Dirac operator

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This article defines a property that can be evaluated for a differential operator on a differential manifold (viz a linear map from the space of differentiable functions to itself)


Given data

Let M be a Riemannian manifold and E a vector bundle over M. Let \Delta denote a (the?) Laplacian on E.

Definition part

A Dirac operator on E is a differential operator from the sheaf of section of E to itself, whose square is \Delta.

In other words, a Dirac operator is a formal squareroot, or a half-iterate, of the Laplacian.