Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -0,0 +1,362 @@+# Vertex algebras: definitions and properties++## 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.++::: 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)++$$+Y\colon V\longrightarrow \operatorname{End}(V)[[z,z^{-1}]],+\qquad+a\longmapsto Y(a,z)=\sum_{n\in\mathbb Z}a_{(n)}z^{-n-1},+$$++satisfying the following conditions for all $a,b\in V$.++1. **Lower truncation.** $Y(a,z)b\in V((z))$; equivalently, $a_{(n)}b=0$ for sufficiently large $n$, with a bound that may depend on $a$ and $b$.+2. **Vacuum and creation.**++   $$+   Y(\mathbf1,z)=\operatorname{id}_V,+   \qquad Y(a,z)\mathbf1\in a+zV[[z]].+   $$++3. **Translation covariance and invariance of the vacuum.**++   $$+   T\mathbf1=0,+   \qquad [T,Y(a,z)]=\partial_zY(a,z).+   $$++4. **Locality.** There exists an integer $N=N(a,b)\geq0$ such that++   $$+   (z-w)^N[Y(a,z),Y(b,w)]=0.+   $$+:::++::: 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++$$+\left.a(z)|0\rangle\right|_{z=0}=|a\rangle,+$$++often written as $\lim_{z\to0}a(z)|0\rangle=|a\rangle$. Here evaluation at zero means taking the constant term of a formal power series; an analytic limit requires a separate analytic interpretation.+:::++The vacuum and translation axioms express the identity field, creation of states from the vacuum, and infinitesimal translations. Locality encodes compatibility between fields, whose commutator is annihilated by a sufficiently high power of $z-w$. But note that this does not require the commutator itself to vanish. We will explain its connection with OPEs below.++## Vacuum and translation++### Creation and translation identities++::: proposition Creation from the vacuum {#prop:creation}+For every $a\in V$,++$$+Y(a,z)\mathbf1=e^{zT}a.+$$++Consequently, for $k\geq0$,++$$+a_{(-k-1)}\mathbf1=\frac{T^ka}{k!},+\qquad a_{(n)}\mathbf1=0\quad(n\geq0).+$$++In particular, $a_{(-1)}\mathbf1=a$ and $a_{(-2)}\mathbf1=Ta$.+:::++::: proof+The creation axiom gives $G(z):=Y(a,z)\mathbf1\in V[[z]]$ with $G(0)=a$. Using $T\mathbf1=0$,++$$+\partial_zG(z)=[T,Y(a,z)]\mathbf1=TG(z).+$$++Coefficient recursion gives $G(z)=\sum_{k\geq0}z^kT^ka/k!$. Comparing modes proves the remaining identities.+:::++Thus inserting a field on the vacuum produces the corresponding translated state.++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}+For a formal parameter $x$,++$$+e^{xT}Y(a,z)e^{-xT}=Y(a,z+x).+$$++The right-hand side means its Taylor expansion in nonnegative powers of $x$:++$$+Y(a,z+x):=\sum_{k\geq0}\frac{x^k}{k!}\partial_z^kY(a,z).+$$+:::++::: proof+Set++$$+F(x):=e^{xT}Y(a,z)e^{-xT}+\in\operatorname{End}(V)[[z,z^{-1}]][[x]].+$$++Translation covariance gives++$$+\partial_xF(x)+=e^{xT}[T,Y(a,z)]e^{-xT}+=\partial_zF(x).+$$++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}+For all $a,b\in V$,++$$+Y(a,z)b=e^{zT}Y(b,-z)a.+$$+:::++::: proof+Choose $N\geq0$ so that locality holds for $Y(a,z)$ and $Y(b,w)$ and, by lower truncation,++$$+u^NY(b,u)a\in V[[u]].+$$++Applying locality to the vacuum gives++$$+\begin{aligned}+(z-w)^NY(a,z)Y(b,w)\mathbf1+&=(z-w)^NY(b,w)Y(a,z)\mathbf1.+\end{aligned}+$$++Using @prop:creation on both sides, we obtain++$$+(z-w)^NY(a,z)e^{wT}b+=(z-w)^NY(b,w)e^{zT}a.+$$++By @prop:field-translation, moving $e^{zT}$ to the left replaces $Y(b,w)$ by $Y(b,w-z)$, initially expanded in nonnegative powers of $z$. Our choice of $N$ removes its poles:++$$+(z-w)^NY(a,z)e^{wT}b+=(z-w)^Ne^{zT}Y(b,w-z)a.+$$++Both sides have only nonnegative powers of $w$. Taking the constant term in $w$ is therefore legitimate and yields++$$+\begin{aligned}+z^NY(a,z)b+&=z^Ne^{zT}Y(b,-z)a.+\end{aligned}+$$++Canceling $z^N$ in $V((z))$ proves the identity.+:::++### Translation of states and differentiation of fields++We now use skew-symmetry to identify the field corresponding to $Ta$.++::: proposition Translation and derivatives {#prop:translation-derivative}+For $a\in V$ and $n\in\mathbb Z$,++$$+Y(Ta,z)=\partial_zY(a,z),+\qquad+(Ta)_{(n)}=[T,a_{(n)}]=-n\,a_{(n-1)}.+$$+:::++::: proof+Fix $b\in V$. By @prop:skew-symmetry and translation covariance,++$$+\begin{aligned}+\partial_zY(a,z)b+&=e^{zT}\bigl(TY(b,-z)-[T,Y(b,-z)]\bigr)a\\+&=e^{zT}Y(b,-z)Ta\\+&=Y(Ta,z)b.+\end{aligned}+$$++The minus sign comes from $\partial_zY(b,-z)=-[T,Y(b,-z)]$; the last equality uses skew-symmetry again. Comparing modes gives $(Ta)_{(n)}=-n a_{(n-1)}$, and translation covariance gives the same formula for $[T,a_{(n)}]$.+:::++## Products and iterates++### Locality and a common expansion++::: proposition Common expansion of two products {#prop:common-expansion}+For $a,b,c\in V$, there is a unique++$$+F(z,w)\in V[[z,w]][z^{-1},w^{-1},(z-w)^{-1}]+$$++such that++$$+\begin{aligned}+Y(a,z)Y(b,w)c&=\iota_{z,w}F(z,w),\\+Y(b,w)Y(a,z)c&=\iota_{w,z}F(z,w).+\end{aligned}+$$+:::++::: proof+The two products lie in $V((z))((w))$ and $V((w))((z))$, respectively. Locality makes them equal after multiplication by $(z-w)^N$. Their common value belongs to++$$+V((z))((w))\cap V((w))((z))+=V[[z,w]][z^{-1},w^{-1}].+$$++Write this common value as $z^{-r}w^{-s}P(z,w)$, with $r,s\geq0$ and $P\in V[[z,w]]$, and set++$$+F(z,w)=\frac{P(z,w)}{z^rw^s(z-w)^N}.+$$++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}+For the same $F$ as in @prop:common-expansion,++$$+Y(Y(a,x)b,w)c=\iota_{w,x}F(w+x,w).+$$++The iterate is understood coefficientwise and belongs to $V((w))((x))$.+:::++::: proof+We consider++$$+\begin{aligned}+\iota_{x,w}\!\left(e^{-wT}F(x+w,w)\right)+&=e^{-wT}Y(a,x+w)Y(b,w)c\\+&=Y(a,x)e^{-wT}Y(b,w)c\\+&=Y(a,x)Y(c,-w)b.+\end{aligned}+$$++The last two equalities use @prop:field-translation and @prop:skew-symmetry. Since $e^{-wT}F(x+w,w)$ belongs to $V[[x,w]][x^{-1},w^{-1},(x+w)^{-1}]$, uniqueness in @prop:common-expansion, applied to $a,c,b$ with variables $x,-w$, identifies it as the common expression for the two product orders. Hence its other expansion is++$$+\iota_{w,x}\!\left(e^{-wT}F(x+w,w)\right)+=Y(c,-w)Y(a,x)b.+$$++Multiplying by $e^{wT}$ and applying skew-symmetry coefficientwise yields++$$+\begin{aligned}+\iota_{w,x}F(x+w,w)+&=e^{wT}Y(c,-w)Y(a,x)b\\+&=Y(Y(a,x)b,w)c.+\end{aligned}+$$++The operators $e^{\pm wT}$ act on vector coefficients and commute with the indicated expansions.+:::++Thus the two products and the iterate are expansions of one expression. Schematically,++$$+Y(a,z)Y(b,w)c\sim Y(b,w)Y(a,z)c\sim Y(Y(a,x)b,w)c,+\qquad z=x+w.+$$++Here $\sim$ means equality of the singular parts at $x=z-w=0$, discarding nonnegative powers of $x$. For the two products, this is understood through their common expression $F(w+x,w)$ expanded in $x$. The common singular part is++$$+\sum_{n\geq0}Y(a_{(n)}b,w)c\,x^{-n-1},+$$++a finite sum by lower truncation. This is the singular operator product expansion (OPE).++### Weak associativity++::: proposition Weak associativity {#prop:weak-associativity}+For $a,b,c\in V$, some integer $L\geq0$ satisfies++$$+\begin{aligned}+&(x+w)^L Y(a,x+w)Y(b,w)c\\+&\qquad=(x+w)^L Y(Y(a,x)b,w)c.+\end{aligned}+$$++On the left, $Y(a,x+w)$ is expanded in nonnegative powers of $w$. After multiplication by the chosen factor, both sides lie in $V[[x,w]][x^{-1},w^{-1}]$.+:::++::: proof+Write the common expression from @thm:iterate-expansion as++$$+F(z,w)=\frac{P(z,w)}{z^rw^s(z-w)^N}.+$$++Take $L=r$. Both sides then expand++$$+\frac{P(x+w,w)}{w^sx^N}.+$$++Its denominator is a monomial, so the two expansion orders agree.+:::++### 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}+For $a,b,c\in V$,++$$+\begin{aligned}+&x^{-1}\delta\!\left(\frac{z-w}{x}\right)Y(a,z)Y(b,w)c\\+&\quad-x^{-1}\delta\!\left(\frac{w-z}{-x}\right)Y(b,w)Y(a,z)c\\+&\qquad=z^{-1}\delta\!\left(\frac{w+x}{z}\right)Y(Y(a,x)b,w)c.+\end{aligned}+$$++The binomials in the three delta kernels are expanded in nonnegative powers of $w$, $z$, and $x$, respectively. Since the identity holds for every $c$, we may omit $c$ and read the operator sums pointwise on $V$.+:::++::: proof+By @thm:iterate-expansion, the three expressions multiplying the delta kernels are the three expansions of++$$+F(z,w)=\frac{P(z,w)}{z^rw^s(z-w)^N},+\qquad P\in V[[z,w]].+$$++Multiply the standard identity++$$+x^{-1}\delta\!\left(\frac{z-w}{x}\right)+-x^{-1}\delta\!\left(\frac{w-z}{-x}\right)+=z^{-1}\delta\!\left(\frac{w+x}{z}\right)+$$++by $P(z,w)/(x^Nz^rw^s)$. This series has a uniform lower bound in each variable, so the products are well-defined. Delta substitution turns the first two terms into the product expansions of $F$ and the third into its iterate expansion, with exactly the conventions stated above. This proves the identity.+:::