On the Stability of Topologically Non-Trivial Vacuum Bubbles in a Three Form Gauge Sector

Abstract: We study a three-form gauge sector in four spacetime dimensions coupled to electrically charged spherical membranes whose worldvolume dynamics are governed by a Dirac--Born--Infeld action. The associated four-form field strength has no local propagating degrees of freedom and contributes a branch-dependent vacuum energy. Motivated by the Hartle--Hawking--Wu selection argument, we restrict attention to the semiclassically admissible four form flux window for which the Hartle-Hawking wave function has support. We then endow the bubble wall with a worldvolume $U(1)$ gauge field carrying quantized monopole flux $n \in \mathbb{Z}$ and evaluate the full DBI energy of the resulting spherical configurations. We show that the energetically preferred branch collapses toward a microscopic core rather than stabilizing at finite radius, but for nonzero monopole flux the energy does not vanish in the collapsed limit. Instead, the bubble relaxes to a finite-energy remnant whose mass is set by the wall scale and the conserved flux. We interpret these objects as stable flux-supported particle-like states, which we call topolons. Within the admissible sector, the effective energy analysis distinguishes stable collapsed remnants from the contrasting runaway vacuum-decay channel, thereby isolating the sector relevant for cosmological relic formation. At macroscopic distances, topolons behave as heavy localized states and provide a concrete microphysical realization of a dark relic candidate. The detailed cosmological abundance and phenomenology are left for future work.

• The paper demonstrates that flux-carrying vacuum bubbles stabilize as finite-energy remnants in a three-form gauge theory.
• It employs a DBI action with quantized monopole flux to regulate energy divergence at small radii and accurately describe bubble energetics.
• The findings suggest that these topolons could serve as viable dark matter candidates produced under early-universe conditions.

Stability of Topologically Non-Trivial Vacuum Bubbles in a Three-Form Gauge Sector

Background and Motivation

This work investigates the nonperturbative excitations of a four-dimensional three-form gauge sector—specifically, the stability and endpoint configuration of electrically charged vacuum bubbles (membranes) coupled to the four-form field strength. Branch transitions between different vacuum energy sectors, induced by such membranes, play a pivotal role in landscape and sequestering approaches to the cosmological constant. The four-form provides discrete, branch-dependent contributions to vacuum energy, with semiclassical support determined by the Hartle–Hawking–Wu (HHW) criterion: only sectors that admit a regular $S^4$ Euclidean cap (i.e., $\Lambda_{\mathrm{eff}} > 0$) are physically realized.

Prior work established that the four-form in four dimensions has no local propagating degrees of freedom and acts as an algebraic variable that scans the effective vacuum energy. However, the nature of non-homogeneous excitations—specifically, whether flux-carrying bubbles necessarily collapse to trivial zero-energy, or may stabilize as finite-energy remnants—has not been systematically characterized. This paper focuses on the energetics and stability of such vacuum bubbles, with special attention to their worldvolume physics as governed by a Dirac–Born–Infeld (DBI) action endowed with quantized monopole flux.

Theoretical Construction

The authors consider a three-form $C_3$ with field strength $F_4 = dC_3$, such that $$F^{\mu\nu\rho\sigma} = q\,\epsilon^{\mu\nu\rho\sigma}$$ in solution space, with spacetime-constant $q$. The effective cosmological constant is $\Lambda_{\textrm{eff}} = \Lambda_0 - 4\pi G q^2$. Within the HHW prescription, the allowed branch window is $|q| < q_0 \equiv \sqrt{\Lambda_0 / 4\pi G}$; outside this, the no-boundary wavefunction has no support.

Membrane nucleation causes discrete jumps $q_{\rm out} - q_{\rm in} = q_b$ across the bubble wall. The membrane charge is quantized: $q_b = N q_{\min}$, where $\Lambda_{\mathrm{eff}} > 0$0 can be of order $\Lambda_{\mathrm{eff}} > 0$1, $\Lambda_{\mathrm{eff}} > 0$2, or $\Lambda_{\mathrm{eff}} > 0$3. The vacuum energy landscape thus forms a lattice of discrete steps, which, depending on $\Lambda_{\mathrm{eff}} > 0$4, may be coarse (for Planck-scale charge) or nearly continuous.

The central focus is on the intrinsic membrane physics. Unlike scalar-induced domain walls, the membrane tension and worldvolume structure are not determined by the bulk three-form sector. Instead, the dynamical degrees of freedom—including a worldvolume $\Lambda_{\mathrm{eff}} > 0$5 gauge sector supporting quantized monopole flux—are described by a DBI action. This nonlinear structure is essential in capturing the endpoint of bubble collapse, especially under large flux.

Energetics and Endpoint Analysis

For a spherically symmetric, static membrane of radius $\Lambda_{\mathrm{eff}} > 0$6 carrying monopole flux $\Lambda_{\mathrm{eff}} > 0$7, the wall energy reduces to

$\Lambda_{\mathrm{eff}} > 0$8

with $\Lambda_{\mathrm{eff}} > 0$9 the membrane tension and $C_3$0 the worldvolume gauge coupling. The DBI form is necessary: in the quadratic Maxwell limit, the energy diverges as $C_3$1, while the full DBI action regulates this, yielding a finite energy at $C_3$2 when $C_3$3:

$C_3$4

The total bubble energy includes a volume term, $C_3$5, with $C_3$6 the vacuum energy difference across the bubble.

A complete small-$C_3$7 expansion demonstrates that for all physically admissible branch transitions ($C_3$8, as enforced by HHW), the collapsed configuration at $C_3$9 is a true global minimum. The wall tension alone would drive the bubble to zero size and energy, but worldvolume monopole flux induces an irreducible energy floor. The presence of a unique non-vanishing mass even in the $F_4 = dC_3$0 limit is a strong claim that distinguishes these flux-supported remnants from wall-only configurations.

The authors further classify possible bubble energetics:

• Class I (Exact sign-flip): $F_4 = dC_3$1; the remnant mass is purely from wall and flux, with no vacuum pressure.
• Class II (Generic admissible): $F_4 = dC_3$2; positive vacuum pressure further reinforces collapse, but the remnant is still the finite-energy flux object.
• Class III (Prohibited, $F_4 = dC_3$3): energetically favors unbounded expansion, corresponding to usual vacuum decay channels rather than stable particles. These are excluded by the HHW admissibility.

Implications and Outlook

The principal outcome is the robust prediction that a flux-carrying vacuum bubble in a three-form gauge sector, with worldvolume DBI action and quantized monopole flux, collapses not to a trivial configuration but to a finite, particle-like soliton ("topolon"). The remnant is stabilized by the interplay of topology (flux conservation), worldvolume gauge dynamics, and the DBI nonlinearity; the bulk three-form dynamics alone are insufficient for this outcome.

Such topolons act as localized, massive states. At cosmological scales, and for sufficiently large $F_4 = dC_3$4, they represent concrete microphysical candidates for cold dark relics. The detailed relic abundance depends on early-universe production mechanisms (gravitational, thermal, or nonthermal) and the initial flux spectrum. At large distances, the topolon behaves as a heavy neutral particle, up to corrections from its internal structure, making it a viable dark matter candidate, but this aspect is left for future work.

Limitations

A number of limitations are noted:

• The results are valid within the regime of the DBI effective action; UV completions or higher-derivative corrections may modify the core structure, but not the general mechanism.
• The $F_4 = dC_3$5 limit should be read as "shrinks below all macroscopic probe scales"; a microscopic core must be resolved by a UV-complete theory or possibly by gravitational physics.
• The branch selection ($F_4 = dC_3$6) is imposed by HHW semiclassical arguments rather than local dynamical constraints.
• Models with many stacked three-forms or in string compactifications (with more complicated flux landscapes) may admit additional dynamics.
• The cosmological constant problem itself is not addressed—this analysis holds for a single sector with (ideally) vanishing background vacuum energy.

Conclusion

The paper demonstrates that, within any three-form gauge sector compatible with semiclassical cosmological selection effects, the endpoint of flux bubble collapse is fundamentally altered by worldvolume physics: nontrivial monopole flux and DBI nonlinearity stabilize finite-mass, particle-like remnants (topolons). These stable topological relics emerge naturally and provide theoretically grounded candidates for nonperturbative cosmological relics, motivating further investigations into their phenomenological signatures and implications for the quantum structure of the vacuum.

Reference: "On the Stability of Topologically Non-Trivial Vacuum Bubbles in a Three Form Gauge Sector" (2604.03363)