Two index theorems in odd dimensions

A recent paper of Hof ava and Witten [HW]-part of the current flurry of activity in string theory-contains an anomaly computation for 5 1 /(r) xE 10 , where (r) is the cyclic group of order two generated by a reflection. It was well established 10 years ago (e.g. [AS1], [Fl]) that anomalies measure nontriviality in the determinant line bundle of a family of Dirac operators, and so can be computed topologically from the Atiyah-Singer index theory. The novelty in the Hofava-Witten computation is a nontrivial index in odd dimensions of a type not seen in standard index theory. We abstract two general theorems which imply the Hof ava-Witten result. (Naturally, we replace M 10 by a compact manifold F 10 ). Theorem A is a Lefschetz formula for an orientation-reversing isometric involution on an odd dimensional manifold. The Atiyah-Bott-Segal-Singer applications of Lefschetz theory [AB1], [ASe], [AS2] all deal with orientation-preserving isometries for which there is no nontrivial Lefschetz formula in odd dimensions [AS2, Proposition 9.3]. Alternatively, we can consider [0,1] x Y in place of S 1 /(T) X Y and then the Hof ava-Witten anomaly computation is a boundary value problem with local boundary conditions. Theorem B generalizes this situation and is closely related to the boundary value problem in the original proof of the Atiyah-Singer index theorem [P].