M-theory, the signature theorem, and geometric invariants

Abstract

The equations of motion and the Bianchi identity of the C-field in M-theory are encoded in terms of the signature operator. We then reformulate the topological part of the action in M-theory using the signature, which leads to connections to the geometry of the underlying manifold, including positive scalar curvature. This results in a variation on the miraculous cancellation formula of Alvarez-Gaum\'e and Witten in 12 dimensions and leads naturally to the Kreck-Stolz $s$-invariant in 11 dimensions. Hence M-theory detects the diffeomorphism type of 11-dimensional (and seven-dimensional) manifolds and in the restriction to parallelizable manifolds classifies topological 11 spheres. Furthermore, requiring the phase of the partition function to be anomaly-free imposes restrictions on allowed values of the $s$-invariant. Relating to string theory in ten dimensions amounts to viewing the bounding theory as a disk bundle, for which we study the corresponding phase in this formulation.