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Cited by 27Black Holes and Theoretical PhysicsParticle physics theoretical and experimental studiesCosmology and Gravitation Theories

The DOI matches its Crossref record; 46 of the 46 references it cites were found in a registry or an index; none of the 28 sources its claims cite is checked for support yet.

Bubbles of nothing and supersymmetric compactifications

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Blanco-Pillado, Jose J., Shlaer, Benjamin, Sousa, Kepa, Urrestilla, Jon·2016·Journal of Cosmology and Astroparticle Physics
DOI 10.1088/1475-7516/2016/10/002
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Abstract

We investigate the non-perturbative stability of supersymmetric compactifications with respect to decay via a bubble of nothing. We show examples where this kind of instability is not prohibited by the spin structure, i.e., periodicity of fermions about the extra dimension. However, such ``topologically unobstructed'' cases do exhibit an extra-dimensional analog of the well-known Coleman-De Luccia suppression mechanism, which prohibits the decay of supersymmetric vacua. We demonstrate this explicitly in a four dimensional Abelian-Higgs toy model coupled to supergravity. The compactification of this model to M 3 × S 1 presents the possibility of vacua with different windings for the scalar field. Away from the supersymmetric limit, these states decay by the formation of a bubble of nothing, dressed with an Abelian-Higgs vortex. We show how, as one approaches the supersymmetric limit, the circumference of the topologically unobstructed bubble becomes infinite, thereby preventing the realization of this decay. This demonstrates the dynamical origin of the decay suppression, as opposed to the more familiar argument based on the spin structure. We conjecture that this is a generic mechanism that enforces stability of any topologically unobstructed supersymmetric compactification.

Overview

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TL;DR

The paper studies whether supersymmetric compactifications can decay via bubbles of nothing when the usual spin-structure obstruction is absent. It finds that topologically unobstructed cases are still stabilized dynamically: as the supersymmetric limit is approached, the bubble becomes infinitely large and flat, suppressing the decay by a Coleman-De Luccia-type mechanism.

Key findings

  • The authors construct a supersymmetric Kaluza-Klein compactification on \(S^1\) with anti-periodic fermions, using a winding scalar and a Wilson line, so Witten’s spin-structure argument does not forbid a bubble-of-nothing decay. This requires \(\gamma \equiv \kappa^2\eta^2 = 1\).
  • In the supersymmetric limit \((\beta=1,\gamma=1)\), the bubble of nothing becomes infinitely large: \(H_0 \to 0\), so the bubble circumference \(2\pi H_0^{-1}\) diverges and the Euclidean action diverges, suppressing the decay. A numerical example near the limit \((\gamma=0.98)\) gives \(H_0^{-1}\approx 256\), compared with the thin-wall prediction \(H_0^{-1}\approx 250\).
  • Away from supersymmetry, bubble-of-nothing solutions exist and are dressed by an Abelian-Higgs vortex. For \(\beta=1,\gamma=10^{-3},R=10\), the numerical deficit angle is \(\Delta_W\approx 2\pi\,10^{-3}\), matching the analytic result \(2\pi\gamma=2\pi\,10^{-3}\). For \(\beta=0.001,\gamma=1,R=40\), the solution gives \(H_0^{-1}\approx 26 > R/2=20\).
  • For the special case \(R=1\) in the supersymmetric limit, the configuration becomes a half-BPS vortex solution interpolating between \(M_3\times S^1\) and a pure magnetic spherical compactification \(M_2\times S^2\), preventing the decay.

Methods

  • The paper uses a four-dimensional \(N=1\) supergravity Abelian-Higgs toy model compactified on a circle, \(M_4 \to M_3\times S^1\), with a complex scalar, \(U(1)\) gauge field, and parameters \(\beta\) and \(\gamma=\kappa^2\eta^2\).
  • It studies bubble-of-nothing instanton geometries, first in a thin-wall/Nambu-Goto approximation with a vortex/string at the bubble tip, then numerically solves the coupled matter and Einstein equations using a multiple-shooting method from the bubble core to asymptotic boundary conditions.
  • Numerical profile functions are compared with analytic thin-wall predictions for the deficit angle and the bubble size \(H_0^{-1}\), including cases with \(R\gg1\), \(R<1\), \(R\sim O(1)\), and \(R=1\).

Limitations

  • The paper states that the thin-wall approximation may not be valid in all regimes and explicitly tests the results numerically outside that approximation.
  • The authors present the stabilization mechanism as a conjecture for generic supersymmetric compactifications, not a general proof; the explicit analysis is in a four-dimensional Abelian-Higgs toy model.
  • The main text treats the Fayet-Iliopoulos term as a continuous parameter, though the appendix argues that its quantization does not affect the conclusions. Evident from the text: the numerical study covers selected parameter choices rather than a complete parameter-space scan.

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Integrity signals

Registry

Crossref · 10.1088/1475-7516/2016/10/002

Cites

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Figures

Figure 1 Representation of the bubble of nothing. Left: this figure shows the two large spatial dimensions of the geometry with the apparent vertical separation representing the Kaluza-Klein extra dimension. Right: this image represents the full extra-dimension but only the radial large direction. We use a thick red region in both images to represent the vortex/string present at the interior edge of lower-dimensional spacetime, where the extra dimension degenerates to zero size, as described in section 4.2.
Figure 1 Representation of the bubble of nothing. Left: this figure shows the two large spatial dimensions of the geometry with the apparent vertical separation representing the Kaluza-Klein extra dimension. Right: this image represents the full extra-dimension but only the radial large direction. We use a thick red region in both images to represent the vortex/string present at the interior edge of lower-dimensional spacetime, where the extra dimension degenerates to zero size, as described in section 4.2.CC-BY · From the PDF
Figure 2 Bubble of nothing in the Abelian-Higgs model with β = 1, γ = 10 -3 and asymptotic radius of the compact dimension R = 10. Note that since R ≫ 1, the vortex is much thinner than the background curvature, and so we can compare the bubble deficit angle ∆ W ≈ 2 π 10 -3 with the flat space deficit angle ∆ = 2 π 10 -3 in eq. (6.2). This is a very light vortex with tension 4 Gµ = 10 -3 , so backreaction is almost negligible. In the left panel, the matter field profiles are shown, together with the metric fields B ′ ( r ) /H 0 and C ( r ) /R . These last two are on top of each other, inferring that we are in a bubble of nothing configuration, as given by (4.6). This can be corroborated by the profiles of B ( r ) and C ( r ) in the right panel.
Figure 2 Bubble of nothing in the Abelian-Higgs model with β = 1, γ = 10 -3 and asymptotic radius of the compact dimension R = 10. Note that since R ≫ 1, the vortex is much thinner than the background curvature, and so we can compare the bubble deficit angle ∆ W ≈ 2 π 10 -3 with the flat space deficit angle ∆ = 2 π 10 -3 in eq. (6.2). This is a very light vortex with tension 4 Gµ = 10 -3 , so backreaction is almost negligible. In the left panel, the matter field profiles are shown, together with the metric fields B ′ ( r ) /H 0 and C ( r ) /R . These last two are on top of each other, inferring that we are in a bubble of nothing configuration, as given by (4.6). This can be corroborated by the profiles of B ( r ) and C ( r ) in the right panel.CC-BY · From the PDF
Figure 3 Bubble of nothing in the Abelian-Higgs model with explicit breaking of supersymmetry, β = 0 . 001, strong gravitational coupling γ = 1, and large asymptotic radius of the compact direction R = 40. Note that the relation (4.6) is satisfied far from the core.
Figure 3 Bubble of nothing in the Abelian-Higgs model with explicit breaking of supersymmetry, β = 0 . 001, strong gravitational coupling γ = 1, and large asymptotic radius of the compact direction R = 40. Note that the relation (4.6) is satisfied far from the core.CC-BY · From the PDF
Figure 4 In the left panel we plot a bubble of nothing configuration near the supersymmetric limit ( γ = 0 . 98). Note that although the matter fields rapidly approach their asymptotic (vacuum) values, in the region plotted the extra dimension has size C ( r ) ≈ 1, and it only very slowly approaches its asymptotic value of 10. The right plot shows a sequence of bubble of nothing geometries for increasing values of γ , fixed β = 1, and R = 10. From top to bottom γ = { 0 . 01 , 0 . 1 , 0 . 2 , 0 . 3 , . . . , 0 . 9 } , showing that for higher values of γ there is a larger region (1 ≲ r ≲ H -1 0 ) where the metric profile functions B ≈ 1 and interestingly, C ≈ 1, regardless of the value of R .
Figure 4 In the left panel we plot a bubble of nothing configuration near the supersymmetric limit ( γ = 0 . 98). Note that although the matter fields rapidly approach their asymptotic (vacuum) values, in the region plotted the extra dimension has size C ( r ) ≈ 1, and it only very slowly approaches its asymptotic value of 10. The right plot shows a sequence of bubble of nothing geometries for increasing values of γ , fixed β = 1, and R = 10. From top to bottom γ = { 0 . 01 , 0 . 1 , 0 . 2 , 0 . 3 , . . . , 0 . 9 } , showing that for higher values of γ there is a larger region (1 ≲ r ≲ H -1 0 ) where the metric profile functions B ≈ 1 and interestingly, C ≈ 1, regardless of the value of R .CC-BY · From the PDF
Figure 5 A bubble of nothing for a winding compactification with KK radius R = 0 . 5 and gravitational coupling γ = 1. The left panel shows the solution for β = 0 . 6. The right panel shows the ( β = 0 . 98) bubble of nothing, close to the supersymmetric case. In this panel the radial distance is arranged on a logarithmic scale so that the figure shows the two regimes: inside/near the core, and asymptotically the compactification boundary conditions.
Figure 5 A bubble of nothing for a winding compactification with KK radius R = 0 . 5 and gravitational coupling γ = 1. The left panel shows the solution for β = 0 . 6. The right panel shows the ( β = 0 . 98) bubble of nothing, close to the supersymmetric case. In this panel the radial distance is arranged on a logarithmic scale so that the figure shows the two regimes: inside/near the core, and asymptotically the compactification boundary conditions.CC-BY · From the PDF
Figure 6 A bubble (solid lines) superimposed with the corresponding asymptotic vacuum solution B ′ vac ( r ) = H 0 C vac ( r ) /R (dotted). Numerically we also find H 0 = 0 . 103 even though the asymptotic vacuum solution would extrapolate to a smaller (Nambu-Goto wrapped) bubble with H δ = 0 . 132. We can characterize this 'thick-vortex' effect as the ratio H δ /H 0 = 1 . 29.
Figure 6 A bubble (solid lines) superimposed with the corresponding asymptotic vacuum solution B ′ vac ( r ) = H 0 C vac ( r ) /R (dotted). Numerically we also find H 0 = 0 . 103 even though the asymptotic vacuum solution would extrapolate to a smaller (Nambu-Goto wrapped) bubble with H δ = 0 . 132. We can characterize this 'thick-vortex' effect as the ratio H δ /H 0 = 1 . 29.CC-BY · From the PDF
Figure 7 Solutions for γ = 1 and R = 2 close to the supersymmetric limit, β = 0 . 98. Note that the initial bubble radius H -1 0 ≫ 1 is very large. The behaviour of the profile functions near the core of the vortex, r ≲ H -1 0 , are shown in the left plot. In the right plot the radial coordinate is represented in logarithmic scale in order to display the behaviour far from the vortex core r ≫ H -1 0 .
Figure 7 Solutions for γ = 1 and R = 2 close to the supersymmetric limit, β = 0 . 98. Note that the initial bubble radius H -1 0 ≫ 1 is very large. The behaviour of the profile functions near the core of the vortex, r ≲ H -1 0 , are shown in the left plot. In the right plot the radial coordinate is represented in logarithmic scale in order to display the behaviour far from the vortex core r ≫ H -1 0 .CC-BY · From the PDF
Figure 8 On the left plot we show a bubble with gravitational coupling γ = 1 and KK radius R = 1 away from the supersymmetric state, β = 0 . 6. Note the important deviation from the thin wall approximation. The right plot shows a bubble with γ = 1 and R = 1 near the supersymmetric limit, β = 0 . 99. The radius of the bubble H -1 0 becomes very large in this case. All indications are that H 0 → 0 as β → 1 from below.
Figure 8 On the left plot we show a bubble with gravitational coupling γ = 1 and KK radius R = 1 away from the supersymmetric state, β = 0 . 6. Note the important deviation from the thin wall approximation. The right plot shows a bubble with γ = 1 and R = 1 near the supersymmetric limit, β = 0 . 99. The radius of the bubble H -1 0 becomes very large in this case. All indications are that H 0 → 0 as β → 1 from below.CC-BY · From the PDF
Figure 9 Solution in the supersymmetric limit γ = 1, β = 1, with asymptotic compactification radius R = 1.
Figure 9 Solution in the supersymmetric limit γ = 1, β = 1, with asymptotic compactification radius R = 1.CC-BY · From the PDF

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Metadata

Authors
Blanco-Pillado, J. J., Shlaer, B., Sousa, K., & Urrestilla, J.
Affiliations
  • Ikerbasque (ES)
  • Tufts University (US)
  • Consejo Superior de Investigaciones Científicas (ES)
  • University of the Basque Country (ES)
ORCID
Venue
Journal of Cosmology and Astroparticle Physics
Publisher
IOP Publishing
Published
2016-10-03
Volume
2016
Issue
10
Pages
002-002
ISSN
1475-7516 (ISSN-L)
DOI
10.1088/1475-7516/2016/10/002
OpenAlex
W2415402231
Cited by
27
References
46
Citations by year
2026: 6 · 2025: 5 · 2024: 6 · 2023: 3 · 2022: 3 · 2021: 1 · 2020: 1 · 2017: 1 · 2016: 1
FWCI
0.25
Type
journal article
Language
en
Open access
Licence
Locations
  • Journal of Cosmology and Astroparticle Physics (publishedVersion, cc-by)
  • arXiv (Cornell University) (submittedVersion)
  • ResearchSpace (University of Auckland) (submittedVersion, cc-by)
  • arXiv (Cornell University)
Funders
Eusko Jaurlaritza (IT-559-10)
Topics
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Particle physics theoretical and experimental studies
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
Keywords
bubble of nothing, supersymmetric compactification, extra dimensions, spin structure, Abelian Higgs model, supergravity, vortex decay, vacuum decay

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