Search

What is the boundary holographic dual of S-duality for gauge theories in asymptotically flat space-times? Celestial amplitudes, by virtue of exhibiting holographic properties of the S-matrix, appear well-suited for studying this question. We scatter electrically and magnetically charged massless scalars off non-trivial electromagnetic potentials such as shockwaves, spin-one conformal primary waves, conformally soft modes and their magnetic duals which we construct. This reveals an intricate relation between conformally soft solutions, descendant CFT three-point functions and, by means of the two-dimensional shadow transform, CFT two-point functions. By comparing celestial amplitudes on electric and magnetic dual backgrounds, we provide evidence that the four-dimensional flat-space holographic dual of S-duality in Abelian gauge theory is two-dimensional T-duality. Moreover, we demonstrate that for two-dimensional boundary actions describing low-energy sectors of the bulk gauge theory, S-duality can be explicitly implemented as a T-duality transformation. We show that the Dirac quantisation condition guarantees gauge invariance in the eikonal re-summation for scattering from potentials which for magnetic scalars can be expressed in terms of 't Hooft loops., Comment: 44 pages, 1 figure

Gauge theories and perturbative gravity in four dimensions are governed by a tower of infinite-dimensional symmetries which arise from tree-level soft theorems. However, aside from the leading soft theorems which are all-loop exact, subleading ones receive loop corrections due to long-range infrared effects which result in new soft theorems with logarithmic dependence on the energy of the soft particle. The conjectured universality of these logarithmic soft theorems to all loop orders cries out for a symmetry interpretation. In this letter we initiate a program to compute long-range infrared corrections to the charges that generate the asymptotic symmetries in (scalar) QED and perturbative gravity. For late-time fall-offs of the electromagnetic and gravitational fields which give rise to infrared dressings for the matter fields, we derive finite charge conservation laws and show that in the quantum theory they correspond precisely to the first among the infinite tower of logarithmic soft theorems. This symmetry interpretation, by virtue of being universal and all-loop exact, is a key element for a holographic principle in spacetimes with flat asymptotics., Comment: 8 pages

The correspondence principle between strings and black holes is a general framework for matching black holes and massive states of fundamental strings at a point where their physical properties (such as mass, entropy and temperature) smoothly agree with each other. This correspondence becomes puzzling when attempting to include rotation: At large enough spins, there exist degenerate string states that seemingly cannot be matched to any black hole. Conversely, there exist black holes with arbitrarily large spins that cannot correspond to any single-string state. We discuss in detail the properties of both types of objects and find that a correspondence that resolves the puzzles is possible by adding dynamical features and non-stationary configurations to the picture. Our scheme incorporates all black hole and string phases as part of the correspondence, save for one outlier which remains enigmatic: the near-extremal Kerr black hole. Along the way, we elaborate on general aspects of the correspondence that have not been emphasized before., Comment: 48 pages, 11 figures. v2: 50 pages, 11 figures. Added discussion of ensembles in the correspondence and other smaller improvements. v3: published version

We study celestial amplitudes in string theory at one-loop. Celestial amplitudes describe scattering processes of boost eigenstates and relate to amplitudes in the more standard basis of momentum eigenstates through a Mellin transform. They are thus sensitive to both the ultraviolet and the infrared, which raises the question of how to appropriately take the field theory limit of string amplitudes in the celestial basis. We address this problem in the context of four-dimensional genus-one scattering processes of gluons in open string theory which reach the two-dimensional celestial sphere at null infinity. We show that the Mellin transform commutes with the adequate limit in the worldsheet moduli space and reproduces the celestial one-loop field theory amplitude expressed in the worldline formalism. The dependence on $\alpha'$ continues to be a simple overall factor in one-loop celestial amplitudes albeit with a power that is shifted by three units with respect to tree-level, thus making manifest that the dimensionless parameter $g_{10}^2/\alpha'^3$ organizes the loop expansion in the celestial basis., Comment: 23 pages, 1 figure

We use tools from conformal representation theory to classify the symmetries associated to conformally soft operators in celestial CFT (CCFT) in general dimensions $d$. The conformal multiplets in $d>2$ take the form of celestial necklaces whose structure is much richer than the celestial diamonds in $d=2$, it depends on whether $d$ is even or odd and involves mixed-symmetric tensor representations of $SO(d)$. The existence of primary descendants in CCFT multiplets corresponds to (higher derivative) conservation equations for conformally soft operators. We lay out a unified method for constructing the conserved charges associated to operators with primary descendants. In contrast to the infinite local symmetry enhancement in CCFT${}_2$, we find the soft symmetries in CCFT${}_{d>2}$ to be finite-dimensional. The conserved charges that follow directly from soft theorems are trivial in $d>2$, while non-trivial charges associated to (generalized) currents and stress tensor are obtained from the shadow transform of soft operators which we relate to (an analytic continuation of) a specific type of primary descendants. We aim at a pedagogical discussion synthesizing various results in the literature., Comment: 56 pages + appendices, 3 figures

We explore the celestial holography proposal for non-trivial asymptotically flat backgrounds including the Coulomb field of a static and spinning point charge, their gravitational counterparts described by the Schwarzschild and Kerr metrics, as well as the Aichelburg-Sexl shockwave and spinning shockwave geometries and their electromagnetic cousins. We compute celestial two-point amplitudes on these Kerr-Schild type backgrounds which have the desirable feature, due to the presence of an external source, that they are non-vanishing for general operator positions and are not constrained by the kinematic delta functions of flat space celestial CFT correlators. Of particular interest is the case of shockwave backgrounds where the two-point scattering amplitude of massless scalars can be interpreted as a standard CFT three-point correlator between two massless asymptotic states and a conformal primary shockwave operator. We furthermore show that the boundary on-shell action for general backgrounds becomes the generating functional for tree-level correlation functions in celestial CFT. Finally, we derive (conformal) Faddeev-Kulish dressings for particle-like backgrounds which remove all infrared divergent terms in the two-point functions to all orders in perturbation theory., Comment: 45 pages + references, 3 figures

The soft limits of scattering amplitudes have been extensively studied due to their essential role in the computation of physical observables in collider physics. The universal factorisation that occurs in these kinematic limits has been shown to be related to conservation laws associated with asymptotic, or large, gauge symmetries. This connection has led to a deeper understanding of the symmetries of gauge and gravitational theories and to a reformulation of scattering amplitudes in a basis of boost eigenstates which makes manifest the two-dimensional global conformal symmetry of the celestial sphere. The recast, or celestial, amplitudes possess many of the properties of conformal field theory correlation functions which has suggested a path towards a holographic description of asymptotically flat spacetimes. In this review we consider these interconnected developments in our understanding of soft theorems, asymptotic symmetries and conformal field theory with a focus on the structure and symmetries of the celestial amplitudes and their holographic interpretation., Comment: 49 pages, 3 figures, see also the overview article arXiv:2203.13011

This is an introduction to, and invitation to read, a series of review articles on scattering amplitudes in gauge theory, gravity, and superstring theory. Our aim is to provide an overview of the field, from basic aspects to a selection of current (2022) research and developments., Comment: 15 pages, overview article. v3: journal version

We consider massless scattering from the point of view of the position, momentum, and celestial bases. In these three languages different properties of physical processes become manifest or obscured. Within the soft sector, they highlight distinct aspects of the infrared triangle: quantum field theory soft theorems arise in the limit of vanishing energy $\omega$, memory effects are described via shifts of fields at the boundary along the null time coordinate $u$, and celestial symmetry algebras are realized via currents that appear at special values of the conformal dimension $\Delta$. We focus on the subleading soft theorems at $\Delta=1-s$ for gauge theory $(s=1)$ and gravity $(s=2)$ and explore how to translate the infrared triangle to the celestial basis. We resolve an existing tension between proposed overleading gauge transformations as examined in the position basis and the `Goldstone-like' modes where we expect celestial symmetry generators to appear. In the process we elucidate various order-of-limits issues implicit in the celestial formalism. We then generalize our construction to the tower of $w_{1+\infty}$ generators in celestial CFT, which probe further subleading-in-$\omega$ soft behavior and are related to subleading-in-$r$ vacuum transitions that measure higher multipole moments of scatterers. In the end we see that the celestial basis is `just right' for identifying the symmetry structure., Comment: 48 pages, 5 figures

Celestial diamonds encode the global conformal multiplets of the conformally soft sector, elucidating the role of soft theorems, symmetry generators and Goldstone modes. Upon adding supersymmetry they stack into a pyramid. Here we treat the soft charges associated to the fermionic layers that tie this structure together. This extends the analysis of conformally soft currents for photons and gravitons which have been shown to generate asymptotic symmetries in gauge theory and gravity to infinite-dimensional fermionic symmetries. We construct fermionic charge operators in 2D celestial CFT from a suitable inner product between 4D bulk field operators and spin $s=\frac{1}{2}$ and $\frac{3}{2}$ conformal primary wavefunctions with definite SL(2,$\mathbb{C}$) conformal dimension $\Delta$ and spin $J$ where $|J|\leq s$. The generator for large supersymmetry transformations is identified as the conformally soft gravitino primary operator with $\Delta=\frac{1}{2}$ and its shadow with $\Delta=\frac{3}{2}$ which form the left and right corners of the celestial gravitino diamond. We continue this analysis to the subleading soft gravitino and soft photino which are captured by degenerate celestial diamonds. Despite the absence of a gauge symmetry in these cases, they give rise to conformally soft factorization theorems in celestial amplitudes and complete the celestial pyramid., Comment: 25 pages + appendix, 5 figures

Celestial diamonds encode the structure of global conformal multiplets in 2D celestial CFT and offer a natural language for describing the conformally soft sector. The operators appearing at their left and right corners give rise to conformally soft factorization theorems, the bottom corners correspond to conserved charges, and the top corners to conformal dressings. We show that conformally soft charges can be expressed in terms of light ray integrals that select modes of the appropriate conformal weights. They reside at the bottom corners of memory diamonds, and ascend to generalized currents. We then identify the top corners of the associated Goldstone diamonds with conformal Faddeev-Kulish dressings and compute the sub-leading conformally soft dressings in gauge theory and gravity which are important for finding nontrivial central extensions. Finally, we combine these ingredients to speculate on 2D effective descriptions for the conformally soft sector of celestial CFT., Comment: 34 pages, 2 figures

We examine the structure of global conformal multiplets in 2D celestial CFT. For a 4D bulk theory containing massless particles of spin $s=\{0,\frac{1}{2},1,\frac{3}{2},2\}$ we classify and construct all SL(2,$\mathbb{C}$) primary descendants which are organized into 'celestial diamonds'. This explicit construction is achieved using a wavefunction-based approach that allows us to map 4D scattering amplitudes to celestial CFT correlators of operators with SL(2,$\mathbb{C}$) conformal dimension $\Delta$ and spin $J$. Radiative conformal primary wavefunctions have $J=\pm s$ and give rise to conformally soft theorems for special values of $\Delta \in \frac{1}{2}\mathbb{Z}$. They are located either at the top of celestial diamonds, where they descend to trivial null primaries, or at the left and right corners, where they descend both to and from generalized conformal primary wavefunctions which have $|J|\leq s$. Celestial diamonds naturally incorporate degeneracies of opposite helicity particles via the 2D shadow transform relating radiative primaries and account for the global and asymptotic symmetries in gauge theory and gravity., Comment: 61 pages, 10 figures

We explore conformal primary wavefunctions for all half integer spins up to the graviton. Half steps are related by supersymmetry, integer steps by the classical double copy. The main results are as follows: we 1) introduce a convenient spin frame and null tetrad to organize all radiative modes of varying spin; 2) identify the massless spin-3/2 conformal primary wavefunction as well as the conformally soft Goldstone mode corresponding to large supersymmetry transformations; 3) indicate how to express a conformal primary of arbitrary spin in terms of differential operators acting on a scalar primary; 4) demonstrate that conformal primary metrics satisfy the double copy in a variety of forms -- operator, Weyl, and Kerr-Schild -- and are exact, albeit complex, solutions to the fully non-linear Einstein equations of Petrov type N; 5) propose a novel generalization of conformal primary wavefunctions; and 6) show that this generalization includes a large class of physically interesting metrics corresponding to ultra-boosted black holes, shockwaves and more., Comment: 42 pages, 8 tables, 1 figure

We study the effect of loop corrections to conformal correlators on the celestial sphere at null infinity. We first analyze finite one-loop celestial amplitudes in pure Yang-Mills theory and Einstein gravity. We then turn to our main focus: infrared divergent loop amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory. We compute the celestial one-loop amplitude in dimensional regularization and show that it can be recast as an operator acting on the celestial tree-level amplitude. This extends to any loop order and the re-summation of all planar loops enables us to write down an expression for the all-loop celestial amplitude. Finally, we show that the exponentiated all-loop expression given by the BDS formula gets promoted on the celestial sphere to an operator acting on the tree-level conformal correlation function, thus yielding, the celestial BDS formula., Comment: 18 pages. Version 2: typos corrected, comments added

Celestial amplitudes which use conformal primary wavefunctions rather than plane waves as external states offer a novel opportunity to study properties of amplitudes with manifest conformal covariance and give insight into a potential holographic celestial CFT at the null boundary of asymptotically flat space. Since translation invariance is obscured in the conformal basis, features of amplitudes that heavily rely on it appear to be lost. Among these are the remarkable relations between gauge theory and gravity amplitudes known as the double copy. Nevertheless, properties of amplitudes reflecting fundamental aspects of the perturbative regime of quantum field theory are expected to survive a change of basis. Here we show that there exists a well-defined procedure for a celestial double copy. This requires a generalization of the usual squaring of numerators which entails first promoting them to generalized differential operators acting on external wavefunctions, and then squaring them. We demonstrate this procedure for three and four point celestial amplitudes, and give an argument for its validity to all multiplicities., Comment: 6 pages; v3: version accepted in PRL

We provide a unified treatment of conformally soft Goldstone modes which arise when spin-one or spin-two conformal primary wavefunctions become pure gauge for certain integer values of the conformal dimension $\Delta$. This effort lands us at the crossroads of two ongoing debates about what the appropriate conformal basis for celestial CFT is and what the asymptotic symmetry group of Einstein gravity at null infinity should be. Finite energy wavefunctions are captured by the principal continuous series $\Delta\in 1+i\mathbb{R}$ and form a complete basis. We show that conformal primaries with analytically continued conformal dimension can be understood as certain contour integrals on the principal series. This clarifies how conformally soft Goldstone modes fit in but do not augment this basis. Conformally soft gravitons of dimension two and zero which are related by a shadow transform are shown to generate superrotations and non-meromorphic diffeomorphisms of the celestial sphere which we refer to as shadow superrotations. This dovetails the Virasoro and Diff(S$^2$) asymptotic symmetry proposals and puts on equal footing the discussion of their associated soft charges, which correspond to the stress tensor and its shadow in the two-dimensional celestial CFT., Comment: 29 pages + appendices, 3 figures

A central feature of scattering amplitudes in gravity or gauge theory is the existence of a variety of energetically soft theorems which put constraints on the amplitudes. Celestial amplitudes which are obtained from momentum-space amplitudes by a Mellin transform over the external particle energies cannot obey the usual energetically soft theorems. Instead, the symmetries of the celestial sphere imply that the scattering of conformally soft particles whose conformal weights under the 4D Lorentz group SL(2,C) are taken to zero obey special relations. Such conformally soft theorems have recently been found for gauge theory. Here, I show conformally soft factorization of celestial amplitudes for gravity and identify it as the celestial analogue of Weinberg's soft graviton theorem., Comment: 16 pages

The four-dimensional $S$-matrix is reconsidered as a correlator on the celestial sphere at null infinity. Asymptotic particle states can be characterized by the point at which they enter or exit the celestial sphere as well as their $SL(2,\mathbb C)$ Lorentz quantum numbers: namely their conformal scaling dimension and spin $h\pm \bar h$ instead of the energy and momentum. This characterization precludes the notion of a soft particle whose energy is taken to zero. We propose it should be replaced by the notion of a conformally soft particle with $h=0$ or $\bar h=0$. For photons we explicitly construct conformally soft $SL(2,\mathbb C)$ currents with dimensions $(1,0)$ and identify them with the generator of a $U(1)$ Kac-Moody symmetry on the celestial sphere. For gravity the generator of celestial conformal symmetry is constructed from a $(2,0)$ $SL(2,\mathbb C)$ primary wavefunction. Interestingly, BMS supertranslations are generated by a spin-one weight $(\frac{3}{2},\frac{1}{2})$ operator, which nevertheless shares holomorphic characteristics of a conformally soft operator. This is because the right hand side of its OPE with a weight $(h,\bar h)$ operator ${\cal O}_{h,\bar h}$ involves the shifted operator ${\cal O}_{h+\frac{1}{2},\bar h+ \frac{1}{2}}$. This OPE relation looks quite unusual from the celestial CFT$_2$ perspective but is equivalent to the leading soft graviton theorem and may usefully constrain celestial correlators in quantum gravity., Comment: 23 pages, 1 figure; corrected typos, added references

We compute the memory effect produced at the black hole horizon by a transient gravitational shockwave. As shown by Hawking, Perry, and Strominger (HPS) such a gravitational wave produces a deformation of the black hole geometry which from future null infinity is seen as a Bondi-Metzner-Sachs (BMS) supertranslation. This results in a diffeomorphic but physically distinct geometry which differs from the original black hole by their charges at infinity. Here we give the complementary description of this physical process in the near-horizon region as seen by an observer hovering just outside the event horizon. From this perspective, in addition to a supertranslation the shockwave also induces a horizon superrotation. We compute the associated superrotation charge and show that its form agrees with the one obtained by HPS at infinity. In addition, there is a supertranslation contribution to the horizon charge, which measures the entropy change in the process. We then turn to electrically and magnetically charged black holes and generalize the near-horizon asymptotic symmetry analysis to Einstein-Maxwell theory. This reveals an additional infinite-dimensional current algebra that acts non-trivially on the horizon superrotations. Finally, we generalize the black hole memory effect to Reissner-Nordstr\"{o}m black holes., Comment: 21 pages, 1 figure

Inspired by the recent work of Bao and Ooguri (BO), we study the distinguishability of the black hole microstates from the thermal state as captured by the average of their relative entropies: the Holevo information. Under the assumption that the vacuum conformal block dominates the entropy calculation, BO find that the average relative entropy vanishes on spatial regions less than half the size of the CFT. However, vacuum block dominance fails for some microstates of the $M=0$ BTZ black hole. We show that this renders the average relative entropy nonzero even on infinitesimal intervals at $O(c^0)$., Comment: 10+7 pages, 4 figures

Supersymmetric microstate geometries with five non-compact dimensions have recently been shown by Eperon, Reall, and Santos (ERS) to exhibit a non-linear instability featuring the growth of excitations at an "evanescent ergosurface" of infinite redshift. We argue that this growth may be treated as adiabatic evolution along a family of exactly supersymmetric solutions in the limit where the excitations are Aichelburg-Sexl-like shockwaves. In the 2-charge system such solutions may be constructed explicitly, incorporating full backreaction, and are in fact special cases of known microstate geometries. In a near-horizon limit, they reduce to Aichelburg-Sexl shockwaves in $AdS_3 \times S^3$ propagating along one of the angular directions of the sphere. Noting that the ERS analysis is valid in the limit of large microstate angular momentum $j$, we use the above identification to interpret their instability as a transition from rare smooth microstates with large angular momentum to more typical microstates with smaller angular momentum. This entropic driving terminates when the angular momentum decreases to $j \sim \sqrt{n_1n_5}$ where the density of microstates is maximal. We argue that, at this point, the large stringy corrections to such microstates will render them non-linearly stable. We identify a possible mechanism for this stabilization and detail an illustrative toy model., Comment: 22 pages, 1 figure. v2: JHEP version with references added

Nice slices have played a pivotal role in the discussion of the black hole information paradox as they avoid regions of strong spacetime curvature and yet smoothly cut through the infalling matter and the outgoing Hawking radiation, thus, justifying the use of low energy field theory. To avoid information loss it has been argued recently, however, that local effective field theory has to break down at the horizon. To assess the extent of this breakdown in a UV complete framework we study string-theoretic effects on nice slices in Schwarzschild black holes. Our purpose is two-fold. First, we use nice slices to address various open questions and caveats of arXiv:1402.1486 where it was argued that boost-enhanced non-adiabatic string-theoretic effects at the horizon could provide a dynamical mechanism for the firewall. Second, we identify two non-adiabatic effects on nice slices in Schwarzschild black holes: pair production of open strings near the horizon enhanced by the presence of the infinite tower of highly excited string states and a late-time non-adiabatic effect intrinsic to nice slices., Comment: 30 pages, 5 figures

Collapsing shells form horizons, and when the curvature is small classical general relativity is believed to describe this process arbitrarily well. On the other hand, quantum information theory based (fuzzball/firewall) arguments suggest the existence of some structure at the black hole horizon. This structure can only form if classical general relativity stops being the correct description of the collapsing shell before it reaches the horizon size. We present strong evidence that classical general relativity can indeed break down prematurely, by explicitly computing the quantum tunneling amplitude of a collapsing shell of branes into smooth horizonless microstate geometries. We show that the amplitude for tunneling into microstate geometries with a large number of topologically non-trivial cycles is parametrically larger than exp(-S), which indicates that the shell can tunnel into a horizonless configuration long before the horizon has any chance to form. We also use this technology to investigate the tunneling of M2 branes into LLM bubbling geometries., Comment: 26 pages + appendix

It was argued in arXiv:1203.4227 that the five-dimensional near-horizon extremal Kerr (NHEK) geometry can be embedded in String Theory as the infrared region of an infinite family of non-supersymmetric geometries that have D1, D5, momentum and KK monopole charges. We show that there exists a method to embed these geometries into asymptotically-AdS_3 x S^3/Z_N solutions, and hence to obtain infinite families of flows whose infrared is NHEK. This indicates that the CFT dual to the NHEK geometry is the IR fixed point of a Renormalization Group flow from a known local UV CFT and opens the door to its explicit construction., Comment: 25 pages

We develop the point of view that brane actions should be understood in the context of effective field theory, and that this is the correct way to treat classical as well as loop divergences. We illustrate this idea in a simple model. We then consider the implications for the dynamics of antibranes in flux backgrounds, focusing on the simplest case of a single antibrane. We argue that there is no tachyonic instability at zero temperature, but there is a nonperturbative process in which an antibrane annihilates with its screening cloud. This is distinct from the NS5-brane instanton decay. Constraints on models of metastable supersymmetry breaking by antibranes may be tightened, but there is no problem of principle with this mechanism., Comment: 20 pages, 3 figures. v2: Error in v1 corrected: antibranes even more stable. v3: Minor clarifications

We study two-charge fuzzball geometries, with attention to the use of the proper duality frame. For zero angular momentum there is an onion-like structure, and the smooth D1-D5 geometries are not valid for typical states. Rather, they are best approximated by geometries with stringy sources, or by a free CFT. For non-zero angular momentum we find a regime where smooth fuzzball solutions are the correct description. Our analysis rests on the comparison of three radii: the typical fuzzball radius, the entropy radius determined by the microscopic theory, and the breakdown radius where the curvature becomes large. We attempt to draw more general lessons., Comment: 22 pages, 1 figure

We study the dynamics of probe M5 branes with dissolved M2 charge in bubbling geometries with SO(4) x SO(4) symmetry. These solutions were constructed by Bena-Warner and Lin-Lunin-Maldacena and correspond to the vacua of the maximally supersymmetric mass-deformed M2 brane theory. We find that supersymmetric probe M2 branes polarize into M5 brane shells whose backreaction creates an additional bubble in the geometry. We explicitly check that the supersymmetric polarization potential agrees with the one found within the Polchinski-Strassler approximation. The main result of this paper is that probe M2 branes whose orientation is opposite to the background flux can polarize into metastable M5 brane shells. These decay to a supersymmetric configuration via brane-flux annihilation. Our findings suggest the existence of metastable states in the mass-deformed M2 brane theory., Comment: 38 pages, 8 figures

The backreactions of anti-branes on a variety of supergravity backgrounds have been shown in a recent series of papers to be riled by some unexplained flux singularities. All of the situations studied so far involve backgrounds with (close to) AdS-asymptotics. It is the purpose of this work to study the backreaction of anti-M2 branes on a background exhibiting a different UV behavior: The so-called $\mathbb{A}_8$ regular solution of eleven-dimensional supergravity that we consider has "Taub-NUT type" asymptotics. As it turns out, some subleading infrared singularities are inevitable; they cannot be naturally ascribed to the anti-branes backreacting on this background. Moreover, our configuration does not involve smeared branes. This lends further credence to the work of Bena et al. suggesting that the singularities encountered are in no way remnants of smearing that would wash away once brane polarization is taken into account., Comment: 25 pages. v2: changes made in Section 2; note and reference added

A key feature of BPS multi-center solutions is that the equations controlling the positions of these centers are not renormalized as one goes from weak to strong coupling. In particular, this means that brane probes can capture the same information as the fully back-reacted supergravity solution. We investigate this non-renormalization property for non-supersymmetric, extremal "almost-BPS" solutions at intermediate coupling when one of the centers is considered as a probe in the background created by the other centers. We find that despite the lack of supersymmetry, the probe action reproduces exactly the equations underlying the fully back-reacted solution, which indicates that these equations also do not receive quantum corrections. In the course of our investigation we uncover the relation between the charge parameters of almost-BPS supergravity solutions and their quantized charges, which solves an old puzzle about the quantization of the charges of almost-BPS solutions., Comment: 29 pages, 1 figure