Spectral Flow & Atiyah-Singer Index Theorem

Spectral Flow and the Index Theorem

The spectral flow sf(D(θ)) counts the net number of eigenvalues crossing zero from below as the parameter θ sweeps from 0 to 2π. The Atiyah-Singer index theorem equates this to the analytical index: ind(D) = dim ker(D) − dim coker(D) = sf.

For a 1D Dirac operator coupled to a U(1) gauge field with winding number w, the spectral flow equals w. Each crossing corresponds to a topological level crossing — a zero mode that can be interpreted as a Jackiw-Rebbi bound state at a domain wall.

The winding number is the topological (geometric) side; zero modes are the analytical side. Their equality is remarkable and survives any smooth deformation.