Duality and cohomology in M-theory with boundary

TL;DR

This paper explores the geometric, topological, and analytical structures of M-theory on manifolds with boundary, focusing on boundary conditions for the C-field, cohomology dualities, and implications for holography and index theory.

Contribution

It introduces a detailed analysis of boundary conditions for the C-field in M-theory, including the role of Poincaré duality angles and the extension of E8 bundles, advancing understanding of boundary effects in M-theory.

Findings

Boundary conditions for the C-field involve mixing Dirichlet and Neumann types.

Poincaré duality angles describe the boundary cohomology mixing.

A gravitational Chern-Simons term influences the boundary dynamics.

Abstract

We consider geometric and analytical aspects of M-theory on a manifold with boundary Y. The partition function of the C-field requires summing over harmonic forms. When Y is closed Hodge theory gives a unique harmonic form in each de Rham cohomology class, while in the presence of a boundary the Hodge-Morrey-Friedrichs decomposition should be used. This leads us to study the boundary conditions for the C-field. The dynamics and the presence of the dual to the C-field gives rise to a mixing of boundary conditions with one being Dirichlet and the other being Neumann. We describe the mixing between the corresponding absolute and relative cohomology classes via Poincar\'e duality angles, which we also illustrate for the M5-brane as a tubular neighborhood. Several global aspects are then considered. We provide a systematic study of the extension of the E8 bundle and characterize obstructions. Considering Y as a fiber bundle, we describe how the phase looks like on the base, hence providing dimensional reduction in the boundary case via the adiabatic limit of the eta invariant. The general use of the index theorem leads to a new effect given by a gravitational Chern-Simons term CS on Y whose restriction to the boundary would be a generalized WZW model. This suggests that holographic models of M-theory can be viewed as a sector within this index-theoretic approach.