Reflections on Parity Breaking

TL;DR

This paper argues that if parity (or CP) is a gauged spacetime symmetry in quantum gravity, spontaneously broken parity produces exactly stable domain walls whose stability is tied to the topology of non-orientable manifolds via the first Stiefel-Whitney class $w_1(TX)$. It shows that parity domain walls carry a novel $(d-2)$-form gauge-like charge, but in a UV-complete theory this symmetry is gauged (or broken) through cobordism constraints, end-of-the-world branes, and related objects, preventing parity vortices from destabilizing walls. The work blends formal QFT/topology (cobordism, Adams spectral sequence) with phenomenological model-building (Nelson–Barr type scenarios) to delineate how a fundamental gauged parity symmetry would shape early-universe cosmology and observable signatures, including gravitational waves from domain-wall dynamics. Overall, it provides a principled framework to reconcile exact generalized topological features with the no-global-symmetry principle in quantum gravity and to guide CP-violation model building under these constraints.

Abstract

Parity and CP symmetries are broken in the world around us. Nonetheless, parity (or CP) may be a gauge symmetry which is higgsed in our universe. This is assumed in many scenarios for physics beyond the Standard Model, including the classic Nelson--Barr proposal for the Strong CP problem. Gauged parity can only arise in quantum gravity, where it corresponds to a path integral over both orientable and non-orientable spacetime manifolds. We show that spontaneous breaking of gauged parity leads to exactly stable domain walls, and describe the implications for the cosmology of models with gauged parity. These domain walls carry an unusual sort of charge, which superficially has features in common with both gauge charges and global charges. We show that these unusual charges are consistent with the expected absence of global symmetries in quantum gravity when there exists a complete spectrum of dynamical objects required by the Swampland Cobordism Conjecture, including end-of-the-world branes.

Figures (19)

• Figure 1: Torus with labeled $a$- and $b$-cycles. In ${\mathbb{Z}}_2$ gauge theory we assign holonomies (Wilson loops) $\pm 1$ on each cycle.
• Figure 2: Nontrivial background gauge field configuration on a torus. The left and right edges are identified, as are the top and bottom edges. We show two coordinate patches ($U_1$, orange; $U_2$, purple), with two disjoint overlaps $I$ and $I'$ with transition elements $-1$ and $+1$ respectively. The $a$-cycle passes through both overlap regions, for a nontrivial Wilson loop $W(a) = -1 \times +1 = -1$. The $b$-cycle does not pass through any overlap region, and has trivial Wilson loop $W(b) = +1$.
• Figure 3: Spontaneous symmetry violating ${\mathbb{Z}}_2$-odd field configuration on the torus with the ${\mathbb{Z}}_2$ Wilson line $W(a) = -1$ introduced in Fig. \ref{['fig:torusgaugefield']}. Beginning with the choice $\phi_1 = +v$ on the left, we encounter a clash in the central region between $\phi_2 = -v$ (approaching from the left, with a sign flip in $I$) and $\phi_2 = +v$ (circling around from the right, with no sign flip in $I'$). This indicates the topological necessity of a domain wall wrapping the $b$-cycle, indicated here by the heavy black line DW.
• Figure 4: For the domain wall to decay, the gauge field configuration must become dynamical, allowing the transition function to change by some novel process in the region marked with the question mark. We will see in §\ref{['subsec:higgsingdiscrete']} that this can actually happen in gauge theories with dynamical vortices.
• Figure 5: A vortex is a codimension-2 dynamical object (black curve) that enforces a discrete holonomy $g \in G$, so that a $G$-charged particle $\psi$ circling the vortex (blue path) comes back to itself only up to a gauge transformation. A familiar example is the $\mathbb{Z}_N$ magnetic string Krauss:1988zc.