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

  title={Cech cocycles for differential characteristic classes -- An infinity-Lie theoretic construction},
  author={Domenico Fiorenza and Urs Schreiber and Jim Stasheff},
  journal={arXiv: Algebraic Topology},
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… 

A higher Chern-Weil derivation of AKSZ sigma-models

Chern–Weil theory provides for each invariant polynomial on a Lie algebra 𝔤 a map from 𝔤-connections to differential cocycles whose volume holonomy is the corresponding Chern–Simons theory action

Spectral sequences in smooth generalized cohomology

We consider spectral sequences in smooth generalized cohomology theories, including differential generalized cohomology theories. The main differential spectral sequences will be of the

String structures associated to indefinite Lie groups

Higher-twisted periodic smooth Deligne cohomology

Degree one twisting of Deligne cohomology, as a differential refinement of integral cohomology, was established in previous work. Here we consider higher degree twists. The Rham complex, hence de

Twisted differential generalized cohomology theories and their Atiyah–Hirzebruch spectral sequence

We construct the Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential generalized cohomology theories. This generalizes to the twisted setting the authors' corresponding earlier

Twisted smooth Deligne cohomology

Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest

Cobordisms of global quotient orbifolds and an equivariant Pontrjagin-Thom construction

We introduce an equivariant Pontrjagin-Thom construction which identifies equivariant cohomotopy classes with certain fixed point bordism classes. This provides a concrete geometric model for

L-infinity algebras of local observables from higher prequantum bundles

To any manifold equipped with a higher degree closed form, one can associate an L-infinity algebra of local observables that generalizes the Poisson algebra of a symplectic manifold. Here, by means

High gauge theory with string 2-groups and higher Poincare lemma

This thesis is concerned with the mathematical formulations of higher gauge theory. Firstly, we develop a complete description of principal 2-bundles with string 2-group model of Schommer-Pries,



Lie theory for nilpotent L∞-algebras

The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic

Geometry of Deligne cohomology

It is well known that degree two Deligne cohomology groups can be identified with groups of isomorphism classes of holomorphic line bundles with connections. There is also a geometric description of

Čech cocycles for characteristic classes

We give general formulae for explicit Čech cocycles representing characteristic classes of real and complex vector bundles, as well as for cocycles representing Chern-Simons classes of bundles with


This is an overview of differential graded manifolds and their homotopy theory. The central topic is the construction of a functor from the category of dg manifolds to the homotopy category of

Twisted Differential String and Fivebrane Structures

In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms

From loop groups to 2-groups

We describe an interesting relation between Lie 2-algebras, the Kac– Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where

Quadratic functions in geometry, topology, and M-theory

We describe an interpretation of the Kervaire invariant of a Riemannian manifold of dimension $4k+2$ in terms of a holomorphic line bundle on the abelian variety $H^{2k+1}(M)\otimes R/Z$. Our results

String connections and Chern-Simons theory

A finite-dimensional and smooth formulation of string structures on spin bundles that enables it to prove that every string structure admits a string connection and that the possible choices form an affine space is presented.

Fivebrane Structures

We study the cohomological physics of fivebranes in type II and heterotic string theory. We give an interpretation of the one-loop term in type IIA, which involves the first and second Pontrjagin