Extended higher cup-product Chern–Simons theories

The proper action functional of (4k+3)(4k+3)-dimensional U(1)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)(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)U(1)-kk-bundle (equivalently: a U(1)U(1)-(k−1)-bundle gerbe) in each codimension kk. 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.