0<〈α,α〉=∫ₓ α ∧ *α=∫ₓ α ∧ τα=∫ₓ α ∧ α.
Hence σ(X)=dimℂ(𝓡 ₊²ᵏ)–dimℂ(𝓡 ₋²ᵏ). We have an elliptic operator d+d*:ΓE₊ → ΓE₋ since its square is the Laplace operator Δ:Γ(∧(T* ⨂ ℂ)) → Γ(∧(T* ⨂ ℂ)) which
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