Extended higher cup-product Chern-Simons theories

TL;DR

This paper refines the mathematical formulation of higher-dimensional U(1)-Chern-Simons theories, extending it to nonabelian gauge fields and exploring their geometric pre-quantization, with applications to string theory and M-branes.

Contribution

It advances the understanding of higher cup product Chern-Simons theories by formulating them on full higher moduli stacks and generalizing to nonabelian gauge fields.

Findings

Refined the action functional on the full higher smooth moduli stack.

Generalized the formulation to nonabelian and higher nonabelian gauge fields.

Discussed off-shell extended geometric pre-quantization and examples like differential T-duality and anomaly line bundles.

Abstract

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.