We explicitly show how moduli stabilization is realized geometrically in M-theory compactified on T^4/ℤ_2 ×K3, by using the Gibbons-Hawking approximation of the K3 metric. By relating this compactification to certain microstate geometries, we present the explicit solutions in which fully backreacted fluxes on certain four-cycles stabilize three of the T^4/ℤ_2 compactification moduli. The minimal tadpole contribution of these fluxes is linear in the number of stabilized moduli, and we argue that this linear relation holds for more general fluxes. We also construct a one-parameter family of supersymmetric eleven-dimensional solutions that break Lorentz invariance and the warped-product structure of the compactification. These solutions are a continuous deformation of the warped-product Lorentz-invariant compactification, to which they reduce when the moduli reach their stabilized values. Away from the Lorentz-invariant locus, the fluxes are no longer self-dual in the internal space, and include fields that do not exist in the corresponding EFT. Remarkably, although these fluxes still stabilize a modulus, it is not the T^4/ℤ_2 modulus that appears stabilized in the Lorentz-invariant solution, but rather a nontrivial combination of the T^4 volume and K3 shape moduli. The existence of these solutions suggests that the EFT description of moduli stabilization can be misleading and does not reflect the moduli-stabilization dynamics of the full eleven-dimensional theory. Our results extend straightforwardly to Type IIB String Theory compactified on orientifolds of T^2 × K3.

We review the 20-year history of the Microstate Geometry Programme and the essential role that supergravity has played, and will continue to play, in the description of black-hole microstructure.

We reveal the supersymmetric brane configurations that give rise to AdS_4×S^2×S^2 supergravity solutions, which are holographic duals to three-dimensional N=4 CFTs or to conformal boundaries and domain walls of four-dimensional N=4 SYM. We show that these solutions preserve the same Killing spinors as orthogonal D3, D5 and NS5 branes in flat space, and that the singular sources of these solutions correspond to semi-infinite D3-D5 and D3-NS5 spikes. We track these solutions all the way from the weak-coupling regime of parameters, where the branes do not backreact, to the supergravity regime. We explain how the AdS_4 factor arises from certain universal self-similar bending regions of the five-branes, whose steepness is the same as the weak-coupling linking numbers. We also propose a brane configuration that gives rise to the Janus interface solutions. Our construction gives a clear geometric explanation of the Gaiotto-Witten "good-bad-ugly" classification of eight-supercharge theories: only good theories have five-branes that do not cross when back-reacting, and end up sourcing an AdS_4×S^2×S^2 solution.

We identify singularity-free Running-Kerr-Taub-Bolt solutions of eleven-dimensional supergravity that descend to four-dimensional rotating solutions with flat-space asymptotics. We compute their spin-induced quadrupole moment and find that for a certain range of charges this quadrupole moment is positive. This behavior differs from the Kerr black hole and from most other spinning objects constructed with ``normal'' four-dimensional matter, and we discuss the top-down physics of these solutions that could be responsible for this unusual behavior.

Flux compactifications that give three- or four-dimensional anti-de Sitter vacua with a parametrically small negative cosmological constant are claimed to be ubiquitous in string theory. However, the 1 + 1 and 2 + 1 dimensional conformal-field-theory (CFT) duals to such vacua should have very large central charges and rather unusual properties. We construct brane configurations that source these would-be anti-de Sitter (AdS) flux compactifications, and identify certain UV AdS geometries that these branes source. The central charge of the CFT duals to these UV AdS geometries place lower bounds on the absolute values of the cosmological constants of the AdS vacua. These bounds are incompatible with the scale separation needed to construct realistic cosmological models.

Abstract AdS3× S3× S3 solutions warped over a Riemann surface, Σ, are indexed by a parameter, γ, that defines the superconformal algebra, D(2, 1; γ) ⨁ D(2, 1; γ) they preserve. We show that these solutions come from multiple back-reacted M2-M5 spikes, and that different values of γ correspond to different scaling limits of the same M2-M5 solutions. We find that when γ switches from positive to negative, the infrared region of the AdS3 switches from the tip of spikes, far from the M5 branes, to the bottom of the spikes, far from the M2 branes. We also explain how the bubbling negative-γ solutions emerge from the geometric transition of multiple M2-M5 spikes.

The Dirac-Born-Infled action that describes the dynamics of D branes also allows one to compute the supersymmetries they preserve using the Kappa-symmetry projector. The ”Lagrangian” expression of this projector depends on the velocity and electric fields of the branes, but not on the corresponding conserved charges. One can also construct the projector in a ”Hamiltonian” approach, by multiplying the conserved string, brane and momentum charges with the corresponding gamma matrix involutions, adding them to the mass of the brane multiplied by the unit matrix, and normalizing the resulting expression. We show that these two procedures are equivalent.

In AdS_3/CFT_2 duality, there are large families of smooth, horizonless microstate geometries that correspond to heavy pure states of the dual CFT. The metric and fluxes are complicated functions of up to five coordinates. There are also many duals of heavy pure states that cannot be described in supergravity, but only admit a worldsheet description. Extracting the physical properties of these solutions is technically challenging. In this paper, we show that there are much simpler effective descriptions of these solutions that capture many of their stringy and geometrical features, at the price of sacrificing supergravity smoothness. In particular, the effective description of some families of superstrata, and of certain worldsheet solutions, is given by easy-to-construct three-center solutions. For example, the effective description of a superstratum with a long AdS_2 throat is a scaling, three-center solution in which the momentum wave is collapsed to a singular source at one of the three centers. This also highlights how momentum migrates away from the supertube locus in the back-reacted geometry. Our results suggest that effective descriptions can be extended to more general microstates, and that many singular multi-center solutions can in fact correspond to effective descriptions of smooth horizonless microstructure.

We construct asymptotically AdS$_3 \times$S$^3 \times$T$^4$ black holes that are localized on the S$^3$ and co-exist with the BTZ black hole at small positive energies. These black holes dominate the microcanonical ensemble for $ E\leq \frac{c}{24}\left(5\sqrt{5}-11\right)$, suggesting they could represent the endpoint of the BTZ instability at low energies. Remarkably, they also exist at negative energies, where pure Einstein gravity predicts no states and the BTZ black hole does not exist. They appear in the spectrum immediately above $-\frac{c}{12}$ (the energy of global AdS$_3$), and their entropy is a significant fraction (up to 1/2) of the entropy of the free orbifold CFT at negative energies. Our solutions exist in an energy window outside the universal predictable range of the modular bootstrap in large-$c$ CFT$_2$ and, despite their microcanonical dominance, do not dominate in the canonical ensemble. To calculate the holographic entanglement entropy of our solutions, we propose the first recipe that can be applied to arbitrary geometries asymptotic to AdS$_3$ times an internal manifold, and depend non-trivially on its coordinates. We find that our new geometries have an entanglement entropy nearly identical to that of the BTZ black hole with the same energy, despite having different horizon structures. However, they can be distinguished by non-minimal extremal surfaces, which unveil finer details of the microstructure.

We show that the near-brane back-reaction of M2 branes ending on M5 branes has a rich “spike structure” that is determined by partitioning the numbers of M2 branes that are terminating on groups of M5 branes. The near-brane limit of the metric describing these branes has an AdS3 factor, implying the existence of a dual CFT. Each partition of the M2 and M5 charges among spikes gives rise to a different “mohawk” revealing a new layer of brane fractionation. We conjecture that all these mohawks are dual to ground states of near-brane-intersection CFT’s. We show that the supergravity solutions describing these mohawks are part of the large families of AdS3 × S3 × S3 solutions described in [1]. We identify precisely which of these families are relevant to brane intersections and show that the AdS3 invariance emerges from the self-similarity of the spikes.

We construct asymptotically AdS3xS3xT4 black holes that are localized on the S3 and co-exist with the BTZ black hole at small positive energies. These black holes dominate the microcanonical ensemble for E <= c24 (55 - 11), suggesting they could represent the endpoint of the BTZ instability at low energies. Remarkably, they also exist at negative energies, where pure Einstein gravity predicts no states and the BTZ black hole does not exist. They appear in the spectrum immediately above - c12 (the energy of global AdS3), and their entropy is a significant fraction (up to 1/2) of the entropy of the free orbifold CFT at negative energies. Our solutions exist in an energy window outside the universal predictable range of the modular bootstrap in large-c CFT2 and, despite their microcanonical dominance, do not dominate in the canonical ensemble.To calculate the holographic entanglement entropy of our solutions, we propose the first recipe that can be applied to arbitrary geometries asymptotic to AdS3 times an internal manifold, and depend non-trivially on its coordinates. We find that our new geometries have an entanglement entropy nearly identical to that of the BTZ black hole with the same energy, despite having different horizon structures. However, they can be distinguished by non-minimal extremal surfaces, which unveil finer details of the microstructure.

Abstract We construct asymptotically AdS3×S3×T4 black holes that are localized on the S3 and co-exist with the BTZ black hole at small positive energies. These black holes dominate the microcanonical ensemble for E ≤ c 24 $$ \frac{c}{24} $$ (5 5 $$ \sqrt{5} $$ − 11), suggesting they could represent the endpoint of the BTZ instability at low energies. Remarkably, they also exist at negative energies, where pure Einstein gravity predicts no states and the BTZ black hole does not exist. They appear in the spectrum immediately above − c 12 $$ \frac{c}{12} $$ (the energy of global AdS3), and their entropy is a significant fraction (up to 1/2) of the entropy of the free orbifold CFT at negative energies. Our solutions exist in an energy window outside the universal predictable range of the modular bootstrap in large-c CFT2 and, despite their microcanonical dominance, do not dominate in the canonical ensemble. To calculate the holographic entanglement entropy of our solutions, we propose the first recipe that can be applied to arbitrary geometries asymptotic to AdS3 times an internal manifold, and depend non-trivially on its coordinates. We find that our new geometries have an entanglement entropy nearly identical to that of the BTZ black hole with the same energy, despite having different horizon structures. However, they can be distinguished by non-minimal extremal surfaces, which unveil finer details of the microstructure.

We show that the supergravity solutions for 1/4-BPS intersecting systems of M2 and M5 branes are completely characterized by a single “maze” function that satisfies a non-linear “maze” equation similar to the Monge-Ampère equation. We also show that the near-brane limit of certain intersections are AdS_3 × S^3 × S^3 solutions warped over a Riemann surface, Σ. There is an extensive literature on these subjects and we construct mappings between various approaches and use brane probes to elucidate the relationships between the M2-M5 and AdS systems. We also use dualities to map our results onto other systems of intersecting branes. This work is motivated by the recent realization that adding momentum to M2-M5 intersections gives a supermaze that can reproduce the black-hole entropy without ever developing an event horizon. We take a step in this direction by adding a certain type of momentum charges that blackens the M2-M5 intersecting branes. The near-brane limit of these solutions is a BTZ^extremal× S^3 × S^3 ×Σ geometry in which the BTZ momentum is a function of the Riemann surface coordinates.

A bstract One way to describe the entropy of black holes comes from partitioning momentum charge across fractionated intersecting brane systems. Here we construct $$ \frac{1}{8} $$ 1 8 -BPS solutions by adding momentum to a maze of M2-brane strips stretched between M5 branes. Before the addition of momentum, the $$ \frac{1}{4} $$ 1 4 -BPS supergravity solution describing the maze is governed by a master function obeying a complicated Monge-Ampère equation. Given such a solution, we show that one can add momentum waves without modifying the $$ \frac{1}{4} $$ 1 4 -BPS M2-M5 background. Remarkably, these excitations are fully determined by a layered set of linear equations. The fields responsible for carrying the momentum are parameterized by arbitrary functions of a null direction, and have exactly the same structure as in brane world-volume constructions. The fact that the momentum and flux excitations of the M2-M5-P system are governed by a linear structure brings us one step closer to using supergravity solutions to capture the entropy of supersymmetric black-holes.

We consider coupled gravitational and electromagnetic perturbations of a family of five-dimensional Einstein-Maxwell solutions that describes both magnetized black strings and horizonless topological stars. We find that the odd perturbations of this background lead to a master equation with five Fuchsian singularities and compute its quasinormal mode spectrum using three independent methods: Leaver, WKB and numerical integration. Our analysis confirms that odd perturbations always decay in time, while spherically symmetric even perturbations may exhibit for certain ranges of the magnetic fluxes instabilities of Gregory-Laflamme type for black strings and of Gross-Perry-Yaffe type for topological stars. This constitutes evidence that topological stars and black strings are classically stable in a finite domain of their parameter space.

The entropy of the supersymmetric D2-D4-P black hole comes at weak coupling from D2-brane strips stretched between parallel D4 branes and carrying momentum waves. We use the DBI action of D4 branes to construct two pieces of plumbing that enter in the construction of these microstates. The first is a semi-infinite D2 brane ending on a D4 brane and carrying a momentum wave along the common D2-D4 direction. The second is a non-Abelian solution to the 5D maximally-supersymmetric SU(2) Super-Yang-Mills theory describing a momentum-carrying D2 strip stretched between two D4 branes. The solution without momentum is the same as the 't Hooft-Polyakov monopole, and the fields that carry the momentum can be added without changing any of the fields of the monopole.

Abstract Anti-D3 branes at the bottom of warped throats, commonly used to uplift the cosmological constant in String-Theory de Sitter proposals, source a plethora of supersymmetry-breaking fluxes, that can interact nontrivially with other ingredients of the flux compactification. In this paper we perform a complex-structure decomposition of these fluxes, and compute the effect of the (0,3) flux component on the stabilization of Kähler moduli via D7-branes gaugino condensation. This allows us to obtain a new constraint on the validity of this stabilization mechanism. This effect does not appear hard to satisfy in de Sitter construction proposals that use long warped throats, but may be problematic in proposals where the warping is small.

Almost all proposals to construct de Sitter vacua with a small cosmological constant involve flux compactifications with stabilized moduli. These give AdS vacua, which are uplifted to de Sitter by adding antibranes in certain regions of the compactification manifold. However, antibranes are charged, singular and interact nontrivially with other ingredients of the compactification; this can invalidate the de Sitter construction. In this Letter, we construct a new ingredient for uplifting AdS solutions to de Sitter, which is neutral, smooth and horizonless, and therefore bypasses some of the problems of antibrane uplift.

We attempt to review all trustworthy and well-controlled de Sitter compactifications of string theory.

Anti-D3 branes at the bottom of warped throats, commonly used to uplift the cosmological constant in String-Theory de Sitter proposals, source a plethora of supersymmetry-breaking fluxes, that can interact nontrivially with other ingredients of the flux compactification. In this paper we perform a complex-structure decomposition of these fluxes, and compute the effect of the (0,3) flux component on the stabilization of Kähler moduli via D7-branes gaugino condensation. This allows us to obtain a new constraint on the validity of this stabilization mechanism. This effect does not appear hard to satisfy in de Sitter construction proposals that use long warped throats, but may be problematic in proposals where the warping is small.