Vertex Operator Algebras: Axioms, Grading, and Locality

A vertex operator algebra packages a chiral state space and all of its formal operator products into a quadruple $(V,Y,\mathbf 1,\omega)$. The decisive requirement is not agreement of a few modes: every pair of fields must obey lower truncation and a uniform formal locality identity, equivalently the Jacobi identity under the remaining axioms. That distinction turns free-field calculations into algebraic structure and also supplies a sharp rejection test for plausible but inconsistent field assignments.

Required background. Free Bosons and Vertex Operators supplies the oscillator construction and current OPE used below. The Virasoro Algebra and the Stress Tensor supplies the conformal vector and Virasoro modes.

Helpful background. Direct Sums, Tensor Products, and Index Structure reviews graded linear algebra. AdS3 Boundary Gravitons and Vacuum Characters gives a useful character-level comparison without replacing the VOA axioms.

State–field correspondence and formal locality

Let $V=\bigoplus_{n\in\mathbb Z}V_n$ be a complex vector space with $\dim V_n<\infty$, $V_n=0$ for sufficiently negative $n$, vacuum $\mathbf1\in V_0$, and conformal vector $\omega\in V_2$. A state $v$ determines a field

$$Y(v,z)=\sum_{m\in\mathbb Z}v_m z^{-m-1}\in (\operatorname{End}V)[[z,z^{-1}]].$$

Lower truncation means $v_m w=0$ for fixed $v,w$ and sufficiently large $m$. The vacuum and creation identities are $Y(\mathbf1,z)=\operatorname{id}_V$ and $Y(v,z)\mathbf1\in v+zV[[z]]$. Writing $T=L_{-1}$ for the translation generator, one requires $[T,Y(v,z)]=\partial_zY(v,z)=Y(Tv,z)$. Finally,

$$Y(\omega,z)=\sum_{n\in\mathbb Z}L_nz^{-n-2}, \qquad L_0v=n v\quad(v\in V_n),$$

and the $L_n$ obey the Virasoro relations with a fixed central charge $c$.

For every $u,v\in V$, formal locality demands an integer $N=N(u,v)\geq0$ such that

$$(z_1-z_2)^N[Y(u,z_1),Y(v,z_2)]=0.$$

Here the equality is coefficientwise in formal variables; it is not an analytic assertion about unbounded operators. Vacuum, translation, truncation, and locality imply the formal Jacobi identity, and conversely the Jacobi identity implies the commutator and iterate formulas. The precise delta-function calculus and this equivalence are developed in Frenkel, Huang, and Lepowsky 1993, Chapter 2, pp. 9–20. A vertex superalgebra changes the commutator to the parity-signed supercommutator. A vertex algebra need not include a conformal vector; a VOA does.

Taking $\operatorname{Res}_z$ means extracting the coefficient of $z^{-1}$. Applying formal residues to Jacobi gives, among other consequences,

$$[u_m,v_n]=\sum_{j\geq0}\binom mj (u_jv)_{m+n-j}.$$

This identity is a stringent all-mode consequence. A finite list of low-mode relations can be evidence for it but cannot establish it.

Rank-one Heisenberg construction

The first QFT application is the rank-one model developed physically in Free Bosons and Vertex Operators. Start from oscillators

$$[a_m,a_n]=m\,\delta_{m+n,0},\qquad a_n\mathbf1=0\quad(n\geq0),$$

and let $M(1)$ be the symmetric algebra generated by $a_{-n}$, $n>0$, acting on $\mathbf1$. Give $a_{-n_1}\cdots a_{-n_r}\mathbf1$ weight $n_1+\cdots+n_r$. Thus the grading is bounded below and each graded piece is finite-dimensional. Set

$$J(z)=Y(a_{-1}\mathbf1,z)=\sum_{n\in\mathbb Z}a_nz^{-n-1}, \qquad \omega=\frac12a_{-1}^2\mathbf1.$$

Normal-ordered products and derivatives of $J$ define $Y(v,z)$ for every Fock state. The oscillator commutator gives the formal distribution identity

$$[J(z_1),J(z_2)]=\partial_{z_2}\!\left(z_1^{-1}\delta\!\left(\frac{z_2}{z_1}\right)\right),$$

so $(z_1-z_2)^2[J(z_1),J(z_2)]=0$. Equivalently, the singular OPE is $J(z_1)J(z_2)\sim(z_1-z_2)^{-2}$. Dong’s lemma propagates mutual locality from the generating current to its derivatives and normal-ordered products; this is the actual mechanism that constructs the full state–field map, not an extrapolation from a truncated mode table. The conformal vector yields $c=1$, $[L_0,a_{-n}]=n a_{-n}$, and $L_{-1}$ differentiates the current.

Two independent checks are immediate. First, the vacuum character is

$$\operatorname{tr}_{M(1)}q^{L_0}=\prod_{n\geq1}(1-q^n)^{-1},$$

whose coefficient of $q^N$ equals the number of oscillator partitions of $N$, matching the graded basis. Second, the coefficient of $z_1^{-m-1}z_2^{-n-1}$ in the current commutator recovers $[a_m,a_n]=m\delta_{m+n,0}$ with the correct normalization.

From generators to every state

The Heisenberg calculation is complete only because the reconstruction theorem for mutually local fields applies. The vacuum field, current, and their translated normal-ordered products are creative, translation covariant, and mutually local; their modes span $M(1)$. The theorem then gives a unique vertex-algebra structure on that spanning space. This uniqueness prevents an unnoticed change to a higher composite field while leaving the current OPE fixed. It also clarifies the role of Dong’s lemma: locality is stable under taking the mode products that generate composite states. Without the spanning and closure conditions, a mutually local family of formal distributions can generate only a vertex subalgebra rather than the proposed entire state space.

Failure boundary: low modes are not locality

Consider a bilinear assignment $Y_0$ that has the correct vacuum action and agrees with the Heisenberg fields through some mode cutoff, but for two states $u,v$ has nonzero commutator coefficients at arbitrarily high pole order. Then no common exponent $N(u,v)$ annihilates $[Y_0(u,z_1),Y_0(v,z_2)]$. The low-mode fixture survives as a finite algebraic approximation, but locality fails and the Jacobi identity rejects $Y_0$ as a VOA state–field map. Nor does formal locality prove analytic convergence, positivity, energy bounds, or strong commutativity of smeared closures; those are separate hypotheses developed later in this chapter.

Exercises

1. Show that $T\mathbf1=0$ and $Y(Tv,z)\mathbf1=\partial_z(Y(v,z)\mathbf1)$ force $Y(v,z)\mathbf1=e^{zT}v$.

Solution

Set $F(z)=Y(v,z)\mathbf1$. Translation gives $F'(z)=Y(Tv,z)\mathbf1=TF(z)$, while creation gives $F(0)=v$. The unique formal-power-series solution is $F(z)=e^{zT}v$.

2. Verify the locality exponent of the Heisenberg current.

Solution

The singular part is $(z_1-z_2)^{-2}$. Multiplication by $(z_1-z_2)^2$ removes both boundary expansions of the commutator, whereas one power leaves a simple-pole formal distribution. Thus exponent $2$ works and exponent $1$ does not.

References

• Borcherds, Richard E. “Vertex Algebras, Kac–Moody Algebras, and the Monster.” Proceedings of the National Academy of Sciences 83 (1986), 3068–3071. DOI.
• Frenkel, Igor B., Yi-Zhi Huang, and James Lepowsky. On Axiomatic Approaches to Vertex Operator Algebras and Modules. Memoirs of the American Mathematical Society 104, no. 494 (1993). DOI.