The geometry of graded cotangent bundles

Abstract

Given a vector bundle A → M we study the geometry of the graded manifoldsincluding 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−1 A * 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. Contents 1. Introduction 1 2. Graded manifolds 3 3. Graded Cotangent bundles 7 4. The symplectic Q-structure 14 5. Semi-direct product of Lie algebroids with 2-term representations up to homotopy 19 6. AKSZ σ-models and integrating objects 25 References 27

Compatible E-Differential Forms on Lie Algebroids Over (Pre-)Multisymplectic Manifolds

“…{{a, pr * e adφ Lie Θ}, b} ∈ E J is equivalent to the homological vector field Q induced from the homological vector field pr * e adφ Lie Θ is tangent to the Lagrangian graded submanifold E J . It is called a Lagrangian Q-submanifold [15].…”