Duality and cohomology in MM-theory with boundary

We consider geometric and analytical aspects of MM-theory on a manifold with boundary Y11Y11. The partition function of the CC-field requires summing over harmonic forms. When Y11Y11 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 CC-field. The dynamics and the presence of the dual to the CC-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 E8E8 bundle and characterize obstructions. Considering Y11Y11 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 CS11CS11 on Y11Y11 whose restriction to the boundary would be a generalized WZW model. This suggests that holographic models of MM-theory can be viewed as a sector within this index-theoretic approach.