A higher Chern-Weil derivation of AKSZ sigma-models

TL;DR

This paper extends Chern-Weil theory to higher structures, showing how AKSZ sigma-models arise naturally from higher Chern-Weil theory within smooth infinity-groupoids, generalizing classical Chern-Simons theory.

Contribution

It provides a higher Chern-Weil framework that recovers and enhances the AKSZ sigma-models as morphisms of higher stacks, generalizing classical Chern-Simons theory.

Findings

Higher Chern-Weil theory recovers AKSZ sigma-models

The framework enhances classical Chern-Simons to higher stacks

Connections between differential cocycles and topological field theories

Abstract

Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connections to differential cocycles whose volume holonomy is the corresponding Chern-Simons theory action functional. Kotov and Strobl have observed that this naturally generalizes from Lie algebras to dg-manifolds and dg-bundles and that the Chern-Simons action functional associated this way to an $n$-symplectic manifold is the action functional of the AKSZ $\sigma$-model whose target space is the given $n$-symplectic manifold (examples of this are the Poisson sigma-model or the Courant sigma-model, including ordinary Chern-Simons theory, or higher dimensional abelian Chern-Simons theory). Here we show how, within the framework of the higher Chern-Weil theory in smooth infinity-groupoids, this result can be naturally recovered and enhanced to a morphism of higher stacks, the same way as ordinary Chern-Simons theory is enhanced to a morphism from the stack of principal G-bundles with connections to the 3-stack of line 3-bundles with connections.