On the structure of $\mathbb{N}$-graded Vertex Operator Algebras

We consider the algebraic structure of $\mathbb{N}$-graded vertex operator algebras with conformal grading $V={\oplus }_{n\ge 0}{V}_n$ and $\dim V_0\ge 1$. We prove several results along the lines that the vertex operators $Y(a,z)$ for $a$ in a Levi factor of the Leibniz algebra $V_1$ generate an affine Kac-Moody subVOA. If $V$ arises as a shift of a self-dual VOA of CFT-type, we show that $V_0$ has a `de Rham structure’ with many of the properties of the de Rham cohomology of a complex connected manifold equipped with Poincar'{e} duality.