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TL;DR

The paper constructs gravitational instantons generalizing Witten’s bubble of nothing in (D+1)-dimensional General Relativity with a cosmological constant, describing non-perturbative decay of constant-curvature vacua via expanding bubbles with dS_d × S^{D-d-1} surfaces. It computes finite Euclidean-action bounces, shows exponential suppression in the semiclassical regime, and discusses covariantly constant spinors as a topological obstruction that can stabilize the vacuum.

Key findings

• The construction generalizes Witten’s 5d Kaluza-Klein bubble to (D+1)-dimensional vacua of the form (3.1), with consistency conditions \(L_{dS_d}=1\), \(L_{S^{D-d-1}}=L_{dS_d}\sqrt{(D-d-2)/(d-1)}\), \(d\ge2\), \(D=d+2\). The Euclidean black-hole metric (3.4) is smooth when \(\ell_{KK}^2=4(D-2)f(\rho_0)^2/(d-1)\) (3.6); for \(\Lambda\le0\) it is non-singular for \(\rho\in(\rho_0,+\infty)\), while \(\Lambda>0\) is excluded because the Euclidean geometry has no asymptotic region.
• The bounce is finite. For zero curvature, \(\Delta S_E^{(\Lambda=0)}\) is given by (3.15); for \(D=4,d=3\) it reduces to Witten’s result \(\Delta S_E=\pi R^2/(4G_N^{(4)})\) (2.6), and scales as \(\Delta S_E^{(\Lambda=0)}\propto (R/\ell_{Pl})^{D-1}\) (3.17). For \(\Lambda<0\), \(\Delta S_E^{(\Lambda<0)}\) is given by (3.18), with scaling (3.19); the decay rate is exponentially suppressed whenever the semiclassical approximation is valid.
• The decay changes topology: the vacuum is foliated by \(S^1\times S^{d-1}\times S^{D-d-1}\), while near the bubble the instanton has topology \(R^2\times S^{d-1}\times S^{D-d-1}\). Covariantly constant spinors require \(\alpha=0\), whereas the unique spin structures on spheres require \(\alpha=\pi\); hence the instanton does not contribute to the path integral for theories with covariantly constant spinors, so suitable fermion boundary conditions can “cure” the instability.
• Appendix B shows the bubble decay can be captured by a tunneling scalar field in one lower dimension via Kaluza-Klein reduction.

Methods

• Theoretical construction: double analytic continuation of non-extremal black holes in (D+1)-dimensional GR with \(\Lambda\), using Wick rotation and continuing \(S^d\to dS^d\) to obtain the instanton geometry (3.7). The Hamilton-Jacobi formulation is used to recast the coupled second-order equations into first-order conditions without supersymmetry.
• Euclidean action analysis: computes \(S_E=S_{E,\text{bulk}}+S_{E,\text{GHY}}\) and the bounce \(\Delta S_E=S_E|_{\text{inst}}-S_E|_{\text{false}}\), evaluating bulk and boundary contributions separately for zero and non-zero curvature.
• No empirical data or sample; the work is an analytic study in arbitrary dimensions, with smoothness and consistency conditions imposed on the solutions.

Limitations

• Stated: the semiclassical approximation is trusted only for \(R\) large compared to the Planck length \(\ell_{Pl}\). For \(\Lambda>0\), the Euclidean geometry does not have an asymptotic region due to a zero in \(f(\rho)\) and the static patch, so the paper specializes to \(\Lambda\le0\).
• Stated: the interpretation of asymptotically AdS decay in holography remains an open problem; it is unknown how to interpret the decay in the dual field theory or whether the decay invalidates the holographic correspondence.
• Text-evident: no explicit string-theory embedding or dual field theory computation is performed; extension to fully stringy decays is left for future work, and the analysis is classical/Euclidean rather than a full quantum-gravity treatment.