AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories

We give a detailed exposition of the Alexandrov-Kontsevich-Schwarz- Zaboronsky 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.Comment: 13 pages, based on lectures at Rencontres mathematiques de Glanon 200

Courant-Dorfman algebras and their cohomology

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 \Theta\in\C^3(\E,\R) whose class $[\Theta]$ 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 \cite{Roy4-GrSymp} using graded manifolds.Comment: Corrected formulas for the brackets in Examples 2.27, 2.28 and 2.29. The corrections do not affect the exposition in any wa