Microstate Geometries Research Articles

Vector superstrata. Part II

Microstate geometries are proposed microstates of black holes which can be described within supergravity. Even though their number may not reproduce the full entropy of black holes with finite-sized horizons, they still offer a glimpse into the microscopic structure of black holes. In this paper we construct a new set of microstate geometries of the supersymmetric D1-D5-P black hole, where the momentum charge is carried by a vector field, as seen from the perspective of six-dimensional supergravity. To aid our construction, we develop an algorithm which solves a complicated partial differential equation using the regularity of the geometries. The new solutions are asymptotically AdS3 × S3, and have a long, but finite AdS2 throat that caps off without ever developing a horizon. These microstate geometries have a holographic interpretation as coherent superpositions of heavy states in the boundary D1-D5 CFT. We identify the states which are dual to our newly constructed solutions and carry out some basic consistency checks to support our identification.

Inscribing geodesic circles on the face of the superstratum

We use families of circular null geodesics as probes of a family of microstate geometries, known as (1, 0, n) superstrata. These geometries carry a left-moving momentum wave and the behavior of some of the geodesic probes is very sensitive to this background wave. The left-moving geodesics behave like BPS particles and so can be placed in circular orbits anywhere in the geometry and actually “float” at fixed radius and angle in the three-dimensional “capped BTZ” geometry. The right-moving geodesics behave like non-BPS particles. We show that they provide a simple geometric characterization of the black-hole bound: when the momentum charge of the geometry is below this bound, such geodesics can be placed anywhere, but exceeding the bound, even by a small amount, means these geodesics are restricted to the deep interior of the geometry. We also show that for left-moving string probes, the tidal forces remain comparable with those of global AdS3. Nevertheless, for some of these probes, the “bumps” in the geometry induce an oscillatory mass term and we discuss how this can lead to chaotic scrambling of the state of the string.

Vector Superstrata

We present the construction of several microstate geometries of the supersymmetric D1-D5-P black hole in which, within six-dimensional supergravity, the momentum charge is carried by a vector field. The fully backreacted geometries are smooth and horizonless: they are asymptotically AdS3 × S3 with an AdS2 throat that smoothly caps off. We propose a holographic dual for these bulk solutions and discuss their extension to asymptotically flat space. In addition, we present several uplifts of the full six-dimensional supersymmetric ansatz to ten-dimensions. In particular, we show that there exists a frame in which geometries based on vector field momentum carriers are entirely in the NS-NS sector, making them possible starting points for the exploration of stringy black-hole microstates.

Linearizing the BPS equations with vector and tensor multiplets

We analyse the BPS equations of mathcal{N} = (1, 0) supergravity theory in six dimensions coupled to a vector and tensor multiplet. We show how these BPS equations can be reduced to a set of linear differential equations. This system is triangular in that each layer of equations, while linear, is quadratically sourced by the solutions of the previous layers. We examine several explicit examples and discuss the construction of new families of microstate geometries. We expect that the result presented here will open up new branches of superstrata in which the momentum is encoded in a new class of charge carriers.

On supersymmetric multipole ratios

Four-dimensional supersymmetric black holes are static and so have all vanishing multipoles (except the mass monopole). Nevertheless, it is possible to define finite multipole ratios for these black holes, by taking the ratio of (finite) multipoles of supersymmetric multicentered geometries and then taking the black hole scaling limit of the multipole ratios within these geometries. An alternative way to calculate these multipole ratios is to deform the supersymmetric black hole slightly into a non-extremal, rotating black hole, calculate the multipole ratios of this altered black hole, and then take the supersymmetric limit of the ratios. Bena and Mayerson observed that for a class of microstate geometries, these two a priori completely different methods give spectacular agreement for the resulting supersymmetric black hole multipole ratios. They conjectured that this agreement is due to the smallness of the entropy parameter for these black holes. We correct this conjecture and give strong evidence supporting a more refined conjecture, which is that the agreement of multipole ratios as calculated with these two different methods is due to both the microstate geometry and its corresponding black hole having a property we call “large dipole”, which can be interpreted as their center of mass being far away from its apparent center.

On the BPS Sector in AdS3/CFT2 Holography

AbstractThe BPS sector in duality has been fertile ground for the exploration of gauge/gravity duality, from the match between black hole entropy and the CFT elliptic genus to the construction of large families of geometrical microstates and the identification of the corresponding states in the CFT. Worldsheet methods provide a tool to further explore the relation between string theory in the bulk and corresponding CFT quantities. We show how to match individual BPS strings to their counterparts in the symmetric product orbifold CFT. In the process, we find an exact match between known constructions of microstate geometries and condensates of BPS supergraviton strings, and discuss their role in the broader collection of BPS states. In particular, we explore how microstate geometries develop singularities; and how string theory resolves these singularities through the appearance of “tensionless” string dynamics, which is the continuation of structures found in the weak‐coupling CFT into the strongly coupled regime described by string theory in the bulk. We argue that such “tensionless” strings are responsible for black hole microstructure in the bulk description.

Resolving black-hole microstructure with new momentum carriers

All known horizonless black-hole microstate geometries correspond to brane sources that acquire a finite size, and hence break the spherical symmetry of the black hole. We construct, for the first time, solutions with zero horizon area that have the same charges as a three-charge F1-NS5-P Type-IIA black hole and preserve this spherical symmetry. The momentum of these solutions is carried by longitudinal D0-D4 density fluctuations inside the NS5-branes. We argue that these solutions should be interpreted as the long-throat limit of a family of smooth, horizonless microstate geometries, called superstrata, where such geometries degenerate. The existence of these geometries indicates that a finite-size horizon does not appear even in the singular corners of the moduli space of three-charge microstate geometries.

Stable bound orbits in microstate geometries

We show the existence of stable bound orbits for the massive and massless particles moving in the simplest microstate geometry spacetime in the bosonic sector of the five-dimensional minimal supergravity. In our analysis, reducing the motion of particles to a two-dimensional potential problem, we numerically investigate whether the potential has a negative local minimum.

Real-time holography and hybrid WKB for BTZ wormholes

We study probe scalar correlation functions in a Solodukhin wormhole corresponding to the non-rotating BTZ black hole, as a toy model for microstate geometries thereof. Using real-time holography, we obtain the retarded scalar correlator in the wormhole geometry and quantitatively compare it to the result of the hybrid WKB method for the same correlator. We also calculate an off-diagonal correlator ∼ 〈HLLH′〉 involving two different (heavy) wormhole states.

New superstrata from three-dimensional supergravity

We find a two-parameter family of generalized superstrata that emerge as smooth, supersymmetric solutions in three-dimensional gauged supergravity coupled to additional scalar fields. This new family of generalized superstrata are smooth microstate geometries and may be thought of as supersymmetric Coulomb-branch extensions of the original superstrata in which the underlying supertube undergoes an elliptical deformation. These solutions had already been obtained numerically, and as series solutions, to the equations of motion, and some of them were conjectured to be supersymmetric. Here we prove the supersymmetry of an entire two-parameter family and we obtain a highly non-trivial analytic and smooth solution for a one-parameter limit in which the global symmetry of the metric is enhanced to SO(3). We also confirm that the other known families of microstrata are not supersymmetric. We conclude with a cursory analysis of some of the singular brane distributions that can be accessed from three-dimensional gauged supergravity while preserving the same supersymmetries as the superstratum, and therefore of the three-charge black hole.

Toroidal tidal effects in microstate geometries

Tidal effects in capped geometries computed in previous literature display no dynamics along internal (toroidal) directions. However, the dual CFT picture suggests otherwise. To resolve this tension, we consider a set of infalling null geodesics in a family of black hole microstate geometries with a smooth cap at the bottom of a long BTZ-like throat. Using the Penrose limit, we show that a string following one of these geodesics feels tidal stresses along all spatial directions, including internal toroidal directions. We find that the tidal effects along the internal directions are of the same order of magnitude as those along other, non-internal, directions. Furthermore, these tidal effects oscillate as a function of the distance from the cap — as a string falls down the throat it alternately experiences compression and stretching. We explain some physical properties of this oscillation and comment on the dual CFT interpretation.

Hbox {AdS}_3 holography for non-BPS geometries

By using the approach introduced in Ganchev et al. (Q-Balls Meet Fuzzballs: Non-BPS Microstate Geometries. arXiv:2107.09677 [hep-th]) we construct non-BPS solutions of 6D (1, 0) supergravity coupled to two tensor multiplets as a perturbation of hbox {AdS}_3times S^3. These solutions are both regular and asymptotically hbox {AdS}_3times S^3, so according to the standard holographic framework they must have a dual CFT interpretation as non-supersymmetric heavy operators of the D1–D5 CFT. We provide quantitative evidence that such heavy CFT operators are bound states of a large number of light BPS operators that are mutually non-BPS.

Q-balls meet fuzzballs: non-BPS microstate geometries

We construct a three-parameter family of non-extremal microstate geometries, or “microstrata”, that are dual to states and deformations of the D1-D5 CFT. These families are non-extremal analogues of superstrata. We find these microstrata by using a Q-ball-inspired Ansatz that reduces the equations of motion to solving for eleven functions of one variable. We then solve this system both perturbatively and numerically and the results match extremely well. We find that the solutions have normal mode frequencies that depend upon the amplitudes of the excitations. We also show that, at higher order in perturbations, some of the solutions, having started with normalizable modes, develop a “non-normalizable” part, suggesting that the microstrata represent states in a perturbed form of the D1-D5 CFT. This paper is intended as a “Proof of Concept” for the Q-ball-inspired approach, and we will describe how it opens the way to many interesting follow-up calculations both in supergravity and in the dual holographic field theory.

Lower bound for angular momenta of microstate geometries in five dimensions

We study the BPS solutions of the asymptotically flat, stationary microstate geometries with bi-axisymmetry and reflection symmetry in the five-dimensional ungauged minimal supergravity. We show that the angular momenta of the microstate geometry with a small number of centers (at least, five centers) have lower bounds, which are slightly smaller than those of the maximally spinning BMPV black hole. Therefore, there exists a certain narrow parameter region such that the microstate geometry with a small number of centers admits the same angular momenta as the BMPV black hole. Moreover, we investigate the dependence of the topological structure of the evanescent ergosurfaces on the magnetic fluxes through the 2-circles between two centers.

Gravitational footprints of black holes and their microstate geometries

We construct a family of non-supersymmetric extremal black holes and their horizonless microstate geometries in four dimensions. The black holes can have finite angular momentum and an arbitrary charge-to-mass ratio, unlike their supersymmetric cousins. These features make them and their microstate geometries astrophysically relevant. Thus, they provide interesting prototypes to study deviations from Kerr solutions caused by new horizon-scale physics. In this paper, we compute the gravitational multipole structure of these solutions and compare them to Kerr black holes. The multipoles of the black hole differ significantly from Kerr as they depend non-trivially on the charge-to-mass ratio. The horizonless microstate geometries (that are comparable in size to a black hole) have a similar multipole structure as their corresponding black hole, with deviations to the black hole multipole values set by the scale of their microstructure.

Interpolating between multi-center microstate geometries

We study interpolation between two multi-center microstate geometries in 4d/5d that represent Lunin-Mathur geometries with circular profiles. The interpolating solution is a Lunin-Mathur geometry with a helical profile, and is represented by a 2-center solution with a codimension-2 source. The interpolating 2-center solution exhibits interesting features such as some of the charges being delocalized, and some of the charges getting transferred from the codimension-2 center to the other, codimension-3 center as the interpolation proceeds. We also discuss the spectral flow of this entire process and speculate on the relevance of such solutions to understanding general microstates of 3-charge black holes.

Black hole microstates from the worldsheet

Recently an exact worldsheet description of strings propagating in certain black hole microstate geometries was constructed in terms of null-gauged WZW models. In this paper we consider a family of such coset models, in which the currents being gauged are specified by a set of parameters that a priori take arbitrary values. We show that consistency of the spectrum of the worldsheet CFT implies a set of quantisation conditions and parity restrictions on the gauging parameters. We also derive these constraints from an independent geometrical analysis of smoothness, absence of horizons and absence of closed timelike curves. This allows us to prove that the complete set of consistent backgrounds in this class of models is precisely the general family of (NS5-decoupled) non-BPS solutions known as the JMaRT solutions, together with their various (BPS and non-BPS) limits. We clarify several aspects of these backgrounds by expressing their six-dimensional solutions explicitly in terms of five non-negative integers and a single length-scale. Finally we study non-trivial two-charge limits, and exhibit a novel set of non-BPS supergravity solutions describing bound states of NS5 branes carrying momentum charge.

Black holes and the swampland: the deep throat revelations

Multi-centered bubbling solutions are black hole microstate geometries that arise as smooth solutions of 5-dimensional mathcal{N} = 2 Supergravity. When these solutions reach the scaling limit, their resulting geometries develop an infinitely deep throat and look arbitrarily close to a black hole geometry. We depict a connection between the scaling limit in the moduli space of Microstate Geometries and the Swampland Distance Conjecture. The naive extension of the Distance Conjecture implies that the distance in moduli space between a reference point and a point approaching the scaling limit is set by the proper length of the throat as it approaches the scaling limit. Independently, we also compute a distance in the moduli space of 3-centre solutions, from the Kähler structure of its phase space using quiver quantum mechanics. We show that the two computations of the distance in moduli space do not agree and comment on the physical implications of this mismatch.

Topological Stars and Black Holes.

We study smooth bubble spacetimes in five-dimensional Einstein-Maxwell theory that resemble four-dimensional magnetic black holes upon Kaluza-Klein reduction. We denote them as topological stars since they have topological cycles supported by magnetic flux. They can be macroscopically large compared to the size of the Kaluza-Klein circle and could describe qualitative properties of microstate geometries for astrophysical black holes. We also describe five-dimensional black strings without curvature singularity, the interior caps as a two-dimensional Milne space with a bubble.

The great escape: tunneling out of microstate geometries

We compute the quasi-normal frequencies of scalars in asymptotically-flat microstate geometries that have the same charge as a D1-D5-P black hole, but whose long BTZ-like throat ends in a smooth cap. In general the wave equation is not separable, but we find a class of geometries in which the non-separable term is negligible and we can compute the quasi-normal frequencies using WKB methods. We argue that our results are a universal property of all microstate geometries with deeply-capped BTZ throats. These throats generate large redshifts, which lead to exceptionally-low-energy states with extremely long decay times, set by the central charge of the dual CFT to the power of twice the dimension of the operator dual to the mode. While these decay times are extremely long, we also argue that the energy decay is bounded, at large t, by (log(t))−2 and is comparable with the behavior of ultracompact stars, as one should expect for microstate geometries.