Differential cohomology in a cohesive infinity-topos

TL;DR

This paper develops an abstract framework for differential cohomology within cohesive infinity-toposes, connecting higher gauge theories, characteristic classes, and quantum field theory applications, including higher Chern-Weil theory and extended prequantization.

Contribution

It introduces a cohesive infinity-topos approach to differential cohomology, unifying higher gauge fields, characteristic classes, and quantum field theory constructs in a new abstract setting.

Findings

Higher principal bundles with connections classified by differential cocycles

Higher Chern-Weil homomorphism refined from secondary characteristic classes

Extended geometric prequantization of higher gauge field theories

Abstract

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. We close with an outlook on the cohomological quantization of such higher boundary prequantum field theories by a kind of cohesive motives.