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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) E 8-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 Hořava–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.

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Correspondence to Urs Schreiber.

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Communicated by H.-T. Yau

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Fiorenza, D., Sati, H. & Schreiber, U. The E 8 Moduli 3-Stack of the C-Field in M-Theory. Commun. Math. Phys. 333, 117–151 (2015).

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  • Gauge Transformation
  • Simons Theory
  • Canonical Morphism
  • High Gauge
  • Bundle Gerbe