Two index theorems in odd dimensions (Q1271392)

The author abstracts from a paper by \textit{P. Hořava} and \textit{E. Witten} [Nucl. Phys. B 460, 506-524 (1996; MR 98c:81167)] two general index theorems, whose novelty is that they involve nontrivial indices for Dirac operators on odd-dimensional manifolds. In one theorem the data includes an orientation-reversing isometric involution that lifts to a spinor involution that anticommutes with the Dirac operator. The theorem gives a Lefschetz fixed-point formula for the index of the Dirac operator as a map from the involution's positive eigenspace to the involution's negative eigenspace. This theorem extends to families only modulo two-torsion. The other theorem solves a boundary-value problem involving local boundary conditions. This theorem extends to families. The author's first theorem arises naturally from Hořava and Witten's interest in the Dirac operator on the Cartesian product of a circle and another manifold, with an involution coming from a reflection of the circle. Motivated by the application to anomalies, the author puts this example in the context of his second theorem, thus providing an index theorem for families of such examples.