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Cited by 87Black Holes and Theoretical PhysicsNoncommutative and Quantum Gravity TheoriesParticle physics theoretical and experimental studies

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Nothing is certain in string compactifications

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Etxebarria, Iñaki García, Montero, Miguel, Sousa, Kepa, Valenzuela, Irene·2020·Journal of High Energy Physics
DOI 10.1007/jhep12(2020)032
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Abstract

A bubble of nothing is a spacetime instability where a compact dimension collapses. After nucleation, it expands at the speed of light, leaving “nothing” behind. We argue that the topological and dynamical mechanisms which could protect a compactification against decay to nothing seem to be absent in string compactifications once supersymmetry is broken. The topological obstruction lies in a bordism group and, surprisingly, it can disappear even for a SUSY-compatible spin structure. As a proof of principle, we construct an explicit bubble of nothing for a T3 with completely periodic (SUSY-compatible) spin structure in an Einstein dilaton Gauss-Bonnet theory, which arises in the low-energy limit of certain heterotic and type II flux compactifications. Without the topological protection, supersymmetric compactifications are purely stabilized by a Coleman-deLuccia mechanism, which relies on a certain local energy condition. This is violated in our example by the nonsupersymmetric GB term. In the presence of fluxes this energy condition gets modified and its violation might be related to the Weak Gravity Conjecture. We expect that our techniques can be used to construct a plethora of new bubbles of nothing in any setup where the low-energy bordism group vanishes, including type II compactifications on CY3, AdS flux compactifications on 5-manifolds, and M-theory on 7-manifolds. This lends further evidence to the conjecture that any non-supersymmetric vacuum of quantum gravity is ultimately unstable.

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Overview

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TL;DR

The paper argues that the usual topological protection against bubbles of nothing is absent for compactifications with three or more internal dimensions, since Ω^Spin_3 = 0, and explicitly constructs a bubble of nothing for T^3 with supersymmetry-preserving periodic spin structure in an Einstein-dilaton-Gauss-Bonnet theory. The decay is dynamically forbidden in the supersymmetric limit α → 0, but a finite-rate instability appears once the Gauss-Bonnet term breaks supersymmetry and violates the Dominant Energy Condition.

Key findings

  • For T^3 with periodic (SUSY-compatible) spin structure, there is no topological obstruction because Ω^Spin_3 = 0; the authors construct an explicit smooth bubble of nothing in an Einstein-dilaton-Gauss-Bonnet model with α > 0.
  • For the T^3 compactification, the nucleation radius and Euclidean action scale as R(α) ∝ 24π² V_T3 α^{-1} and S_BON(α) ∝ 24π² V_T3 α^{-(D-5)}, so the decay rate vanishes as α → 0 via a Coleman-de Luccia mechanism.
  • For the 27 non-supersymmetric T^3/Γ quotients, bubbles exist with finite rate even at α = 0; the radius and action are R = 6(D−5)/(12−N) R_kk and S_BON = A_{D−4}/(8πG_{D−3}) R^{D−5}/2, with the missing G3 case corresponding to N = 8.
  • The construction is embedded in heterotic/type I string theory, where the required flux breaks supersymmetry; the authors argue this supports the conjecture that any non-supersymmetric vacuum of quantum gravity is ultimately unstable.

Methods

  • The authors construct Euclidean bubble-of-nothing solutions with ansatz M_D = B_4 ×_W S^{D−4}, where B_4 is a nulbordism of T^3 or T^3/Γ, described by a Weierstrass elliptic fibration with N degenerations.
  • They solve the Einstein-dilaton-Gauss-Bonnet equations using a layered approximation: perturbative treatment in α near the bubble core, semi-flat and hyperkähler/Taub-NUT descriptions around degenerations, numerical integration of the outer-bubble ODEs, and gluing across layers.
  • A numerical example for D = 7 gives R = 32.2 for 24π² V_T3 α = 0.02 and V*_T3/V_T3 = 0.5; the paper also discusses string embeddings via heterotic on T^4 and type I/IIB duals.

Limitations

  • The paper states that the solutions are approximate and perturbative in α, that the fluctuation spectrum is not computed, and that quantum effects are not computed; the resulting expressions provide an upper bound on the actual decay rate.
  • The construction relies on a degenerate/large-volume limit, a small T^2 fibre, well-separated degenerations, and small α; the string embedding turns on flux that produces runaway potentials and vacuum energy at higher order in α.
  • The paper does not provide a general construction for arbitrary compactifications; it focuses on T^3 and T^3/Γ, with explicit numerical checks mainly in D = 7, and leaves broader flux/AdS/CY cases for future work.

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Crossref · 10.1007/jhep12(2020)032

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Figures

Figure 1 Two d -dimensional manifolds C A and C B are equivalent in bordism if together they form the boundary of a ( d +1) -dimensional manifold B .
Figure 1 Two d -dimensional manifolds C A and C B are equivalent in bordism if together they form the boundary of a ( d +1) -dimensional manifold B .CC-BY · From the PDF
Figure 2 Flowchart illustrating when can one get a bubble of nothing. Given a compactification manifold C , one first checks that there is no topological obstruction (that the manifold is trivial in bordism). Assuming this is the case, one must make sure there are either no covariantly constant spinors in the compactification manifold, or that the relevant energy condition is violated. If either of these happens, there can be a bubble of nothing. As we will see in the paper, our expectation is that if it can be there, it will be.
Figure 2 Flowchart illustrating when can one get a bubble of nothing. Given a compactification manifold C , one first checks that there is no topological obstruction (that the manifold is trivial in bordism). Assuming this is the case, one must make sure there are either no covariantly constant spinors in the compactification manifold, or that the relevant energy condition is violated. If either of these happens, there can be a bubble of nothing. As we will see in the paper, our expectation is that if it can be there, it will be.CC-BY · From the PDF
Figure 3 Schematic representation of the Weierstrass fibration over a disc (3.1): there is a T 2 , which can pinch off at a discrete set of points. The bubbles we will consider in this paper all share this general topological structure.
Figure 3 Schematic representation of the Weierstrass fibration over a disc (3.1): there is a T 2 , which can pinch off at a discrete set of points. The bubbles we will consider in this paper all share this general topological structure.CC-BY · From the PDF
Figure 5 Layered structure of the fibration B 4 ∼ = D × T 2 for the T 3 BON. From left to right the diagram displays the different regimes of the manifold B 4 : the outer-bubble regime ( I. ), whose asymptotic boundary matches the compact space T 3 , and where only the T 3 volume is dynamical; the semi-flat regime ( II.a ), valid away from the degeneration points ( Deg. ), and where the complex structure of the T 2 fibre becomes dynamical; the hyperkähler regime ( II.b ) describing the neighbourhood of the degenerations; and the KK monopoles ( KKM ) ( II.c ) describing the BON cores, where the compact space is smoothly sealed off, and the metric is locally R 4 .
Figure 5 Layered structure of the fibration B 4 ∼ = D × T 2 for the T 3 BON. From left to right the diagram displays the different regimes of the manifold B 4 : the outer-bubble regime ( I. ), whose asymptotic boundary matches the compact space T 3 , and where only the T 3 volume is dynamical; the semi-flat regime ( II.a ), valid away from the degeneration points ( Deg. ), and where the complex structure of the T 2 fibre becomes dynamical; the hyperkähler regime ( II.b ) describing the neighbourhood of the degenerations; and the KK monopoles ( KKM ) ( II.c ) describing the BON cores, where the compact space is smoothly sealed off, and the metric is locally R 4 .CC-BY · From the PDF
Figure 6 Representation of the semi-flat geometry j (i) (5.18) with N = 6 , and the monodromy paths. The fundamental domain is mapped six times to the disc D via (5.18), and the shaded region represents a single image of F . The T 2 geometry on adjacent regions (e.g. A and B ) is related by a S transformation, and the gluing across the common boundary (dotted line) is done with the diffeomorphism in (5.17). The points z ∞ represent the degenerations points, and z i , z ρ are the locations where τ attains the values τ ( z i ) = i and τ ( z ρ ) = ρ respectively
Figure 6 Representation of the semi-flat geometry j (i) (5.18) with N = 6 , and the monodromy paths. The fundamental domain is mapped six times to the disc D via (5.18), and the shaded region represents a single image of F . The T 2 geometry on adjacent regions (e.g. A and B ) is related by a S transformation, and the gluing across the common boundary (dotted line) is done with the diffeomorphism in (5.17). The points z ∞ represent the degenerations points, and z i , z ρ are the locations where τ attains the values τ ( z i ) = i and τ ( z ρ ) = ρ respectivelyCC-BY · From the PDF
Figure 7 An image of the fundamental domain F and its boundary (left) on the disc D (right) under the mapping (5.18) with arg( z 0 ) = π/ 2 .
Figure 7 An image of the fundamental domain F and its boundary (left) on the disc D (right) under the mapping (5.18) with arg( z 0 ) = π/ 2 .CC-BY · From the PDF
Figure 8 Outer-bubble regime of the T 3 BON with deformation parameter 24 π 2 V T 3 α = 0 . 005 . The solid line is related to the warp factor on the sphere, R W ′ ( ρ ) , and the dashed line represents the volume of the T 3 compact space, V T 3 ( ρ ) / V ∞ T 3 , with the radial coordinate ρ displayed in logarithmic scale. The bubble nucleation radius is R = 32 . 2 , and the three-torus volume at the boundary of the inner-bubble regime is given by V ∗ T 3 = 0 . 5 ×V ∞ T 3 , where V ∞ T 3 is the asymptotic T 3 volume, denoted simply by V T 3 in the text.
Figure 8 Outer-bubble regime of the T 3 BON with deformation parameter 24 π 2 V T 3 α = 0 . 005 . The solid line is related to the warp factor on the sphere, R W ′ ( ρ ) , and the dashed line represents the volume of the T 3 compact space, V T 3 ( ρ ) / V ∞ T 3 , with the radial coordinate ρ displayed in logarithmic scale. The bubble nucleation radius is R = 32 . 2 , and the three-torus volume at the boundary of the inner-bubble regime is given by V ∗ T 3 = 0 . 5 ×V ∞ T 3 , where V ∞ T 3 is the asymptotic T 3 volume, denoted simply by V T 3 in the text.CC-BY · From the PDF
Figure 9 Dependence of the outer-bubble geometry on the value of the T 3 volume at the matching point, V ∗ T 3 = V ∞ T 3 e -∆ ϕ . The plot displays the T 3 volume V T 3 / V ∞ T 3 as a function of the radial coordinate ρ for a fixed parameter 24 π 2 V T 3 α = 0 . 005 , and varying values of V ∗ T 3 / V ∞ T 3 = { 0 . 7 , 0 . 8 , 0 . 9 , 0 . 99 } (from bottom to top). The corresponding radii are respectively R = { 455 . 4 , 231 . 5 , 95 . 1 , 8 . 1 } .
Figure 9 Dependence of the outer-bubble geometry on the value of the T 3 volume at the matching point, V ∗ T 3 = V ∞ T 3 e -∆ ϕ . The plot displays the T 3 volume V T 3 / V ∞ T 3 as a function of the radial coordinate ρ for a fixed parameter 24 π 2 V T 3 α = 0 . 005 , and varying values of V ∗ T 3 / V ∞ T 3 = { 0 . 7 , 0 . 8 , 0 . 9 , 0 . 99 } (from bottom to top). The corresponding radii are respectively R = { 455 . 4 , 231 . 5 , 95 . 1 , 8 . 1 } .CC-BY · From the PDF
Figure 10 Flowchart illustrating when one can get a bubble of nothing; a more detailed version of figure 2. Given a manifold with bundles M , one first checks that there is no topological obstruction in Ω Spin G d . Assuming this is the case, one must make sure there are either no covariantly constant spinors in the compactification manifold, or that the relevant energy condition is violated.
Figure 10 Flowchart illustrating when one can get a bubble of nothing; a more detailed version of figure 2. Given a manifold with bundles M , one first checks that there is no topological obstruction in Ω Spin G d . Assuming this is the case, one must make sure there are either no covariantly constant spinors in the compactification manifold, or that the relevant energy condition is violated.CC-BY · From the PDF
Figure 11 Construction of a bordism between a S 3 with 24 units of H -flux and a T 3 with 12 units of H fl ux. One starts with K3 in the stable degeneration limit plus 24 NS5 branes, and then removes the part of the geometry encircled by dashed lines.
Figure 11 Construction of a bordism between a S 3 with 24 units of H -flux and a T 3 with 12 units of H fl ux. One starts with K3 in the stable degeneration limit plus 24 NS5 branes, and then removes the part of the geometry encircled by dashed lines.CC-BY · From the PDF
Figure 12 Outer-bubble geometry for the model (A.1) with D = 7 , ∆ β = 8 . 61 · 10 -5 , AdS scale L = 6 . 78 · 10 3 , and the scale of α ′ corrections given by 24 π 2 α ′ V T 3 = 1 . 8 · 10 -4 . The solid line represents the warp factor on the sphere LW ′ /W , and the dashed line the three torus volume V T 3 ( ρ ) / V ∞ T 3 . The BON radius is R = 5 . 46 · 10 3 , just slightly smaller that the AdS radius L . The plot displays how the non-compact space approaches to the AdS 4 for large ρ →∞ , i.e. LW ′ /W → 1 .
Figure 12 Outer-bubble geometry for the model (A.1) with D = 7 , ∆ β = 8 . 61 · 10 -5 , AdS scale L = 6 . 78 · 10 3 , and the scale of α ′ corrections given by 24 π 2 α ′ V T 3 = 1 . 8 · 10 -4 . The solid line represents the warp factor on the sphere LW ′ /W , and the dashed line the three torus volume V T 3 ( ρ ) / V ∞ T 3 . The BON radius is R = 5 . 46 · 10 3 , just slightly smaller that the AdS radius L . The plot displays how the non-compact space approaches to the AdS 4 for large ρ →∞ , i.e. LW ′ /W → 1 .CC-BY · From the PDF
Figure 13 Outer-bubble geometry for the model (A.1) with D = 7 in the limit β → 1 with R /L = 8 . 05 · 10 -5 . The AdS scale is L = 4 . 57 · 10 7 , and the scale of α ′ corrections is given by 24 π 2 α ′ V T 3 = 2 . 68 · 10 -8 . The solid line represents the warp factor on the sphere, expressed in terms of R W ′ ( ρ ) , and the dashed line is the three torus volume V T 3 ( ρ ) / V ∞ T 3 . The BON nucleation radius R = 3 . 68 · 10 3 ≪ L is much smaller that the AdS scale. In the regime R ≪ ρ ≪ L the bubble spacetime approaches a configuration M 4 × T 3 , i.e. W ′ ( ρ ) →R -1 , mimicking the Minkowski bubbles we discussed in the main text.
Figure 13 Outer-bubble geometry for the model (A.1) with D = 7 in the limit β → 1 with R /L = 8 . 05 · 10 -5 . The AdS scale is L = 4 . 57 · 10 7 , and the scale of α ′ corrections is given by 24 π 2 α ′ V T 3 = 2 . 68 · 10 -8 . The solid line represents the warp factor on the sphere, expressed in terms of R W ′ ( ρ ) , and the dashed line is the three torus volume V T 3 ( ρ ) / V ∞ T 3 . The BON nucleation radius R = 3 . 68 · 10 3 ≪ L is much smaller that the AdS scale. In the regime R ≪ ρ ≪ L the bubble spacetime approaches a configuration M 4 × T 3 , i.e. W ′ ( ρ ) →R -1 , mimicking the Minkowski bubbles we discussed in the main text.CC-BY · From the PDF
Image 13
CC-BY · From the PDF
Figure 15 Schematic construction of our bubble solutions with negative mass. The innermost circle, in the core, with radius x ∼ x c , corresponds to the radius of the critical bubble - the radius of the bubble of nothing in the instant it nucleates. The circle at the solid line corresponds to the physical radius of the bubble at some time τ > 0 , when it has been expanding for a while. The starting point of our construction is to take this τ > 0 'snapshot' of the bubble and use it as an initial condition for GR evolution, while modifying the time derivative of the metric according to (C.22) outside of the dashed circle at x = x 0 , far away from the core of the bubble. This asymptotic modification is designed to take away some positive energy from the bubble, and therefore the resulting object has negative mass. As proven in the main text, this can be made arbitrarily negative, thereby establishing the instability of the vacuum even if the original bubble was not an exact (only approximate) solution to Einstein's equations.
Figure 15 Schematic construction of our bubble solutions with negative mass. The innermost circle, in the core, with radius x ∼ x c , corresponds to the radius of the critical bubble - the radius of the bubble of nothing in the instant it nucleates. The circle at the solid line corresponds to the physical radius of the bubble at some time τ > 0 , when it has been expanding for a while. The starting point of our construction is to take this τ > 0 'snapshot' of the bubble and use it as an initial condition for GR evolution, while modifying the time derivative of the metric according to (C.22) outside of the dashed circle at x = x 0 , far away from the core of the bubble. This asymptotic modification is designed to take away some positive energy from the bubble, and therefore the resulting object has negative mass. As proven in the main text, this can be made arbitrarily negative, thereby establishing the instability of the vacuum even if the original bubble was not an exact (only approximate) solution to Einstein's equations.CC-BY · From the PDF

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Metadata

Authors
Etxebarria, I. G., Montero, M., Sousa, K., & Valenzuela, I.
Affiliations
  • Harvard University (US)
  • Charles University (CZ)
Corresponding author
  • Montero, Miguel
  • Sousa, Kepa
  • Valenzuela, Irene
ORCID
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2020-12
Volume
2020
Issue
12
ISSN
1029-8479 (ISSN-L)
DOI
10.1007/jhep12(2020)032
OpenAlex
W3024522407
Semantic Scholar
f04bc50c30f8c73faf01aa0e86378dfadec020ac
Cited by
87
References
143
Citations by year
2026: 10 · 2025: 12 · 2024: 17 · 2023: 14 · 2022: 19 · 2021: 10 · 2020: 5
FWCI
4.70
Type
journal article
Language
en
Open access
Licence
Locations
  • Journal of High Energy Physics (publishedVersion, cc-by)
  • Durham Research Online (Durham University) (acceptedVersion, other-oa)
  • arXiv (Cornell University) (submittedVersion)
  • Durham Research Online (Durham University) (submittedVersion, cc-by)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • arXiv (Cornell University)
Funders
  • National Science Foundation (ST/P000371/1, 602883, 1748958)
  • Simons Foundation (602883, PHY-1748958)
  • Grantová Agentura České Republiky (GA19-01850S)
  • Fonds Wetenschappelijk Onderzoek (PHY-1748958)
  • KU Leuven
  • Science and Technology Facilities Council (ST/P000371/, ST/P000371/1)
  • Kavli Institute for Theoretical Physics, University of California, Santa Barbara (1748958, PHY-1748958)
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 › Noncommutative and Quantum Gravity Theories
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Particle physics theoretical and experimental studies
Keywords
bubble of nothing, string compactification, bordism groups, supersymmetry breaking, spin structure, energy condition violation, flux compactification, heterotic compactifications, weak gravity conjecture, vacuum instability, M-theory

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