Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -204,12 +204,186 @@    Thus a regular VOA product of composite states need not be their ordinary polynomial product. ::: -## Lattice VOAs--- Construction from an even positive-definite lattice.-- Conformal structure and basic properties.- ## Virasoro minimal models -- Construction and basic properties.-- Relation to suitable coset constructions.+The Heisenberg VOA is generated by a weight-one current. We now construct a VOA generated by its conformal vector alone. At certain central charges, its vacuum descendants satisfy additional relations; imposing these relations gives the Virasoro minimal models.++### The universal vacuum VOA++::: definition [Virasoro vacuum representation] {@def:virasoro-vacuum}+Fix $c\in\mathbb C$. Let $V^c$ be the representation of the Virasoro algebra generated by a vector $\mathbf1$, subject only to the Virasoro relations with central charge $c$ and++$$+L_n\mathbf1=0\qquad(n\geq-1).+$$++By the Poincaré–Birkhoff–Witt theorem, a basis is++$$+L_{-n_1}\cdots L_{-n_s}\mathbf1,+\qquad n_1\geq\cdots\geq n_s\geq2,+$$++including the empty product $\mathbf1$. Set++$$+\omega=L_{-2}\mathbf1,\qquad+\mathcal T(z)=\sum_{n\in\mathbb Z}L_nz^{-n-2}.+$$+:::++The condition $L_{-1}\mathbf1=0$ is essential. A highest-weight vector of weight zero only satisfies $L_n\mathbf1=0$ for $n\geq0$; the vacuum additionally has zero translation. Thus $V^c$ is the weight-zero Verma module with the submodule generated by $L_{-1}\mathbf1$ removed.++::: proposition [Universal Virasoro VOA] {@prop:universal-virasoro}+The space $V^c$ has a unique VOA structure with conformal vector $\omega$ and $Y(\omega,z)=\mathcal T(z)$. It is of CFT type, has central charge $c$, and satisfies++$$+\begin{aligned}+Y(L_{-n_1}\cdots L_{-n_s}\mathbf1,z)+&=:\prod_{i=1}^s+\frac{\partial_z^{n_i-2}\mathcal T(z)}{(n_i-2)!}:,\\+\operatorname{wt}(L_{-n_1}\cdots L_{-n_s}\mathbf1)+&=n_1+\cdots+n_s.+\end{aligned}+$$++The colons again mean right nesting. For any VOA $W$ of central charge $c$, there is a unique VOA homomorphism++$$+V^c\longrightarrow W,\qquad \mathbf1\longmapsto\mathbf1_W,+\quad \omega\longmapsto\omega_W.+$$++Its image is the vertex subalgebra generated by $\omega_W$.+:::++::: proof+The Virasoro relations give $(z-w)^4[\mathcal T(z),\mathcal T(w)]=0$ and $[L_{-1},\mathcal T(z)]=\partial_z\mathcal T(z)$. Also $\mathcal T(z)\mathbf1\in\omega+zV^c[[z]]$, and the coefficients of iterated fields on $\mathbf1$ span $V^c$. The reconstruction theorem therefore gives the vertex algebra with translation $T=L_{-1}$. The normal-ordering formula follows from @cor:normal-ordered-products, using $L_{-n}=\omega_{(-n+1)}$. The Virasoro relations and the finite-dimensional PBW grading verify the conformal axioms. The map to $W$ sends each vacuum descendant to the same descendant of $\mathbf1_W$; the field formula shows that it preserves $Y$. See [Keller, §§3.2 and 5.2](https://archive.math.arizona.edu/cakeller/VertexOperatorAlgebras.pdf).+:::++### Singular vectors and the simple quotient++::: definition [Virasoro singular vector] {@def:virasoro-singular}+A nonzero homogeneous vector $\chi\in V^c$ of positive weight is **singular** if++$$+L_n\chi=0\qquad(n>0).+$$+:::++Such a vector generates a proper Virasoro submodule: its descendants have weight at least $\operatorname{wt}(\chi)$, so they cannot contain the vacuum. Since $V^c$ is generated by $\omega$, this submodule is also a VOA ideal. In the quotient, the state $\chi$ and its entire field $Y(\chi,z)$ vanish.++The following representation-theoretic result supplies the central charges and the defining relation.++::: theorem [Virasoro vacuum singular-vector theorem] {@thm:virasoro-minimal-ideal}+The VOA $V^c$ is not simple precisely when++$$+c=c_{p,q}:=1-\frac{6(p-q)^2}{pq},+\qquad 2\leq p<q,\quad \gcd(p,q)=1.+$$++At this central charge, its unique maximal proper ideal is generated by a singular vector $\chi_{p,q}$ of weight++$$+N_{p,q}=(p-1)(q-1).+$$++This is the first positive weight at which $V^{c_{p,q}}$ has a singular vector, and $\chi_{p,q}$ is unique up to nonzero scalar.+:::++We use this standard theorem without proof; see [Ridout–Wood, §4](https://arxiv.org/pdf/1409.4847#page=10). The general result uses the structure of Virasoro highest-weight representations. Once $p,q$ are fixed, the vector itself can be found by a finite calculation: write a linear combination of the PBW basis at weight $N_{p,q}$ and solve $L_1\chi=L_2\chi=0$. These two equations imply $L_n\chi=0$ for every $n>0$, since $L_1,L_2$ generate the positive Virasoro modes.++::: definition [Virasoro minimal model] {@def:virasoro-minimal-model}+For coprime $2\leq p<q$, the **Virasoro minimal-model VOA** is++$$+\operatorname{Vir}_{p,q}+:=V^{c_{p,q}}/(\chi_{p,q}),+$$++where $(\chi_{p,q})$ is the VOA ideal generated by $\chi_{p,q}$. The quotient vacuum and conformal vector retain the notation $\mathbf1,\omega$. This simple VOA is also denoted $L(c_{p,q},0)$.+:::++Thus specifying $c_{p,q}$ in the Virasoro commutation relations constructs $V^{c_{p,q}}$; passing to the minimal model additionally imposes the null-field relation $Y(\chi_{p,q},z)=0$.++### A defining relation at weight four++::: example [The Yang–Lee minimal model] {@ex:yang-lee-voa}+For $(p,q)=(2,5)$,++$$+c_{2,5}=-\frac{22}{5},\qquad N_{2,5}=4.+$$++The weight-four space of $V^c$ has basis $L_{-2}^2\mathbf1,L_{-4}\mathbf1$. Take++$$+\chi=(L_{-2}^2+\alpha L_{-4})\mathbf1.+$$++The Virasoro relations give++$$+\begin{aligned}+L_1\chi&=(3+5\alpha)L_{-3}\mathbf1,\\+L_2\chi&=(8+c+6\alpha)L_{-2}\mathbf1.+\end{aligned}+$$++The first equation forces $\alpha=-3/5$; the second then forces $c=-22/5$. Hence++$$+\chi_{2,5}=\left(L_{-2}^2-\frac35L_{-4}\right)\mathbf1,+\qquad+\operatorname{Vir}_{2,5}=V^{-22/5}/(\chi_{2,5}).+$$++The state–field formula converts this relation into the exact field identity++$$+:\mathcal T(z)^2:\;=\frac{3}{10}\partial_z^2\mathcal T(z).+$$++Indeed, $Y(L_{-2}^2\mathbf1,z)=:\mathcal T(z)^2:$ and $Y(L_{-4}\mathbf1,z)=\tfrac12\partial_z^2\mathcal T(z)$. This equality holds in the quotient; its difference is a nonzero field in the universal VOA.+:::++### Basic properties and further examples++::: proposition [Vacuum states of the minimal models] {@prop:minimal-model-vacuum}+Every $\operatorname{Vir}_{p,q}$ is simple, of CFT type, and generated by its conformal vector. The quotient map is an isomorphism on weight spaces below $N_{p,q}$; at weight $N_{p,q}$ its kernel is $\mathbb C\chi_{p,q}$.++Except for $(p,q)=(2,3)$,++$$+(\operatorname{Vir}_{p,q})_1=0,\qquad+(\operatorname{Vir}_{p,q})_2=\mathbb C\omega.+$$++For $(p,q)=(2,3)$, $c=0$, $\chi_{2,3}$ is proportional to $\omega$, and the quotient is the one-dimensional VOA $\mathbb C\mathbf1$.+:::++::: proof+Simplicity follows from @thm:virasoro-minimal-ideal. The ideal is spanned by negative-mode descendants of $\chi_{p,q}$, so it has no states below weight $N_{p,q}$ and has a one-dimensional component at that weight. Apply this to the PBW grading, whose first spaces are $V^c_0=\mathbb C\mathbf1$, $V^c_1=0$, and $V^c_2=\mathbb C\omega$. If $N_{p,q}=2$, the ideal contains the generator $\omega$, leaving only the vacuum.+:::++Some small examples are:++| $(p,q)$ | Central charge | First vacuum null weight | Model |+| --- | --- | --- | --- |+| $(2,5)$ | $-22/5$ | $4$ | Yang–Lee |+| $(3,4)$ | $1/2$ | $6$ | Ising |+| $(4,5)$ | $7/10$ | $12$ | Tricritical Ising |++These VOAs have no weight-one currents. Their nontrivial conformal structure begins in weight two.++::: exercise [The Ising vacuum relation] {@ex:ising-null-vector}+At $c=1/2$, write a weight-six vector as++$$+\chi=A L_{-2}^3\mathbf1+B L_{-3}^2\mathbf1++C L_{-4}L_{-2}\mathbf1+D L_{-6}\mathbf1.+$$++Solve $L_1\chi=L_2\chi=0$ for a nonzero vector, up to scale. Translate the resulting relation into an identity between right-nested normal-ordered products and derivatives of $\mathcal T(z)$.+:::