Čech cocycles for differential characteristic classes -- An infinity-Lie theoretic construction

Abstract

What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and from Lie groups to higher connected covers of Lie groups by smooth infinity-groups, i.e., by smooth groupal A-infinity-spaces. Namely, we realize differential characteristic classes as morphisms from infinity-groupoids of smooth principal infinity-bundles with connections to infinity-groupoids of higher U(1)-gerbes with connections. This allows us to study the homotopy fibers of the differential characteristic maps thus obtained and to show how these describe differential obstruction problems. This applies in particular to the higher twisted differential spin structures called twisted differential string structures and twisted differential fivebrane structures.

2026-09-29 Erratum (v3: metadata only; file unchanged): The sitting-instants condition (4.2.1) does not justify horn-filling in 4.2.10. We thank Emilio Verdooren for identifying the defect and proving the Kan property using extended smooth simplices without sitting instants: "Higher Lie Integration", PhD thesis, Texas Tech (2026), Thm. 6.2.3, Sec. 6.3, https://hdl.handle.net/2346/109591

References in corpus (1)

• Four Equivalent Versions of Non-Abelian Gerbes

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