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--- Base version+++ Proposed version@@ -1,18 +1,21 @@ # About this note  - Scope: foundations of vertex operator algebras and their applications.-- Prerequisites and conventions: formal calculus; compact notation; purely even vertex algebras over $\mathbb C$.-- Roadmap: VA axioms and properties → conformal structure → chiral correlation functions → modules → tensor categories → extensions and quantum Galois theory.-- Formal identities and their analytic interpretation.---# Vertex algebras: definitions and properties+- Prerequisites: formal calculus.+- Roadmap: VA axioms and properties → conformal structure → tensor products, orbifolds, and cosets → Heisenberg VOA and Virasoro minimal models → modules → tensor categories → extensions and quantum Galois theory.++The material on vertex operator superalgebras, the Majorana fermion, and Pfaffian amplitudes is developed separately in [Vertex Operator Superalgebras and the Majorana Fermion](https://peerpedia.dev/papers/3e440f74-ebe3-4097-a243-7b6bf74f1c87/), by Chenqi Meng.+++# Vertex algebras  ## 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}+For endomorphisms $A,B\in\operatorname{End}(V)$, the bracket $[A,B]=AB-BA$ denotes their commutator; for fields it is interpreted coefficientwise.++::: 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)  $$@@ -45,7 +48,7 @@    $$ ::: -::: 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  $$@@ -61,7 +64,7 @@  ### Creation and translation identities -::: proposition Creation from the vacuum {#prop:creation}+::: proposition [Creation from the vacuum] {@prop:creation} For every $a\in V$,  $$@@ -92,7 +95,7 @@  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 $x$,  $$@@ -125,9 +128,9 @@ Writing $F(x)=\sum_{k\geq0}F_kx^k$ yields $(k+1)F_{k+1}=\partial_zF_k$ with $F_0=Y(a,z)$, hence the Taylor expansion above. ::: -### Skew-symmetry {#sec:skew-symmetry}--::: proposition Skew-symmetry {#prop:skew-symmetry}+### Skew-symmetry {@sec:skew-symmetry}++::: proposition [Skew-symmetry] {@prop:skew-symmetry} For all $a,b\in V$,  $$@@ -181,7 +184,7 @@  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 $n\in\mathbb Z$,  $$@@ -210,7 +213,7 @@  ### Locality and a common expansion -::: proposition Common expansion of two products {#prop:common-expansion}+::: proposition [Common expansion of two products] {@prop:common-expansion} For $a,b,c\in V$, there is a unique  $$@@ -244,7 +247,7 @@ 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,  $$@@ -303,7 +306,7 @@  ### Weak associativity -::: proposition Weak associativity {#prop:weak-associativity}+::: proposition [Weak associativity] {@prop:weak-associativity} For $a,b,c\in V$, some integer $L\geq0$ satisfies  $$@@ -332,11 +335,11 @@ 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 $a,b,c\in V$,  $$@@ -369,23 +372,1441 @@ by $P(z,w)/(x^Nz^rw^s)$. This series has a uniform lower bound in each variable, so the products are well-defined. Delta substitution turns the first two terms into the product expansions of $F$ and the third into its iterate expansion, with exactly the conventions stated above. This proves the identity. ::: +## Fields of products++::: proposition [Iterate formula] {@prop:field-iterate}+For $a,b,c\in V$ and $m\in\mathbb Z$,++$$+\begin{aligned}+Y(a_{(m)}b,z)c+=\operatorname{Res}_u\Bigl(&\iota_{u,z}(u-z)^m+Y(a,u)Y(b,z)c\\+&-\iota_{z,u}(u-z)^m+Y(b,z)Y(a,u)c\Bigr).+\end{aligned}+$$++The first kernel is expanded in nonnegative powers of $z$, and the second in nonnegative powers of $u$.+:::++::: proof+Apply @prop:jacobi-identity with field coordinates $u,z$ and relative coordinate $x$. Multiply by $x^m$ and take $\operatorname{Res}_x\operatorname{Res}_u$. For the iterate term, the $u$-residue of $u^{-1}\delta((z+x)/u)$ is $1$, leaving++$$+\begin{aligned}+\operatorname{Res}_x x^mY(Y(a,x)b,z)c+&=\operatorname{Res}_x x^m+\sum_{j\in\mathbb Z}Y(a_{(j)}b,z)c\,x^{-j-1}\\+&=Y(a_{(m)}b,z)c.+\end{aligned}+$$++For the two product terms,++$$+\begin{aligned}+\operatorname{Res}_x x^{m-1}\delta\!\left(\frac{u-z}{x}\right)+&=\iota_{u,z}(u-z)^m,\\+\operatorname{Res}_x x^{m-1}\delta\!\left(\frac{z-u}{-x}\right)+&=\iota_{z,u}(u-z)^m.+\end{aligned}+$$++Taking the remaining $u$-residue proves the formula.+:::++For $m\ge0$, the kernel is a polynomial, so its two expansions agree and the formula simplifies to++$$+Y(a_{(m)}b,z)+=\operatorname{Res}_u (u-z)^m[Y(a,u),Y(b,z)].+$$++These are the singular OPE coefficients. Negative products are expressed by normal ordering as follows.++::: definition [Normal ordering] {@def:normal-ordering}+For a field $A(z)=\sum_j A_{(j)}z^{-j-1}$, write++$$+A_+(z)=\sum_{j<0}A_{(j)}z^{-j-1},\qquad+A_-(z)=\sum_{j\ge0}A_{(j)}z^{-j-1}.+$$++Here $A_+$ is the creation part and $A_-$ the annihilation part. The **normal-ordered product** with a field $B(z)$ is++$$+:A(z)B(z):=A_+(z)B(z)+B(z)A_-(z).+$$++For several fields, **multiple normal ordering** means right nesting:++$$+:A_1(z)\cdots A_k(z):+:=:A_1(z)\bigl(:A_2(z)\cdots A_k(z):\bigr):,+\qquad :A_1(z):=A_1(z).+$$++Thus $:ABC:=:A(:BC:):$. This is a choice of parentheses, not an associativity identity; in general $:A(:BC:):\ne:(:AB:)C:$.+:::++::: corollary [Fields of negative products] {@cor:normal-ordered-products}+For $a,b\in V$ and $k\ge0$,++$$+Y(a_{(-k-1)}b,z)+=\frac1{k!}:\bigl(\partial_z^kY(a,z)\bigr)Y(b,z):.+$$++In particular, $Y(a_{(-1)}b,z)=:Y(a,z)Y(b,z):$.+:::++::: proof+For $m=-1$, the kernels in @prop:field-iterate are++$$+\iota_{u,z}(u-z)^{-1}=\sum_{j\ge0}z^ju^{-j-1},\qquad+\iota_{z,u}(u-z)^{-1}=-\sum_{j\ge0}u^jz^{-j-1}.+$$++Their residues give, respectively, the creation part of $Y(a,z)$ on the left and the annihilation part on the right:++$$+Y(a_{(-1)}b,z)+=Y(a,z)_+Y(b,z)+Y(b,z)Y(a,z)_-.+$$++Translation gives $Y(T^ka,z)=\partial_z^kY(a,z)$ and, on comparing the coefficient of $z^0$, $(T^ka)_{(-1)}=k!a_{(-k-1)}$. Apply the case $k=0$ to $T^ka,b$ and divide by $k!$.+:::++With the right-nesting convention, repeated application gives, for $k_1,\ldots,k_s\ge0$,++$$+Y\bigl(a^1_{(-k_1-1)}\cdots a^s_{(-k_s-1)}b,z\bigr)+=\frac{:\bigl(\partial_z^{k_1}Y(a^1,z)\bigr)\cdots+\bigl(\partial_z^{k_s}Y(a^s,z)\bigr)Y(b,z):}+{k_1!\cdots k_s!}.+$$++Thus fields of states built from negative products are nested normal-ordered derivatives of the original fields. The general iterate formula also handles nonnegative products through their singular OPE coefficients. These identities hold in any vertex algebra, without a conformal vector; see [Nozaradan, §4.3.3](https://arxiv.org/pdf/0809.1380).+  # Vertex operator algebras -- Conformal vector and Virasoro modes.-- Virasoro relations and compatibility with translation.-- Conformal weights and grading restrictions.-- Additional CFT-type assumptions.-- Examples of VOAs.---# Chiral correlation functions--- Matrix coefficients of products and iterates.-- Formal expansions and analytic convergence.-- OPE data and Ward identities.-- Explicit genus-zero examples.-- Reconstruction statements and their hypotheses.+Although vertex algebra contains many familiar properties a chiral CFT constains, it does not contain the most important structure: the energy-momentum tensor and Virasoro algebra.+A vertex operator algebra (VOA) is a vertex algebra with so-called conformal structure, which is essentially the energy-momentum tensor and Virasoro algebra. We now define VOA.++## Definition of VOA+++::: definition [Vertex operator algebra]{@definition:1ff90b75}+Let $(V,Y,\mathbf1,T)$ be a vertex algebra. A **vertex operator algebra** (VOA) structure on $V$ is specified by a **conformal vector** $\omega\in V$ whose field++$$+Y(\omega,z)=\sum_{n\in\mathbb Z}L_nz^{-n-2},+\qquad L_n=\omega_{(n+1)},+$$++satisfies the following conditions.++1. **Virasoro relations.** For some $c\in\mathbb C$, called the **central charge**,++   $$+   [L_m,L_n]=(m-n)L_{m+n}+   +\frac{c}{12}(m^3-m)\delta_{m+n,0}\operatorname{id}_V.+   $$++2. **Translation compatibility.** $L_{-1}=T$.++3. **Conformal grading.** $L_0$ acts diagonally with integral eigenvalues,++   $$+   V=\bigoplus_{h\in\mathbb Z}V_h,+   \qquad V_h=\{a\in V:L_0a=ha\},+   $$++   where $\dim V_h<\infty$ and $V_h=0$ for sufficiently negative $h$. A state in $V_h$ has **conformal weight** $h$.++We denote the VOA by $(V,Y,\mathbf1,\omega)$. The VOA is called **CFT type** if they satisfy additional conditions $V_h=0$ for $h<0$ and $V_0=\mathbb C\mathbf1$.+:::++## Vacuum and conformal vector++::: proposition [Vacuum and conformal weights] {@prop:voa-vacuum}+The vacuum and conformal vector satisfy++$$+L_n\mathbf1=0\quad(n\geq-1),+\qquad \omega=L_{-2}\mathbf1,+\qquad \mathbf1\in V_0,\quad\omega\in V_2.+$$+:::++::: proof+Apply @prop:creation to $\omega$ and compare coefficients in $Y(\omega,z)\mathbf1=e^{zT}\omega$. Then $L_0\mathbf1=0$ and++$$+L_0\omega=[L_0,L_{-2}]\mathbf1=2\omega.+$$+:::++## Virasoro action on fields++::: proposition [Virasoro covariance] {@prop:virasoro-covariance}+For $a\in V$ and $m\in\mathbb Z$,++$$+[L_m,Y(a,z)]+=\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}Y(L_{j-1}a,z).+$$++The sum is finite by lower truncation. In particular,++$$+\begin{aligned}+[L_{-1},Y(a,z)]&=\partial_zY(a,z),\\+[L_0,Y(a,z)]&=z\partial_zY(a,z)+Y(L_0a,z),\\+[L_1,Y(a,z)]&=z^2\partial_zY(a,z)+2zY(L_0a,z)+Y(L_1a,z).+\end{aligned}+$$+:::++::: proof+Use @prop:jacobi-identity with first state $\omega$ and field coordinates $u,z$. Taking $\operatorname{Res}_x$ turns each of the first two delta kernels into $1$. Multiplying by $u^{m+1}$ and taking $\operatorname{Res}_u$ therefore gives++$$+\operatorname{Res}_u u^{m+1}[Y(\omega,u),Y(a,z)]+=[L_m,Y(a,z)].+$$++On the right-hand side, the $u$-residue of the delta kernel is++$$+\operatorname{Res}_u u^{m+1}u^{-1}+\delta\!\left(\frac{z+x}{u}\right)+=\iota_{z,x}(z+x)^{m+1}+=\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}x^j.+$$++Thus the remaining $x$-residue is++$$+\begin{aligned}+[L_m,Y(a,z)]+&=\operatorname{Res}_x\left(+\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}x^j\right)+\left(\sum_{n\in\mathbb Z}Y(\omega_{(n)}a,z)x^{-n-1}\right)\\+&=\sum_{j\geq0}\binom{m+1}{j}z^{m+1-j}Y(\omega_{(j)}a,z).+\end{aligned}+$$++The residue selects $n=j$. Since $\omega_{(j)}=L_{j-1}$, this is the claimed formula; lower truncation makes the sum finite even for $m<-1$. The three special cases use $Y(L_{-1}a,z)=\partial_zY(a,z)$.+:::++::: remark [Global conformal transformations] {@rem:global-conformal}+The modes $L_{-1},L_0,L_1$ span an $\mathfrak{sl}_2$ subalgebra with no central term. Their actions on fields describe infinitesimal translations, dilations, and special conformal transformations, respectively. The corresponding coordinate flows are++$$+z\longmapsto z+t,\qquad z\longmapsto e^t z,+\qquad z\longmapsto\frac{z}{1-tz}.+$$++Together they generate the Möbius transformations, the global holomorphic conformal transformations of the Riemann sphere. For $a\in V_h$ with $L_1a=0$ (a quasiprimary state), the three commutators act on $Y(a,z)$ as++$$+\partial_z,\qquad z\partial_z+h,\qquad z^2\partial_z+2hz.+$$++The terms involving $h$ account for the conformal weight of the field. For a general homogeneous state, the additional term $Y(L_1a,z)$ describes mixing with another field. See [Keller, §7.3](https://archive.math.arizona.edu/cakeller/VertexOperatorAlgebras.pdf#page=25).+:::++## Weights and dilations++::: proposition [Weight compatibility] {@prop:voa-weights}+For $a\in V_h$, $b\in V_k$, and $n\in\mathbb Z$,++$$+[L_0,a_{(n)}]=(h-n-1)a_{(n)},+\qquad a_{(n)}b\in V_{h+k-n-1}.+$$++In particular, $T V_h\subseteq V_{h+1}$ and $L_nV_h\subseteq V_{h-n}$. For $q\in\mathbb C^\times$,++$$+q^{L_0}Y(a,z)q^{-L_0}=q^hY(a,qz).+$$+:::++::: proof+Compare coefficients in $[L_0,Y(a,z)]=(z\partial_z+h)Y(a,z)$. Applying the resulting commutator to $b$ gives the weight of $a_{(n)}b$. The statements for $T=L_{-1}$ and $L_n$ follow from the Virasoro relations. Conjugation by $q^{L_0}$ multiplies $a_{(n)}$ by $q^{h-n-1}$, proving the last identity.+:::++Thus $L_0$ generates dilations, and conformal weights determine the powers allowed in each OPE. If $V_\ell=0$ for $\ell<B$, then $a_{(n)}b=0$ whenever $n>h+k-B-1$.++## Products involving the conformal vector++::: proposition [Products of the conformal vector] {@prop:stress-tensor-ope}+The nonnegative products of $\omega$ with itself are++$$+\omega_{(0)}\omega=T\omega,\qquad+\omega_{(1)}\omega=2\omega,\qquad+\omega_{(2)}\omega=0,\qquad+\omega_{(3)}\omega=\frac c2\mathbf1,+$$++and $\omega_{(n)}\omega=0$ for $n\geq4$. Equivalently, the exact identity in $V((x))$ is++$$+Y(\omega,x)\omega+=\frac{c\mathbf1}{2x^4}+\frac{2\omega}{x^2}++\frac{T\omega}{x}++\sum_{k\geq0}L_{-k-2}\omega\,x^k.+$$+:::++::: proof+For $n\geq0$, use $\omega=L_{-2}\mathbf1$ and $L_n\mathbf1=0$ to obtain++$$+\begin{aligned}+L_n\omega+&=[L_n,L_{-2}]\mathbf1\\+&=(n+2)L_{n-2}\mathbf1++\frac{c}{12}(n^3-n)\delta_{n,2}\mathbf1.+\end{aligned}+$$++For $n=0,1,2$, this gives $2\omega,0,c\mathbf1/2$, respectively; for $n\geq3$ it vanishes. Together with $L_{-1}\omega=T\omega$ and $\omega_{(j)}=L_{j-1}$, these are the claimed products. Expanding $Y(\omega,x)\omega=\sum_nL_n\omega\,x^{-n-2}$ gives the full identity, including its regular terms.+:::++Applying the state-field map coefficientwise gives++$$+\begin{aligned}+Y(Y(\omega,x)\omega,w)+&=\frac{c\,\operatorname{id}_V}{2x^4}++\frac{2Y(\omega,w)}{x^2}++\frac{\partial_wY(\omega,w)}{x}\\+&\quad+\sum_{k\geq0}Y(L_{-k-2}\omega,w)x^k.+\end{aligned}+$$++After acting on any $b\in V$, @thm:iterate-expansion identifies this with the expansion at $x=0$ of the common expression for the two products of $Y(\omega,u)$ and $Y(\omega,w)$, with $u=w+x$. This is the precise expansion underlying the stress-tensor OPE.++::: definition [Primary and quasiprimary states] {@def:primary-state}+A homogeneous state $a\in V_h$ is **primary** if $L_na=0$ for all $n>0$, and **quasiprimary** if $L_1a=0$.+:::++::: proposition [Primary-field covariance] {@prop:primary-covariance}+If $a\in V_h$ is primary, then++$$+Y(\omega,x)a+=\frac{ha}{x^2}+\frac{Ta}{x}++\sum_{k\geq0}L_{-k-2}a\,x^k,+$$++and, for every $m\in\mathbb Z$,++$$+[L_m,Y(a,z)]+=\bigl(z^{m+1}\partial_z+(m+1)hz^m\bigr)Y(a,z).+$$++For quasiprimary $a$, the latter identity still holds for $m=-1,0,1$.+:::++::: proof+In $Y(\omega,x)a=\sum_nL_na\,x^{-n-2}$, the primary-state conditions leave only $L_0a=ha$ and $L_{-1}a=Ta$ among the singular terms. The same conditions in @prop:virasoro-covariance give the commutator. For $m=-1,0,1$, only $L_{-1}a,L_0a,L_1a$ can contribute.+:::++The three modes $L_{-1},L_0,L_1$ describe infinitesimal translations, dilations, and special conformal transformations. The conformal vector itself is quasiprimary since $L_1\omega=0$; it is not primary when $c\ne0$, since $L_2\omega=c\mathbf1/2$.++## The weight-one Lie algebra++::: proposition [The weight-one Lie algebra] {@prop:voa-weight-one}+If $V$ is of CFT type, then $V_1$ is a Lie algebra with bracket++$$+[a,b]:=a_{(0)}b,\qquad a,b\in V_1.+$$+:::++::: proof+The bracket is bilinear, and weight compatibility gives $a_{(0)}b\in V_1$. Taking the $z^{-1}$ coefficient of skew-symmetry gives++$$+a_{(0)}b=-b_{(0)}a+T(b_{(1)}a).+$$++Higher terms vanish by nonnegative weights. Since $b_{(1)}a\in V_0=\mathbb C\mathbf1$ and $T\mathbf1=0$, the bracket is antisymmetric.++Now we prove the Jacobi identity holds for the Lie algebra. Taking $\operatorname{Res}_x\operatorname{Res}_z\operatorname{Res}_w$ in @prop:jacobi-identity gives++$$+[a_{(0)},b_{(0)}]=(a_{(0)}b)_{(0)},+$$++where the bracket on the left is the commutator of endomorphisms. Applying this identity to $d\in V_1$ yields++$$+a_{(0)}(b_{(0)}d)-b_{(0)}(a_{(0)}d)+=(a_{(0)}b)_{(0)}d,+$$++or, in terms of the bracket on states,++$$+[a,[b,d]]-[b,[a,d]]=[[a,b],d].+$$++Together with antisymmetry, this is the Jacobi identity.+:::++::: proposition [The invariant form on weight one] {@prop:voa-weight-one-form}+Let $V$ be of CFT type. The bilinear form on $V_1$ defined by++$$+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.+$$++The action is $a\otimes t^m\mapsto a_{(m)}$ and $K\mapsto\operatorname{id}_V$. In particular, the zero modes give an action of $\mathfrak g$ on each graded piece of $V$. Their exponentials will give VOA automorphisms in @prop:voa-inner-automorphisms.++## References++- Christoph A. Keller, [Introduction to Vertex Operator Algebras](https://archive.math.arizona.edu/cakeller/VertexOperatorAlgebras.pdf), §5.+- Christophe Nozaradan, [Introduction to Vertex Algebras](https://arxiv.org/pdf/0809.1380), §3.7.+++# Basic structure of VOAs++We review subalgebras, homomorphisms, ideals, quotients, extensions, and tensor products before turning to orbifolds and cosets in separate chapters. Throughout, tensor products are over $\mathbb C$.++## Vertex subalgebras and conformal vectors++::: definition [Vertex subalgebra] {@def:vertex-subalgebra}+A **vertex subalgebra** of a vertex algebra $V$ is a linear subspace $U\subseteq V$ such that++$$+\mathbf1\in U,\qquad Y_V(u,x)v\in U((x))+\quad(u,v\in U).+$$++Its fields are the restrictions $Y_U(u,x)=Y_V(u,x)|_U$.+:::++::: proposition [Inherited vertex-algebra structure] {@prop:inherited-va}+A vertex subalgebra $U\subseteq V$ is a vertex algebra with vacuum $\mathbf1$ and translation operator $T|_U$. If $V$ is a VOA and $\omega_V\in U$, then $U$ is a VOA with the same conformal vector and central charge, graded by++$$+U_h=U\cap V_h.+$$++If $V$ is of CFT type, so is $U$.+:::++**Exercise.** Prove @prop:inherited-va. *Hint:* Use $Y(u,x)\mathbf1=e^{xT}u$ to check $T$-stability, and $L_0$-stability to recover the grading.++::: definition [VOA subalgebra] {@def:voa-subalgebra}+In this chapter, a **VOA subalgebra** $(U,\omega_U)$ of $V$ means a graded vertex subalgebra $U=\bigoplus_h(U\cap V_h)$ with $\omega_U\in U\cap V_2$ making $U$ a VOA, whose grading agrees with the ambient grading:++$$+L_0^U=L_0^V|_U.+$$++It is **full** if $\omega_U=\omega_V$.+:::++The translation operators necessarily agree on $U$, since both are determined by $\left.\partial_xY_U(u,x)\mathbf1\right|_{x=0}$. The conformal vectors need not agree. Keeping them distinct is essential for the coset construction, where their difference will supply the complementary conformal structure under suitable hypotheses. Terminology for subalgebras varies; the convention above follows [Yamauchi, §3.8](https://www.math.twcu.ac.jp/~yamauchi/math/phd/myphd.pdf#page=52).++## Homomorphisms and automorphisms++::: definition [Homomorphisms and automorphisms] {@def:voa-homomorphism}+A **vertex-algebra homomorphism** $f:U\to V$ is a linear map satisfying++$$+f(\mathbf1_U)=\mathbf1_V,\qquad+f\bigl(Y_U(u,x)v\bigr)=Y_V(f(u),x)f(v)+\quad(u,v\in U).+$$++For VOAs, a **VOA homomorphism** additionally satisfies $f(\omega_U)=\omega_V$. A bijective VOA homomorphism $V\to V$ is a **VOA automorphism**; these form the group $\operatorname{Aut}(V)$.+:::++::: proposition [Compatibility with translation and grading] {@prop:homomorphism-compatibility}+A vertex-algebra homomorphism satisfies $fT_U=T_Vf$. A VOA homomorphism also satisfies++$$+fY_U(\omega_U,x)=Y_V(\omega_V,x)f,+\qquad f(U_h)\subseteq V_h,+\qquad c_U=c_V.+$$++Consequently, for $g\in\operatorname{Aut}(V)$,++$$+gY(a,x)g^{-1}=Y(ga,x),\qquad+[g,Y(\omega,x)]=0,\qquad g(V_h)=V_h.+$$+:::++**Exercise.** Prove @prop:homomorphism-compatibility. *Hint:* Apply the homomorphism identity to $Y(u,x)\mathbf1$ and $Y(\omega,x)\omega$.++The inclusion of a VOA subalgebra is always a vertex-algebra homomorphism; it is a VOA homomorphism in the above sense precisely when the subalgebra is full.++### Inner automorphisms++::: proposition [Exponentiating weight-one zero modes] {@prop:voa-inner-automorphisms}+Suppose $V$ is of CFT type. For $a\in V_1$, the operator $a_{(0)}$ preserves every $V_h$ and satisfies++$$+[a_{(0)},Y(b,x)]=Y(a_{(0)}b,x)\quad(b\in V),+\qquad a_{(0)}\mathbf1=a_{(0)}\omega=0.+$$++Its exponential, defined on each finite-dimensional $V_h$ by the ordinary matrix exponential, is a VOA automorphism:++$$+e^{a_{(0)}}Y(b,x)e^{-a_{(0)}}=Y(e^{a_{(0)}}b,x).+$$+:::++::: proof+The grading assertion follows from the weight of $a$, and the commutator identity follows from the Jacobi identity. Creation gives $a_{(0)}\mathbf1=0$. To check the conformal vector, skew-symmetry gives++$$+a_{(0)}\omega=-Ta+T L_0a-\frac12T^2L_1a=0:+$$++here $L_0a=a$, $L_1a\in\mathbb C\mathbf1$, and $L_ja=0$ for $j\ge2$. Exponentiating the commutator identity proves the field identity. All exponentials are taken on finite-dimensional graded pieces, and the vacuum and conformal vector are fixed.+:::++::: definition [Inner automorphism group] {@def:voa-inner-automorphisms}+For a VOA of CFT type, its **inner automorphism group** is++$$+G_{\mathrm{inn}}(V)+:=\big\langle e^{a_{(0)}}\mid a\in V_1\big\rangle+\subseteq\operatorname{Aut}(V).+$$+:::++More precisely, the Lie algebra action $a\mapsto a_{(0)}$ integrates on each $V_h$ to the simply connected complex Lie group $\widetilde G$ with Lie algebra $V_1$. These actions give a homomorphism $\rho:\widetilde G\to\operatorname{Aut}(V)$ with image $G_{\mathrm{inn}}(V)$. Thus++$$+G_{\mathrm{inn}}(V)\cong\widetilde G/\ker\rho.+$$++The action need not be faithful: even a nonzero $a\in V_1$ may have $a_{(0)}=0$ on all of $V$. Consequently, one should distinguish $V_1$ from its image in the derivations of $V$, and $\widetilde G$ from its image in $\operatorname{Aut}(V)$. See [van Ekeren–Möller–Scheithauer, §1.6](https://www.mathematik.tu-darmstadt.de/media/algebra/homepages/scheithauer/publications/Dimension_formulae_in_genus_zero.pdf#page=8).++## Ideals and simplicity++::: definition [Ideal and simple VOA] {@def:simple-voa}+An **ideal** of a VOA $V$ is a linear subspace $I\subseteq V$ such that++$$+Y(a,x)b\in I((x))\qquad(a\in V,\ b\in I).+$$++A nonzero VOA $V$ is **simple** if its only ideals are $0$ and $V$.+:::++::: proposition [Properties of ideals] {@prop:ideal-properties}+An ideal $I\subseteq V$ is stable under the Virasoro operators, in particular under $T$, and is graded:++$$+I=\bigoplus_h(I\cap V_h).+$$++It is also two-sided:++$$+Y(b,x)a\in I((x))\qquad(b\in I,\ a\in V).+$$++An ideal containing $\mathbf1$ equals $V$. Sums and intersections of ideals are ideals.+:::++::: proof+Apply the ideal condition to $Y(\omega,x)$; stability under the diagonalizable operator $L_0$ gives the grading. Skew-symmetry and $T$-stability give++$$+Y(b,x)a=e^{xT}Y(a,-x)b\in I((x)).+$$++If $\mathbf1\in I$, creation gives $a=\left.Y(a,x)\mathbf1\right|_{x=0}\in I$ for every $a\in V$. The assertions about sums and intersections follow from linearity.+:::++::: remark [The ordering of ideals] {@rem:ideal-lattice}+Ideals are partially ordered by inclusion. By @prop:ideal-properties, they form a complete lattice: the greatest lower bound of a family is its intersection, and its least upper bound is its sum,++$$+\bigwedge_\alpha I_\alpha=\bigcap_\alpha I_\alpha,+\qquad+\bigvee_\alpha I_\alpha=\sum_\alpha I_\alpha.+$$++The least and greatest ideals are $0$ and $V$. Among proper ideals, **maximal** means that no strictly larger proper ideal contains it, whereas **largest** means that it contains every proper ideal. A largest proper ideal need not exist merely because sums of ideals exist: their sum might be $V$.+:::++## Quotients and simple quotients++::: proposition [Quotient VOA] {@prop:quotient-voa}+Let $I\subsetneq V$ be an ideal and write $[a]=a+I$. Then $V/I$ is a VOA with++$$+Y_{V/I}([a],x)[b]=[Y_V(a,x)b],\qquad+\mathbf1_{V/I}=[\mathbf1],\qquad \omega_{V/I}=[\omega].+$$++Its grading and central charge are++$$+(V/I)_h\simeq V_h/(I\cap V_h),\qquad c_{V/I}=c_V.+$$++The projection $V\to V/I$ is a surjective VOA homomorphism. If $V$ is of CFT type, so is $V/I$.+:::++**Exercise.** Use two-sidedness to check independence of representatives, then descend the VOA axioms. Properness ensures $[\mathbf1]\ne0$.++::: proposition [Kernels and the first isomorphism theorem] {@prop:voa-isomorphism}+For a VOA homomorphism $f:V\to W$, the kernel is an ideal and $f$ induces a VOA isomorphism++$$+V/\ker f\xrightarrow{\sim}\operatorname{im}f,\qquad [a]\mapsto f(a).+$$++In particular, every VOA homomorphism from a simple VOA to a nonzero VOA is injective.+:::++**Exercise.** Apply $f$ to $Y(a,x)b$ for $b\in\ker f$, and use preservation of the vacuum. For the general quotient and kernel constructions, see [Joyce, lectures 1–2](https://people.maths.ox.ac.uk/joyce/VertexAlgebras2021/VA1%2B2.pdf).++::: proposition [Largest proper ideal and simple quotient for CFT type] {@prop:unique-simple-quotient}+Let $V$ be a nonzero VOA of CFT type, so that $V=\bigoplus_{h\geq0}V_h$ and $V_0=\mathbb C\mathbf1$. Then++$$+J(V)=\sum_{\substack{I\subsetneq V\\I\text{ an ideal}}}I+$$++is the largest proper ideal of $V$: every proper ideal $I$ satisfies $I\subseteq J(V)$. In particular, $J(V)$ is the unique maximal proper ideal, and $V/J(V)$ is the unique simple quotient, up to the isomorphism induced by a quotient map. Moreover, $J(V)=0$ if and only if $V$ is simple.+:::++::: proof+By @prop:ideal-properties, every proper ideal is graded and excludes $\mathbf1$. Since $V_0=\mathbb C\mathbf1$, every proper ideal is contained in $\bigoplus_{h>0}V_h$. Their sum $J(V)$ is therefore an ideal with zero weight-zero part, so it is proper. By construction it contains every proper ideal.++The inverse image of a nonzero proper ideal of $V/J(V)$ would be a proper ideal strictly containing $J(V)$, a contradiction. Thus $V/J(V)$ is simple. Conversely, the kernel of any surjective VOA homomorphism from $V$ onto a nonzero simple VOA is a maximal proper ideal, hence equals $J(V)$. The uniqueness follows from @prop:voa-isomorphism. Finally, $J(V)=0$ precisely when $V$ has no nonzero proper ideals.+:::++::: remark [What CFT type guarantees] {@rem:cft-type-largest-ideal}+This argument applies to every nonzero VOA of CFT type, including universal Virasoro VOAs; it is not specific to minimal-model central charges. A unique largest proper ideal may itself contain smaller nonzero ideals, so the proposition does not assert that there is only one nonzero proper ideal.++For comparison, let $W$ be a nonzero simple VOA of CFT type and equip $V=W\oplus W$ with componentwise fields, vacuum $(\mathbf1_W,\mathbf1_W)$, and conformal vector $(\omega_W,\omega_W)$. The proper ideals $W\oplus0$ and $0\oplus W$ sum to $V$, so no largest proper ideal exists. Here $\dim V_0=2$, and $V$ is not of CFT type.+:::+++## Extensions++::: definition [VOA extension] {@def:voa-extension}+An **extension** of a VOA $U$ is a VOA $V$ together with an injective VOA homomorphism $\iota:U\hookrightarrow V$. Identifying $U$ with its image, this means that $U$ is a full VOA subalgebra of $V$:++$$+\mathbf1_U=\mathbf1_V,\qquad+\omega_U=\omega_V,\qquad+Y_V(u,x)|_U=Y_U(u,x)\quad(u\in U).+$$++The extension is **proper** if $U\ne V$.+:::++By @prop:homomorphism-compatibility, an extension has $U_h=U\cap V_h$ and $c_U=c_V$. Thus a proper extension adds states while retaining the original conformal structure.++## Tensor products++::: proposition [Tensor product of VOAs] {@prop:tensor-product-voa}+Let $U,W$ be VOAs. Their tensor product is a VOA with++$$+\begin{aligned}+\mathbf1_{U\otimes W}&=\mathbf1_U\otimes\mathbf1_W,\\+Y_{U\otimes W}(u\otimes v,x)(u'\otimes v')+&=Y_U(u,x)u'\otimes Y_W(v,x)v',\\+\omega_{U\otimes W}&=\omega_U\otimes\mathbf1_W++\mathbf1_U\otimes\omega_W.+\end{aligned}+$$++The product on the second line uses the same formal variable in both factors. Its grading and central charge are++$$+\begin{aligned}+(U\otimes W)_h&=\bigoplus_{r+s=h}U_r\otimes W_s,\\+c_{U\otimes W}&=c_U+c_W.+\end{aligned}+$$++If both factors are of CFT type, so is their tensor product.+:::++::: proof+For fixed $u,u',v,v'$, both $Y_U(u,x)u'$ and $Y_W(v,x)v'$ are lower-bounded Laurent series. Their tensor product in the same variable is therefore well-defined and lower truncated. The vacuum and creation axioms follow factorwise. Translation covariance follows from the Leibniz rule with++$$+T_{U\otimes W}=T_U\otimes\operatorname{id}_W++\operatorname{id}_U\otimes T_W.+$$++For locality, choose exponents $N$ and $M$ for the pairs $u,u'$ and $v,v'$, respectively. Multiplication by $(x-y)^{N+M}$ allows us to exchange the two $U$-fields using the first factor $(x-y)^N$, and then the two $W$-fields using $(x-y)^M$. Thus the tensor-product fields are local.++The conformal field is++$$+Y_{U\otimes W}(\omega_{U\otimes W},x)+=Y_U(\omega_U,x)\otimes\operatorname{id}_W++\operatorname{id}_U\otimes Y_W(\omega_W,x).+$$++The two summands act on separate factors, so their Virasoro commutators add, with central charge $c_U+c_W$; their translation operators add to the operator above. The sum of the two $L_0$ operators gives the grading. Since both gradings are bounded below, only finitely many pairs $r+s=h$ occur at each weight, and each summand is finite-dimensional. The CFT-type assertion follows from the same grading formula.+:::++::: remark [The two tensor factors] {@rem:tensor-factors}+The maps $u\mapsto u\otimes\mathbf1_W$ and $v\mapsto\mathbf1_U\otimes v$ identify the factors with VOA subalgebras carrying their own conformal vectors. Their fields commute:++$$+[Y(u\otimes\mathbf1_W,x),Y(\mathbf1_U\otimes v,y)]=0.+$$++Indeed, these fields act on separate tensor factors. They recover every pure tensor through++$$+\left.Y(u\otimes\mathbf1_W,x)(\mathbf1_U\otimes v)\right|_{x=0}+=u\otimes v.+$$++The ambient conformal vector is the sum of those of the two subalgebras.+:::++## References++- Christoph A. Keller, [Introduction to Vertex Operator Algebras](https://archive.math.arizona.edu/cakeller/VertexOperatorAlgebras.pdf), §5.3: tensor products.+- Hiroshi Yamauchi, [A Theory of Simple Current Extensions of Vertex Operator Algebras and Applications to the Moonshine Vertex Operator Algebra](https://www.math.twcu.ac.jp/~yamauchi/math/phd/myphd.pdf), §3.8: subalgebras with their own conformal vectors and cosets.+++- Dominic Joyce, [Vertex Algebras, lectures 1–2](https://people.maths.ox.ac.uk/joyce/VertexAlgebras2021/VA1%2B2.pdf): ideals, quotients, and morphisms.+++# G-orbifolds++We construct VOAs by taking fixed points of automorphism groups, using the subalgebra and ideal properties from the preceding chapter.++## The fixed-point VOA++::: definition [Fixed-point subalgebra and orbifold] {@def:orbifold}+Let $G\leq\operatorname{Aut}(V)$. The **fixed-point subalgebra** is++$$+V^G=\{a\in V:ga=a\text{ for every }g\in G\},+\qquad Y_{V^G}(a,x)=Y_V(a,x)|_{V^G}.+$$++When $G$ is finite, we call $V^G$ the **$G$-orbifold** of $V$.+:::++::: proposition [Inherited conformal structure] {@prop:orbifold-structure}+For any $G\leq\operatorname{Aut}(V)$, $V^G$ is a full VOA subalgebra of $V$, with++$$+\mathbf1_{V^G}=\mathbf1_V,\qquad+\omega_{V^G}=\omega_V,\qquad+(V^G)_h=(V_h)^G,\qquad c_{V^G}=c_V.+$$++CFT type is inherited. In particular, $V$ is an extension of $V^G$, proper whenever $G\ne\{1\}$.+:::++**Exercise.** Prove @prop:orbifold-structure using @prop:inherited-va and the identity++$$+g\bigl(Y(a,x)b\bigr)=Y(ga,x)gb=Y(a,x)b+\quad(a,b\in V^G).+$$++Thus taking fixed points restricts the states while retaining the conformal field; it does not divide the central charge by $|G|$.++## Averaging++For the rest of this chapter, assume $G$ is finite.++::: proposition [Averaging projection] {@prop:orbifold-averaging}+The **averaging operator**++$$+R_G=\frac1{|G|}\sum_{g\in G}g+$$++preserves weights and satisfies++$$+R_G^2=R_G,\qquad \operatorname{im}R_G=V^G,+\qquad V=V^G\oplus\ker R_G.+$$++For $u\in V^G$ and $v\in V$,++$$+R_G\bigl(Y(u,x)v\bigr)=Y(u,x)R_Gv,+\qquad+R_G\bigl(Y(v,x)u\bigr)=Y(R_Gv,x)u.+$$++Consequently, both summands are preserved by the fields of $V^G$.+:::++**Exercise.** Prove @prop:orbifold-averaging by averaging the automorphism identity. The projection $R_G$ is generally not a VOA homomorphism: the identities above require one state to be invariant.++## Simplicity++::: theorem [Simplicity of finite orbifolds] {@thm:orbifold-simple}+If $V$ is simple and $G\leq\operatorname{Aut}(V)$ is finite, then $V^G$ is simple.+:::++::: proof+Complicated.+:::++## Fixed points in stages++::: proposition [Subgroups and successive orbifolds] {@prop:orbifold-stages}+If $H\leq G$, then++$$+V^G\subseteq V^H\subseteq V+$$++are full VOA inclusions. If $H\trianglelefteq G$, the quotient $G/H$ acts on $V^H$ by VOA automorphisms through $(gH)a=ga$, and++$$+(V^H)^{G/H}=V^G.+$$+:::++**Exercise.** Check that normality makes the quotient action well-defined and that being fixed by $G/H$ inside $V^H$ is equivalent to being fixed by $G$. The quotient action need not be faithful for a general $V$; the displayed equality uses its fixed vectors.++++## References++- Chongying Dong, Haisheng Li, Geoffrey Mason, [Compact automorphism groups of vertex operator algebras](https://arxiv.org/abs/q-alg/9608009), Proposition 2.1 and Theorem 2.4: cyclicity and simplicity of fixed-point VOAs.+++# Commutants and cosets++The coset construction keeps the states whose fields commute with those of a chosen subalgebra. Its conformal vector will be the difference of the ambient and subalgebra conformal vectors, under the compatibility condition below.++## The commutant++::: definition [Commutant] {@def:commutant}+Let $U\subseteq V$ be a vertex subalgebra. Its **commutant** in $V$ is++$$+\operatorname{Com}_V(U)+=\{a\in V:Y(u,x)a\in V[[x]]\text{ for every }u\in U\}.+$$++Thus $a$ belongs to the commutant precisely when $Y(u,x)a$ has no singular terms for every $u\in U$.+:::++::: proposition [Commuting fields] {@prop:commutant-fields}+For $a\in V$, the following are equivalent:++$$+\begin{aligned}+a\in\operatorname{Com}_V(U)+&\quad\Longleftrightarrow\quad+[Y(u,x),Y(a,y)]=0\quad(u\in U)\\+&\quad\Longleftrightarrow\quad+Y(a,x)u\in V[[x]]\quad(u\in U).+\end{aligned}+$$+:::++::: proof+Taking the residue in the iterate variable in @prop:jacobi-identity gives++$$+[Y(u,x),Y(a,y)]+=\operatorname{Res}_t\,y^{-1}\delta\!\left(\frac{x-t}{y}\right)+Y(Y(u,t)a,y).+$$++The delta kernel is expanded in nonnegative powers of $t$. If $Y(u,t)a$ is regular in $t$, the integrand has no $t^{-1}$ term, so the commutator vanishes.++Conversely, apply the commuting-field identity to $\mathbf1$ and take the constant term in $y$. On the left this gives $Y(u,x)a$. On the right, $Y(a,y)Y(u,x)\mathbf1=Y(a,y)e^{xT}u$ has no negative powers of $x$, so $Y(u,x)a$ is regular. The last equivalence follows from skew-symmetry, $Y(a,x)u=e^{xT}Y(u,-x)a$.+:::++::: proposition [Vertex-algebra structure of the commutant] {@prop:commutant-va}+The space $C=\operatorname{Com}_V(U)$ is a vertex subalgebra of $V$, with vacuum $\mathbf1$, translation $T|_C$, and fields $Y_C(a,x)=Y_V(a,x)|_C$.+:::++::: proof+Creation gives $\mathbf1\in C$. For $a,b\in C$ and $u\in U$, @prop:commutant-fields gives++$$+Y(u,z)Y(a,x)b=Y(a,x)Y(u,z)b.+$$++The right-hand side has no negative powers of $z$, because $b\in C$. Taking any coefficient in $x$ shows that every coefficient of $Y(a,x)b$ belongs to $C$. Thus $Y(a,x)b\in C((x))$, and @prop:inherited-va applies.+:::++## The conformal vector of the coset++Now let $(U,\omega_U)$ be a VOA subalgebra of $(V,\omega_V)$ in the sense of @def:voa-subalgebra. The ambient field $Y_V(\omega_U,x)$ acts on all of $V$, even though $\omega_U$ is the conformal vector of $U$.++::: theorem [Coset VOA] {@thm:coset-voa}+Let $U\subseteq V$ be a VOA subalgebra and set $C=\operatorname{Com}_V(U)$. Assume++$$+\omega_C:=\omega_V-\omega_U\in C.+$$++Then $C$ is a VOA with conformal vector $\omega_C$, central charge and grading++$$+c_C=c_V-c_U,\qquad C_h=C\cap V_h.+$$++We call $(C,Y_C,\mathbf1,\omega_C)$ the **coset VOA**. If $V$ is of CFT type, so is $C$.+:::++::: proof+Write the conformal fields on $V$ as++$$+\begin{aligned}+Y_V(\omega_V,x)&=\sum_n L_n^Vx^{-n-2},\\+Y_V(\omega_U,x)&=\sum_n L_n^Ux^{-n-2},\\+L_n^C&=L_n^V-L_n^U.+\end{aligned}+$$++The operators $L_n^U$ satisfy the Virasoro relations on $V$: this follows from Jacobi and the conformal-vector products in $U$, just as in @prop:virasoro-covariance. Since $\omega_U\in U$ and $\omega_C\in C$, their fields commute, so $[L_m^U,L_n^C]=0$. Consequently,++$$+\begin{aligned}+[L_m^C,L_n^C]+&=[L_m^V,L_n^V]-[L_m^U,L_n^U]\\+&=(m-n)L_{m+n}^C++\frac{c_V-c_U}{12}(m^3-m)\delta_{m+n,0}\operatorname{id}_V.+\end{aligned}+$$++For $a\in C$, the series $Y_V(\omega_U,x)a$ is regular. Its $x^{-1}$ and $x^{-2}$ coefficients therefore vanish: $L_{-1}^Ua=L_0^Ua=0$. Hence++$$+L_{-1}^C|_C=T|_C,\qquad L_0^C|_C=L_0^V|_C.+$$++The modes of $\omega_C\in C$ preserve $C$, so these relations give translation compatibility and the inherited grading. Finite-dimensional weight spaces, the lower bound on weights, and CFT type follow from those of $V$.+:::++::: remark [Full subalgebras] {@rem:full-subalgebra-coset}+If $U$ is full, then $\omega_U=\omega_V$. For $a\in\operatorname{Com}_V(U)$, regularity of $Y(\omega_V,x)a$ forces $L_0^Va=0$. Thus, when $V$ is of CFT type,++$$+\operatorname{Com}_V(U)=\mathbb C\mathbf1.+$$++A nontrivial coset in this setting therefore requires a subalgebra with its own conformal vector.+:::++## Elementary commutant relations++::: proposition [Inclusions and double commutants] {@prop:double-commutant}+For vertex subalgebras $U_1\subseteq U_2\subseteq V$,++$$+\operatorname{Com}_V(U_2)\subseteq\operatorname{Com}_V(U_1).+$$++For every vertex subalgebra $U\subseteq V$,++$$+U\subseteq\operatorname{Com}_V(\operatorname{Com}_V(U)).+$$+:::++**Exercise.** Use the definition for the first assertion and the symmetry of commuting fields for the second. Equality in the double-commutant inclusion is an additional property, not part of the coset construction.++## References++- Hiroshi Yamauchi, [A Theory of Simple Current Extensions of Vertex Operator Algebras and Applications to the Moonshine Vertex Operator Algebra](https://www.math.twcu.ac.jp/~yamauchi/math/phd/myphd.pdf), §3.8: commutants and cosets.+- Yanjun Chu and Zongzhu Lin, [The varieties of Heisenberg vertex operator algebras](https://arxiv.org/abs/1508.02963), §§1–2: semi-conformal subalgebras and complementary conformal vectors.+++# Examples of VOAs++## The Heisenberg VOA++The rank-one Heisenberg VOA $M(1)$ describes the current algebra of one chiral free boson. We construct its states and fields explicitly, then verify its conformal structure.++### The oscillator space++::: definition [Heisenberg vacuum space] {@def:heisenberg-vacuum}+Let++$$+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.+   $$++   Thus a regular VOA product of composite states need not be their ordinary polynomial product.+:::++## Virasoro minimal models++The Heisenberg VOA is generated by a weight-one current. We now construct a VOA generated by its conformal vector alone. At certain central charges, its vacuum descendants satisfy additional relations; imposing these relations gives the Virasoro minimal models.++### The universal vacuum VOA++::: definition [Virasoro vacuum representation] {@def:virasoro-vacuum}+Fix $c\in\mathbb C$. Let $V^c$ be the representation of the Virasoro algebra generated by a vector $\mathbf1$, subject only to the Virasoro relations with central charge $c$ and++$$+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 $\mathcal T(z)$.+:::   # Modules and intertwining operators@@ -412,7 +1833,7 @@ - Finite-group actions and fixed-point subalgebras. - Quantum Galois correspondence and its hypotheses. -## References for expanding the outline {#sec:extension-references}+## References for expanding the outline {@sec:extension-references}  - Yi-Zhi Huang, Alexander Kirillov Jr., and James Lepowsky, [Braided tensor categories and extensions of vertex operator algebras](https://arxiv.org/abs/1406.3420). - Akihide Hanaki, Masahiko Miyamoto, and Daisuke Tambara, [Quantum Galois theory for finite groups](https://arxiv.org/abs/q-alg/9709041).