Proposition: The scalar VOA

For $V=\mathbb{C}\Omega$ with conformal vector zero, its vacuum net has $A(I)=\mathbb{C}$ on the one-dimensional vacuum Hilbert space. It is local, irreducible, split, strongly additive, and has two-interval index one.

The grading-restricted finite-length module category is the category of finite-dimensional complex vector spaces; after positive forms are chosen, its unitary version is finite-dimensional Hilbert spaces with ordinary tensor product and flip. The additive dualizable representation category of the scalar net is the same Hilbert-space category.

Proof. All vacuum fields are scalar multiples of identity, so their local algebras are $\mathbb{C}$. Every interval and two-interval inclusion is equality. The asserted net properties and index follow directly.

On a module, the conformal operator is zero because the conformal vector is zero. Grading restriction therefore makes the whole module finite dimensional. Fusion is tensor product over $\mathbb{C}$.

A normal representation of $\mathbb{C}$ is scalar multiplication on a Hilbert multiplicity space. A finite-dimensional multiplicity space has its usual conjugate dual. Conversely, a coevaluation vector and evaluation satisfying the conjugate equations would represent the identity operator as a compact operator on that Hilbert space; hence dualizability forces finite dimension.