Vertex Operator Algebra

Vertex Algebra

A vertex algebra is a collection of data:

1. A vector space $V$ (usually over $\mathbb{C}$), called the state space.
2. An element $|0⟩\in V$, called the vacuum vector.
3. An endomorphism $T:V\to V$, called the translation operator.
4. A map \newcommand{\(}{(\!(}\newcommand{\)}{)\!)} Y\colon V \otimes V \to V\(z\), where V\(z\) denotes the space of formal Laurent series in $z$ with coefficients in $V$. This map is called the state-field correspondence.

These data must satisfy the following axioms:

1. Vacuum Axiom: $Y(|0⟩,z)={\text{id}}_V$ and $Y(a,z)|0⟩\in a+zV[[z]]$.
2. Translation Axiom: $[T,Y(a,z)]={\partial }_{z}Y(a,z)$ and $T|0⟩=0$.
3. Locality Axiom: For all $a,b\in V$, there exists an integer $N>0$ such that $(z-w{)}^N[Y(a,z),Y(b,w)]=0$ in $V[[z,w]]$.