The index theorem for an elliptic operator gives an integrality theorem; a characteristic integral over the manifold is an integer because it is the index of the operator. I will review some geometric examples: the Dirac operator on a spin manifold, the spin_C Dirac operator, and their analogues for complex manifolds.

When the manifold has no spin_C structure, these operators do not exist. Nevertheless one can define a 'projective' Dirac operator which has an analytic index with values in the rationals. This fractional analytic index can also be expressed as a characteristic integral. I'll describe a possible application to string theory. [Joint work with V. Mathai and R.B. Melrose, J. Diff Geom 74 no 2 (2006) math.DG/0206002]