Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -271,6 +271,89 @@ Together with antisymmetry, this is the Jacobi identity. ::: +::: proposition [The invariant form on weight one] {@prop:voa-weight-one-form}+Let $V$ be of CFT type. The bilinear form on $V_1$ defined by++$$+a_{(1)}b=\langle a,b\rangle\mathbf1+$$++is symmetric and invariant under the Lie bracket:++$$+\langle a,b\rangle=\langle b,a\rangle,\qquad+\langle[a,b],d\rangle=\langle a,[b,d]\rangle+\quad(a,b,d\in V_1).+$$+:::++::: proof+Weight compatibility gives $a_{(1)}b\in V_0=\mathbb C\mathbf1$, and the $x^{-2}$ coefficient of skew-symmetry gives $a_{(1)}b=b_{(1)}a$. The zero-mode commutator identity, obtained from @prop:jacobi-identity, is++$$+[a_{(0)},Y(b,x)]=Y(a_{(0)}b,x).+$$++Applying it to $d$ and comparing coefficients of $x^{-2}$ gives++$$+0=a_{(0)}(b_{(1)}d)+=\bigl(\langle[a,b],d\rangle+\langle b,[a,d]\rangle\bigr)\mathbf1.+$$++Using this identity with $a,b$ interchanged proves invariance.+:::++The form may be degenerate; the term **quadratic Lie algebra** is usually reserved for a Lie algebra with a nondegenerate invariant symmetric bilinear form. Its normalization here is fixed by $Y$ and the vacuum. See [Lam, *Vertex operator algebras and finite groups*](https://ems.press/content/serial-article-files/46563).++## The weight-one current algebra++::: proposition [Current algebra] {@prop:voa-weight-one-currents}+Let $V$ be of CFT type. For $a,b\in V_1$,++$$+Y(a,x)b-\frac{\langle a,b\rangle\mathbf1}{x^2}+-\frac{[a,b]}x\in V[[x]].+$$++The corresponding operators on $V$ satisfy++$$+[a_{(m)},b_{(n)}]+=[a,b]_{(m+n)}++m\langle a,b\rangle\delta_{m+n,0}\operatorname{id}_V+\qquad(m,n\in\mathbb Z).+$$+:::++::: proof+The state $a_{(j)}b$ has weight $1-j$, so it vanishes for $j\ge2$. This proves the first assertion. Taking coefficients in the Jacobi identity gives++$$+[a_{(m)},b_{(n)}]+=\sum_{j\ge0}\binom mj(a_{(j)}b)_{(m+n-j)}.+$$++Only $j=0,1$ remain. Substitute $a_{(0)}b=[a,b]$, $a_{(1)}b=\langle a,b\rangle\mathbf1$, and $Y(\mathbf1,x)=\operatorname{id}_V$.+:::++Thus the fields $Y(a,x)$, $a\in\mathfrak g:=V_1$, give $V$ a representation of the **affine current algebra**++$$+\widehat{\mathfrak g}+=(\mathfrak g\otimes\mathbb C[t,t^{-1}])\oplus\mathbb C K,+$$++with $K$ central and bracket++$$+[a\otimes t^m,b\otimes t^n]+=[a,b]\otimes t^{m+n}++m\langle a,b\rangle\delta_{m+n,0}K.+$$++The action is $a\otimes t^m\mapsto a_{(m)}$ and $K\mapsto\operatorname{id}_V$. In particular, the zero modes give an action of $\mathfrak g$ on each graded piece of $V$. Their exponentials will give VOA automorphisms in @prop:voa-inner-automorphisms.+ ## References  - Christoph A. Keller, [Introduction to Vertex Operator Algebras](https://archive.math.arizona.edu/cakeller/VertexOperatorAlgebras.pdf), §5.