Duality and cohomology in M-theory with boundary

Abstract

We consider geometric and analytical aspects of M-theory on a manifold with boundary Y 11 . The partition function of the C-field requires summing over harmonic forms. When Y 11 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 11 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 CS11 on Y 11 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. *

Framed M-branes, corners, and topological invariants

“…The f -invariant appears in [66] in the description of anomalies in heterotic string theory. Combining that with the approach of [59] [63] leads naturally to our discussion on the relation of the M-branes to the f -invariant. This generalizes previous connections to the Adam's e-invariant and the Atiyah-Patodi-Singer η-invariant [58] [57].…”

“…so that the normal bundle N M m is stably trivial if and only if the worldvolume M m is a framed manifold. Next we consider the disk bundle, needed for the description of tubular neighborhood for the M5-brane (see [18] [59]). Let X be the total space of the disk bundle over the worldvolume M m , which we take to be a framed manifold, associated to a vector bundle η.…”

“…We now consider corner structures on the M2-brane and the M5-brane, as in [63] [59]. This uncovers some structures and connections to invariants, such as the Z 2 Hopf invariant and the f -invariant.…”

Super-Lie n-algebra extensions, higher WZW models and super-p-branes with tensor multiplet fields

“…Such hierarchies of higher order corner field theories have previously been recognized and amplified in string theory and M-theory [42,43,44]. More discussion of the above formalization of these hierarchies in local (multi-tiered) prequantum field theory is in [47].…”

Extended higher cup-product Chern–Simons theories

“…With the local formula ch ni = dCS 2ni+1 and passing to differential cohomology, we can write each of the factors in (3.3.4) in terms of CS 2ni+1 , for i = 1, · · · , k. This involves using a type of Stokes formula for various faces in codimension k, in the setting advocated in [50,54,55]. That is, we take Z n1+n2+···+n k to admit a codimension-k corner X n1+n2+···+n k −k , on which the action schematically takes the form (3.3.5) S X = X n 1 +n 2 +···+n k −k CS 2n1+1 (A 1 ) ∧ CS 2n2+1 (A 2 ) ∧ · · · ∧ CS 2n k +1 (A k ) .…”