Chiral--Maxwell Cavity EFT: Reduced Gauge-Mode Condensation and Quantum-Optics Limits
We develop a finite-volume Chiral--Maxwell effective field theory in which a topologically nontrivial hadronic sector acts as a nonlinear medium for a reduced cavity gauge mode. Starting from the SU(2) nonlinear sigma model minimally coupled to Maxwell theory, we construct an equivariant low-mode sector that retains the topological charge structure while projecting the dynamics onto an effective 1+1 theory. Both mass-like scales in the reduced action are finite-volume kinematic gaps---arising from transverse winding energy and holonomy curvature, respectively---with no explicit symmetry breaking, in direct analogy with the Hosotani mechanism on compact cycles. Both gaps vanish in the infinite-volume limit, consistently with massless photons and pions in bulk ChPT. In one hierarchy, Coleman duality yields a gauge-invariant one-loop effective potential for the reduced gauge coordinate; we identify its domain of validity and establish that the locally stable nontrivial branch satisfies the EFT hierarchy condition. In the complementary hierarchy, integrating out the gauge mode produces a one-loop deformed sine-Gordon EFT for the chiral sector with a closed-form kink tension correction. Upon quantization, the reduced theory maps onto nonlinear cavity Hamiltonians whose trivial and locally condensed branches obey distinct selection rules, providing experimentally accessible diagnostics.