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Domain walls, near-BPS bubbles, and probabilities in the landscape

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Anna Ceresole, Gianguido Dall'Agata, Alexander Giryavets, Renata Kallosh, Andrei Linde

High Energy Physics - Theory · Submitted 29 May 2006 · 71 references

Pith's verdict: Section 2's multi-moduli BPS machinery and the e^{-S0/2} Minkowski decay rate are solid and will be cited; the dramatic AdS decay rate rests on an unquantified assumption about the wall tension, and Eq. 5.19 has an algebra slip.

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CitationAlexander Giryavets, Andrei Linde, Anna Ceresole, Gianguido Dall'Agata, Renata Kallosh. "Domain walls, near-BPS bubbles, and probabilities in the landscape." arXiv:hep-th/0605266 (2006). https://pith.science/paper/ZWNXRHIX
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The paper

The paper claims

Static supersymmetric walls between string vacua, once uplifted to de Sitter, become near-critical bubble walls whose one-way decay channels act as sinks and can invert the usual landscape probability distribution.

The reading

The paper asks how the many vacua of the string landscape are separated from one another, and what those separations imply about where a typical observer should expect to find themselves. It first builds an explicit catalogue of static, half-supersymmetric (BPS) domain walls — in KKLT, in STU-type toroidal orbifolds, in a weighted projective Calabi–Yau hypersurface that admits two distinct vacua for a single flux choice, and in racetrack models with one or two AdS minima — and computes the wall tension exactly, either $2(|Z|_{+}-|Z|_{-})$ when the superpotential keeps its sign or $2(|Z|_{+}+|Z|_{-})$ when it crosses zero. It then argues that after the anti-D3 uplifting that turns AdS or Minkowski minima into de Sitter, these walls are the near-critical walls of Coleman–De Luccia bubbles, so the decay rate can be extracted from the small deviation from the BPS limit rather than from a full instanton calculation. The payoff is a claim about probability: decay into a collapsing negative-energy universe, or into Minkowski space, is one-way, so those channels are sinks, and sinks change which vacuum dominates the comoving volume. If the argument holds, the familiar picture of a universe settling into its lowest de Sitter state is replaced by one in which the lowest state is steadily drained.

What carries the argument

The load-bearing device is the pair of first-order BPS flow equations for a flat domain wall in $d=4$, $N=1$ supergravity with arbitrary Kähler potential and any number of moduli: $a'/a = ±|Z|$ and $z^{i}{}' = ∓2 g^{i j̄} ∂_{j̄}|Z|$, where $Z = e^{K/2}W$ is the covariantly holomorphic superpotential. Rewriting the supergravity action plus Gibbons–Hawking boundary terms as a sum of squares of these equations turns the on-shell action into a boundary term, which yields the wall tension directly and reproduces the known critical and super-critical tensions without a Nester construction. The second device is the near-BPS expansion: in the Parke thin-wall formulas, written in the variables $x = (ρ_0/2Λ)^2$ and $y = Λ_2/λ^2$, the BPS limit sits at the singular point $x=1$, $y=-1$ where the bubble radius and the tunneling exponent diverge, and expanding about that point in the small uplifting $V_0$ produces both the universal $S_0/2$ result and the gravitino-mass expression. The third device is the replacement of detailed balance by current equations $ḢP_0 = -J_{0s} - J_{01} + J_{10}$ and $ḢP_1 = -J_{1s} - J_{10} + J_{01}$, whose stationary ratio $p = P_1/P_0$ is set by whichever of the sink rate and the upward rate is larger.

Extended reading notes

Core claim

The central claim is that near-BPS reasoning fixes the decay rate of uplifted de Sitter vacua, and that the resulting rates are large enough to restructure landscape probabilities. For a de Sitter vacuum obtained by uplifting a supersymmetric Minkowski minimum, the paper finds a model-independent tunneling exponent $B = 12π^2/V_0 = S_0/2$, i.e. half the de Sitter entropy, so the decay time is shorter than the recurrence time $t_r ∼ e^{S_0}$ by a factor $e^{S_0/2}$ — for our vacuum, roughly $e^{10^{120}/2}$. For a de Sitter vacuum obtained by uplifting a supersymmetric AdS minimum of depth $|V_0|$, the exponent is instead of order $8π^2/m_{3/2}^2$ times an $O(1)$ factor built from the ratio $C^2 = |V_1|/|V_0|$, tying the decay rate to the gravitino mass rather than to the cosmological constant; since $|V_0|$ before uplifting can exceed the present vacuum energy by a hundred orders of magnitude, this rate can far exceed $e^{-S_0}$. Because a piece of space that tunnels to a collapsing AdS region or to Minkowski never returns, these channels are sinks, and solving the resulting current equations rather than the detailed-balance equation gives, in the wide-sink regime, an inverted distribution $P_1/P_0 = e^{S_1 - |S(φ)| - C_0} ≫ 1$: most of the de Sitter comoving volume sits in the higher-energy vacuum, and the upward current is suppressed relative to the sink current by $e^{-S_0}$.

Load-bearing premise

The claim that uplifted AdS vacua decay quickly rests on the expectation that uplifting lowers the bubble wall tension; if the tension stays at or above its supersymmetric value, the gravitational suppression factor $(1 - 9σ^4/(16|V_1|^2))^{-1}$ diverges or turns negative and tunneling is forbidden, and the paper supports the expectation with only one numerical check in the KL model. The universal Minkowski result $B = S_0/2$ does not need this premise, but the gravitino-mass result does.

Editorial extensions

If this is right

  • The decay rate of an uplifted Minkowski vacuum is fixed at $e^{-S_0/2}$ independent of wall tension and of the depth of the AdS minimum, so it is a universal prediction of KKLT-type uplifting rather than a model-by-model number.
  • For uplifted AdS vacua the rate scales as $\exp(-O(m_{3/2}^{-2}))$, which for a gravitino mass above roughly $10^2$ GeV exceeds $e^{-S_0} \sim e^{-10^{120}}$ by an enormous factor, making such vacua short-lived on cosmological timescales.
  • Because the decay channels to collapsing space and to Minkowski are irreversible, the comoving volume in de Sitter vacua decreases in time as $\exp(-e^{-C_0}t/(1+p))$, so no true stationary distribution exists in the landscape.
  • In the wide-sink regime the standard Hartle–Hawking weighting $P \propto e^{24π^2/V}$ is replaced by an inverted distribution peaked at larger vacuum energy, and the upward current ratio $J_{01}/J_{0s}$ becomes of order $e^{-S_0}$.
  • The lifetime bound that our own vacuum has survived longer than its age becomes a selection rule on the depth of the pre-uplifting AdS minimum and hence on the gravitino mass.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The universal $S_0/2$ exponent suggests a general principle worth stating separately: whenever a de Sitter vacuum is produced by a small explicit breaking of a supersymmetric pair of vacua, the tunneling exponent is half the entropy rather than the full entropy, because the near-BPS wall supplies one of the two factors of the CDL suppression. It would be worth checking whether the same halving app
  • The sink argument implicitly assumes that a bubble interior, once collapsed, is removed from the probability accounting rather than recycled through a quantum jump back to de Sitter; the paper notes this option in a footnote and sets it aside. A reader could test how sensitive the inverted distribution is to allowing even a strongly suppressed return channel.
  • Because the AdS decay rate tracks $m_{3/2}$, the sink mechanism converts a cosmological-constant question into a supersymmetry-breaking question. That suggests an observable-side consequence: if the landscape picture is right, the gravitino mass and the depth of the pre-uplifting AdS minimum are not independent parameters but are linked by the requirement that our vacuum has not already drained.
  • The paper keeps fluxes fixed across every wall it constructs, and says so explicitly in the introduction and discussion. Extending the sink analysis to transitions that change flux would be the natural next step, since those transitions are the ones that connect different area codes rather than different points within one.
Original abstract

We develop a theory of static BPS domain walls in stringy landscape and present a large family of BPS walls interpolating between different supersymmetric vacua. Examples include KKLT models, STU models, type IIB multiple flux vacua, and models with several Minkowski and AdS vacua. After the uplifting, some of the vacua become dS, whereas some others remain AdS. The near-BPS walls separating these vacua may be seen as bubble walls in the theory of vacuum decay. As an outcome of our investigation of the BPS walls, we found that the decay rate of dS vacua to a collapsing space with a negative vacuum energy can be quite large. The parts of space that experience a decay to a collapsing space, or to a Minkowski vacuum, never return back to dS space. The channels of irreversible vacuum decay serve as sinks for the probability flow. The existence of such sinks is a distinguishing feature of the landscape. We show that it strongly affects the probability distributions in string cosmology.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,745 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.08 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v5. Results and the main claim are underlined; .

We develop a theory of static BPS domain walls in stringy landscape and present a large family of BPS walls interpolating between different supersymmetric vacua. Examples include KKLT models, STU models, type IIB multiple flux vacua, and models with several Minkowski and AdS vacua. After the uplifting, some of the vacua become dS, whereas some others remain AdS. The near-BPS walls separating these vacua may be seen as bubble walls in the theory of vacuum decay. As an outcome of our investigation of the BPS walls, we found that the decay rate of dS vacua to a collapsing space with a negative vacuum energy can be quite large. The parts of space that experience a decay to a collapsing space, or to a Minkowski vacuum, never return back to dS space. The channels of irreversible vacuum decay serve as sinks for the probability flow. The existence of such sinks is a distinguishing feature of the landscape. We show that it strongly affects the probability distributions in string cosmology.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We develop a theory of static BPS domain walls in stringy landscape and present a large family of BPS walls interpolating between different supersymmetric vacua. contribution 1.00 0.96 0.11 asserted 1.00
s1 Examples include KKLT models, STU models, type IIB multiple flux vacua, and models with several Minkowski and AdS vacua. contribution 0.88 0.21 0.03 asserted 0.99
s2 After the uplifting, some of the vacua become dS, whereas some others remain AdS. background 0.77 0.20 0.03 asserted 0.94
s3 The near-BPS walls separating these vacua may be seen as bubble walls in the theory of vacuum decay. interpretation 0.97 0.26 0.08 hedged possibility 1.00
s4 As an outcome of our investigation of the BPS walls, we found that the decay rate of dS vacua to a collapsing space with a negative vacuum energy can be quite large. result 1.00 0.95 0.42 asserted 0.86
s5 The parts of space that experience a decay to a collapsing space, or to a Minkowski vacuum, never return back to dS space. result 0.77 0.60 0.14 asserted 1.00
s6 The channels of irreversible vacuum decay serve as sinks for the probability flow. interpretation 0.94 0.51 0.18 asserted 1.00
s7 The existence of such sinks is a distinguishing feature of the landscape. interpretation 0.93 0.41 0.17 asserted 1.00
s8 We show that it strongly affects the probability distributions in string cosmology. result 0.56 0.90 0.36 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read
  • arXiv listing abstract v5
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. KKLT potential V (blue upper curve) and Z (red curve below) as functions of the modulus. One can see that the potential has an AdS minimum at φ = φ ∗ ≃ 4.09854 and tends to Minkowski limit at large φ. Z is always negative in the interval between AdS and Minkowski vacua. Potential energy density V is shown in units 10−15M4 p , whereas the covariantly holomorphic superpotential Z is shown in units 10−8M3 p . One finds that for these parameters during the total flow the covariantly holomorphic su￾perpotential Z is negative and approaches zero at infinity. We can reduce the problem to a
Figure 2. The KKLT wall interpolating between the AdS and flat Minkowski space which are at infinite distance from each other in the moduli space. We plot the scalar φ, the covariantly holomorphic superpotential Z (in units 10−9 ), the warp factor a and the curvature R (in units 10−15), all as functions of r. Here r is the coordinate of the domain wall configuration given in Eq. (2.6); we show r in units 107 . In
Figure 3. Here is the contour plot of the covariantly holomorphic superpotential Z(φ1, φ2) and the gradient flow between the two critical points. Z(φ1, φ2) is everywhere positive between AdS critical points, it vanishes only at the asymptotically Minkowski vacuum. on
Figure 4. Flow of the potential V (φ1(r), φ2(r)). domain wall interpolating between them. A similar phenomenon occurred in the realization of the Randall–Sundrum scenario with a thick brane in [23]. We will have examples of such domain walls in Sec. 4.2.2. From
Figure 5. Flow on ψ complex plane. 0.1 0.2 0.3 0.4 0.5 ReHΤL 0.2 0.4 0.6 0.8 1 ImHΤL
Figure 6. Flow on τ complex plane. 2.5 5 7.5 10 12.5 15 17.5 20 u -1280 -1260 -1240 -1220 -1200 V
Figure 7. Flow of the potential V (ψ(u), τ (u)). sometimes repellers (for saddle points or maxima of the potential) and sometimes attractors, in particular stable attractors in case of minima of the potential. In this case, the flow tends to escape from the LG point towards the conifold. 23
Figure 8. The potential energy density V , blue upper curve, and the covariantly holomorphic super￾potential Z, the red curve below, as the functions of the modulus φ for the KL model. The potential is shown in units 10−8 , and Z is in units 10−4 . Both V and Z have extrema at the same point, which is a reflection of the fact that supersymmetry is unbroken in the minima of V . This plot is for A = 1, B = −2, W0 = −0.125, a = π/100 and b = π/50. 5000 1000015000200002500030000r 3.4 3.5 3.6 3.7 3.8 3.9 φ 5000 10000 15000 20000 25000 30000r -3 -2.5 -2 -1.5 -1 -0.5 Z 5000 10000 15000 20000 25000…
Figure 9. KL wall between M1 and AdS. We plot the scalar φ, Z (in units 10−5 ), warp factor a and the curvature scalar R (in units 10−8 ) as functions of the coordinate r. The curvature R is equal to zero in the Minkowski vacuum, has a peak at the domain wall, and approaches a constant negative value in the AdS vacuum. M1 and stays negative all the way towards AdS. The warp factor on the bottom left starts as a constant at M1 and interpolates into an exponent of r near the AdS. Finally, the curvature jumps from zero at M1 to some positive value and falls down to a negative constant at AdS. It
Figure 10. The potential V (upper curve, in units 10−8 ) and the covariantly holomorphic superpo￾tential Z (lower curve, in units 10−4 ) for the KL model, as function of the modulus. Notice that we took A = 0.96 instead of A = 1, all other parameters being the same as for the potential in
Figure 11. KL wall with a critical tension between AdS1 and AdS2. We plot the scalar φ and the covariantly holomorphic superpotential Z (in units 10−5 ), which is always negative, warp factor a, and the curvature scalar R (in units 10−8 ), as a function of the coordinate r. The curvature R has a peak at the domain wall, and approaches different constant negative values in two different AdS minima. tunneling since these supersymmetric walls saturate the CDL bound. We will discuss this issue later when we will study tunneling, see Section 5. 4.2.2 BPS walls with super-critical tension In the…
Figure 12. The potential V (upper curve, in units 10−8 ) and the covariantly holomorphic superpo￾tential Z (lower curve, in units 10−4 ) for the KL model as function of the modulus. Z is positive at φ0 and negative at φ1. Here we took A = 1.05 instead of A = 1, all other parameters being the same as for the potential in
Figure 13. A BPS wall between two AdS domains in the version of the KL model where Z changes the sign on the domain wall solution. We plot the scalar φ, the covariantly holomorphic superpotential Z (in units 10−5 ), the warp factor a, and the curvature R (in units 10−8 ). Notice the characteristic peak of the scale factor at the point where Z changes sign. 28
Figure 14. A slightly uplifted KL potential, shown in units 10−12 . the investigation of its decay, we will consider potentials such that the difference between the depth of the two minima is much smaller than the height of the barrier. An example of such a potential is provided by the KL model with parameters A = 0.1, B = −10, W0 = −0.00025, a = 2π/50 and b = 2π/25. For these values of parameters the potential has a supersymmetric Minkowski minimum at φ ∼ 4.6 and an AdS minimum at φ ∼ 5. If one slightly changes the parameters, e.g. takes A = 0.095, then the Minkowski minimum becomes an AdS…

Figures reproduced from arXiv:hep-th/0605266 by the authors.

References
How the literature uses it

34 arXiv papers cite this in 102 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 70 sentences without a confident label

Builds on it 2 papers
Uses its method or tool 7 papers
  • To compute the full tension, including the effect of the closed string moduli, we use the an approximation similar to that of [ref] as described in section [ref].

    Evidence for a bound on the lifetime of de Sitter space · Corrections to the tension · 1.00

  • Around a straight enough membrane, one can assume such flow to be governed by the BPS flow equations discussed in [ref] .

    Swampland Conjectures for Strings and Membranes · Membranes · 0.91

  • With its help the flow equations take the form Ȧ=- , ϕ̇^i = 2 g^i ∂_ , whereas the equation for $ϑ$ in [ref] is automatically satisfied, cf. [ref] .

    Domain walls from ten dimensions · Ten–dimensional flow from four–dimensional effective theory · 0.85

Compares with it 3 papers
  • As compared to the tension obtained purely in supergravity [ref] , [ref] additionally contains the explicit contribution from the branes [ref].

    Domain walls from ten dimensions · The D–brane modified c-theorem · 0.80

  • The path we found numerically is a spatial planar interpolation between two vacua that minimizes the action integral, [equation] Dropping the (2+1) planar integral, we get the tension of this transition domain wall, [equation] Comparing to the similar problem in supergravity [ref] , we dropped the Kahler sector and 4D gravity.

    Conifolds and Tunneling in the String Landscape · The BPS Path · 0.96

  • This reduced action is identical in form to the one obtained in [ref] in the absence of membranes, basically because of the observed reciprocal cancellations of various terms localised on the membrane.

    Three-forms, dualities and membranes in four-dimensional supergravity · BPS action and domain wall tension · 0.90

Cites it as support 1 paper
  • An investigation of the rates of tunneling in the landscape performed in [ref] suggests that this condition most probably is met for many of such vacua.

    Stationary Measure in the Multiverse · No Boltzmann brains · 0.97

Cites it as background 8 papers
  • The decay to AdS space (or, more accurately, a decay to a collapsing open universe with a negative cosmological constant) was studied in Ref. [ref] .

    Boltzmann brains and the scale-factor cutoff measure of the multiverse · Vacuum Decay Rates · 0.88

  • This has been extensively studied in the literature starting in [ref] (see also a more recent discussion about this topic in [ref] and references therein).

    Brane Nucleation in Supersymmetric Models · Flat membrane solutions in supergravity models · 0.85

  • The main application of the logical independence of the probability rules is to the measure problem in cosmology [ref] , the problem of how to make statistical predictions for observations in a universe that may be so large that almost all theoretically possible observations actually occur somewhere.

    The Born Rule Dies · Introduction · 0.87

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