Vertex operator algebra

In mathematics, a vertex operator algebra (abbreviated: VOA) is a certain kind of algebra that plays a key part in conformal field theory and other fields of study in physics, and has also proven useful in purely mathematical contexts such as moonshine theory.

Vertex operator algebras were first introduced by Richard Borcherds in 1986; important examples include the Virasoro VOAs (i.e., VOAs corresponding to representations of the Virasoro algebra L) and the moonshine VOA V^♮.

Formal definition

A vertex operator is a Z₊-graded vector space

$V=\bigoplus _{n=0}^{\infty }V_{n}$

with a sequence of linear operators

$\left\{{}_{a}(n)|n\in \mathbb {Z} \right\}\subset \mathrm {End} (V)$

associated to every a ∈ V such that for fixed a, b ∈ V, _a(n)_b = 0 for n sufficiently large. The generating function

$Y(a,z)=\sum _{n\in \mathbb {Z} }{}_{a}(n)_{z^{-n-1}}\in \mathrm {End} (V)[[z,z^{-1}]]$

is called the vertex operator of a, and the following conditions hold:

1. Y(a, z) = 0 iff a = 0.

2. There is a distinguished element, called the vacuum and denoted 1, that satisfies

   $Y(\mathbf {1} ,z)=I_{V}$

   where I_V is the identity in End(V)).

3. There is a distinguished element, called the Virasoro element and denoted ω, the corresponding vertex operator of which can be written as follows:

   $Y(\omega ,z)=\sum _{n\in \mathbb {Z} }{}_{\omega }(n)_{z^{-n-1}}=\sum _{n\in \mathbb {Z} }L_{n}z^{-n-2}$

   such that (for every a ∈ V)

   $L_{0}|{}_{V_{n}}=nI_{V_{n}}$

   $Y(L_{-1}a,z)={\frac {d}{dz}}Y(a,z)$

   $[L_{m},L_{n}]=(m-n)L_{m+n}+\delta _{m+n,0}{\frac {m^{3}-m}{12}}c$

   where c ∈ C is a constant called the rank of V.

4. The Jacobi identity holds; that is, for any m, n ∈ Z,

   	$\mathrm {Res} _{z-w}(Y(Y(a,z-w)b,w)\iota _{w,z-w}((z-w)^{m}z^{n})$
   $=$	$\mathrm {Res} _{z}(Y(a,z)Y(b,w)\iota _{z,w}(z-w)^{m}z^{n})-\mathrm {Res} _{z}(Y(b,w)Y(a,z)\iota _{w,z}(z-w)^{m}z^{n})$

   where for a rational function f(z₁, z₂) with poles only possible at z₁ = z₂, z₁ = 0 and z₂ = 0

   $\iota _{z_{1},z_{2}}$

   denotes the power series expansion of f(z₁, z₂) in the domain |z₁| > |z₂|.

References

• Richard Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA. 83 (1986) 3068-3071
• Weiqiang Wang, Rationality of Virasoro Vertex Operator Algebras, Intern. Math. Res. Notices, No. 7 (1993), 197-211, online