Introduction

From mathematical point of view the BV formalism may be defined as the theory of odd symplectic supermanifolds.

Consider a supermanifold with an odd 2-form satisfying .

It is not true that defines an antisymmetric bivector. (The symmetry properties are not of a bivector.)

Instead, it defines a second order differential operator on half-densities.

A half-density defines a measure on every Lagrangian submanifold . The volume of the Lagrangian submanifold with respect to this measure is a mathematical model for the path integral in a gauge theory.