Abstract
Given a vector bundle A→M we study the geometry of the graded manifolds T∗[k]A[1], including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these graded objects to classical structures, such as higher Courant algebroids on A⊕⋀k−1A∗ and higher Dirac structures therein, semi-direct products of Lie algebroid structures on A with their coadjoint representations up to homotopy, and branes on certain AKSZ σ-models.
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