De Sitter Vacua & pUniverses: Spontaneous Symmetry Breaking in the p-Schwinger Model

TL;DR

Can a quantum field theory maintain multiple, distinct "universes" or vacuum states in the rapidly expanding background of de Sitter space? This paper answers with a definitive "yes." By analyzing the p-Schwinger model—where fermions carry non-unit charge $p$—the authors prove that discrete global symmetries are spontaneously broken in 2D de Sitter space, leading to $p$ locally indistinguishable but globally distinct Hadamard vacua.

Background: The Symmetry Map

Spontaneous Symmetry Breaking (SSB) in de Sitter (dS) space is a subtle topic. While continuous symmetries in 2D are generally not broken due to infrared fluctuations, discrete symmetries can persist.

The authors choose the p-Schwinger model as their laboratory. In Minkowski space, this model is known to flow to a topological BF theory in the infrared. The core innovation here is extending this analysis to the round two-sphere ($S^2$), which is the Euclidean continuation of $d{S}_2$.

Methodology: Bosonization and Topological Operators

To solve an interacting theory of fermions non-perturbatively, the authors employ Abelian bosonization. The fermionic currents are mapped to a compact scalar field $B(x)$:

• Vector Current: $j_V^{\mu }o-\frac{1}{2\pi }{ϵ}^{\mu u}{\partial }_{u}B$
• Axial Current: $j_A^{\mu }o-\frac{i}{2\pi }{\partial }^{\mu }B$

The presence of charge-$p$ fermions means that the axial $U(1)$ symmetry is not entirely destroyed by the anomaly; a discrete ${\mathbb{Z}}_p^{(0)}$ subgroup survives. Simultaneously, the theory possesses a ${\mathbb{Z}}_p^{(1)}$ one-form symmetry because unit-charged Wilson lines cannot be screened by charge-$p$ matter.

The Construction of States

Using the Hartle-Hawking construction, the authors define the Hilbert space by performing path integrals over the northern hemisphere.

• Vacua: States carrying definite ${\mathbb{Z}}_p^{(0)}$ charge.
• pUniverses: States carrying definite ${\mathbb{Z}}_p^{(1)}$ charge.

Figure: The path integral representation of vacuum overlaps using topological operator insertions.

Key Insights: Why It Works

1. Topological Robustness: The ${\mathbb{Z}}_p$ symmetries participate in a mixed 't Hooft anomaly. This ensures that the vacuum cannot be unique; even if the theory is gapped, the degenerate vacua must exist.
2. Hadamard Property: Unlike the famous "$\alpha$-vacua" which have singularities at antipodal points, the $p$ states found here are Hadamard, meaning they look like the standard Minkowski vacuum at short distances.
3. Locally Indistinguishable: For a local observer, any of these $p$ states would look identical in terms of local correlation functions (like the electric field two-point function), yet they are globally distinct.

Experiments & Results

The paper provides exact, closed-form expressions for $n$-point functions of "meson" operators (fermion bilinears).

Figure: Representation of Wilson line screening (or lack thereof) in the charge-p theory.

Key numerical and analytical findings:

• Mass Generation: Interactions generate a mass $m^2=\frac{p^2{q}^2{N}_f}{\pi }$, where $q$ is the coupling.
• Vacuum Overlaps: Overlaps between different vacuum sectors are strictly zero ($⟨m|n⟩={\delta }_{mn}$), confirming the SSB.
• Gravitational Coupling: By adding $N_f$ flavors and coupling to 2D gravity, they find a semiclassical dS saddle where the horizon size scales as ${\ell }_{cl}\propto \sqrt{N_f}$.

Critical Analysis & Future Outlook

The "pUniverses" provide a fascinating perspective on de Sitter microstates. If we view the dS horizon as a thermodynamic object, these degenerate states could contribute to the total entropy.

Limitations:

• The 2D nature of the model simplifies the gravity sector significantly (Liouville-like). Extending this to 4D is the "holy grail" for dS physics.
• The authors touch upon the "no global symmetries in gravity" conjecture, suggesting that wormholes or topology changes might eventually lift the degeneracy of these pUniverses.

Conclusion: This paper bridges the gap between topological quantum field theory (TQFT) and de Sitter gravity, showing that even in the most volatile backgrounds, discrete topological information remains remarkably resilient.