Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -4,7 +4,7 @@  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. -::: definition Vertex algebra {#def:vertex-algebra}+::: 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)  $$@@ -37,7 +37,7 @@    $$ ::: -::: remark State-field correspondence in physics {#rem:state-field}+::: remark [State-field correspondence in physics] {@rem:state-field} In physics, the map $Y$ is often left implicit by writing $a(z)=Y(a,z)$, $|0\rangle=\mathbf1$, and $|a\rangle=a$. The creation condition in @def:vertex-algebra then becomes  $$@@ -53,7 +53,7 @@  ### Creation and translation identities -::: proposition Creation from the vacuum {#prop:creation}+::: proposition [Creation from the vacuum] {@prop:creation} For every $a\in V$,  $$@@ -84,7 +84,7 @@  The operator $T$ acts on states, while its commutator acts on fields. Exponentiating this commutator translates the insertion coordinate. -::: proposition Formal translation of fields {#prop:field-translation}+::: proposition [Formal translation of fields] {@prop:field-translation} For a formal parameter $x$,  $$@@ -117,9 +117,9 @@ Writing $F(x)=\sum_{k\geq0}F_kx^k$ yields $(k+1)F_{k+1}=\partial_zF_k$ with $F_0=Y(a,z)$, hence the Taylor expansion above. ::: -### Skew-symmetry {#sec:skew-symmetry}--::: proposition Skew-symmetry {#prop:skew-symmetry}+### Skew-symmetry {@sec:skew-symmetry}++::: proposition [Skew-symmetry] {@prop:skew-symmetry} For all $a,b\in V$,  $$@@ -173,7 +173,7 @@  We now use skew-symmetry to identify the field corresponding to $Ta$. -::: proposition Translation and derivatives {#prop:translation-derivative}+::: proposition [Translation and derivatives] {@prop:translation-derivative} For $a\in V$ and $n\in\mathbb Z$,  $$@@ -202,7 +202,7 @@  ### Locality and a common expansion -::: proposition Common expansion of two products {#prop:common-expansion}+::: proposition [Common expansion of two products] {@prop:common-expansion} For $a,b,c\in V$, there is a unique  $$@@ -236,7 +236,7 @@ Expanding $(z-w)^{-N}$ in the two orders recovers the products. Injectivity of either expansion map gives uniqueness. ::: -::: theorem The iterate is the third expansion {#thm:iterate-expansion}+::: theorem [The iterate is the third expansion] {@thm:iterate-expansion} For the same $F$ as in @prop:common-expansion,  $$@@ -295,7 +295,7 @@  ### Weak associativity -::: proposition Weak associativity {#prop:weak-associativity}+::: proposition [Weak associativity] {@prop:weak-associativity} For $a,b,c\in V$, some integer $L\geq0$ satisfies  $$@@ -324,11 +324,11 @@ Its denominator is a monomial, so the two expansion orders agree. ::: -### Jacobi identity {#sec:jacobi-identity}+### Jacobi identity {@sec:jacobi-identity}  The two products and the iterate are now three expansions of one expression. The formal delta identity combines them into the VA Jacobi identity. -::: proposition Jacobi identity {#prop:jacobi-identity}+::: proposition [Jacobi identity] {@prop:jacobi-identity} For $a,b,c\in V$,  $$