<p>We propose an entropy function for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></math> BPS black holes in M theory with general magnetic charges, resolving in particular a long-standing puzzle about baryonic charges. The entropy function is constructed from a gravitational block defined solely in terms of topological data of the internal manifold. We show that the entropy of twisted black holes can always be reformulated as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">I</mi></math>-extremization problem—even in cases where existing large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi></math> field theory computations fail to provide an answer. Furthermore, we correctly reproduce the entropy for a class of known black holes with purely baryonic magnetic charges. Our results offer both a conjecture for the general gravitational block for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></math> black holes in M theory and a prediction for the large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi></math> limit of several partition functions whose saddle points have yet to be found.</p>

United Kingdom

Seyed Morteza Hosseini

Italy

Alberto Zaffaroni

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">I</mi></mrow></math> Extremization with Baryonic Charges

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.

Edoardo Colombo

Dario Martelli

Antonio Pittelli

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 study the large N limit of the refined topologically twisted index of three-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 gauge theories with a holographic dual, which is relevant for counting the number of microstates of rotating, magnetically charged and twisted AdS4 black holes. As a difference with previous computations, we perform a direct saddle point analysis of the relevant matrix model. Among other things we explicitly identify the distribution of gauge magnetic fluxes that contributes at large N. The index exhibits a large N factorized form in agreement with the supergravity expectations based on gravitational blocks and correctly reproduces the entropy of magnetically charged rotating AdS4 × S 7 black holes.

Seyed Hosseini

The large N limit of topologically twisted indices: a direct approach

We discuss the large N factorization properties of five-dimensional supersymmetric partition functions for CFT with a holographic dual. We consider partition functions on manifolds of the form   <math> <mi>ℳ</mi> <mo>=</mo> <msub> <mi>ℳ</mi> <mn>3</mn> </msub> <mo>×</mo> <msubsup> <mi>S</mi> <mi>ϵ</mi> <mn>2</mn> </msubsup> </math>  $$ \mathrm{\mathcal{M}}={\mathrm{\mathcal{M}}}_3\times {S}_{\upepsilon}^2 $$ , where ϵ is an equivariant parameter for rotation. We show that, when ℳ3 is a squashed three-sphere, the large N partition functions can be obtained by gluing elementary blocks associated with simple physical quantities. The same is true for various observables of the theories on   <math> <msub> <mi>ℳ</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>Σ</mi> <mi>g</mi> </msub> <mo>×</mo> <msup> <mi>S</mi> <mn>1</mn> </msup> </math>  $$ {\mathrm{\mathcal{M}}}_3={\Sigma}_{\mathfrak{g}}\times {S}^1 $$ , where   <math> <msub> <mi>Σ</mi> <mi>g</mi> </msub> </math>  $$ {\Sigma}_{\mathfrak{g}} $$  is a Riemann surface of genus 𝔤, and, with a natural assumption on the form of the saddle point, also for the partition function, corresponding to either the topologically twisted index or a mixed one. This generalizes results in three and four dimensions and correctly reproduces the entropy of known black objects in AdS6 ×  w  S 4 and AdS7 × S 4. We also provide the supersymmetric background and explicitly perform localization for the mixed index on   <math> <msub> <mi>Σ</mi> <mi>g</mi> </msub> <mo>×</mo> <msup> <mi>S</mi> <mn>1</mn> </msup> <mo>×</mo> <msubsup> <mi>S</mi> <mi>ϵ</mi> <mn>2</mn> </msubsup> </math>  $$ {\Sigma}_{\mathfrak{g}}\times {S}^1\times {S}_{\upepsilon}^2 $$ , filling a gap in the literature.

Japan

Itamar Yaakov

The joy of factorization at large N: five-dimensional indices and AdS black holes

Some AdS3 × M 7 type IIB vacua have been recently proposed to arise from D3-branes wrapped on a spindle, a sphere with conical singularities at the poles. We explicitly construct a generalization of these solutions corresponding to a class of electrically charged and rotating supersymmetric black strings in AdS5 × S 5 with general magnetic fluxes on the spindle. We then perform a counting of their microstates using the charged Cardy formula. To this purpose, we derive the general form of the anomaly polynomial of the dual   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (0, 2) CFT in two dimensions and we show that it can be obtained via a simple gluing procedure.

Bulgaria

Kiril Hristov

Rotating multi-charge spindles and their microstates

We provide a universal microscopic counting for the microstates of the asymptotically AdS black holes and black strings that arise as solutions of the half-maximal gauged supergravity in 4 and 5 dimensions. These solutions can be embedded in all M-theory and type II string backgrounds with an AdS vacuum and 16 supercharges and provide an infinite set of examples dual to <math><mi>N</mi><mo>=</mo><mn>2</mn></math> and <math><mi>N</mi><mo>=</mo><mn>4</mn></math> conformal field theories in four and three dimensions, respectively. The counting is universal and it is performed by either studying the large <math><mi>N</mi></math> limit of the relevant supersymmetric index of the dual field theory or by using the charged Cardy formula.

Universal AdS Black Holes in Theories with 16 Supercharges and Their Microstates

We derive the general anomaly polynomial for a class of two-dimensional CFTs arising as twisted compactifications of a higher-dimensional theory on compact manifolds ℳ d , including the contribution of the isometries of ℳ d . We then use the result to per- form a counting of microstates for electrically charged and rotating supersymmetric black strings in AdS5 × S 5 and AdS7 × S 4 with horizon topology BTZt⋉S 2 and BTZt⋉S 2 ×   <math> <msub> <mi>Σ</mi> <mi>g</mi> </msub> </math>  $$ {\Sigma}_{\mathfrak{g}} $$ , respectively, where   <math> <msub> <mi>Σ</mi> <mi>g</mi> </msub> </math>  $$ {\Sigma}_{\mathfrak{g}} $$  is a Riemann surface. We explicitly construct the latter class of solutions by uplifting a class of four-dimensional rotating black holes. We provide a microscopic explanation of the entropy of such black holes by using a charged version of the Cardy formula.

Yuji Tachikawa

Anomalies, black strings and the charged Cardy formula

We provide a unifying entropy functional and an extremization principle for black holes and black strings in AdS4 × S 7 and AdS5 × S 5 with arbitrary rotation and generic electric and magnetic charges. This is done by gluing gravitational blocks, basic building blocks that are directly inspired by the holomorphic blocks appearing in the factorization of supersymmetric partition functions in three and four dimensions. We also provide an explicit realization of the attractor mechanism by identifying the values of the scalar fields at the horizon with the critical points of the entropy functional. We give examples based on dyonic rotating black holes with a twist in AdS4 × S 7, rotating black strings in AdS5 × S 5, dyonic Kerr-Newman black holes in AdS4 × S 7 and Kerr-Newman black holes in AdS5 × S 5. In particular, our entropy functional extends existing results by adding rotation to the twisted black holes in AdS4 and by adding flavor magnetic charges for the Kerr-Newman black holes in AdS4. We also discuss generalizations to higher-dimensional black objects.

Gluing gravitational blocks for AdS black holes

The entropy of a class of asymptotically AdS4 magnetically charged BPS black holes can be obtained by extremizing the topologically twisted index of the dual three-dimensional field theory. This principle is known as   <math> <mi>I</mi> </math>  $$ \mathcal{I} $$ -extremization. A gravitational dual of   <math> <mi>I</mi> </math>  $$ \mathcal{I} $$ -extremization for a class of theories obtained by twisted compactifications of M2-branes living at a Calabi-Yau four-fold has been recently proposed. In this paper we investigate the relation between the two extremization principles. We show that the two extremization procedures are equivalent for theories without baryonic symmetries, which include ABJM and the theory dual to the non-toric Sasaki-Einstein manifold V 5,2. We then consider a class of quivers dual to M2-branes at toric Calabi-Yau four-folds for which the   <math> <mi>I</mi> </math>  $$ \mathcal{I} $$ -functional can be computed in the large N limit, and depends on three mesonic fluxes. We propose a gravitational dual for this construction, that we call mesonic twist, and we show that the gravitational extremization problem and   <math> <mi>I</mi> </math>  $$ \mathcal{I} $$ -extremization are equivalent. We comment on more general cases.