Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== How do we get a complex phase (not merely a magnitude) from geometry? two routes: ===

A. Wilson-line / holonomy phase. Suppose the boundary supports a nontrivial gauge holonomy (integral of gauge potential around some closed loop on the shell). A boundary mode transporting around that loop picks up a phase:

Φgeom=exp⁡ ⁣(i∮CA⋅dl).\Phi_{\rm geom} = \exp\!\Big(i\oint_{\mathcal{C}} A\cdot dl \Big).Φgeom=exp(i∮CA⋅dl).
If different flavor components pick up different holonomies, the relative phases feed into the effective Yukawa/coupling matrices and generate complex entries ⇒ CP violation.

B. Berry / adiabatic geometric phases. If internal parameters (shape, local EM field) slowly change during the “formation” of the boundary state, spinor modes acquire geometric phases (Berry phases) that are complex and gauge invariant. These complex phases multiply transition amplitudes just like the CKM phase.

Topological boundary terms (WZW-like or effective θ-terms) can also produce parity / time asymmetries when coupled to gauge fields and curvature. The important conceptual point: on a curved spherical shell with nontrivial gauge backgrounds, geometric phases naturally appear and those phases are complex numbers that alter interference between amplitudes → that is exactly the structure needed for CP asymmetry.