$L_\infty$-Algebra Models and Higher Chern-Simons Theories

TL;DR

This paper explores the connection between $L_$-algebra models, higher Chern-Simons theories, and multisymplectic geometry, revealing new algebraic structures and their physical implications in field theories.

Contribution

It introduces new $L_$-algebra models related to higher Chern-Simons theories and links multisymplectic manifolds with algebraic structures like Heisenberg and Lie $p$-algebras.

Findings

Higher Chern-Simons theories are derived from $L_$-algebra models.

Nambu-Poisson and multisymplectic manifolds are connected via Heisenberg algebras.

Quantized multisymplectic manifolds emerge as vacuum solutions in these models.

Abstract

We continue our study of zero-dimensional field theories in which the fields take values in a strong homotopy Lie algebra. In a 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 $L_\infty$-algebra models. We then show how to describe superconformal field theories and how to perform dimensional reductions in this context. In a 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 $p$-algebra extensions of $\mathfrak{so}(p+2)$. Finally, we study a number of $L_\infty$-algebra models which are physically interesting and which exhibit quantized multisymplectic manifolds as vacuum solutions.