Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -1,4 +1,208 @@ # Examples of VOAs++## The Heisenberg VOA++The rank-one Heisenberg VOA $M(1)$ describes the current algebra of one chiral free boson. We construct its states and fields explicitly, then verify its conformal structure.++### The oscillator space++::: definition [Heisenberg vacuum space] {@def:heisenberg-vacuum}+Let++$$+M(1)=\mathbb C[x_1,x_2,\ldots],\qquad \mathbf1=1.+$$++Every element is a polynomial in finitely many variables. Define operators, for $n>0$, by++$$+J_{-n}=x_n\cdot,\qquad J_n=n\frac{\partial}{\partial x_n},\qquad J_0=0.+$$++They satisfy the **Heisenberg relations**++$$+[J_m,J_n]=m\delta_{m+n,0}\operatorname{id},\qquad+J_n\mathbf1=0\quad(n\geq0).+$$++Equivalently, $M(1)$ has basis++$$+J_{-n_1}\cdots J_{-n_s}\mathbf1,+\qquad n_1\geq\cdots\geq n_s\geq1,+$$++including the empty product $\mathbf1$. Set $j=J_{-1}\mathbf1$ and++$$+J(z)=\sum_{n\in\mathbb Z}J_nz^{-n-1}.+$$+:::++The polynomial realization specifies the action of every mode, not just its commutation relations. In particular, $J(z)v\in M(1)((z))$ for every $v\in M(1)$.++### The state–field correspondence++::: proposition [Heisenberg vertex algebra] {@prop:heisenberg-va}+There is a unique vertex algebra structure on $M(1)$ with vacuum $\mathbf1$ and $Y(j,z)=J(z)$. Its fields are++$$+Y(J_{-n_1}\cdots J_{-n_s}\mathbf1,z)+=:\prod_{i=1}^s\frac{\partial_z^{n_i-1}J(z)}{(n_i-1)!}:,+\qquad Y(\mathbf1,z)=\operatorname{id}.+$$++The colons use the right-nested convention of @def:normal-ordering. For these factors, this amounts to moving all creation modes $J_{-n}$ to the left of all annihilation modes $J_n$, $n>0$. The translation operator is++$$+T=\sum_{n\geq1}n x_{n+1}\frac{\partial}{\partial x_n}.+$$+:::++::: proof+The Heisenberg relations give++$$+[J(z),J(w)]=\partial_w\bigl(z^{-1}\delta(w/z)\bigr)\operatorname{id},+\qquad (z-w)^2[J(z),J(w)]=0.+$$++Also $T\mathbf1=0$, $[T,J_n]=-nJ_{n-1}$, and $J(z)\mathbf1\in j+zM(1)[[z]]$. Thus $[T,J(z)]=\partial_zJ(z)$. Derivatives and normal-ordered products preserve mutual locality by Dong's lemma. The displayed fields consequently satisfy locality and translation covariance; their action on the vacuum is regular with constant term $J_{-n_1}\cdots J_{-n_s}\mathbf1$. Lower truncation follows from the oscillator action. These are the vertex algebra axioms. Uniqueness follows from @cor:normal-ordered-products, which forces the displayed formula. See [Keller, §3.2](https://archive.math.arizona.edu/cakeller/VertexOperatorAlgebras.pdf#page=12).+:::++For example,++$$+Y(J_{-2}\mathbf1,z)=\partial_zJ(z),\qquad+Y(J_{-1}^2\mathbf1,z)=:J(z)^2:.+$$++The complete product of the current with its state is++$$+Y(j,z)j=z^{-2}\mathbf1++\sum_{n\geq1}z^{n-1}J_{-n}J_{-1}\mathbf1.+$$++Thus $j_{(1)}j=\mathbf1$ and $j_{(0)}j=0$. This exact series specifies the regular terms as well as the pole.++### The conformal vector++::: proposition [Heisenberg conformal structure] {@prop:heisenberg-voa}+The vector++$$+\omega=\frac12J_{-1}^2\mathbf1+$$++makes $M(1)$ a VOA of CFT type and central charge $1$. Its conformal field and modes are++$$+Y(\omega,z)=\frac12:J(z)^2:+=\sum_{m\in\mathbb Z}L_mz^{-m-2},\qquad+L_m=\frac12\sum_{r\in\mathbb Z}:J_{m-r}J_r:.+$$++Here mode normal ordering puts negative-index modes to the left. Each sum is finite on a fixed state. Moreover,++$$+[L_m,J_n]=-nJ_{m+n},\qquad+L_{-1}=T,\qquad+L_0=\sum_{n>0}J_{-n}J_n.+$$++In particular,++$$+\operatorname{wt}(J_{-n_1}\cdots J_{-n_s}\mathbf1)+=n_1+\cdots+n_s.+$$+:::++::: proof+Using the Heisenberg relations in the quadratic expression for $L_m$ gives++$$+\begin{aligned}+[L_m,J_n]+&=\frac12\sum_r\left(+r\delta_{r+n,0}J_{m-r}++(m-r)\delta_{m-r+n,0}J_r\right)\\+&=-nJ_{m+n}.+\end{aligned}+$$++The same quadratic expression yields++$$+L_{-1}=\sum_{n>0}J_{-n-1}J_n=T,\qquad+L_0=\sum_{n>0}n x_n\frac{\partial}{\partial x_n}.+$$++It remains to check the Virasoro relations, including the central term. By the commutator just proved,++$$+D_{m,n}:=[L_m,L_n]-(m-n)L_{m+n}+$$++commutes with every $J_r$. Any such operator is scalar: commuting with all derivatives forces its value on $\mathbf1$ to be constant, and commuting with all multiplications determines its value on every polynomial. Since $D_{m,n}$ has degree $-m-n$ for the grading above, it vanishes unless $m+n=0$.++For $m>0$, $L_m\mathbf1=L_0\mathbf1=0$ and++$$+\begin{aligned}+L_{-m}\mathbf1+&=\frac12\sum_{r=1}^{m-1}J_{-r}J_{-(m-r)}\mathbf1,\\+L_mL_{-m}\mathbf1+&=\frac12\sum_{r=1}^{m-1}r(m-r)\mathbf1+=\frac{m^3-m}{12}\mathbf1.+\end{aligned}+$$++Consequently,++$$+[L_m,L_n]=(m-n)L_{m+n}++\frac{m^3-m}{12}\delta_{m+n,0}\operatorname{id}.+$$++Finally, the weight-$h$ basis is indexed by partitions of $h$, so every weight space is finite dimensional, negative weights vanish, and $M(1)_0=\mathbb C\mathbf1$. All VOA axioms follow.+:::++### Basic properties++::: proposition [Simplicity] {@prop:heisenberg-simple}+The VOA $M(1)$ is simple.+:::++::: proof+An ideal is stable under every $J_n$, hence under all polynomial multiplications and partial derivatives. Given a nonzero polynomial in the ideal, differentiate according to a monomial of maximal total degree. The result is a nonzero constant. Thus the ideal contains $\mathbf1$ and equals $M(1)$.+:::++The first weight spaces are++$$+\begin{aligned}+M(1)_0&=\mathbb C\mathbf1,\\+M(1)_1&=\mathbb Cj,\\+M(1)_2&=\mathbb C J_{-2}\mathbf1\oplus\mathbb C J_{-1}^2\mathbf1+=\mathbb C Tj\oplus\mathbb C\omega.+\end{aligned}+$$++The weight-one Lie algebra is abelian with $\langle j,j\rangle=1$, yet $j_{(0)}=J_0=0$ on all of $M(1)$. Thus its zero-mode action, and the associated inner automorphism group, are trivial even though its current algebra is nontrivial.++::: exercise [Reflection and composite products] {@ex:heisenberg-properties}+1. Show that $\theta(J_{-n_1}\cdots J_{-n_s}\mathbf1)=(-1)^sJ_{-n_1}\cdots J_{-n_s}\mathbf1$ is a VOA automorphism and that $\operatorname{Aut}(M(1))=\{1,\theta\}$. Identify the fixed-point space $M(1)^\theta$ in the oscillator basis.+2. For $u=J_{-1}^2\mathbf1$, use $Y(u,z)=:J(z)^2:$ to verify++   $$+   u_{(-1)}u=J_{-1}^4\mathbf1+4J_{-3}J_{-1}\mathbf1.+   $$++   Thus a regular VOA product of composite states need not be their ordinary polynomial product.+:::  ## Lattice VOAs