Vertex Poisson algebras associated with Courant algebroids and their deformations; I

This is the first of two papers on vertex Poisson algebras associated with Courant algebroids, and their deformations. In this work, we study relationships between vertex Poisson algebras and Courant algebroids. For any $\mathbb{N}$-graded vertex Poisson algebra $A={\coprod }_{n\in \mathbb{N}}{A}_{(n)}$, we show that $A_{(1)}$ is a Courant $A_{(0)}$-algebroid. On the other hand, for any Courant $\mathcal{A}$-algebroid $\mathcal{B}$, we construct an $\mathbb{N}$-graded vertex Poisson algebra $A={\coprod }_{n\in \mathbb{N}}{A}_{(n)}$ such that $A_{(0)}$ is $\mathcal{A}$ and the Courant $\mathcal{A}$-algebroid $A_{(1)}$ is isomorphic to $\mathcal{B}$ as a Courant $\mathcal{A}$-algebroid.