Chapter 3 of 7

§Vertex operator algebras

Although vertex algebra contains many familiar properties a chiral CFT constains, it does not contain the most important structure: the energy-momentum tensor and Virasoro algebra. A vertex operator algebra (VOA) is a vertex algebra with so-called conformal structure, which is essentially the energy-momentum tensor and Virasoro algebra. We now define VOA.

§3.1 Definition of VOA

Definition 3.1 (Vertex operator algebra)

Let $(V,Y,\mathbf1,T)$ be a vertex algebra. A vertex operator algebra (VOA) structure on $V$ is specified by a conformal vector $\omega\in V$ whose field

Y(\omega,z)=\sum_{n\in\mathbb Z}L_nz^{-n-2}, \qquad L_n=\omega_{(n+1)},

satisfies the following conditions.

Virasoro relations. For some $c\in\mathbb C$ $C$ , called the central charge,

$[L_m,L_n]=(m-n)L_{m+n} +\frac{c}{12}(m^3-m)\delta_{m+n,0}\operatorname{id}_V.$
Translation compatibility. $L_{-1}=T$ $T$ .
Conformal grading. $L_0$ acts diagonally with integral eigenvalues,

$V=\bigoplus_{h\in\mathbb Z}V_h, \qquad V_h=\{a\in V:L_0a=ha\},$

where $\dim V_h<\infty$ and $V_h=0$ for sufficiently negative $h$ $h$ . A state in $V_h$ has conformal weight $h$ $h$ .

We denote the VOA by $(V,Y,\mathbf1,\omega)$ $(V, Y, 1, ω)$ . The VOA is called CFT type if they satisfy additional conditions $V_h=0$ for $h<0$ and $V_0=\mathbb C\mathbf1$ $C 1$ .

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