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We show that the 2d Poisson Sigma Model on a Poisson groupoid arises as an effective theory of the 3d Courant Sigma Model associated with the double of the underlying Lie bialgebroid. This field-theoretic result follows from a Lie-theoretic one involving a coisotropic reduction of the odd cotangent bundle by a generalized space of algebroid paths. We also provide several examples, including the case of symplectic groupoids in which we relate the symplectic realization construction of Crainic–Marcut to a particular gauge fixing of the 3d theory.

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

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