Extended higher cup-product Chern-Simons theories

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 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 and M-branes [FSS12a, FSS12b]. We discuss aspects of the off-shell extended geometric pre-quantization (in the sense of extended or multi-tiered QFT) of these theories [∞Quant, Sc12b], where there is a prequantum circle k-bundle (equivalently: (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, as line bundles with connection on the smooth higher moduli stacks field configurations.