We provide a first principles derivation of the microscopic entropy of a very general class of supersymmetric, rotating, and accelerating black holes in <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></math>. This is achieved by analyzing the large-<math><mi>N</mi></math> limit of the spindle index and completes the construction of the first example of a holographic duality involving supersymmetric field theories defined on orbifolds with conical singularities.

Italy

Edoardo Colombo

United Kingdom

Seyed Morteza Hosseini

Dario Martelli

Antonio Pittelli

Alberto Zaffaroni

Microstates of Accelerating and Supersymmetric <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></math> Black Holes from the Spindle Index

In a previous paper two of us (D.M. and A.Z.) proposed that a vast class of gravitational extremization problems in holography can be formulated in terms of the equivariant volume of the internal geometry, or of the cone over it. We substantiate this claim by analysing supergravity solutions corresponding to branes partially or totally wrapped on a four-dimensional orbifold, both in M-theory as well as in type II supergravities. We show that our approach recovers the relevant gravitational central charges/free energies of several known supergravity solutions and can be used to compute these also for solutions that are not known explicitly. Moreover, we demonstrate the validity of previously conjectured gravitational block formulas for M5 and D4 branes. In the case of M5 branes we make contact with a recent approach based on localization of equivariant forms, constructed with Killing spinor bilinears.

Federico Faedo

Equivariant volume extremization and holography

We introduce an entropy function for supersymmetric accelerating black holes in four-dimensional anti–de Sitter space that uplift on general Sasaki-Einstein manifolds <math><msub><mi>X</mi><mn>7</mn></msub></math> to solutions of <math><mi>M</mi></math> 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.

Andrea Boido

Jerome P. Gauntlett

James Sparks

Entropy Functions for Accelerating Black Holes

We construct a family of multidyonically charged and rotating supersymmetric <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><mi>Σ</mi></mrow></math> solutions of <math><mi>D</mi><mo>=</mo><mn>4</mn></math>, <math><mi>N</mi><mo>=</mo><mn>4</mn></math> gauged supergravity, where <math><mi>Σ</mi></math> 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 <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></math> 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 <math><msup><mi>S</mi><mn>7</mn></msup></math>, or its <math><mi>N</mi><mo>=</mo><mn>4</mn></math> orbifolds <math><msup><mi>S</mi><mn>7</mn></msup><mo>/</mo><mi>Γ</mi></math>, to smooth supersymmetric solutions to <math><mi>D</mi><mo>=</mo><mn>11</mn></math> supergravity, where the entropy is expected to count microstates of the theory on <math><mi>N</mi></math> M2-branes wrapped on a spinning spindle, in the large <math><mi>N</mi></math> limit.

Pietro Ferrero

Matteo Inglese

Multicharge accelerating black holes and spinning spindles

We construct supersymmetric   solutions of D = 6 gauged supergravity, where    is a two-dimensional orbifold known as a spindle. These uplift to solutions of massive type IIA supergravity using a general prescription, that we describe. We argue that these solutions correspond to the near-horizon limit of a system of N  f  D8-branes, together with N D4-branes wrapped on a spindle, embedded as a holomorphic curve inside a Calabi-Yau three-fold. The dual field theories are d = 3,   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFTs that arise from a twisted compactification of the d = 5,   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 USp(2N) gauge theory. We show that the holographic free energy associated to these solutions is reproduced by extremizing an off- shell free energy, that we conjecture to arise in the large N limit of the localized partition function of the d = 5 theories on   . We formulate a universal proposal for a class of off-shell free energies, whose extremization reproduces all previous results for branes wrapped on spindles, as well as on genus g Riemann surfaces Σg. We further illustrate this proposal discussing D4-branes wrapped on   , for which we present a supersymmetric    solution of D = 6 gauged supergravity along with the associated entropy function.

D4-branes wrapped on a spindle

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,   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 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.

Jerome Gauntlett

M5-branes wrapped on a spindle

We study the thermodynamics of <math><msub><mi>AdS</mi><mn>4</mn></msub></math> 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 <math><mi>D</mi><mo>=</mo><mn>11</mn></math> 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.

Davide Cassani

Thermodynamics of accelerating and supersymmetric <math><msub><mi>AdS</mi><mn>4</mn></msub></math> black holes

We study solutions in the Plebański–Demiański family which describe an accelerating, rotating, and dyonically charged black hole in <math><msub><mi>AdS</mi><mn>4</mn></msub></math>. These are solutions of <math><mrow><mi>D</mi><mo>=</mo><mn>4</mn></mrow></math> Einstein-Maxwell theory with a negative cosmological constant and hence minimal <math><mi>D</mi><mo>=</mo><mn>4</mn></math> gauged supergravity. It is well known that when the acceleration is nonvanishing the <math><mi>D</mi><mo>=</mo><mn>4</mn></math> black hole metrics have conical singularities. By uplifting the solutions to <math><mi>D</mi><mo>=</mo><mn>11</mn></math> supergravity using a regular Sasaki-Einstein seven-manifold, <math><mi>S</mi><msub><mi>E</mi><mn>7</mn></msub></math>, we show how the free parameters can be chosen to eliminate the conical singularities. Topologically, the <math><mi>D</mi><mo>=</mo><mn>11</mn></math> solutions incorporate an <math><mi>S</mi><msub><mi>E</mi><mn>7</mn></msub></math> fibration over a two-dimensional weighted projective space, <math><mrow><mi>W</mi><mi>C</mi><msubsup><mrow><mi>P</mi></mrow><mrow><mo>[</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>]</mo></mrow><mrow><mn>1</mn></mrow></msubsup></mrow></math>, also known as a spindle, which is labeled by two integers that determine the conical singularities of the <math><mi>D</mi><mo>=</mo><mn>4</mn></math> metrics. We also discuss the supersymmetric and extremal limit and show that the near horizon limit gives rise to a new family of regular supersymmetric <math><msub><mi>AdS</mi><mn>2</mn></msub><mo>×</mo><msub><mi>Y</mi><mn>9</mn></msub></math> solutions of <math><mi>D</mi><mo>=</mo><mn>11</mn></math> supergravity, which generalize a known family by the addition of a rotation parameter. We calculate the entropy of these black holes and argue that it should be possible to derive this from certain <math><mi>N</mi><mo>=</mo><mn>2</mn></math>, <math><mi>d</mi><mo>=</mo><mn>3</mn></math> quiver gauge theories compactified on a spinning spindle with the appropriate magnetic flux.

Juan Manuel Pérez Ipiña

Accelerating black holes and spinning spindles

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

D3-Branes Wrapped on a Spindle

We systematically analyze the large-N limit of the superconformal index of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 superconformal theories having a quiver description. The index of these theories is known in terms of unitary matrix integrals, which we calculate using the recently-developed technique of elliptic extension. This technique allows us to easily evaluate the integral as a sum over saddle points of an effective action in the limit where the rank of the gauge group is infinite. For a generic quiver theory under consideration, we find a special family of saddles whose effective action takes a universal form controlled by the anomaly coefficients of the theory. This family includes the known supersymmetric black hole solution in the holographically dual AdS5 theories. We then analyze the index refined by turning on flavor chemical potentials. We show that, for a certain range of chemical potentials, the effective action again takes a universal cubic form that is controlled by the anomaly coefficients of the theory. Finally, we present a large class of solutions to the saddle-point equations which are labelled by group homomorphisms of finite abelian groups of order N into the torus.