ABSTRACT
Recall that a module W over a Clifford algebra Cl(Vr) is said to be graded if it is provided with a decomposition W — W+ 0 W-such that Clifford multiplication by any v £ Vr interchanges the summands W+ and W-. A Clifford bundle 5 on a Riemannian manifold is graded if it is provided with a decomposition S = 5+ 0 S_ which respects the metric and connection and makes each fiber Sx a graded Clifford module over C\(TXM). It is equivalent to say that S is provided with an involution £ (the grading operator) which is self-adjoint, parallel1 and such that ec(v) + c(v)e = 0 for every tangent vector v. The sub-bundles S± are the ±1 eigenspaces of
If 5 is a graded Clifford bundle, then the algebra of bounded operators on L2(S) is a superalgebra in the sense of definition 4.1.
