  By using help of algebraic operad theory, Leibniz algebra theory and
symplectic-Poisson geometry are connected. We introduce the notion of
cohomological vector field defined on nongraded symplectic plane. It will be
proved that the cohomological vector fields induce the finite dimensional
Leibniz algebras by the derived bracket construction. This proposition is a
Leibniz analogue of the cohomological field theory in the category of Lie
algebras. The basic properties of the cohomological fields will be studied, in
particular, we discuss a factorization problem with the cohomological fields
and introduce the notion of double-algebra in the category of Leibniz algebras.
