Dirac operators and Clifford modules

For a real Riemannian cotangent vector v, choose Clifford multiplication c(v) to be skew-Hermitian and to square to minus its squared norm. The anticommutator of c(v) and c(w) is therefore minus twice their metric pairing. A Hermitian Clifford module is a complex vector bundle carrying this action smoothly. It need not be an irreducible spinor bundle: differential forms with exterior multiplication minus contraction already give a Clifford module even when the manifold has no spin structure.

A compatible Hermitian connection satisfies the Leibniz identity relating differentiation of c(v) to the Levi-Civita derivative of v. Contracting that connection with Clifford multiplication defines D. Using the symbol convention ∂ maps to iξ, the principal symbol is i c(ξ), a Hermitian endomorphism whose square is |ξ|². Thus D is elliptic away from the zero section. The factor i is essential: c(ξ) itself has negative square under our convention.

On an oriented even-dimensional spin manifold of dimension 2m, the complex spinor bundle has the grading operator Γ=i^m c(e¹)···c(e^(2m)). It is frame-independent, self-adjoint, squares to one and anticommutes with Clifford multiplication. Its plus and minus eigenspaces define S⁺ and S⁻, and a compatible Dirac operator exchanges them. The chiral map D⁺ goes from positive spinors to negative spinors, while D⁻ is its formal adjoint on a closed manifold.

The full self-adjoint D has index zero as an ungraded map. The interesting integer is instead the difference between the two chiral zero-mode counts. An orientation reversal changes the sign of the Clifford volume element, interchanges the chiral labels and reverses that chiral index when the underlying ungraded data are identified. An arbitrary graded Clifford module may come with a specified compatible grading; its grading should not automatically be identified with the irreducible spinor volume grading without checking the construction.

For the untwisted spin Dirac operator, the Lichnerowicz identity gives D²=∇*∇+scal/4. On a closed manifold, strictly positive scalar curvature forces every harmonic spinor to vanish by integrating this identity. Twisting by an auxiliary bundle contributes an additional curvature endomorphism, so positive scalar curvature alone no longer proves vanishing for every twisted operator. For a general Clifford module the analogous Weitzenböck curvature term likewise depends on more than scalar curvature.

In a flat trivialization the connection and curvature disappear, leaving constant Clifford matrices times ordinary derivatives. Fourier modes then reduce the operator to finite matrices, which are excellent checks of conventions. Such a local or flat calculation does not construct a spinor bundle on an arbitrary curved manifold. The next lesson supplies the global lifting obstruction that determines whether these irreducible matrices can be assembled consistently around every loop and overlapping coordinate chart.