An article that we have written:
Domenico Fiorenza, Hisham Sati, Urs Schreiber,
Extended higher cup-product Chern-Simons theories
Journal of Geometry and Physics
Volume 74, 2013, Pages 130–163
in the context of differential cohomology in a cohesive topos on aspects of ∞-Chern-Simons theory.
Abstract. It is well known that the proper action functional of (4k+3)-dimensional U(1)-Chern-Simons theory including the instanton sectors is given on gauge equivalence classes of gauge fields by the fiber integration of the cup product square of classes in degree-$(2k+2)$ differential cohomology.
We first refine this statement from gauge equivalence classes to the full higher smooth moduli stack of fields, to which the higher-order-ghost BRST complex is the infinitesimal approximation. Then we generalize the refined formulation to cup product Chern-Simons theories of nonabelian and higher nonabelian gauge fields, such as the nonabelian String 2-connections appearing in quantum-corrected 11-dimensional supergravity (FSSa, FSSb).
We discuss aspects of the off-shell extended geometric quantization (in the sense of extended or multi-tiered QFT) of these theories (FRS), with prequantum circle k-bundles ($(k-1)$-bundle gerbes) in each codimension $k$.
As an example we discuss the anomaly line bundle of heterotic supergravity as a line bundle on the smooth moduli stack of gravity-, $E_8$-gauge field- and twisted B-field configurations.
differential cohomology in a cohesive topos
Extended higher cup product Chern-Simons theories