Dirac spectrum under adiabatic gauge field evolution — level crossing & anomaly inflow
Spectral flow (Atiyah–Patodi–Singer index theorem):
A 1D Dirac fermion on a circle has spectrum E_n = (n + θ/2π) where θ is the gauge flux.
As θ increases by 2π (one flux quantum), every eigenvalue shifts by one unit — levels flow continuously
from positive to negative, crossing zero. The spectral flow equals the Chern number.
This is the chiral anomaly in 1+1D: charge is not conserved quantum-mechanically
(ΔQ = Chern number per flux quantum). In 3+1D the anomaly coefficient ∝ Tr[Q³] constrains particle
physics — the Standard Model is anomaly-free only because quarks and leptons conspire to cancel it.
The Chern–Simons term in the bulk generates the anomaly at the boundary (anomaly inflow).