Edge physics and the Casimir interaction in Maxwell--Chern--Simons theory on a strip

We study Maxwell--Chern--Simons theories on a strip $\mathcal M=\mathbb R^{1,1}\times[0,h]$ within the Symanzik framework for quantum field theory with boundaries. We add the most general local boundary functional within the quadratic one-tangential-derivative truncation adopted throughout the paper and derive the admissible boundary conditions from locality and the variational principle, with gauge symmetry implemented through boundary Ward identities. The strip supports two boundary $U(1)$ current algebras, with opposite levels fixed by the Chern--Simons coupling $\kappa$ and by boundary orientation, while edge velocities depend on local boundary couplings. They are equal and opposite on the flip-symmetric branch, which is the natural choice for a spatially symmetric strip. The same boundary data determine the reflection coefficients for bulk modes and yield a closed determinant (scattering) representation of the finite interaction energy. In the Maxwell limit ($\kappa=0$) the inter-edge interaction is long-ranged and produces a power-law Casimir force, whereas in Maxwell--Chern--Simons theory the topological mass $m=\kappa g^2$ generates Yukawa suppression at large separations $mh\gg1$. The determinant form also gives analytic control of the asymptotic regimes and separates topological boundary data (levels and anomaly inflow) from dynamical bulk effects (dispersion, boundary mixing and Casimir interaction).