Extended higher cup-Product Chern-Simons theories

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 ^Stringc-2-connections appearing in quantum-corrected 11-dimensional supergravity and M-branes. We discuss aspects of the off-shell extended geometric prequantization (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.