Understand
We formulate differential cohomology and Chern-Weil theory -- the theory of connections on fiber bundles and of gauge fields -- abstractly in the context of a certain class of higher toposes that we call "cohesive".
- Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive structure (topological, smooth, synthetic differential, supergeometric, etc.) and equipped with connections, hence higher gauge fields.
- We discuss various models of the axioms and applications to fundamental notions and constructions in quantum field theory and string theory.
- In particular we show that the cohesive and differential refinement of universal characteristic cocycles constitutes a higher Chern-Weil homomorphism refined from secondary caracteristic classes to morphisms of higher moduli stacks of higher gauge fields, and at the same time constitutes extended geometric prequantization -- in the sense of extended/multi-tiered quantum field theory -- of hierarchies of higher dimensional Chern-Simons-type field theories, their higher Wess-Zumino-Witten-type boundary field theories and all further higher codimension defect field theories.