Dirac operator on a spin manifold

Summary

The Dirac operator of a Riemannian spin manifold, also called the Atiyah–Singer operator, acts on sections of the spinor bundle. It combines the spin connection, induced by the Levi-Civita connection, with Clifford multiplication by tangent vectors. It is a first-order elliptic and formally self-adjoint operator whose square is the rough Laplacian plus a quarter of the scalar curvature. On a closed spin manifold of dimension divisible by $4$, the index of its restriction to positive spinors is the $\hat{A}$-genus. This ties the existence of metrics of positive scalar curvature to topology. It is the main example of a Dirac operator.