Preliminary draftTHE DIRAC OPERATOR ON PIN MANIFOLDSANDRZEJ

Odd-dimensional Riemannian spaces that are non-orientable, but have a pin structure, require the consideration of the twisted adjoint representation of the corresponding pin group. It is shown here how the Dirac operator should be modiied, also on even-dimensional spaces, to make it equivariant with respect to the action of that group when the twisted adjoint representation is used in the deenition of the pin structure. An explicit description of a pin structure on a hypersurface, deened by its immersion in a Euclidean space, is used to derive a simple formula for the Dirac operator in that case. 1. Introduction Most of the research on the Dirac operator on Riemannian spaces is restricted to the case of orientable manifolds. It is of some interest to treat also the non-orientable case that requires the introduction of pin structures. In physics, even in the orientable case, one considers spinor elds transforming under space and time reeections, which are covered by elements of a suitable pin group. The generalization to the non-orientable case involves interesting subtleties. First of all, for a real vector space with a quadratic form of signature (k; l), the Cliiord construction yields two groups Pin k;l and Pin l;k , which need not be isomorphic; see Ka] and Section 3 for a precise statement. This fact is of interest also to physics CDeW]. There are non-orientable spaces with a metric tensor eld of signature (k; l) admitting either a pin k;l-structure or a pin l;k-structure. If a space admits a spin k;l-structure, then it is orientable and admits both these structures. Real projective spaces and quadrics provide the simplest examples of such situations DaT1, CaGuT]. If the dimension k +l is even, then one can use either the adjoint or the twisted adjoint representation of Pin k;l. If one uses the twisted adjoint representation, as one has to do when k + l is odd, then the classical Dirac operator (see, e.g., ABP, LM, BoWo]) needs to be modiied to make it equivariant with respect to the action of the pin group T1, T2].