Nothing is certain in string compactifications

TL;DR

The work investigates nonperturbative instabilities of string compactifications by constructing explicit bubbles of nothing (BON) in a setting with three compact extra dimensions. It shows that topological protection from spin bordism can fail in higher dimensions, and that dynamical protection via the Positive Energy Theorem can be evaded when a SUSY-violating higher-derivative Gauss–Bonnet term is present, enabling BON with periodic spin structures. The authors develop a layered, controlled approach combining an inner Calabi–Yau/bordism geometry (with hyperkähler and semi-flat regimes) and an outer region, glued perturbatively, to produce smooth BON instantons for $T^3$ and its freely-acting quotients, and they compute the decay rate $S_{BON}$ scaling as $(\frac{24\pi^2}{\mathcal{V}_{T^3}}\alpha)^{-(D-5)}$ in the small-$\alpha$ limit. They also embed the construction in string theory, discuss flux and Spin$^c$ generalizations, connect to energy-condition violations and the Weak Gravity Conjecture, and argue that many non-supersymmetric vacua may be inherently unstable. The results lend support to the conjecture that non-supersymmetric quantum gravities generically harbor instabilities, with implications for the Swampland program and AdS/CFT stability analyses.

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 $T^3$ 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 $CY_3$, 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.

Figures (5)

• Figure 1: Two $d$-dimensional manifolds $\mathcal{C}_A$ and $\mathcal{C}_B$ are equivalent in bordism if together they form the boundary of a $(d+1)$-dimensional manifold $\mathcal{B}$.
• Figure 2: Flowchart illustrating when can one get a bubble of nothing. Given a compactification manifold $\mathcal{C}$, 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 3: Schematic representation of the Weierstrass fibration over a disk (\ref{['eq:Weierstrass']}): 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 4: Starting with a noncompact manifold $\mathcal{B}_4$ with an infinite tube, one can construct an auxiliary compact manifold ${\cal S}$ by taking two copies of $\mathcal{B}_4$, reversing orientation of one copy, and gluing them along their common boundary $\mathcal{C}_3$. ${\cal S}$ is not a complete manifold with respect to the induced metric, but this can be easily fixed as described in the main text.
• Figure 5: Layered structure of the fibration ${\cal B}_4\cong {\cal D} \times T^2$ for the $T^3$ BON. From left to right the diagram displays the different regimes of the manifold ${\cal B}_4$: 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 $\mathbb{R}^4$.