Spin manifold

A spin manifold is an oriented Riemannian manifold whose orthonormal frame bundle lifts from $SO(n)$ to its double cover $\mathrm{Spin}(n)$. Such a lift defines a spinor bundle.

A spinor field is a section of a spinor bundle. Clifford multiplication by tangent vectors acts fiberwise on spinor fields.

The Dirac operator is the first-order elliptic operator obtained by composing the spin connection with Clifford multiplication. On an even-dimensional compact spin manifold it exchanges positive and negative chirality spinors.

An equivariant index theorem expresses the supertrace of a symmetry on the kernel and cokernel of an elliptic operator as an integral of local characteristic data over the symmetry's fixed-point set.