M-theory, the signature theorem, and geometric invariants

TL;DR

This paper connects M-theory's equations and topological actions to geometric invariants like the signature and s-invariant, revealing how M-theory detects manifold diffeomorphism types and imposes anomaly constraints.

Contribution

It reformulates M-theory's topological action using the signature, linking it to geometric invariants and manifold classification, including the Kreck-Stolz s-invariant.

Findings

M-theory encodes equations via the signature operator.

The topological action relates to the Kreck-Stolz s-invariant.

Anomaly cancellation constrains the s-invariant values.

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 twelve dimensions and leads naturally to the Kreck-Stolz s-invariant in eleven dimensions. Hence M-theory detects diffeomorphism type of eleven-dimensional (and seven-dimensional) manifolds, and in the restriction to parallelizable manifolds classifies topological eleven-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.