Dirac Operator Eigenvalues · Spectral Flow = Analytical Index
Parameters
Index Theorem
sf(D_A) = index(D⁺) = dim ker D⁺ − dim ker D⁻
index(D) = ∫ Â(M) ∧ ch(E)
spectral flow = Q (topological charge)
Spectral flow count...
Theory
As a parameter θ varies from 0 to 2π, eigenvalues of the Dirac operator D(θ) flow through zero. The spectral flow — net signed count of level crossings — equals the analytical index of D⁺. The Atiyah-Singer theorem identifies this with a topological invariant (Chern character). Zero modes of D are topologically protected.