Bubbles of nothing and supersymmetric compactifications

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.