An article of ours:
Domenico Fiorenza, Hisham Sati, Urs Schreiber,
The $E_8$ moduli 3-stack of the C-field
Communications in Mathematical Physics,
Volume 333, Issue 1 (2015), Page 117-151
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 ordinary 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 field of gravity and the (auxiliary) E8-Yang-Mills 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 (HS02) to formalize Witten’s argument (Wi96) 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 (SSS12). Finally we show how to naturally encode the Hořava-Witten boundary condition on the level of moduli 3-stacks, and refine it from a condition on 3-forms to a condition on full differential cocycles.
This may be read as a companion article to