# Schreiber Higher Chern-Weil Derivation of AKSZ Sigma-Models

An article of ours

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

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

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 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 $L_\infty$-algebroid (for instance the Poisson sigma-model or the Courant sigma-model).

## 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

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 $L_\infty$-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 (89.204.138.150)