The projective Dirac operator and its fractional analytic index
                         V. Mathai, R.B. Melrose   and I.M. Singer

Abstract:
  For a finite-rank projective bundle over a compact manifold, so associated to a torsion Dixmier-Douady class w on the manifold, we define the ring of differential operators "acting on sections of the principle 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 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 heat kernel method for the twisted, projective spin Dirac operator, we show that this index is given by the usual formula, now in terms 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 a rational number but in general it is not an integer.

For the 22-page arXiv preprint, see Fractional analytic index.