Geometry-Induced Vacuum Polarization and Mode Shifts in Maxwell-Klein-Gordon Theory
Geometric confinement is known to modify single-particle dynamics through effective potentials, yet its imprint on the interacting quantum vacuum remains largely unexplored. In this work, we investigate the Maxwell--Klein--Gordon system constrained to curved surfaces and demonstrate that the geometric potential $Σ_{\mathrm{geom}}(\mathbf{r})$ acts as a local renormalization environment. We show that extrinsic curvature modifies the scalar loop spectrum, entering the vacuum polarization as a position-dependent mass correction $M^2(\mathbf{r}) \to m^2 + Σ_{\mathrm{geom}}(\mathbf{r})$. This induces a finite, gauge-invariant ``geometry-induced running'' of the electromagnetic response. In the long-wavelength regime ($|{\bf Q}|R \ll 1$), we derive a closed-form expression for the relative frequency shift $Δω/ω$, governed by the overlap between the electric energy density and the geometric potential. Applying this formalism to Gaussian bumps, cylindrical shells, and tori, we identify distinct spectral signatures that distinguish these quantum loop corrections from classical geometric optics. Our results suggest that spatial curvature can serve as a tunable knob for ``vacuum engineering,'' offering measurable shifts in high-$Q$ cavities and plasmonic systems.