Dirac Operators on 4-manifolds

Dirac operators are important geometric operators on a manifold. The Dirac operator DA on the four dimensional Euclidean space M = R is the order one differential operator whose square DA ◦ DA is the Euclidean Laplacian − ∑4 i=1 ∂ψ ∂xi . However, this is not possible unless we allow coefficients for this linear operator to be matrix-valued. Let M = R be the four dimensional Euclidean space with global Euclidean coordinate (x1, x2, x3, x4) and W ⊗ L = M × C be the trivial 4-dimensional complex vector bundle over M , then the Dirac operator DA : C ∞(R4,C4)→ C∞(R4,C4) is given by