Dirac-type operator - Imported Knowlpedia Content

Definition

Let $M$ be a Riemannian manifold and $E\to M$ a Hermitian Clifford module with Clifford multiplication $c:T^*M\to\operatorname{End}(E)$ . A Dirac-type operator on $E$ is a first-order differential operator

D:\Gamma^\infty(E)\longrightarrow\Gamma^\infty(E)

whose principal symbol satisfies

\sigma_1(D)(x,\xi)=c_x(\xi).

Here the symbol convention has no factor of $i$ , so $c_x(\xi)^2=-|\xi|^2\operatorname{id}_{E_x}$ . Therefore the symbol is invertible for $\xi\neq0$ , and every Dirac-type operator is an elliptic differential operator. Formal self-adjointness is included only when explicitly stated.

Construction from a Clifford connection

If $\nabla^E$ is a metric Clifford connection, then

D_{\nabla^E}=c\circ\nabla^E=\sum_j c(e^j)\nabla^E_{e_j}

is Dirac type. Adding any smooth bundle endomorphism preserves its principal symbol, so every $D_{\nabla^E}+\Phi$ is also Dirac type. Conversely, a Dirac-type operator determines a compatible connection plus a zero-order endomorphism after the metric data are fixed; see Berline, Getzler, and Vergne, chapter 3.

Examples and consequences

The spin Dirac operator, the Hodge–de Rham operator $d+d^*$ , and the signature operator are standard examples. Squaring a Dirac-type operator produces a second-order operator of Laplace type; for a Clifford connection, the Weitzenböck formula separates this square into a connection Laplacian and a curvature endomorphism.

Conventions and scope

References

N. Berline, E. Getzler, and M. Vergne, Heat Kernels and Dirac Operators, Springer, 1992. Publisher record. Relevant: chapter 3, Clifford modules and generalized Dirac operators.
H. B. Lawson Jr. and M.-L. Michelsohn, Spin Geometry, Princeton University Press, 1989. Publisher record. Relevant: chapters II–III, Dirac operators and their symbols.