L∞-algebra models and higher Chern–Simons theories

Abstract

We continue our study of zero-dimensional field theories in which the fields take values in a strong homotopy Lie algebra. In the first part, we review in detail how higher Chern–Simons theories arise in the AKSZ-formalism. These theories form a universal starting point for the construction of [Formula: see text]-algebra models. We then show how to describe superconformal field theories and how to perform dimensional reductions in this context. In the second part, we demonstrate that Nambu–Poisson and multisymplectic manifolds are closely related via their Heisenberg algebras. As a byproduct of our discussion, we find central Lie [Formula: see text]-algebra extensions of [Formula: see text]. Finally, we study a number of [Formula: see text]-algebra models which are physically interesting and which exhibit quantized multisymplectic manifolds as vacuum solutions.