Čech cocycles for differential characteristic classes: an ∞-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 ∞-groups, i.e., by smooth groupal A∞spaces. Namely, we realize differential characteristic classes as morphisms from ∞-groupoids of smooth principal ∞-bundles with connections to ∞-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.

The Rational Higher Structure of M‐theory

“…If here are higher connections on higher bundles (see []), then this precisely reproduces the familiar notion of higher gauge transformation between higher gauge fields (see [] for review); for example of the combined gauge field and B‐field in heterotic string theory, or the C‐field in 11d supergravity …”

“…The super algebraic version of these differential‐graded algebras (or dg‐algebras for short) have come to be known as “FDAs” in the supergravity literature ([], following []). Under Koszul duality, we may equivalently think of these as being the Chevalley–Eilenberg algebras of super ‐algebras (or, more generally, super ‐algebroids), see []: In accord with the super ‐algebraic interpretation of “FDAs”, the Sullivan construction of rational homotopy theory may naturally be enhanced to a higher super analog of the process of Lie integration of Lie algebras to Lie groups [, Sec. 3.]…”

“… Rational approximation and higher Lie integration in super homotopy theory [, Sec. 3.1] (following [], see []). Shown is the .…”

Twisted Weil Algebras for the String Lie 2‐Algebra

“…Furthermore, gauge transformations can be encoded in flat homotopies between two such gauge configurations, i.e. in morphisms where denotes the interval and vanishes on those additional directions.…”

Higher gauge theories from Lie n-algebras and off-shell covariantization

“…The QP-structure on this space includes the differential crossed module and the semistrict Lie 2-algebra. Gauge fields, gauge transformations and field strengths are constructed by associating supercoordinates on the QP-manifold to fields on the spacetime Σ [21,25,26]. Consistent field strengths and gauge symmetries are determined by a geometric datum of the corresponding QP-manifold, which is called Hamiltonian function (also called homological function).…”