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

United Kingdom

Sameer Murthy

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

In AdS/CFT, two-sided black holes are described by states in the tensor product of two Hilbert spaces associated with the two asymptotic boundaries of the spacetime. Understanding how such a tensor product arises from the bulk perspective is an important open problem in holography, known as the factorisation puzzle. In this paper, we show how the Hilbert space of bulk states factorises due to non-perturbative contributions of spacetime wormholes: the trace over two-sided states with different particle excitations behind the horizon factorises into a product of traces of the left and right sides. This precisely occurs when such states form a complete basis for the bulk Hilbert space. We prove that the factorisation of the trace persists to all non-perturbative orders in 1$\textit{/G}$ , consequently providing a possible resolution to the factorisation puzzle from the gravitational path integral. In the language of von Neumann algebras, our results provide strong evidence that the algebra of one-sided observables transitions from a Type II or Type III algebra, depending on whether or not perturbative gravity effects are included, to a Type I factor when including non-perturbative corrections in the bulk.

Guanda Lin

Cynthia Yan

How the Hilbert space of two-sided black holes factorises

One of the fundamental problems in quantum gravity is to describe the experience of a gravitating observer in generic spacetimes. In this paper, we develop a framework for describing non-perturbative physics relative to an observer using the gravitational path integral. We apply our proposal to an observer that lives in a closed universe and one that falls behind a black hole horizon. We find that the Hilbert space that describes the experience of the observer is much larger than the Hilbert space in the absence of an observer. In the case of closed universes, the Hilbert space is not one-dimensional, as calculations in the absence of the observer suggest. Rather, its dimension scales exponentially with $${G}_{N}^{-1}$$ . Similarly, from an observer’s perspective, the dimension of the Hilbert space in a two-sided black hole is increased. We compute various observables probing the experience of a gravitating observer in this Hilbert space. We find that an observer experiences non-trivial physics in the closed universe in contrast to what it would see in a one-dimensional Hilbert space. In the two-sided black hole setting, our proposal implies that non-perturbative corrections to effective field theory for an infalling observer are suppressed until times exponential in the black hole entropy, resolving a recently-raised puzzle in black hole physics. While the framework that we develop is exemplified in the toy-model of JT gravity, most of our analysis can be extended to higher dimensions and, in particular, to generic spacetimes not admitting a conventional holographic description, such as cosmological universes or black hole interiors.

Ahmed Abdalla

Stefano Antonini

Adam Levine

The gravitational path integral from an observer’s point of view

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

What is the bulk Hilbert space of quantum gravity? In this paper, we resolve this problem in 2d JT gravity, both with and without matter, providing an explicit definition of a non-perturbative Hilbert space specified in terms of metric variables. The states are wavefunctions of the length and matter state, but with a non-trivial and highly degenerate inner product. We explicitly identify the null states, and discuss their importance for defining operators non-perturbatively. To highlight the power of the formalism we developed, we study the non-perturbative effects for two bulk linear operators that may serve as proxies for the experience of an observer falling into a two-sided black hole: one captures the length of an Einstein-Rosen bridge and the other captures the center-of-mass collision energy between two particles falling from opposite sides. We track the behavior of these operators up to times of order   <math> <msup> <mi>e</mi> <msub> <mi>S</mi> <mi>BH</mi> </msub> </msup> </math>  $$ {e}^{S_{\textrm{BH}}} $$ , at which point the wavefunction spreads to the complete set of eigenstates of these operators. If these observables are indeed good proxies for the experience of an infalling observer, our results indicate an O(1) probability of detecting a firewall at late times that is self-averaging and universal.

Henry Lin

Henry Maxfield

Márk Mezei

On the non-perturbative bulk Hilbert space of JT gravity

Understanding how to prepare and count black hole micro-states by using the gravitational path integral is one of the most important problems in quantum gravity. Nevertheless, a state-by-state count of black hole microstates is difficult because the apparent number of degrees of freedom available in the gravitational effective theory can vastly exceed the entropy of the black hole, even in the special case of BPS black holes. In this paper, we show that we can use the gravitational path integral to prepare a basis for the Hilbert space of all BPS black hole microstates. We find that the dimension of this Hilbert space computed by an explicit state count is in complete agreement with the degeneracy obtained from the Gibbons-Hawking prescription. Specifically, this match includes all non-perturbative corrections in 1/G  N . Such corrections are, in turn, necessary in order for this degeneracy of BPS states to match the non-perturbative terms in the 1/G  N  expansion in the string theory count of such microstates.

Constructing all BPS black hole microstates from the gravitational path integral

The extent to which quantum mechanical features of black holes can be understood from the Euclidean gravity path integral has recently received significant attention. In this paper, we examine this question for the calculation of the supersymmetric index. For concreteness, we focus on the case of charged black holes in asymptotically flat four-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 ungauged supergravity. We show that the gravity path integral with supersymmetric boundary conditions has an infinite family of Kerr-Newman classical saddles with different angular velocities. We argue that fermionic zero-mode fluctuations are present around each of these solutions making their contribution vanish, except for a single saddle that is BPS and gives the expected value of the index. We then turn to non-perturbative corrections involving spacetime wormholes and show that a fermionic zero mode is present in all these geometries, making their contribution vanish once again. This mechanism works for both single- and multi-boundary path integrals. In particular, only disconnected geometries without wormholes contribute to the gravitational path integral which computes the index, and the factorization puzzle that plagues the black hole partition function is resolved for the supersymmetric index. Finally, we classify all other single-centered geometries that yield non-perturbative contributions to the gravitational index of each boundary.

Murat Koloğlu

Supersymmetric indices factorize

We perform a bootstrap analysis of a mixed system of four-point functions of bosonic and fermionic operators in parity-preserving 3d CFTs with O(N) global symmetry. Our results provide rigorous bounds on the scaling dimensions of the O(N)-symmetric Gross-Neveu-Yukawa (GNY) fixed points, constraining these theories to live in isolated islands in the space of CFT data. We focus on the cases N = 1, 2, 4, 8, which have applications to phase transitions in condensed matter systems, and compare our bounds to previous analytical and numerical results.

Rajeev Erramilli

Petr Kravchuk

Aike Liu

David Poland

David Simmons-Duffin

The Gross-Neveu-Yukawa archipelago

We find models of two dimensional gravity that resolve the factorization puzzle and have a discrete spectrum, whilst retaining a semiclassical description. A novelty of these models is that they contain non-trivially correlated spacetime branes or, equivalently, nonlocal interactions in their action. Such nonlocal correlations are motivated in the low-energy gravity theory by integrating out UV degrees of freedom. Demanding factorization fixes almost all brane correlators, and the exact geometric expansion of the partition function collapses to only two terms: the black hole saddle and a subleading “half-wormhole” geometry, whose sum yields the desired discrete spectrum. By mapping the insertion of correlated branes to a certain double-trace deformation in the dual matrix integral, we show that factorization and discreteness also persist non-perturbatively. While in our model all wormholes completely cancel, they are still computationally relevant: self-averaging quantities, like the Page curve, computed in the original theory with wormholes, accurately approximate observables in our theory, which accounts for UV corrections. Our models emphasize the importance of correlations between different disconnected components of spacetime, providing a possible resolution to the factorization puzzle in any number of dimensions.