In a previous paper, we proposed an entropy function for AdS BPS black holes in M-theory with general magnetic charges, resolving a long-standing puzzle about baryonic charges in three-dimensional holography and offering a prediction for the large-$\textit{N}$ limit of several partition functions whose saddle points have yet to be found. The entropy function is constructed from the master volume of the internal manifold. In this paper, we prove that the entropy of a general class of black holes based on toric geometry can indeed be reformulated as an $$ \mathcal{I} $$ -extremization problem, and we provide a set of examples. As an aside, we also simplify existing proofs of the equivalence between $\textit{a}$-, $\textit{c}$-, and $\textit{F}$-extremizations and their gravitational duals.

United Kingdom

Seyed Hosseini

Italy

Alberto Zaffaroni

$$ \mathcal{I} $$ -extremization for AdS

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.

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 perform a consistent reduction of 6d matter-coupled F(4) supergravity on a compact Riemann surface   <math> <msub> <mi>Σ</mi> <mi>g</mi> </msub> </math>  $$ {\Sigma}_{\mathfrak{g}} $$  of genus   <math> <mi>g</mi> </math>  $$ \mathfrak{g} $$ , at the level of the bosonic action. The result is an   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 gauged supergravity coupled to two vector multiplets and a single hypermultiplet. The four-dimensional model is holographically dual to the 3d superconformal field theories of class ℱ, describing different brane systems in massive type IIA and IIB wrapped on   <math> <msub> <mi>Σ</mi> <mi>g</mi> </msub> </math>  $$ {\Sigma}_{\mathfrak{g}} $$ . The naive reduction leads to a non-standard 4d mixed duality frame with both electric and magnetic gauge fields, as well as a massive tensor, that can be only described in the embedding tensor formalism. Upon a chain of electromagnetic dualities, we are able to determine the scalar manifolds and electric gaugings that uniquely specify the model in the standard supergravity frame. We then use the result to construct the first examples of static dyonic black holes in AdS6 and perform a microscopic counting of their entropy via the 5d topologically twisted index. Finally, we show the existence of further subtruncations to the massless sector of the 4d theory, such as the Fayet-Iliopoulos gauged T 3 model and minimal gauged supergravity. We are in turn able to find new asymptotically AdS4 solutions, providing predictions for the squashed S 3 partition functions and the superconformal and refined twisted indices of class ℱ theories.

4d F(4) gauged supergravity and black holes of class ℱ

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

The supersymmetric Rényi entropy across a spherical entangling surface in a d-dimensional SCFT with flavor defects is equivalent to a supersymmetric partition function on ℍ d−1 × 𝕊1, which can be computed exactly using localization. We consider the holographically dual BPS solutions in (d + 1)-dimensional matter coupled supergravity (d = 3, 5), which are charged hyperbolically sliced AdS black holes. We compute the renormalized on-shell action and the holographic supersymmetric Rényi entropy and show a perfect match with the field theory side. Our setup allows a direct map between the chemical potentials for the global symmetries of the field theories and those of the gravity solutions. We also discuss a simple case where angular momentum is added.

United States

France

Chiara Toldo

Supersymmetric Rényi entropy and charged hyperbolic black holes

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.

Geometry of   <math> <mi>I</mi> </math>  $$ \mathcal{I} $$ -extremization and black holes microstates

We provide a general formula for the refined topologically twisted index of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 gauge theories living on the world-volume of D3-branes at conical Calabi-Yau singularities in the Cardy limit. The index is defined as the partition function on T 2 ×   <math> <msubsup> <mi>S</mi> <mi>ω</mi> <mn>2</mn> </msubsup> </math>  $$ {S}_{\omega}^2 $$ , with a partial topological twist and a Ω-deformation along S , in the presence of background magnetic fluxes and fugacities for the global symmetries and can be used to study the properties of a class of BPS black strings. To this purpose, we find rotating domain- wall solutions of five-dimensional gauged supergravity interpolating between AdS5 and a near horizon region consisting of a warped fibration of BTZ over a sphere. We explicitly construct rotating domain-walls that can be embedded in AdS5 × S 5 by uplifting a class of four-dimensional rotating black holes. We then provide a microscopic explanation of the entropy of such black holes by using the refined topologically twisted index of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super Yang-Mills.