Extended higher cup-Product Chern-Simons theories

Abstract The proper action functional of ( 4 k + 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- ( 2 k + 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 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.

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