Geodesic
Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications
Archivum mathematicum, Tome 47 (2011) no. 5, pp. 415-471.

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These notes are intended to provide a self-contained introduction to the basic ideas of finite dimensional Batalin-Vilkovisky (BV) formalism and its applications. A brief exposition of super- and graded geometries is also given. The BV–formalism is introduced through an odd Fourier transform and the algebraic aspects of integration theory are stressed. As a main application we consider the perturbation theory for certain finite dimensional integrals within BV-formalism. As an illustration we present a proof of the isomorphism between the graph complex and the Chevalley-Eilenberg complex of formal Hamiltonian vectors fields. We briefly discuss how these ideas can be extended to the infinite dimensional setting. These notes should be accessible to both physicists and mathematicians.
Classification : 16E45, 58A50, 97K30
Keywords: Batalin-Vilkovisky formalism; graded symplectic geometry; graph homology; perturbation theory 
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Qiu, Jian; Zabzine, Maxim. Introduction to Graded Geometry,  Batalin-Vilkovisky Formalism and their Applications. Archivum mathematicum, Tome 47 (2011) no. 5, pp. 415-471. http://geodesic.mathdoc.fr/item/ARM_2011__47_5_a8/