Introduction to Vertex Operator Algebras

--- Base version+++ Proposed version@@ -0,0 +1,181 @@+# 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++::: 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 $Y_U(u,z)=Y_V(u,z)|_U$.+:::++::: 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 $U$.+:::++::: 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 $h<0$.+:::++::: 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 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++::: 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 $\operatorname{Aut}(V)$.+:::++::: proposition [Compatibility with translation and grading] {@prop:homomorphism-compatibility}+A vertex-algebra homomorphism satisfies $fT_U=T_Vf$. A VOA homomorphism also satisfies++$$+fL_n^U=L_n^Vf\quad(n\in\mathbb Z),+\qquad f(U_h)\subseteq V_h,+\qquad c_U=c_V.+$$++Consequently, for $g\in\operatorname{Aut}(V)$,++$$+gY(a,z)g^{-1}=Y(ga,z),\qquad+[g,L_n]=0,\qquad g(V_h)=V_h.+$$+:::++::: 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++::: 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 $V$:++$$+\mathbf1_U=\mathbf1_V,\qquad+\omega_U=\omega_V,\qquad+Y_V(u,z)|_U=Y_U(u,z)\quad(u\in U).+$$++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++::: proposition [Tensor product of VOAs] {@prop:tensor-product-voa}+Let $U,W$ be VOAs. Their tensor product is a VOA with++$$+\begin{aligned}+\mathbf1_{U\otimes W}&=\mathbf1_U\otimes\mathbf1_W,\\+Y_{U\otimes W}(u\otimes v,z)(u'\otimes v')+&=Y_U(u,z)u'\otimes Y_W(v,z)v',\\+\omega_{U\otimes W}&=\omega_U\otimes\mathbf1_W++\mathbf1_U\otimes\omega_W.+\end{aligned}+$$++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++$$+\begin{aligned}+(U\otimes W)_h&=\bigoplus_{r+s=h}U_r\otimes W_s,\\+L_n^{U\otimes W}&=L_n^U\otimes\operatorname{id}_W++\operatorname{id}_U\otimes L_n^W,\\+c_{U\otimes W}&=c_U+c_W.+\end{aligned}+$$++If both factors are of CFT type, so is their tensor product.+:::++::: 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++$$+T_{U\otimes W}=T_U\otimes\operatorname{id}_W++\operatorname{id}_U\otimes T_W.+$$++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.+:::++::: 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:++$$+[Y(u\otimes\mathbf1_W,z),Y(\mathbf1_U\otimes v,w)]=0.+$$++Indeed, these fields act on separate tensor factors. They recover every pure tensor through++$$+(u\otimes\mathbf1_W)_{(-1)}(\mathbf1_U\otimes v)=u\otimes v.+$$++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.