N-manifolds of degree 2 and metric double vector bundles

This paper shows the equivalence of the categories of $N$-manifolds of degree $2$ with the category of double vector bundles endowed with a linear metric. Split Poisson $N$-manifolds of degree $2$ are shown to be equivalent to self-dual representations up to homotopy. As a consequence, the equivalence above induces an equivalence between so called metric VB-algebroids and Poisson $N$-manifolds of degree $2$. Then a new description of split Lie $2$-algebroids is given, as well as their "duals", the Dorfman $2$-representations. We show that Dorfman $2$-representations are equivalent in a simple manner to Lagrangian splittings of VB-Courant algebroids. This yields the equivalence of the categories of Lie $2$-algebroids and of VB-Courant algebroids. We give several natural classes of examples of split Lie $2$-algebroids and of the corresponding VB-Courant algebroids. We then show that a split Poisson Lie $2$-algebroid is equivalent to the "matched pair" of a Dorfman $2$-representation with a self-dual representation up to homotopy. We deduce a new proof of the equivalence of categories of LA-Courant algebroids and Poisson Lie $2$-algebroids. We show that the core of an LA-Courant algebroid inherits naturally the structure of a degenerate Courant algebroid. This yields a new formula to retrieve in a direct manner the Courant algebroid found by Roytenberg to correspond to a symplectic Lie $2$-algebroid. Finally we study VB- and LA-Dirac structures in VB- and LA-Courant algebroids. As an application, we extend Li-Bland's results on pseudo-Dirac structures and we construct a Manin pair associated to an LA-Dirac structure.