  The proper action functional of (4k+3)-dimensional U(1)-Chern-Simons theory
including the instanton sectors has a well known description: it is given on
the moduli space of fields by the fiber integration of the cup product square
of classes in degree-(2k+2) differential cohomology. We first refine this
statement from the moduli space 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^c-2-connections appearing in quantum-corrected
11-dimensional supergravity and M-branes. We discuss aspects of the off-shell
extended geometric pre-quantization (in the sense of extended or multi-tiered
QFT) of these theories, where there is a prequantum U(1)-k-bundle
(equivalently: a U(1)-(k-1)-bundle gerbe) in each codimension k. Examples we
find include moduli stacks for differential T-duality structures as well as the
anomaly line bundles of higher electric/magnetic charges, such as the 5-brane
charges appearing in heterotic supergravity, appearing as line bundles with
connection on the smooth higher moduli stacks of field configurations.
