# Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications

## Jian Qiu and Maxim Zabzine

Address:

J. Qiu, I.N.F.N. and Dipartimento di Fisica, Via G. Sansone 1, 50019 Sesto Fiorentino – Firenze, Italy

M. Zabzine, Department of Physics and Astronomy, Uppsala university, Box 516, SE-75120 Uppsala, Sweden

E-mail: Maxim.Zabzine@fysast.uu.se

Abstract: 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.

AMSclassification: primary 58A50; secondary 16E45, 97K30.

Keywords: Batalin-Vilkovisky formalism, graded symplectic geometry, graph homology, perturbation theory.