Corners in M-theory

TL;DR

This paper explores the role of manifolds with corners in M-theory, analyzing their geometric and analytical implications, especially in relation to boundary conditions, branes, and the phase of the partition function.

Contribution

It introduces the natural appearance of manifolds with corners in M-theory and studies their impact on the phase of the partition function using advanced index theorems.

Findings

Manifolds with corners appear naturally in M-theory contexts.

The phase of the partition function is affected by corner contributions.

Extension of index theorems to manifolds with corners is developed.

Abstract

M-theory can be defined on closed manifolds as well as on manifolds with boundary. As an extension, we show that manifolds with corners appear naturally in M-theory. We illustrate this with four situations: The lift to bounding twelve dimensions of M-theory on Anti de Sitter spaces, ten-dimensional heterotic string theory in relation to twelve dimensions, and the two M-branes within M-theory in the presence of a boundary. The M2-brane is taken with (or as) a boundary and the worldvolume of the M5-brane is viewed as a tubular neighborhood. We then concentrate on (variant) of the heterotic theory as a corner and explore analytical and geometric consequences. In particular, we formulate and study the phase of the partition function in this setting and identify the corrections due to the corner(s). The analysis involves considering M-theory on disconnected manifolds, and makes use of the extension of the Atiyah-Patodi-Singer index theorem to manifolds with corners and the b-calculus of Melrose.