Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -1,8 +1,10 @@-# Vertex algebras: definitions and properties+# Vertex algebras  ## Definition and properties of a vertex algebra  We first give the definition in terms of locality, then derive its consequences and compare equivalent formulations. Throughout, we work over $\mathbb C$ with ordinary, purely even vertex algebras. We assume familiarity with formal calculus; all variables are formal.++For endomorphisms $A,B\in\operatorname{End}(V)$, the bracket $[A,B]=AB-BA$ denotes their commutator; for fields it is interpreted coefficientwise.  ::: definition [Vertex algebra] {@def:vertex-algebra} A **vertex algebra** is a tuple $(V,Y,\mathbf1,T)$ consisting of a complex vector space $V$, a nonzero **vacuum vector** $\mathbf1\in V$, a linear **translation operator** $T\colon V\to V$, and a linear **state-field correspondence** (also called the state-operator correspondence)