Higher Chern-Weil Derivation of AKSZ Sigma-Models

An article of ours

in the context of ∞-Chern-Simons theory.


Chern-Weil theory provides for each invariant polynomial on a Lie algebra 𝔀\mathfrak{g} a map from 𝔀\mathfrak{g}-connections to differential cocycles whose volume holonomy is the the corresponding Chern-Simons theory action 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 ∞L_\infty-algebroid (for instance the Poisson sigma-model or the Courant sigma-model).


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

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

and conference meetings

The corresponding preprint appeared a little later:

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 ∞L_\infty-algebraic constructions to circle n-bundles with connection on smooth ∞-stacks was finally accomplished in

This is what the above article is based on.

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

Revised on February 23, 2015 17:58:55 by Urs Schreiber (