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 classical master equation. Geometrically such a solution can be considered as a QP -manifold, i.e. a supermanifold 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 ΠG. (Here G stands for a Lie algebra and Π denotes parity inversion.) 1 E-mail: [email protected] 2 Research supported in part by NSF grant DMS-9322519. E-mail: [email protected] 3 Research supported in part by NSF grant DMS-9201366. E-mail: [email protected] 4 On leave from the Institute of Theoretical and Experimental Physics, Moscow, Russia. E-mail: [email protected]