Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -1,20 +1,18 @@-# Constructions from existing VOAs--We begin with subalgebras, structure-preserving maps, and tensor products. These provide the notation and elementary results needed for fixed-point and coset constructions. Throughout, tensor products are over $\mathbb C$.--## Preliminaries--### Vertex subalgebras and conformal vectors+# 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  $$-\mathbf1\in U,\qquad u_{(n)}v\in U-\quad(u,v\in U,\ n\in\mathbb Z).-$$--Its fields are the restrictions $Y_U(u,z)=Y_V(u,z)|_U$.+\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}@@ -27,11 +25,7 @@ If $V$ is of CFT type, so is $U$. ::: -::: proof-Since $Tu=u_{(-2)}\mathbf1$, closure under products implies $T(U)\subseteq U$. Truncation, creation, translation covariance, and locality restrict from $V$ to $U$.--If $\omega_V\in U$, its modes preserve $U$ and satisfy the same Virasoro relations. In particular, $U$ is $L_0$-stable. As $L_0$ is diagonalizable, $U=\bigoplus_h(U\cap V_h)$: each homogeneous component of a vector is obtained by a polynomial in $L_0$ on its finite set of weights. Finite-dimensionality and the lower bound on weights are inherited. Finally, for CFT type, $U_0=\mathbb C\mathbf1$ and $U_h=0$ for $h<0$.-:::+**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:@@ -43,27 +37,27 @@ It is **full** if $\omega_U=\omega_V$. ::: -The translation operators necessarily agree on $U$, since both send $u$ to $u_{(-2)}\mathbf1$. 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+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(u_{(n)}v)=f(u)_{(n)}f(v)-\quad(u,v\in U,\ n\in\mathbb Z).-$$--Equivalently, it preserves the vacuum and satisfies $f(Y_U(u,z)v)=Y_V(f(u),z)f(v)$ coefficientwise. For VOAs, a **VOA homomorphism** additionally satisfies $f(\omega_U)=\omega_V$. A bijective VOA homomorphism $V\to V$ is a **VOA automorphism**; these form the group $\operatorname{Aut}(V)$.+f\bigl(Y_U(u,x)v\bigr)=Y_V(f(u),x)f(v)+\quad(u,v\in U).+$$++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  $$-fL_n^U=L_n^Vf\quad(n\in\mathbb Z),+fY_U(\omega_U,x)=Y_V(\omega_V,x)f, \qquad f(U_h)\subseteq V_h, \qquad c_U=c_V. $$@@ -71,18 +65,96 @@ Consequently, for $g\in\operatorname{Aut}(V)$,  $$-gY(a,z)g^{-1}=Y(ga,z),\qquad-[g,L_n]=0,\qquad g(V_h)=V_h.-$$+gY(a,x)g^{-1}=Y(ga,x),\qquad+[g,Y(\omega,x)]=0,\qquad g(V_h)=V_h.+$$+:::++**Exercise.** Prove @prop:homomorphism-compatibility. *Hint:* Apply the homomorphism identity to $Y(u,x)\mathbf1$ and $Y(\omega,x)\omega$.++The inclusion of a VOA subalgebra is always a vertex-algebra homomorphism; it is a VOA homomorphism in the above sense precisely when the subalgebra is full.++## Ideals and simplicity++::: definition [Ideal and simple VOA] {@def:simple-voa}+An **ideal** of a VOA $V$ is a linear subspace $I\subseteq V$ such that++$$+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).+$$++An ideal containing $\mathbf1$ equals $V$. Sums and intersections of ideals are ideals. :::  ::: proof-The translation identity follows from $T_Uu=u_{(-2)}\mathbf1_U$. Preservation of $\omega$ and its products gives $fL_n^U=L_n^Vf$, and $n=0$ gives preservation of weights. Applying $f$ to $(\omega_U)_{(3)}\omega_U=c_U\mathbf1_U/2$ gives $c_U=c_V$, since $\mathbf1_V\ne0$. The automorphism identities follow by composing with $g^{-1}$.-:::--The inclusion of a VOA subalgebra is always a vertex-algebra homomorphism; it is a VOA homomorphism in the above sense precisely when the subalgebra is full.--### Extensions+Apply the ideal condition to $Y(\omega,x)$; stability under the diagonalizable operator $L_0$ gives the grading. Skew-symmetry and $T$-stability give++$$+Y(b,x)a=e^{xT}Y(a,-x)b\in I((x)).+$$++If $\mathbf1\in I$, creation gives $a=\left.Y(a,x)\mathbf1\right|_{x=0}\in I$ for every $a\in V$. The assertions about sums and intersections follow from linearity.+:::++## Quotients and simple quotients++::: proposition [Quotient VOA] {@prop:quotient-voa}+Let $I\subsetneq V$ be an ideal and write $[a]=a+I$. Then $V/I$ is a VOA with++$$+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).+$$++In particular, every VOA homomorphism from a simple VOA to a nonzero VOA is injective.+:::++**Exercise.** Apply $f$ to $Y(a,x)b$ for $b\in\ker f$, and use preservation of the vacuum. For the general quotient and kernel constructions, see [Joyce, lectures 1–2](https://people.maths.ox.ac.uk/joyce/VertexAlgebras2021/VA1%2B2.pdf).++::: proposition [Unique simple quotient for CFT type] {@prop:unique-simple-quotient}+If $V$ is of CFT type, the sum $J$ of all proper ideals is its unique maximal proper ideal, and $V/J$ is simple.+:::++::: proof+Every proper ideal is graded and excludes $\mathbf1$, so its weight-zero part is zero because $V_0=\mathbb C\mathbf1$. Their sum $J$ still has zero weight-zero part and is therefore proper. It contains every proper ideal. The inverse image of a nonzero proper ideal of $V/J$ would be a proper ideal strictly containing $J$, a contradiction.+:::+++## Extensions  ::: definition [VOA extension] {@def:voa-extension} An **extension** of a VOA $U$ is a VOA $V$ together with an injective VOA homomorphism $\iota:U\hookrightarrow V$. Identifying $U$ with its image, this means that $U$ is a full VOA subalgebra of $V$:@@ -90,15 +162,15 @@ $$ \mathbf1_U=\mathbf1_V,\qquad \omega_U=\omega_V,\qquad-Y_V(u,z)|_U=Y_U(u,z)\quad(u\in U).+Y_V(u,x)|_U=Y_U(u,x)\quad(u\in U). $$  The extension is **proper** if $U\ne V$. ::: -By @prop:homomorphism-compatibility, an extension has $U_h=U\cap V_h$ and $c_U=c_V$. Thus the larger VOA contains additional states while retaining the original conformal structure.--### Tensor products+By @prop:homomorphism-compatibility, an extension has $U_h=U\cap V_h$ and $c_U=c_V$. Thus a proper extension adds states while retaining the original conformal structure.++## Tensor products  ::: proposition [Tensor product of VOAs] {@prop:tensor-product-voa} Let $U,W$ be VOAs. Their tensor product is a VOA with@@ -106,27 +178,18 @@ $$ \begin{aligned} \mathbf1_{U\otimes W}&=\mathbf1_U\otimes\mathbf1_W,\\-Y_{U\otimes W}(u\otimes v,z)(u'\otimes v')-&=Y_U(u,z)u'\otimes Y_W(v,z)v',\\+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} $$ -The product on the second line uses the same formal variable in both factors. Explicitly,--$$-(u\otimes v)_{(n)}(u'\otimes v')-=\sum_{i+j=n-1}(u_{(i)}u')\otimes(v_{(j)}v').-$$--Its grading, Virasoro modes, and central charge are+The product on the second line uses the same formal variable in both factors. Its grading and central charge are  $$ \begin{aligned} (U\otimes W)_h&=\bigoplus_{r+s=h}U_r\otimes W_s,\\-L_n^{U\otimes W}&=L_n^U\otimes\operatorname{id}_W-+\operatorname{id}_U\otimes L_n^W,\\ c_{U\otimes W}&=c_U+c_W. \end{aligned} $$@@ -135,47 +198,47 @@ :::  ::: proof-For fixed $u,u',v,v'$, lower truncation bounds $i$ and $j$ above, so only finitely many pairs contribute to each mode sum. The product is therefore well-defined and lower truncated. The vacuum and creation axioms follow factorwise. Translation covariance follows from the Leibniz rule with+For fixed $u,u',v,v'$, both $Y_U(u,x)u'$ and $Y_W(v,x)v'$ are lower-bounded Laurent series. Their tensor product in the same variable is therefore well-defined and lower truncated. The vacuum and creation axioms follow factorwise. Translation covariance follows from the Leibniz rule with  $$ T_{U\otimes W}=T_U\otimes\operatorname{id}_W +\operatorname{id}_U\otimes T_W. $$ -For locality, choose exponents $N$ and $M$ for the pairs $u,u'$ and $v,v'$, respectively. Multiplication by $(z-w)^{N+M}$ allows us to exchange the two $U$-fields using the first factor $(z-w)^N$, and then the two $W$-fields using $(z-w)^M$. Thus the tensor-product fields are local.--The displayed conformal vector gives the stated $L_n^{U\otimes W}$. Operators on different factors commute, so their Virasoro commutators add, with central charge $c_U+c_W$; the $n=-1$ mode is the translation operator above. The sum of the two $L_0$ operators gives the grading. Since both gradings are bounded below, only finitely many pairs $r+s=h$ occur at each weight, and each summand is finite-dimensional. The CFT-type assertion follows from the same grading formula.+For locality, choose exponents $N$ and $M$ for the pairs $u,u'$ and $v,v'$, respectively. Multiplication by $(x-y)^{N+M}$ allows us to exchange the two $U$-fields using the first factor $(x-y)^N$, and then the two $W$-fields using $(x-y)^M$. Thus the tensor-product fields are local.++The conformal field is++$$+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,z),Y(\mathbf1_U\otimes v,w)]=0.+[Y(u\otimes\mathbf1_W,x),Y(\mathbf1_U\otimes v,y)]=0. $$  Indeed, these fields act on separate tensor factors. They recover every pure tensor through  $$-(u\otimes\mathbf1_W)_{(-1)}(\mathbf1_U\otimes v)=u\otimes v.+\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. :::--## Fixed-point subalgebras and G-orbifolds--- Definition of $V^G$ and inheritance of the VOA structure.-- Grading, central charge, and the averaging projection for finite $G$.-- Further properties with their hypotheses.--## Commutants and cosets--- Definition of $\operatorname{Com}_V(U)$ and closure under vertex-algebra products.-- Compatibility of conformal vectors and the coset VOA structure.-- Central charge and further properties with their hypotheses.  ## 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.