Clifford module - Imported Knowlpedia Content

Definition

Let $\operatorname{Cl}(V,q)$ be a Clifford algebra. A Clifford module is a module $E$ together with a unital algebra homomorphism

c:\operatorname{Cl}(V,q)\longrightarrow \operatorname{End}(E).

Equivalently, it is a linear map $c:V\to\operatorname{End}(E)$ , called Clifford multiplication, satisfying

c(v)^2=-q(v)\operatorname{id}_E

under the Riemannian sign convention. Polarization yields

c(v)c(w)+c(w)c(v)=-2g(v,w)\operatorname{id}_E

when $q(v)=g(v,v)$ . A different Clifford-algebra sign convention changes both displayed signs. The scalar ring and module category are understood to be the same as those used to construct $\operatorname{Cl}(V,q)$ .

Graded modules

If $E=E^0\oplus E^1$ is $\mathbb Z/2$ -graded, it is a graded Clifford module when Clifford multiplication by every $v\in V$ is odd:

c(v):E^j\longrightarrow E^{j+1\!\!\pmod 2}.

Equivalently, the algebra representation preserves total degree: the even part of the Clifford algebra acts evenly and the odd part acts oddly. Ungraded Clifford modules are also standard, especially in odd dimension.

Clifford module bundles

Let $(M,g)$ be a Riemannian manifold. A Clifford module bundle is a vector bundle $E\to M$ with a smooth bundle-algebra action

c:\operatorname{Cl}(T^*M,g)\longrightarrow \operatorname{End}(E).

Fiberwise, each $E_x$ is a module over $\operatorname{Cl}(T_x^*M,g_x)$ . If $E$ is Hermitian, one usually also requires $c(\xi)^*=-c(\xi)$ for real covectors $\xi$ ; this compatibility is an extra metric condition, not part of the purely algebraic definition.

Role in Dirac operators

A compatible connection $\nabla^E$ on a Clifford module bundle allows the composition

\Gamma^\infty(E) \xrightarrow{\ \nabla^E\ } \Gamma^\infty(T^*M\otimes E) \xrightarrow{\ c\ } \Gamma^\infty(E).

This first-order differential operator is of Dirac type. Clifford multiplication supplies its principal symbol, whereas the choice of compatible connection supplies additional geometric data.

References

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989. DOI record. Relevant: Chapters I–II, Clifford modules and spinors.
Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat Kernels and Dirac Operators, Springer, 1992. DOI record. Relevant: Chapter 3, Clifford modules and Dirac-type operators.