Fractional Analytic Index

For a finite rank projective bundle over a compact manifold, so associated to a torsion, Dixmier-Douady, 3-class, w, on the manifold, we define the ring of differential operators ‘acting on sections of the bundle’ in a formal sense. In particular any oriented even-dimensional manifold carries a projective spin Dirac operator in this sense. More generally the corresponding space of pseudodifferential operators is defined, with supports sufficiently close to the diagonal, i.e. the identity relation. For such elliptic operators we define the numerical index in an essentially analytic way, as the trace of the commutator of the operator and a parametrix and show that this is homotopy invariant. Using the Dirac case we show that this index is given by the usual formula in terms now of the twisted Chern character of the symbol, which in this case defines an element of K-theory twisted by w; hence the index is rational but in general it is not an integer. The Atiyah-Singer index theorem for an elliptic (pseudodifferential) operator gives an integrality theorem; namely a certain characteristic integral is an integer because it is the index of an elliptic operator. Notably, for a closed spin manifold Z, the  genus, ∫ Z Â(Z) is an integer because it is equal to the index of the Dirac operator on Z. When Z is not a spin manifold, the spin bundle S does not exist, as a vector bundle, and when Z has no spinC structure, there is no global vector bundle resulting from the patching of the local bundles S ⊗ Li, where the Li are line bundles. However, as we show below, S is a always a projective vector bundle associated to the the Clifford algebra of the cotangent bundle T ∗Z, Cl(Z), which is an Azumaya bundle cf. [8]. Such a (finite rank) projective vector bundle, E, over a compact manifold has local trivializations which may fail to satisfy the cocycle condition on triple overlaps by a scalar factor; this defines the Dixmier-Douady invariant in H(Z,Z). If this torsion twisting is non-trivial there is no, locally spanning, space of global sections. The Dixmier-Douady invariant for Cl(Z) is the third integral Stieffel-Whitney class, W3(Z). In particular, the spin Dirac operator does not exist when Z is not a spin manifold. Correspondingly the  genus is a rational number, but not necessarily an integer. In this paper we show that, in the oriented even-dimensional case, one can nevertheless define a projective spin Dirac operator, with an analytic index valued in the rational numbers, and prove the analogue of the Atiyah-Singer index theorem for this operator twisted by a general projective bundle. In fact we establish the analogue of the Atiyah-Singer index theorem for a general projective elliptic pseudodifferential operator. In a subsequent paper the families case will be discussed. For a compact manifold, Z, and vector bundles E and F over Z the Schwartz kernel theorem gives a one-to-one correspondence between continuous linear operators from C∞(Z,E) to C−∞(Z,F ) and distributions in C−∞(Z2,Hom(E,F )⊗ΩR). Here