Schreiber Higher Chern-Weil Derivation of AKSZ Sigma-Models

An article of ours

• Domenico Fiorenza, Chris Rogers, Urs Schreiber

  A higher Chern-Weil derivation of AKSZ $\sigma$-models

  Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250078 (36 pages)

  (arXiv:1108.4378, publisher)

in the context of ∞-Chern-Simons theory.

Abstract

Chern-Weil theory provides for each

invariant polynomial on a Lie algebra $\mathfrak{g}$ a map from $\mathfrak{g}$-connections to differential cocycles whose volume holonomy is the the corresponding

Chern-Simons theoryaction functional.

We observe that in the context of higher Chern-Weil theory in smooth ∞-groupoids this statement generalizes from Lie algebras to L-∞ algebras and further to L-∞ algebroids. It turns out that the symplectic form on a symplectic higher Lie algebroid (for instance a Poisson Lie algebroid or a Courant Lie 2-algebroid) is ∞-Lie-theoretically an invariant polynomial. We show that the higher Chern-Simons action functional associated to this by higher Chern-Weil theory is the action functional of the AKSZ sigma-model with target space the given $L_\infty$-algebroid (for instance the Poisson sigma-model or the Courant sigma-model).

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 – theory, presentations and applications

    • 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

    • nonabelian 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

References

The ideas of ∞-Chern-Weil theory that this is based on, the notion of invariant polynomials and Chern-Simons elements on $L_\infty$-structures had been presented from spring 2007 on in blog posts

• Chern Lie $(2n+1)$-Algebras, (May 2007) (blog, pdf)

and conference meetings

• String and Chern-Simons Lie 3-algebras (August 2007) (blog pdf slides)

• On String- and Chern-Simons n-Transport (October 2007) (blog, expanded talk notes, workshop page)

The corresponding preprint appeared a little later:

• Hisham Sati, Urs Schreiber, Jim Stasheff, L-∞ algebra connections (arXiv:0801.3480)

Meanwhile also (arXiv:0711.4106) had appeared, with similar observations.

The construction of the refined $\infty$-Chern-Weil homomorphism, hence the Lie integration of these $L_\infty$-algebraic constructions to circle n-bundles with connection on smooth ∞-stacks was finally accomplished in

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, ?ech cocycles for differential characteristic classes? (arXiv:1011.4735)

This is what the above article is based on.

For more related references see also differential cohomology in a cohesive topos – references.