We propose a heuristic for the brick wall in AdS/CFT: the location where a boundary mode’s local bulk energy reaches a (Planckian) UV cut-off. This accomplishes two things: (a) the brick wall is framed as a breakdown criterion for bulk effective field theory, and (b) the definition is boundary-anchored rather than horizon-anchored, aligning it with holography. Near the horizon, spacetime effectively gets cut-off due to blueshift relative to the boundary, and leads to normal modes. By directly computing these new modes for the BTZ black hole, we show that they are qualitatively unchanged from conventional ’t Hooftian brick wall normal modes in the relevant part of the spectrum — successfully reproducing black hole thermodynamics and exterior smooth-horizon correlators, under similar approximations. However, unlike ’t Hooft’s (and our own previous) calculations, we also do an $\textit{exact}$ numerical evaluation of the normal mode partition function. This allows us to identify a “little hierarchy” problem in the brick wall paradigm, irrespective of whether it is horizon-anchored or boundary-anchored: because the modes are not exactly degenerate in the $\textit{J}$-direction, the coefficient of the area law is slightly subleading, unless the brick wall is slightly trans-Planckian. One way to evade the problem is to increase the number of active species. While this is certainly a possibility in string theory, we argue that a natural resolution is to take into account the degrees of freedom $\textit{intrinsic}$ to the (stretched) horizon, as suggested by the recent results in [1]. We argue that this will lead to a dominant contribution from a quantum number associated to the $\textit{radial}$ direction, while retaining the successes of the $\textit{J}$-degenerate toy model. We discuss the possible significance of these observations for (a) quantum chaos in black holes, and (b) the fuzzball program.

India

Vishal Gayari

Chethan Krishnan

Pradipta Pathak

Holography, brick wall and a little hierarchy problem

Black hole normal modes have intriguing connections to logarithmic spectra, and the spectral form factor (SFF) of $\textit{E}$ = log $\textit{n}$ is the mod square of the Riemann zeta function $\textit{ζ}$($\textit{s}$). In this paper, we first provide an analytic understanding of the dip-ramp-plateau structure of $\textit{ζ}$($\textit{s}$) and show that the ramp at $$ \beta \equiv \mathfrak{R}(s)=0 $$ has a slope precisely equal to 1. The $\textit{s}$ = 1 pole of $\textit{ζ}$($\textit{s}$) can be viewed as due to a Hagedorn transition in this setting, and Riemann’s analytic continuation to $$ \mathfrak{R}(s)&lt;1 $$ provides the quantum contribution to the truncated log $\textit{n}$ partition function. This perspective yields a precise definition of $\textit{ζ}$($\textit{s}$) as the “full ramp after removal of the dip”, and allows an unambiguous determination of the Thouless time. For black hole microstates, the Thouless time is expected to be $$ \mathcal{O}(1) $$ — remarkably, the $\textit{ζ}$($\textit{s}$) also exhibits this behavior. To our knowledge, this is the first black hole-inspired toy model that has a demonstrably $$ \mathcal{O}(1) $$ Thouless time. In contrast, it is $$ \mathcal{O}\left(\log N\right) $$ in the SYK model and expected to be $$ \mathcal{O}\left({N}^{\#}\right) $$ in supergravity fuzzballs. We trace the origins of the ramp to a certain reflection property of the functional equation satisfied by $\textit{ζ}$($\textit{s}$), and suggest that it is a general feature of $\textit{L}$-functions — we find evidence for ramps in large classes of $\textit{L}$-functions. As an aside, we also provide an analytic determination of the slopes of (non-linear) ramps that arise in power law spectra using Poisson resummation techniques.

South Africa

Pallab Basu

Suman Das

An analytic zeta function ramp at the black hole Thouless time

<p>In the semiclassical (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi><mo stretchy="false">→</mo><mi>∞</mi></math>) limit, 4-point heavy-light-light-heavy correlators in 2D conformal field theory exhibit periodic Euclidean singularities. Periodic singularities in Euclidean time are a general feature of thermal correlators, even at weak coupling. Therefore, the bulk significance of this observation (in particular, the role of the horizon) is somewhat obscure. Explicit numerical computations of finite-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi></math> Virasoro blocks furthermore suggest that their departure from semiclassical blocks may begin already at half the period. In this paper, we provide a bulk understanding of these facts and clarify the role of the horizon. We present a bulk geodesic Witten diagram calculation of semiclassical Virasoro blocks in coordinates that are naturally adapted to the Bañados-Teitelboim-Zanelli black hole. This allows a bulk geometric interpretation for boundary time separation. In this language, half of a thermal time period is the boundary timescale at which the light operator geodesic straddles the Euclidean horizon, capturing both the role of the horizon and the associated timescale. This timescale arises in a calculation that does <i>not</i> involve a periodic thermal circle on the bulk or the boundary.</p>

Aaditya Datar

Virasoro blocks and trouble at the Euclidean horizon

The double null form of the Schwarzschild metric is usually arrived at by demanding Eddington-Finkelstein (EF) conditions at the horizon. This leads to certain logarithmic fall-offs that are too slow along null directions at $$ \mathcal{I} $$ , resulting in divergences in the covariant surface charges. These coordinates are therefore $\textit{not}$ asymptotically flat in a conventional sense. In this paper, we find a natural alternative double null form for Schwarzschild that is adapted to $$ {\mathcal{I}}^{+} $$ or $$ {\mathcal{I}}^{-} $$ instead of the horizon. In its final form, the metric has only power law fall-offs and fits into the recently introduced Special Double Null (SDN) gauge, with finite surface charges. One remarkable feature of SDN gauge is that spherical symmetry and vacuum Einstein equations allow an infinite number of asymptotic integration constants in the metric, on top of the mass. This is an apparent violation of Birkhoff’s theorem. We note however that all except two of these new parameters are absent in the charges, and therefore correspond to trivial hair. The remaining two parameters do show up in the charges, depending on the choice of allowed fall-offs. We provide an understanding of this observation — Birkhoff’s theorem fixes Schwarzschild only $\textit{up to diffeomorphisms}$, but diffeomorphisms need not vanish at infinity and can in principle become global symmetries. If such asymptotic diffeomorphisms are spherically symmetric, their associated soft modes can become Birkhoff hair. The relevant global symmetries here are certain hypertranslation shifts in the $\textit{v}$-coordinate at $$ {\mathcal{I}}^{+} $$ (and $\textit{u}$ at $$ {\mathcal{I}}^{-} $$ ), which are inaccessible in other gauges.

United States

Jude Pereira

Soft hair on Schwarzschild: a wrinkle in Birkhoff ’s theorem

We synthesize and sharpen various observations about sub-AdS holography in the literature to associate a central charge to sub-AdS scales of AdS×$\textit{X}$ geometries. A key ingredient in our proposal is the idea that the AdS length is the length of the long string in a stack of $\textit{N backreacted}$ D-branes. This allows us to make statements about sub-AdS scales by considering long strings in sub-stacks of $\textit{M}$ < $\textit{N}$ branes. Our proposal applies in general dimensions, connects with and refines previous ideas about sub-matrix deconfinement & long strings, makes crucial use of the compact space, and is consistent with the expected thermodynamics of small black holes localized on $\textit{X}$. Some of our arguments draw intuition from separating branes, and have natural connections to (heating up) the Coulomb branch of the gauge theory. We apply related ideas to non-conformal D-branes at zero and finite temperature, discuss the holographic bound and IR/UV duality in theories with 16 supercharges, and find broad consistency.

Abir Ghosh

Adinath Mahalingam

A central charge for sub-AdS holography

In 1+1 dimensions, the equations of motion of the Horowitz-Polchinski (HP) effective string have a re-writing in terms of a first order system. This is attributed to FZZ duality. In this note, we observe that a similar re-writing exists in higher dimensions, so that the degree of the dilaton-winding subsystem reduces to first order. The 1+1 first order equations emerge as a natural limit of the higher dimensional HP system in the cap region of the cigar. As a result, there is a critical value of the winding amplitude that matches with the 1+1 coset SCFT prediction. At this critical point, the cigar has a puncture at the Euclidean horizon and the $\textit{higher dimensional}$ black hole entropy is correctly reproduced by the winding condensate.

Sarthak Talukdar

A vestige of FZZ duality in higher dimensions

Defining finite entanglement entropy for a subregion in quantum field theory requires the introduction of two logically independent scales: an IR scale that controls the size of the subregion, and a UV cut-off. In AdS/CFT, the IR scale is the AdS lengthscale, the UV cut-off is the bulk radial cut-off, and the subregion is specified by dimensionless angles. This is the data that determines Ryu-Takayanagi surfaces and their areas in AdS/CFT. We argue that in asymptotically flat space there exists the notion of a “spi-subregion” that one can associate to spatial infinity (spi). Even though geometrically quite different from an AdS subregion, this angle data has the crucial feature that it allows an interpretation as a bi-partitioning of spi. Therefore, the area of the RT surface associated to the spi-subregion can be interpreted as the entanglement entropy of the reduced density matrix of the bulk state under this bi-partition, as in AdS/CFT. For symmetric spi-subregions, these RT surfaces are the waists of Asymptotic Causal Diamonds. In empty flat space they reduce to Rindler horizons, and are analogues of the AdS-Rindler horizons of Casini, Huerta &amp; Myers. We connect these results to previous work on minimal surfaces anchored to screens in empty space, but also generalize the discussion to the case where there are black holes in the bulk. The phases of black hole RT surfaces as the spi-subregion is varied, naturally connect with those of black holes (small and large) in AdS. A key observation is that the radial cut-off is associated to an IR scale in flat space — and in fact there are no UV divergences. We argue that this is consistent with previous suggestions that in sub-AdS scales the holographic duality is an IR/IR correspondence and that the degrees of freedom are not those of a local QFT, but those of long strings. Strings are of course, famously UV finite.

A holographic entanglement entropy at spi

Opinion is divided about the nature of state dependence in the black hole interior. Some argue that it is a necessary feature, while others argue it is a bug. In this paper, we consider the extended half-sided modular translation U (s 0) (with s 0 &gt; 0) of Leutheusser and Liu that takes us inside the horizon. We note that we can use this operator to construct a modular Hamiltonian H and a conjugation J on the infalling time-evolved wedges. The original thermofield double translates to a new cyclic and separating vector in the shifted algebra. We use these objects and the Connes’ cocycle to repeat Witten’s crossed product construction in this new setting, and to obtain a Type II ∞  algebra that is independent of the various choices, in particular that of the cyclic separating vector. Our emergent times are implicitly boundary-dressed. But if one admits an “extra” observer in the interior, we argue that the (state-independent) algebra can be Type I or Type II1 instead of Type II ∞ , depending on whether the observer’s light cone contains an entire Cauchy slice or not. Along with these general considerations, we present some specific calculations in the setting of the Poincare BTZ black hole. We identify a specific pointwise (as opposed to non-local) modular translation in BTZ-Kruskal coordinates that is analytically tractable, exploiting a connection with AdS-Rindler. This modular translation can reach the singularity.

Iceland

Vyshnav Mohan

State-independent black hole interiors from the crossed product

In 1984, ’t Hooft famously used a brickwall (aka stretched horizon) to compute black hole entropy up to a numerical pre-factor. This calculation is sometimes interpreted as due to the entanglement of the modes across the horizon, but more operationally, it is simply an indirect count of the semi-classical modes trapped between the stretched horizon and the angular momentum barrier. Because the calculation was indirect, it needed both the mass and the temperature of the black hole as inputs, to reproduce the area. A more conventional statistical mechanics calculation should be able to get the entropy, once the ensemble is specified (say via the energy, in a microcanonical setting). In this paper, we explicitly compute black hole normal modes in various examples, numerically as well as (in various regimes) analytically. The explicit knowledge of normal modes allows us to reproduce both the Hawking temperature as well as the entropy, once the charges are specified, making this a conventional statistical mechanics calculation. A quasi-degeneracy in the angular quantum numbers is directly responsible for the area scaling of the entropy, and is the key distinction between the Planckian black body calculation (volume scaling) and the ’t Hooftian calculation (area scaling). We discuss the (rotating) BTZ case in detail and match the thermodynamic quantities exactly. Schwarzschild and Kerr normal modes are discussed in less detail using near-horizon approximations. Our calculations reveal a new hierarchy in the angular quantum numbers, which we speculate is related to string theory.

Normal modes of the stretched horizon: a bulk mechanism for black hole microstate level spacing

Recent observations on type III algebras in AdS/CFT raise the possibility that smoothness of the black hole horizon is an emergent feature of the large-N limit. In this paper, we present a bulk model for the finite-N mechanism underlying this transition. We quantize a free scalar field on a BTZ black hole with a Planckian stretched horizon placed as a Dirichlet boundary for the field. This is a tractable model for the stretched horizon that does not ignore the angular directions, and it defines a black hole vacuum which has similarities to (but is distinct from) the Boulware state. Using analytic approximations for the normal modes, we first improve upon ’t Hooft’s brick wall calculation: we are able to match both the entropy and the temperature, exactly. Emboldened by this, we compute the boundary Wightman function of the scalar field in a typical pure state built on our stretched horizon vacuum, at an energy sliver at the mass of the black hole. A key result is that despite the manifest lack of smoothness, this single-sided pure state calculation yields precisely the Hartle-Hawking thermal correlator associated to the smooth horizon, in the small-G  N  limit. At finite G  N , there are variance corrections that are suppressed as   <math> <mi>O</mi> <mfenced> <msup> <mi>e</mi> <mrow> <mo>−</mo> <msub> <mi>S</mi> <mi>BH</mi> </msub> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfenced> </math>  $$ \mathcal{O}\left({e}^{-{S}_{BH}/2}\right) $$ . They become important at late times and resolve Maldacena’s information paradox. Highly excited typical pure states on the stretched horizon vacuum are therefore models for black hole microstates, while the smooth horizon describes the thermal state. We note that heavy excited states on the stretched horizon are better defined than the vacuum itself. These results suggest that complementarity in the bulk EFT could arise from a UV complete bulk description in which the black hole interior is not manifest.