This paper is devoted to a study of geometric structures expressible in terms of graded symplectic supermanifolds. We extend the classical BRST formalism to arbitrary pseudo-Euclidean vector bundles E → M 0 by canonically associating to such a bundle a graded symplectic supermanifold (M, Ω), with deg(Ω) = 2. Conversely, every such manifold arises in this way. We describe the algebra of functions on M in terms of E and show that "BRST charges" on M correspond to Courant algebroid structures on E, thereby constructing the standard complex for the latter as a generalization of the classical BRST complex. As an application of these ideas, we prove the acyclicity of "higher de Rham complexes", a generalization of a classic result of Fröhlicher-Nijenhuis, and derive several easy but useful corollaries.1991 Mathematics Subject Classification. Primary 53D05, 81T70; Secondary 51P05, 81T45.

Abstract. We give a detailed exposition of the Alexandrov-Kontsevich-SchwarzZaboronsky superfield formalism using the language of graded manifolds. As a main illustarting example, to every Courant algebroid structure we associate canonically a three-dimensional topological sigma-model. Using the AKSZ formalism, we construct the Batalin-Vilkovisky master action for the model.

A Lie 2-algebra is a linear category equipped with a functorial bilinear operation satisfying skew-symmetry and Jacobi identity up to natural transformations which themselves obey coherence laws of their own. Functors and natural transformations between Lie 2-algebras can also be defined, yielding a 2-category. Passing to the normalized chain complex gives an equivalence of 2-categories between Lie 2-algebras and 2-term "homotopy everything" Lie algebras; for strictly skew-symmetric Lie 2-algebras, these reduce to $L_\infty$-algebras, by a result of Baez and Crans. Lie 2-algebras appear naturally as infinitesimal symmetries of solutions of the Maurer--Cartan equation in some differential graded Lie algebras and $L_\infty$-algebras. In particular, (quasi-) Poisson manifolds, (quasi-) Lie bialgebroids and Courant algebroids provide large classes of examples.Comment: Based on my talk at the Bialowieza workshop, July 200

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Abstract. We introduce a new type of algebra, the Courant-Dorfman algebra. These are to Courant algebroids what Lie-Rinehart algebras are to Lie algebroids, or Poisson algebras to Poisson manifolds. We work with arbitrary rings and modules, without any regularity, finiteness or non-degeneracy assumptions. To each Courant-Dorfman algebra (R, E) we associate a differential graded algebra C(E, R) in a functorial way by means of explicit formulas. We describe two canonical filtrations on C(E, R), and derive an analogue of the Cartan relations for derivations of C(E, R); we classify central extensions of E in terms of H 2 (E, R) and study the canonical cocycle ∈ C 3 (E, R) whose class [ ] obstructs re-scalings of the Courant-Dorfman structure. In the nondegenerate case, we also explicitly describe the Poisson bracket on C(E, R); for Courant-Dorfman algebras associated to Courant algebroids over finite-dimensional smooth manifolds, we prove that the Poisson dg algebra C(E, R) is isomorphic to the one constructed in Roytenberg (On the structure of graded symplectic supermanifolds and Courant algebroids. American Mathematical Society, Providence, 2002) using graded manifolds.