An article of ours
Domenico Fiorenza, Chris Rogers, Urs Schreiber
A higher Chern-Weil derivation of AKSZ $\sigma$-models
Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250078 (36 pages)
in the context of β-Chern-Simons theory.
Chern-Weil theory provides for each invariant polynomial on a Lie algebra $\mathfrak{g}$ a map from $\mathfrak{g}$-connections to differential cocycles whose volume holonomy is the the corresponding Chern-Simons theory action functional. We observe that in the context of higher Chern-Weil theory in smooth β-groupoids this statement generalizes from Lie algebras to L-β algebras and further to L-β algebroids. It turns out that the symplectic form on a symplectic higher Lie algebroid (for instance a Poisson Lie algebroid or a Courant Lie 2-algebroid) is β-Lie-theoretically an invariant polynomial. We show that the higher Chern-Simons action functional associated to this by higher Chern-Weil theory is the action functional of the AKSZ sigma-model with target space the given $L_\infty$-algebroid (for instance the Poisson sigma-model or the Courant sigma-model).
differential cohomology in a cohesive topos
Higher Chern-Weil Derivation of AKSZ Sigma-Models
The ideas of β-Chern-Weil theory that this is based on, the notion of invariant polynomials and Chern-Simons elements on $L_\infty$-structures had been presented from spring 2007 on in blog posts
and conference meetings
String and Chern-Simons Lie 3-algebras (August 2007) (blog pdf slides)
On String- and Chern-Simons n-Transport (October 2007) (blog, expanded talk notes, workshop page)
The corresponding preprint appeared a little later:
Meanwhile also (arXiv:0711.4106) had appeared, with similar observations.
The construction of the refined $\infty$-Chern-Weil homomorphism, hence the Lie integration of these $L_\infty$-algebraic constructions to circle n-bundles with connection on smooth β-stacks was finally accomplished in
This is what the above article is based on.
For more related references see also differential cohomology in a cohesive topos -- references.