Corners in m-theory

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 12 dimensions of M-theory on anti-de Sitter spaces, ten-dimensional heterotic string theory in relation to 12 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 the (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.