5 and D= 4 STU gauged supergravity theories that are dual to D3-branes and M2-branes wrapping spindles, respectively, which realize both the anti-twist, as seen before, but also the twist. For the wrapped D3-brane solutions we reproduce the central charge of the gravity solution from the dual field theory by analysing the anomaly polynomial of= 4 SYM theory. We also discuss …

We study the thermodynamics of AdS 4 black hole solutions of Einstein-Maxwell theory that are accelerating, rotating, and carry electric and magnetic charges. We focus on the class for which the black hole horizon is a spindle and can be uplifted on regular Sasaki-Einstein spaces to give solutions of D= 11 supergravity that are free from conical singularities. We use holography to calculate the Euclidean on-shell action and to define a set of conserved charges which give rise to a first law. We identify a complex locus of supersymmetric and nonextremal solutions, defined through an analytic continuation of the parameters, upon which we obtain a simple expression for the on-shell action. A Legendre transform of this action combined with a reality constraint then leads to the Bekenstein-Hawking entropy for the class of supersymmetric and extremal black holes.

We study solutions in the Plebański–Demiański family which describe an accelerating, rotating, and dyonically charged black hole in AdS 4. These are solutions of D= 4 Einstein-Maxwell theory with a negative cosmological constant and hence minimal D= 4 gauged supergravity. It is well known that when the acceleration is nonvanishing the D= 4 black hole metrics have conical singularities. By uplifting the solutions to D= 11 supergravity using a regular Sasaki-Einstein seven-manifold, S E 7, we show how the free parameters can be chosen to eliminate the conical singularities. Topologically, the D= 11 solutions incorporate an S E 7 fibration over a two-dimensional weighted projective space, W C P [n−, n+] 1, also known as a spindle, which is labeled by two integers that determine the conical singularities of the D= 4 metrics. We also discuss the supersymmetric and extremal limit and show that the near horizon limit …

We construct families of supersymmetric AdS 3× Y 7 and AdS 3× Y 8 solutions to type IIB string theory and M-theory, respectively. Here Y 7 is an S 5 fibration over Σ, while Y 8 is an S 4 fibration over Σ g× Σ, where Σ g is a Riemann surface of genus g> 1 and Σ is a two-dimensional orbifold known as a spindle. We interpret the solutions as near-horizon limits of N D3-branes wrapped on Σ and N M5-branes wrapped on Σ g× Σ, respectively. These are holographically dual to d= 2,(0, 2) SCFTs, and we show that the central charge and superconformal R-symmetry of the gravity solutions agree with dual field theory calculations.

We construct supersymmetric AdS 3× Σ solutions of minimal gauged supergravity in D= 5, where Σ is a two-dimensional orbifold known as a spindle. Remarkably, these uplift on S 5, or more generally on any regular Sasaki-Einstein manifold, to smooth solutions of type IIB supergravity. The solutions are dual to d= 2, N=(0, 2) SCFTs and we show that the central charge for the gravity solution agrees with a field theory calculation associated with D3-branes wrapped on Σ.

We construct supersymmetric AdS 5× Σ solutions of D= 7 gauged supergravity, where Σ is a two-dimensional orbifold known as a spindle. These uplift on S 4 to solutions of D= 11 supergravity which have orbifold singularites. We argue that the solutions are dual to d= 4,= 1 SCFTs that arise from N M5-branes wrapped on a spindle, embedded as a holomorphic curve inside a Calabi-Yau three-fold. In contrast to the usual topological twist solutions, the superconformal R-symmetry mixes with the isometry of the spindle in the IR, and we verify this via a field theory calculation, as well as reproducing the gravity formula for the central charge.

We study the holographic dual to cc-extremization for 2d (0, 2)(0, 2) superconformal field theories (SCFTs) that have an AdS_33 dual realized in Type IIB with varying axio-dilaton, ie F-theory. M/F-duality implies that such AdS_33 solutions can be mapped to AdS_22 solutions in M-theory, which are holographically dual to superconformal quantum mechanics (SCQM), obtained by dimensional reduction of the 2d SCFTs. We analyze the corresponding map between holographic cc-extremization in F-theory and\mathcal {I} ℐ-extremization in M-theory, where in general the latter receives corrections relative to the F-theory result.

While the study of bordered (pseudo-) holomorphic curves with boundary on Lagrangian submanifolds has a long history, a similar problem that involves (special) Lagrangian submanifolds with boundary on complex surfaces appears to be largely overlooked in both physics and math literature. We relate this problem to geometry of coassociative submanifolds in G 2 holonomy spaces and to Spin (7) metrics on 8-manifolds with T 2 fibrations. As an application to physics, we propose a large class of brane models in type IIA string theory that generalize brane brick models on the one hand and 2d theories T [M 4] on the other.

We study the holographic dual to -extremization for 2d superconformal field theories (SCFTs) that have an AdS dual realized in Type IIB with varying axio-dilaton, ie F-theory. M/F-duality implies that such AdS solutions can be mapped to AdS solutions in M-theory, which are holographically dual to superconformal quantum mechanics (SCQM), obtained by dimensional reduction of the 2d SCFTs. We analyze the corresponding map between holographic -extremization in F-theory and -extremization in M-theory, where in general the latter receives corrections relative to the F-theory result.