The Geometry of the Master Equation and Topological Quantum Field Theory

In Batalin-Vilkovisky formalism a classical mechanical system is specified by means of a solution to the {\sl classical master equation}. Geometrically such a solution can be considered as a $QP$-manifold, i.e. a super\m equipped with an odd vector field $Q$ obeying $\{Q,Q\}=0$ and with $Q$-invariant odd symplectic structure. We study geometry of $QP$-manifolds. In particular, we describe some construction of $QP$-manifolds and prove a classification theorem (under certain conditions). We apply these geometric constructions to obtain in natural way the action functionals of two-dimensional topological sigma-models and to show that the Chern-Simons theory in BV-formalism arises as a sigma-model with target space $\Pi {\cal G}$. (Here ${\cal G}$ stands for a Lie algebra and $\Pi$ denotes parity inversion.)Comment: 29 pages, Plain TeX, minor modifications in English are made by Jim Stasheff, some misprints are corrected, acknowledgements and references adde