Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -1,7 +1,39 @@ # Vertex operator algebras -- Conformal vector and Virasoro modes.-- Virasoro relations and compatibility with translation.-- Conformal weights and grading restrictions.-- Additional CFT-type assumptions.-- Examples of VOAs.+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.++## Definition of VOA+++::: definition [Vertex operator algebra]{@definition:1ff90b75}+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.++1. **Virasoro relations.** For some $c\in\mathbb C$, called the **central charge**,++   $$+   [L_m,L_n]=(m-n)L_{m+n}+   +\frac{c}{12}(m^3-m)\delta_{m+n,0}\operatorname{id}_V.+   $$++2. **Translation compatibility.** $L_{-1}=T$.++3. **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 $h$. A state in $V_h$ has **conformal weight** $h$.++We denote the VOA by $(V,Y,\mathbf1,\omega)$. The VOA is called **CFT type** if they satisfy additional conditions $V_h=0$ for $h<0$ and $V_0=\mathbb C\mathbf1$ define.+:::+