We study quantum gravity corrections to the no-boundary wavefunction describing a universe with spatial topology $\textit{S}$ × $\textit{S}$ . It has been suggested that quantum effects become increasingly important when the size of the circle is large relative to the sphere. In this paper, we confirm this claim by an explicit four-dimensional one-loop calculation of the gravitational path integral preparing such a state. In the process, we clarify some aspects of the gravitational path integral on complex spacetimes. These quantum corrections play a crucial role in ensuring that the norm of the wavefunction is naturally expressed in terms of a path integral over $\textit{S}$ × $\textit{S}$ at the classical level. We extend some of the analysis to more general spatial topologies, as well as to the inclusion of fermions.

United States

Gustavo Turiaci

Chih-Hung Wu

The wavefunction of a quantum $\textit{S}$

The gravitational path integral on $\textit{S}$ × $\textit{S}$ can be interpreted either as evaluating a contribution to the norm of the Hartle-Hawking wavefunction conditional on spatial $\textit{S}$ × $\textit{S}$ topology, or the pair creation rate of black holes in de Sitter. Both interpretations are distinguished at the quantum level. The former requires the path integral to be real and the latter to be imaginary. We develop a formalism to efficiently compute the phase of the gravitational path integral on Einstein spaces. We apply it to a broad class of spacetimes and in particular $\textit{S}$ × $\textit{S}$ , finding it to be real and positive. We generalize some of the analysis to cases with charge and rotation.

Xiaoyi Shi

The phase of the gravitational path integral

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.

Luca Iliesiu

United Kingdom

Sameer Murthy

Revisiting the logarithmic corrections to the black hole entropy

The spectrum and dynamics of near-extremal black holes is strongly modified by quantum effects at low temperatures. When the extremal limit does not preserve any supersymmetry, the density of states goes to zero at extremality and no extremal black holes remain. However, when the extremal limit is supersymmetric, a large microscopic degeneracy survives and there is a gap to the first excited black hole visible from gravity. In this article we study large $\textit{N}$ quantum mechanical models where supersymmetry is explicitly broken, allowing us to interpolate between these two qualitatively different pictures. We propose and analyze deformations of $$ \mathcal{N} $$ = 2 SYK models with such a pattern of (super) symmetry breaking which violate the U(1) $\textit{R}$-symmetry. These models feature a lifting of the BPS degeneracy and a closing of the spectral gap, and we further show that the large $\textit{N}$ soft effective action is given by a modification of the $$ \mathcal{N} $$ = 2 Schwarzian theory in which the U(1) mode becomes massive.

Matthew Heydeman

Yifei Liu

Supersymmetry breaking in SYK and the black hole spectrum

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

Jan Boruch

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

Near-extremal black holes are subject to large quantum effects, which modify their low-temperature thermodynamic behavior. Hitherto, these quantum effects were analyzed by separating the geometry into the near-horizon region and its exterior. It is desirable to understand and reproduce such corrections from the full higher-dimensional asymptotically flat or AdS geometry’s perspective. We address this question in this article and fill this gap. Specifically, we find off-shell eigenmodes of the quadratic fluctuation operator of the Euclidean gravitational dynamics, with eigenvalues that vanish linearly with temperature. We illustrate this for BTZ and neutral black holes with hyperbolic horizons in AdS in Einstein-Hilbert theory, and for the charged black holes in Einstein-Maxwell theory. The linear scaling with Matsubara frequency, which is a distinctive feature of the modes, together with the fact that their wavefunctions localize close to the horizon as we approach extremality, identifies them as responsible for the aforementioned quantum effects. We provide a contour prescription to deal with the sign indefiniteness of the Euclidean Einstein-Maxwell action, which we derive to aid our analysis. We also resolve a technical puzzle regarding modes associated with rotational isometries in stationary black hole spacetimes.

Maciej Kolanowski

Donald Marolf

Ilija Rakic

Mukund Rangamani

Looking at extremal black holes from very far away

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 examine the thermodynamics of a near-extremal Kerr black hole, and demonstrate that the geometry behaves as an ordinary quantum system with a vanishingly small degeneracy at low temperatures. This is in contrast with the classical analysis, which instead predicts a macroscopic entropy for the extremal Kerr black hole. Our results follow from a careful analysis of the gravitational path integral. Specifically, the low temperature canonical partition function behaves as   <math> <mi>Z</mi> <mo>∼</mo> <msup> <mi>T</mi> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </msup> <mspace width="0.25em"></mspace> <msup> <mi>e</mi> <mrow> <msub> <mi>S</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>c</mi> <mo>log</mo> <msub> <mi>S</mi> <mn>0</mn> </msub> </mrow> </msup> </math>  $$ Z\sim {T}^{\frac{3}{2}}\ {e}^{S_0+c\log {S}_0} $$ , with S 0 the classical degeneracy and c a numerical coefficient we compute. This is in line with the general expectations for non-supersymmetric near-extremal black hole thermodynamics, as has been clarified in the recent past, although cases without spherical symmetry have not yet been fully analyzed until now. We also point out some curious features relating to the rotational zero modes of the near-extremal Kerr black hole background that affects the coefficient c. This raises a puzzle when considering similar black holes in string theory. Our results generalize to other rotating black holes, as we briefly exemplify.

Thermodynamics of the near-extremal Kerr spacetime

The gravitational path integral can be used to compute the number of black hole states for a given energy window, or the free energy in a thermal ensemble. In this article we explain how to use the gravitational path integral to compute the separate number of bosonic and fermionic black hole microstates. We do this by comparing the partition function with and without the insertion of (−1)F. In particular we introduce a universal rotating black hole that contributes to the partition function in the presence of (−1)F. We study this problem for black holes in asymptotically flat space and in AdS, putting constraints on the high energy spectrum of holographic CFTs (not necessarily supersymmetric). Finally, we analyze wormhole contributions to related quantities.

Spin-statistics for black hole microstates

Generalizing previous results for   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 0 and   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1, we analyze   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 JT supergravity on asymptotically AdS2 spaces with arbitrary topology and show that this theory of gravity is dual, in a holographic sense, to a certain random matrix ensemble in which supermultiplets of different R-charge are statistically independent and each is described by its own   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 random matrix ensemble. We also analyze the case with a time-reversal symmetry, either commuting or anticommuting with the R-charge. In order to compare supergravity to random matrix theory, we develop an   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 analog of the recursion relations for Weil-Petersson volumes originally discovered by Mirzakhani in the bosonic case.