## Advances in Theoretical and Mathematical Physics

- Adv. Theor. Math. Phys.
- Volume 16, Number 1 (2012), 149-250.

### Čech cocycles for differential characteristic classes: an ∞-Lie theoretic construction

Domenico Fiorenza, Urs Schreiber, and Jim Stasheff

#### 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 $\infty$-groups, i.e., by smooth groupal $A_\infty$-
spaces. Namely, we realize differential characteristic classes as morphisms
from $\infty$-groupoids of smooth principal $\infty$-bundles with connections to
$\infty$-groupoids of higher $U(1)$-gerbes with connections. This allows us to
study the homotopy fibres 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*.

#### Article information

**Source**

Adv. Theor. Math. Phys., Volume 16, Number 1 (2012), 149-250.

**Dates**

First available in Project Euclid: 23 January 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.atmp/1358950853

**Mathematical Reviews number (MathSciNet)**

MR3019405

**Zentralblatt MATH identifier**

06171205

#### Citation

Fiorenza, Domenico; Schreiber, Urs; Stasheff, Jim. Čech cocycles for differential characteristic classes: an ∞-Lie theoretic construction. Adv. Theor. Math. Phys. 16 (2012), no. 1, 149--250. https://projecteuclid.org/euclid.atmp/1358950853