Higher Form Brackets for even Nambu-Poisson Algebras

Let $\mathit{k}$ be a field of characteristic zero and $A=\mathit{k}[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]\mathrm{/}I$ with $I=(f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_k)$ be an affine algebra. We study Nambu-Poisson brackets on $A$ of arity $m\ge 2$, focusing on the case when $m$ is even. We construct an $L_{\mathrm{\infty }}$-algebroid on the cotangent complex ${\mathbb{L}}_{A\mathrm{\mid }\mathit{k}}$, generalizing previous work on the case when $A$ is a Poisson algebra. This structure is referred to as the higher form brackets. The main tool is a $P_{\mathrm{\infty }}$-structure on a resolvent $R$ of $A$. These $P_{\mathrm{\infty }}$- and $L_{\mathrm{\infty }}$-structures are merely ${\mathbb{Z}}_2$-graded for $m\mathrm{\ne }2$. We discuss several examples and propose a method to obtain new ones that we call the outer tensor product. We compare our higher form brackets with the form bracket of Vaisman. We introduce the notion of a Lie-Rinehart $m$-algebra, the form bracket of a Nambu-Poisson bracket of even arity being an example. We find a flat Nambu connection on the conormal module.