Vertex algebra in the state–field convention

A (non-super) vertex algebra over $\mathbb{C}$ is a complex vector space $V$ equipped with a nonzero vector $\mathbf{1}\in V$, a linear operator $T:V\to V$, and a linear state–field map $Y(-,z):V⟶({\mathrm{End}}_{\mathbb{C}}(V){)}^{\mathbb{Z}},a⟼Y(a,z)={\sum }_{n\in \mathbb{Z}}{a}_{(n)}{z}^{-n-1},$ where the codomain denotes all endomorphism-valued coefficient families indexed by $\mathbb{Z}$. satisfying the following axioms.

1. Field and truncation. For each $a,v\in V$, there is an integer $N(a,v)$ such that $a_{(n)}v=0$ for all $n\ge N(a,v)$. Thus $Y(a,z)v\in V((z))$ in the lower-bounded support sense of Formal Laurent series $K((x))$, their order, derivative, and residue.

2. Vacuum and creation. $Y(\mathbf{1},z)={\mathrm{id}}_V$, $T\mathbf{1}=0$, and $Y(a,z)\mathbf{1}=e^{zT}a:={\sum }_{r\ge 0}\frac{z^r{T}^{r}a}{r!}\mathrm{.}$ This is a formal power series; no analytic convergence is asserted.

3. Translation covariance. For every $a\in V$, $[T,Y(a,z)]={\mathrm{\partial }}_{z}Y(a,z)\mathrm{.}$

4. Locality. For every $a,b\in V$, some integer $N\ge 0$ satisfies $(z-w{)}^N[Y(a,z),Y(b,w)]=0.$

The coefficient $a_{(n)}b$ is called the $n$th product of the states $a,b$. The mode and formal-distribution conventions in Formal distributions and fields on a vector space are in force. The axioms are algebraic and do not impose a grading, conformal vector, simplicity, unitarity, or analytic OPE convergence.

The creation axiom is the strong form of the vacuum condition: it gives a formal power series in $z$ with coefficients in $V$ and constant coefficient $a$. Together with translation covariance and $T\mathbf{1}=0$, it fixes every coefficient of that vacuum series by the recursion $(r+1)[z^{r+1}]Y(a,z)\mathbf{1}=T([z^r]Y(a,z)\mathbf{1})$.