Gauge-Invariant Obstruction and Particle Production of Ultra-light Axion-like Fields in Kerr-Spacetime Environments

Ultra-light axion-like fields provide a theoretically well-motivated dark-matter candidate and a sensitive probe of quantum field theory in strong gravitational backgrounds. This talk develops a gauge-invariant framework for studying their nonlinear effects in rotating Kerr black-hole environments. The central question is whether a perturbative but spatially structured axion condensate can generate physical signatures that go beyond coordinate-dependent metric deformations and reveal genuine deviations from the Kerr paradigm. I first discuss how the stress-energy distribution of an ultra-light axion field modifies the Kerr geometry at the perturbative level. Particular emphasis is placed on the hidden symmetry structure responsible for the separability of geodesic motion and field equations in exact Kerr spacetime. I argue that the axion-induced deformation generically produces a gauge-invariant obstruction to this integrability structure, leading to measurable consequences such as orbital-frequency shifts, phase drift in extreme-mass-ratio inspirals, and deformation of photon trajectories near the black hole. The second part of the talk connects this geometric obstruction with quantum particle production. By quantizing field perturbations on the corrected rotating background, one can analyze how frame dragging, ergoregion dynamics, superradiant amplification, and non-adiabatic axion evolution contribute to particle creation in curved spacetime. This provides a unified theoretical bridge between ultra-light dark matter, gauge-controlled perturbation theory, and quantum field production in strong gravity. The proposed framework suggests that future gravitational-wave observations and horizon-scale black-hole imaging may offer complementary probes of axion-like dark sectors and nonlinear quantum effects around rotating black holes.