$L_{\infty }$-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 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.

44 pages, minor corrections, published version

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• Algebraic structures on parallel M2-branes
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Cited by in corpus (11)

• Towards an M5-Brane Model I: A 6d Superconformal Field Theory
• Leibniz Gauge Theories and Infinity Structures
• Towards an M5-Brane Model II: Metric String Structures
• Infinity-enhancing of Leibniz algebras
• Supersymmetric Yang-Mills Theory as Higher Chern-Simons Theory
• Algebraic formulation of higher gauge theory
• Higher Chern-Simons-Antoniadis-Savvidy forms based on crossed modules
• (2-)Drinfel'd Double and (2-)BF Theory
• Higher Chern-Simons based on (2-)crossed modules
• Extended Cartan homotopy formula for higher Chern-Simons-Antoniadis-Savvidy theory
• Higher (gauged) Wess--Zumino--Witten terms based on Lie crossed modules