Schreiber Extended higher cup-product Chern-Simons theories

An article that we have written:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber:

  
  

  Extended higher cup-product Chern-Simons theories

  
  

  Journal of Geometry and Physics

  Volume 74, 2013, Pages 130-163

  arXiv:1207.5449,

  doi:10.1016/j.geomphys.2013.07.011

  spire:1123556

in the context of differential cohomology in a cohesive topos on aspects of ∞-Chern-Simons theory.

Abstract. 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 gauge 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 (FSSa, FSSb).

We discuss aspects of the off-shell extended geometric quantization (in the sense of extended or multi-tiered QFT) of these theories (FRS), with prequantum circle k-bundles ($(k-1)$-bundle gerbes) in each codimension $k$.

As an example we discuss the anomaly line bundle of heterotic supergravity as a line bundle on the smooth moduli stack of gravity-, $E_8$-gauge field- and twisted B-field configurations.

Related projects

• differential cohomology in a cohesive topos

  • Synthetic Quantum Field Theory

    • Quantum gauge field theory in Cohesive homotopy type theory

    • Type-semantics for quantization

  • cohomology

    • Principal ∞-bundles – models and general theory

    • Fivebrane structures

  • ∞-Chern-Weil theory

    • L-∞ algebra connections

    • Cech cocycles for differential characteristic classes

    • Twisted differential structures

      • Twisted Differential String and Fivebrane Structures

      • The moduli 3-stack of the C-field

  • ∞-Chern-Simons theory

    • A higher stacky perspective on Chern-Simons theory

    • Higher Chern-Weil Derivation of AKSZ Sigma-Models

    • 7d Chern-Simons theory and the 5-brane

    • Extended higher cup product Chern-Simons theories

  • ∞-Wess-Zumino-Witten theory

    • The brane bouquet

    • Obstruction theory for parameterized higher WZW terms

  • Higher geometric prequantum theory

    • Classical field theory via higher differential geometry

    • Local prequantum field theory

    • L-∞ algebras of local observables from higher prequantum bundles

    • Higher extensions of mapping class groups

  • Motivic quantization of local prequantum field theory

    • Geometric quantization of symplectic and Poisson manifolds

    • Cohomological quantization of local prequantum boundary field theory