The $$ \frac{1}{2} $$ -BPS indices of $$ \mathcal{N} $$ = 4 Super Yang-Mills theory with unitary, orthogonal, and symplectic groups all admit $\textit{q}$-expansions suggesting an interpretation in terms of D- branes in the dual bulk AdS string theories. We present a derivation of these expansions in the corresponding bulk duals by quantizing the moduli space of $$ \frac{1}{2} $$ -BPS giant gravitons using supersymmetric localization, extending and clarifying our study in arxiv:2312.14921. We perform a detailed analysis of the one-loop fluctuations around the maximal giants (the fixed points), and show how the Hamiltonian analysis is recovered from the functional integral for the equivariant index. We show that the analytic continuation for these giant graviton expansions observed in the literature maps precisely to a wall-crossing phenomenon for the index. In the case of orthogonal and symplectic gauge groups, the $\textit{ℤ}$ quotient in the bulk leads to a corresponding projection in the $\textit{q}$-expansion. Additional terms in the expansion related to the Pfaffian operator arise from topologically stable branes in the bulk dual on AdS × $$ {\mathbbm{RP}}^5 $$ .

United Kingdom

Giorgos Eleftheriou

Sameer Murthy

Taiwan

Martí Rosselló

Localization and wall-crossing of giant graviton expansions in AdS

Logarithmic corrections to the entropy of extremal black holes have been successfully used to accurately match degeneracies from microscopic constructions to calculations of the gravitational path integral. In this paper, we revisit the problem of deriving such corrections for the case of extremal black holes, either non-supersymmetric or supersymmetric, and for near-extremal black holes. The zero-modes that are present at extremality are crucial, since their path integral cannot be treated quadratically and needs to be regulated. We show how the regulated result can be obtained by taking the zero-temperature limit of either the 4$\textit{d}$ Einstein-Maxwell or 4$\textit{d}$ supergravity path integral to find the Schwarzian or super-Schwarzian theories. This leads to drastically different estimates for the degeneracy of extremal black hole: it vanishes when the extremal limit does not preserve supersymmetry, while it reproduces Bekenstein-Hawking in the BPS case. In a companion paper, we discuss how such zero-modes affect the calculation of BPS black holes degeneracies, using supersymmetric localization for an exact computation of the gravitational path integral.

United States

Luca Iliesiu

Gustavo Turiaci

Revisiting the logarithmic corrections to the black hole entropy

The supersymmetric index of 5$\textit{d}$ black strings and spinning black holes in M-theory is related to that of 4$\textit{d}$ black holes in type IIA supergravity when both theories are compactified on the same Calabi-Yau threefold. We find the finite-temperature saddles for the 5$\textit{d}$ gravitational supersymmetric index by uplifting the recently found attractor saddles of the corresponding 4$\textit{d}$ index. We study uplifts for two types of geometries: 5$\textit{d}$ black holes and 5$\textit{d}$ black strings. For 5$\textit{d}$ black holes, the uplift guarantees that the index of 4$\textit{d}$ and 5$\textit{d}$ black holes match. For 5$\textit{d}$ black strings, the saddle reproduces the microscopic index at leading order in $\textit{G}$ , even without the conventional decoupling limit taken in AdS/CFT. In particular, when the temperature is set to be finite in the 5$\textit{d}$ flat space region, the black string index is computed from an asymptotically flat solution where the AdS throat is absent. Further, as the temperature is lowered and eventually becomes infinitesimally small in the flat space region, the solution admits a novel decoupling limit in which the AdS throat takes the form of a finite-temperature BTZ black hole that is known to compute the index in AdS /CFT . This represents the first step towards understanding holography for supersymmetric observables in flat space, away from the decoupling limit.

Jan Boruch

Spain

Roberto Emparan

The gravitational index of 5d black holes and black strings

We revisit the computation of the string partition function in AdS focussing on the appearance of spacetime (super) symmetries. We show how the asymptotic symmetries of the AdS spacetime, which generate the boundary (super) Virasoro currents, are captured by the one-loop partition sum. We use this to argue that the recent understanding of near-extremal black hole thermodynamics based on the gravitational path integral continues to hold for finite string length. Along the way we clarify some aspects of the AdS /CFT duality and, in particular, deduce which bulk gauge fields lead to boundary currents. We also explain how one can interpolate between supersymmetric and thermal (Atick-Witten) fermion boundary conditions in the target space by suitably tuning rotational chemical potentials in the string partition function.

Christian Ferko

Mukund Rangamani

Strings in AdS

The count of microstates for supersymmetric black holes is typically obtained from a supersymmetric index in weakly-coupled string theory. We find the saddles in the gravitational path integral corresponding to this index in a general theory of $$ \mathcal{N} $$ = 2 supergravity in asymptotically flat space. This saddle exhibits a new attractor mechanism which explains the agreement between the string theory index and the macroscopic entropy. These saddles are smooth, complex Euclidean spinning black holes that are supersymmetric but not extremal, i.e. they are formally finite-temperature solutions. With this new mechanism, the scalars and the electromagnetic fields get attracted to temperature- and moduli-independent values at the north and south poles of the rotating black hole, although they vary along the Euclidean horizon in a non-universal way. Further, although the area and the spin of the black hole depend non-trivially on the temperature and on the moduli, the free energy is essentially a function only of the black hole charges (apart from a trivial dependence on the temperature and the moduli through the BPS mass), and agrees with the string theory index.

Poland

New forms of attraction: attractor saddles for the black hole index

The fundamental heterotic string has a tower of BPS states whose supersymmetric index has an exponential growth in the charges. We construct the saddle-point of the gravitational path integral corresponding to this index. The saddle-point configuration is a supersymmetric rotating non-extremal Euclidean black hole. This configuration is singular in the two-derivative theory. We show that the addition of higher-derivative terms in four-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 supergravity resolves the singularity. In doing so, we extend the recently-developed “new attractor mechanism” to include the effect of higher-derivative terms. Remarkably, the one-loop, four-derivative F-term contribution to the prepotential leads to a precise match of the gravitational and microscopic index. We also comment, using the effective theory near the horizon, on the possibility of a string-size near-extremal black hole. Our results clarify the meaning of different descriptions of this system in the literature. The thermal state transitions to a winding condensate and a gas of strings without ever reaching a small black hole, while the index is captured by the rotating Euclidean black hole solution and is constant and thus smoothly connected to the microscopic ensemble.

Yiming Chen

Gravitational index of the heterotic string

We begin to develop the formalism of localization for the functional integral of supergravity on AdS3 × S2. We show how the condition of supersymmetry in the Euclidean ℍ3 /ℤ × S2 geometry naturally leads to a twist of the S2 around the time direction of AdS3. The twist gives us a five-dimensional Euclidean supergravity background dual to the elliptic genus of (0, 4) SCFT2 at the semiclassical level. On this background we set up the off-shell BPS equations for one of the Killing spinors, such that the functional integral of five-dimensional Euclidean supergravity on ℍ3 /ℤ × S2 localizes to its space of solutions. We obtain a class of solutions to these equations by lifting known off-shell BPS solutions of 4-dimensional supergravity on AdS2 × S2. In order to do this consistently, we construct and use a Euclidean version of the off-shell 4d/5d lift of  arXiv:1112.5371  , which could be of independent interest.

Axel Ciceri

South Korea

Imtak Jeon

Localization on AdS3 × S2. Part I. The 4d/5d connection in off-shell Euclidean supergravity

We exhibit an infinite family of supersymmetric phases in the three-dimensional ABJM superconformal field theory and the dual asymptotically AdS4 gravity. They are interpreted as partially deconfined phases which generalize the confined/pure AdS phase and deconfined/supersymmetric black hole phase. Our analysis involves finding a family of saddle-points of the superconformal index labelled by rational points (equivalently, roots of unity), separately in the bulk and boundary theories. In the ABJM theory we calculate the free energy of each saddle by the large-N asymptotic expansion of the superconformal index to all orders in perturbation theory near the saddle-point. We find that this expansion terminates at finite order. In the gravitational theory we show that there is a corresponding family of solutions, constructed by orbifolding the eleven-dimensional uplift of the supersymmetric black hole. The on-shell gravitational action of each orbifold agrees with the free energy of the corresponding saddle in the SCFT. We find that there are two saddles in the ABJM theory with the same entropy as the supersymmetric black hole, corresponding to the two primitive fourth-roots of unity, which causes macroscopic oscillations in the microcanonical index.

Pietro Genolini

Italy

Alejandro Cabo-Bizet

Supersymmetric phases of AdS4/CFT3

We present a formula for the quantum entropy of supersymmetric five-dimensional spinning black holes in M-theory compactified on CY 3, i.e., BMPV black holes. We use supersymmetric localization in the framework of off-shell five dimensional N = 2 supergravity coupled to I = 1, . . . , N  V  + 1 off-shell vector multiplets. The theory is governed at two-derivative level by the symmetric tensor   <math> <msub> <mi>C</mi> <mi>IJK</mi> </msub> </math>  $$ {\mathcal{C}}_{IJK} $$  (the intersection numbers of the Calabi-Yau) and at four-derivative level by the gauge-gravitational Chern-Simons coupling c  I  (the second Chern class of the Calabi-Yau). The quantum entropy is an N  V  +2-dimensional integral parameterised by one real parameter φ  I  for each vector multiplet and an additional parameter φ 0 for the gravity multiplet. The integrand consists of an action governed completely by   <math> <msub> <mi>C</mi> <mi>IJK</mi> </msub> </math>  $$ {\mathcal{C}}_{IJK} $$  and c  I , and a one-loop determinant. Consistency with the on-shell logarithmic corrections to the entropy, the symmetries of the very special geometry of the moduli space, and an assumption of analyticity constrains the one-loop determinant up to a scale-independent function g(φ 0). For g = 1 our result agrees completely with the topological M-theory conjecture of Dijkgraaf, Gukov, Neitzke, and Vafa for static black holes at two derivative level, and provides a natural extension to higher derivative corrections. For rotating BMPV black holes, our result differs from the DGNV conjecture at the level of the first quantum corrections.

India

Rajesh Gupta

Manya Sahni

Quantum entropy of BMPV black holes and the topological M-theory conjecture

We study the microcanonical superconformal index of <math><mfrac><mn>1</mn><mn>16</mn></mfrac></math>-supersymmetric operators in 4d <math><mi>N</mi><mo>=</mo><mn>4</mn></math> <math><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> super Yang-Mills (SYM) theory. We show, numerically for <math><mi>N</mi><mo>≤</mo><mn>10</mn></math>, that the large-charge asymptotics are consistent with the entropy of supersymmetric black holes in the dual anti de Sitter space. We then prove, using representation theory, that the index agrees precisely with the multigraviton index, when the charge is less than <math><mn>2</mn><mo>(</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo>)</mo></math>, and begins to deviate for larger values of charge. Thus the <math><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> SYM index interpolates between multigraviton values at small charges and black hole growth at large charges.

Growth of the <math><mfrac><mn>1</mn><mn>16</mn></mfrac></math>-BPS index in 4d <math><mi>N</mi><mo>=</mo><mn>4</mn></math> supersymmetric Yang-Mills theory