The E_8 moduli 3-stack of the C-field in M-theory

The higher gauge field in 11-dimensional supergravity—the C-field—is 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 integralWu structures that Hopkins–Singer had shown to formalizeWitten’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 showhowto naturally encode the Horava–Witten boundary condition on the level ofmoduli 3-stacks, and refine it from a condition on 3-forms to a condition on the corresponding full differential cocycles.