The higher gauge field in 11-dimensional supergravity -- the C-field -- isView original: http://arxiv.org/abs/1202.2455
constrained by quantum effects to be a cocycle in some twisted version of
differential cohomology. We argue that it should indeed be a cocycle in a
certain twisted nonabelian differential cohomology. We give a simple and
natural characterization of the full smooth moduli 3-stack of configurations of
the C-field, the gravitational field/background, and the (auxiliary) E8-field.
We show that the truncation of this moduli 3-stack to a bare 1-groupoid of
field configurations reproduces the differential integral Wu structures that
Hopkins-Singer had shown to formalize Witten's argument on the nature of the
C-field. We give a similarly simple and natural characterization of the moduli
2-stack of boundary C-field configurations and show that it is equivalent to
the moduli 2-stack of anomaly free heterotic supergravity field configurations.
Finally we show how to naturally encode the Horava-Witten boundary condition on
the level of moduli 3-stacks, and refine it from a condition on 3-forms to a
condition on the corresponding full differential cocycles.