Atiyah–Singer Index Theorem
Heat kernel proof: Index = Tr(e
−tD*D
) − Tr(e
−tDD*
) → topological invariant as t→0
Heat kernel K(t,x,y) on 2D torus
Spectral asymptotic expansion
Heat time t:
0.10
Genus g (surface):
2
N eigenvalues:
40
Curvature κ:
−1.00
−2
Index (analytical)
−2
χ = 2−2g (topological)
—
Heat trace difference
Atiyah–Singer:
Index(D) = (2π)⁻ⁿ/² ∫_M Â(M) ∧ ch(E)
Heat kernel:
K(t,x,x) ~ (4πt)⁻ⁿ/² [a₀ + a₁t + a₂t² + …]
For Dirac on surface:
Index(D) = χ(M) = 2 − 2g (Gauss-Bonnet)