Clifford module - Knowlpedia

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. In categorical language, a graded Clifford module is precisely a supermodule over the naturally graded superalgebra $\operatorname{Cl}(V,q)$ . Its ordinary morphisms are even intertwining maps; odd intertwiners belong to the internal Hom.

Ungraded Clifford modules are also standard, especially in odd dimension. They are modules over the underlying ungraded algebra and should not be confused with graded modules whose grading has merely gone unmentioned.

Spinor modules and representations

A spinor module is a selected Clifford module, usually irreducible in a stated real, complex, graded, or ungraded module category, whose restriction to the spin group gives a spin representation. Consequently, “Clifford representation” is another name for a Clifford module, while “spin representation” refers to the restricted group action and includes additional choices.

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.