On $\mathbb{N}$-graded vertex algebras associated with Gorenstein algebras

This paper investigates the algebraic structure of indecomposable $\mathbb{N}$-graded vertex algebras $V = \bigoplus_{n=0}^{\infty} V_n$, emphasizing the intricate interactions between the commutative associative algebra $V_0$, the Leibniz algebra $V_1$ and how non-degenerate bilinear forms on $V_0$ influence their overall structure. We establish foundational properties for indecomposability and locality in $\mathbb{N}$-graded vertex algebras, with our main result demonstrating the equivalence of locality, indecomposability, and specific structural conditions on semiconformal-vertex algebras. The study of symmetric invariant bilinear forms of semiconformal-vertex algebra is investigated. We also examine the structural characteristics of $V_0$ and $V_1$, demonstrating conditions under which certain $\mathbb{N}$-graded vertex algebras cannot be quasi vertex operator algebras, semiconformal-vertex algebras, or vertex operator algebras, and explore $\mathbb{N}$-graded vertex algebras $V=\bigoplus_{n=0}^{\infty}V_n$ associated with Gorenstein algebras. Our analysis includes examining the socle, Poincar\'{e} duality properties, and invariant bilinear forms of $V_0$ and their influence on $V_1$, providing conditions for embedding rank-one Heisenberg vertex operator algebras within $V$. Supporting examples and detailed theoretical insights further illustrate these algebraic structures.