Observables on multisymplectic manifolds and higher Courant algebroids

Abstract

Let $\omega$ be a closed, non-degenerate differential form of arbitrary degree. Associated to it there are an $L_{\infty }$-algebra of observables, and an $L_{\infty }$-algebra of sections of the higher Courant algebroid twisted by $\omega$. Our main result is the existence of an $L_{\infty }$-embedding of the former into the latter. We display explicit formulae for the embedding, involving the Bernoulli numbers. When $\omega$ is an integral symplectic form, the embedding can be realized geometrically via the prequantization construction, and when $\omega$ is a 3-form the embedding was found by Rogers in 2010. Further, in the presence of homotopy moment maps, we show that the embedding is compatible with gauge transformations.

48 pages