Duality and cohomology in M-theory with boundary

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 Poincaré 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.

19 pages

References in corpus (5)

• The M-theory 3-form and E8 gauge theory
• The complete Dirichlet-to-Neumann map for differential forms
• M-theory, the signature theorem, and geometric invariants
• The Loop Group of E8 and Targets for Spacetime
• Four dimensional Abelian duality and SL(2,Z) action in three dimensional conformal field theory

Cited by in corpus (10)

• Differential cohomology in a cohesive infinity-topos
• Extended higher cup-product Chern-Simons theories
• Framed M-branes, corners, and topological invariants
• The E8 moduli 3-stack of the C-field in M-theory
• M-Theory with Framed Corners and Tertiary Index Invariants
• Flux Quantization on 11-dimensional Superspace
• Entanglement, space-time and the Mayer-Vietoris theorem
• Super Lie n-algebra extensions, higher WZW models, and super p-branes with tensor multiplet fields
• General Boundary Formulation for $n$-Dimensional Classical Abelian Theory with Corners
• Constraints on heterotic M-theory from s-cobordism