Schreiber A higher stacky perspective on Chern-Simons theory

An article that we wrote for a book on mathematical aspects of quantum field theory:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber,

  A higher stacky perspective on Chern-Simons theory

  (arXiv:1301.2580)

  in

  Damien Calaque et al. (eds.)

  Mathematical Aspects of Quantum Field Theories

  Mathematical Physics Studies, Springer 2014

  pages 153-211

  doi:10.1007/978-3-319-09949-1

on Chern-Simons theory viewed in higher geometry with an outlook on ∞-Chern-Simons theory and its ∞-geometric prequantization.

Abstract

The first part of this text is a gentle exposition of some basic constructions and results in the extended prequantum theory of Chern-Simons-type gauge field theories. We explain in some detail how the action functional of ordinary 3d Chern-Simons theory is naturally localized (“extended”, “multi-tiered”) to a map on the universal moduli stack of principal connections, a map that itself modulates a circle-principal 3-connection on that moduli stack, and how the iterated transgressions of this extended Lagrangian unify the action functional with its prequantum bundle and with the WZW-functional. In the second part we provide a brief review and outlook of the higher prequantum field theory of which this is a first example. This includes a higher geometric description of supersymmetric Chern-Simons theory, Wilson loops and other defects, generalized geometry, higher Spin-structures, anomaly cancellation and various other aspects of quantum field theory.

Related work

This article is part of the research laid out in the following related articles.

• 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

Further references

For more discussion of higher category theory and physics see also the collection

• Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics, volume 83 AMS (2011)

For lecture notes on material closely related to the above article see also

• Urs Schreiber, twisted smooth cohomology in string theory ESI (June 2012)

For further discussion of applications in string theory/M-theory see

• Hisham Sati, Geometric and topological structures related to M-branes