\documentclass[11pt]{article} \usepackage[letterpaper,top=0.89in,bottom=0.91in,left=0.96in,right=0.96in]{geometry} \usepackage[T1]{fontenc} \usepackage{lmodern,microtype,amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage[hidelinks]{hyperref} \usepackage{bookmark} \allowdisplaybreaks[1] \setlength{\parskip}{1.5pt} \setlength{\parindent}{1.15em} \setlength{\emergencystretch}{1.5em} \newcommand{\C}{\mathbb C} \newcommand{\1}{\mathbf1} \newcommand{\Span}{\operatorname{span}} \newcommand{\wt}{\operatorname{wt}} \newcommand{\gr}{\operatorname{gr}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Supp}{\operatorname{Supp}} \newcommand{\Ann}{\operatorname{Ann}} \newcommand{\id}{\operatorname{id}} \newcommand{\m}{\mathfrak m} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \hypersetup{pdftitle={C2-cofiniteness of finite-group orbifolds}, pdfauthor={Andrew Riesen}, pdfsubject={Finite-group fixed points of graded vertex algebras}} \title{$C_2$-cofiniteness of finite-group orbifolds} \author{Andrew Riesen} \date{28 October 2026} \begin{document} \maketitle \begin{abstract} Let $V$ be a simple vertex algebra with a connected nonnegative integer grading, and let $G$ be a finite group of grading-preserving automorphisms. We prove that $C_2$-cofiniteness of $V$ implies $C_2$-cofiniteness of $V^G$. The proof bounds the length and height of monomials in strong generators representing classes in a relative $C_2$ quotient. The length bound follows from a good filtration and finite-group invariant theory. The height bound follows from differential equations for products of fields, together with identities for invariants and vacuum coefficients. The height bound also shows that finite strong generation passes to $V^G$, without assuming $C_2$-cofiniteness. For strongly rational vertex operator algebras, the cofiniteness result and McRae's orbifold theorem imply strong rationality of every finite-group fixed-point algebra. \end{abstract} \section{Introduction}\label{sec:intro} Taking invariants under a finite group of automorphisms, or forming \emph{orbifolds}, is one of the most basic constructions in the theory of vertex operator algebras and conformal field theory \cite{DVVV,DLM,FLM,GR}. A longstanding conjecture predicts that finite-group fixed-point subalgebras of strongly rational vertex operator algebras are again strongly rational. The $C_2$ condition, introduced by Zhu in his work on modular invariance \cite{Zhu}, is central to this conjecture. For a vertex algebra $A$, write \[ C_2(A)=\Span_{\C}\{a_{-2}b:a,b\in A\},\qquad R_A=A/C_2(A). \] The quotient $R_A$ is a commutative algebra under $[a][b]=[a_{-1}b]$, and $A$ is \emph{$C_2$-cofinite} when $R_A$ is finite dimensional. For vertex operator algebras, this condition implies that Zhu's algebra is finite dimensional \cite[Proposition~3.6]{DLM}, and hence that there are only finitely many irreducible admissible modules up to isomorphism \cite[Theorem~2.2.2]{Zhu}. Substantial progress on the conjecture was made by Miyamoto, who proved $C_2$-cofiniteness for cyclic orbifolds of simple $C_2$-cofinite vertex operator algebras of CFT type, and hence for all finite solvable groups \cite{Miyamoto}. Carnahan and Miyamoto subsequently established regularity of fixed-point subalgebras under finite solvable groups for simple nonnegatively graded regular vertex operator algebras with a nondegenerate invariant bilinear form \cite{CM}. These results settled the solvable case of the conjecture, while the general nonsolvable case remained open. For arbitrary finite groups, McRae proved that a fixed-point algebra of a strongly rational vertex operator algebra is strongly rational as soon as it is $C_2$-cofinite \cite[Corollary 4.23]{McRae}. This reduces the strong-rationality problem to $C_2$-cofiniteness. We prove that $C_2$-cofiniteness passes to fixed points for every finite group. We work over $\C$ throughout. More precisely, let $V$ be a simple vertex algebra with a \emph{connected nonnegative integer grading}, meaning \[ V=\bigoplus_{n\ge0}V_n,\qquad V_0=\C\1,\qquad a_mV_b\subseteq V_{h+b-m-1}\quad(a\in V_h), \] with $V_n=0$ for $n<0$. Let $G$ be a finite group of grading-preserving automorphisms. Under these hypotheses, we prove the following. \begin{theorem}\label{thm:main} If $V$ is $C_2$-cofinite, then $V^G$ is $C_2$-cofinite. \end{theorem} In particular, the theorem applies to simple $C_2$-cofinite vertex operator algebras of CFT type. McRae's theorem then implies that every finite-group orbifold of a strongly rational vertex operator algebra is strongly rational (Corollary~\ref{cor:rational}). We also prove that finite strong generation passes to fixed points, without assuming $C_2$-cofiniteness. \begin{theorem}\label{thm:sfg} Let $V$ be a simple, strongly finitely generated vertex algebra with a connected nonnegative integer grading. For every finite group $G$ of grading-preserving automorphisms, $V^G$ is strongly finitely generated. \end{theorem} The proof is given in Section~\ref{sec:finish}. This resolves the finite-group case of the vertex algebra Hilbert problem in this setting \cite[Problem 7.1]{LianLinshaw}. \paragraph{Notation.} For a formal power or Laurent series with vector coefficients, we write \[ [z^\alpha]\left(\sum_\gamma v_\gamma z^\gamma\right)=v_\alpha, \qquad z^\alpha=z_1^{\alpha_1}\cdots z_m^{\alpha_m}. \] Coefficient extraction leaves any unextracted variables unchanged. In particular, $[t^j]$ extracts the coefficient of $t^j$, and $[d^0]$ extracts the constant coefficient in $d$. For $\alpha\in\mathbb Z_{\ge0}^m$, we use $|\alpha|=\sum_i\alpha_i$, $\alpha!=\prod_i\alpha_i!$, and $\partial_z^\alpha=\prod_i\partial_{z_i}^{\alpha_i}$; inequalities between multi-indices are componentwise. Write $[n]=\{1,\ldots,n\}$ and $I^c=[n]\setminus I$ for $I\subseteq[n]$. Linear maps on coefficient spaces act coefficientwise on formal series. \paragraph{Proof strategy.} The proof has four steps. \begin{enumerate} \item \textbf{Pass to a relative $C_2$-quotient.} Regard $V$ as a $V^G$-module and consider \[ Q=R_{V^G}(V) =V/\Span\{b_{-2}v:b\in V^G,\ v\in V\}. \] Taking $G$-invariants gives \[ Q^G\simeq R_{V^G}(V^G)=V^G/C_2(V^G). \] Thus it suffices to prove that $Q$ is finite dimensional. \item \textbf{Choose homogeneous strong generators.} Since $V$ is $C_2$-cofinite, choose homogeneous positive-weight strong generators $u^1,\ldots,u^d$ with $G$-stable span $U$. Their monomials give the spanning classes \[ [u^{i_1}_{-n_1}\cdots u^{i_\ell}_{-n_\ell}\1]\in Q, \qquad \ell\ge0,\quad 1\le i_j\le d,\quad n_j\ge1, \] where $\ell=0$ gives the vacuum class. We need two bounds: \begin{enumerate} \item \textbf{Length.} Find $L$ such that classes with $\ell\le L$ already span $Q$. \item \textbf{Height.} For nonempty monomials of length at most $L$, find $H$ such that their classes vanish whenever $\max_j n_j>H$. \end{enumerate} \item \textbf{Bound the length.} This is the more direct part of the argument. A good filtration, defined by the sum of the generating states' weights and independent of the mode indices, reduces it to a calculation in a commutative differential algebra. Modulo the ideal generated by derivatives of invariants, integration by parts moves all derivatives off a chosen factor in each projected monomial. Nilpotence supplied by $C_2$-cofiniteness, together with finite-group invariant theory, then gives the length bound. \item \textbf{Bound the height.} This is the main difficulty. We proceed in the following order. \begin{enumerate} \item \textbf{Differential equations.} For $n\ge1$, consider the $n$-point functions $Y(a_1,z_1)\cdots Y(a_n,z_n)\1$, with the fields labeled by $[n]:=\{1,\cdots,n\}$. Multiplying by suitable powers of $z_i-z_j$ and passing to $Q$ gives a power series $H_Q(z_1,\ldots,z_n)$. For each nonempty subset $I\subseteq[n]$, we separate the fields labeled by $I$ from those labeled by $I^c$. Weak associativity combines the selected fields at a common coordinate $t$, with their relative positions fixed. The factors joining $I$ to $I^c$ remain as a scalar polynomial, which we call the \emph{cross factor}. After taking invariants in the selected factors, the relative $C_2$ relation kills the positive powers of $t$ in the iterated field. The cross factor therefore bounds the powers of $t$ in the full expression from above, while regularity excludes negative powers. Its degree determines the differential equation \[ \left(\sum_{i\in I}\partial_{z_i}\right)^{d_I} (H_Q\circ P_I)=0, \] where $P_I$ projects onto invariants in the selected factors and $d_I$ is one more than the $t$-degree of the cross factor. \item \textbf{Vacuum pairs.} We may take the $G$-action to be faithful. For each $g\ne1$, choose homogeneous $a_g$ with $u_g=(1-g)a_g\ne0$. Simplicity gives a homogeneous $w_g$ with $(u_g)_{\wt u_g+\wt w_g-1}w_g=\1$. To each tensor product of homogeneous components of $U$, we adjoin sufficiently many copies of the $G$-spans of $a_g$ and $w_g$. \item \textbf{A polynomial module and its support.} Form the universal module $M$ for the differential relations on this enlarged tensor product over $R=\C[p_1,\ldots,p_n]$, with $p_i$ representing differentiation. Let $\mathcal N$ be the submodule generated by tensors with the chosen auxiliary states fixed. Finite-group identities show that $\mathcal N$ is supported at the origin. Since $\mathcal N$ is finitely generated, some power of the maximal ideal at that point annihilates it: \[ \m_0^{q_0}\mathcal N=0,\qquad \m_0=(p_1,\ldots,p_n). \] \item \textbf{Recover the height bound.} The differential equations and this annihilation relation force the enlarged series to have bounded total degree when evaluated on tensors with the auxiliary states fixed. Taking the vacuum term of each auxiliary pair removes the extra fields and transfers the resulting weight bound to the original products. The mode weight formula then bounds $\max_j n_j$. \end{enumerate} \end{enumerate} The length and height bounds leave only finitely many choices of $\ell$, $i_j$, and $n_j$. Hence $Q$ is finite dimensional, and taking invariants gives $C_2$-cofiniteness of $V^G$. \paragraph{Acknowledgements and use of generative AI.} Generative AI, including ChatGPT, was used extensively under the author's direction for proof exploration, calculations, checking arguments and references, and drafting and revising the exposition. This assistance contributed substantial portions of the initial arguments and text. The author takes responsibility for the mathematical claims and final text. \section{The length bound}\label{sec:length} Throughout, $V$ is a simple vertex algebra with a connected nonnegative integer grading, and $G$ is a finite group of grading-preserving automorphisms. We may replace $G$ by its image in $\operatorname{Aut}(V)$ without changing $V^G$, and hence assume the action is faithful. Write $r=|G|$, $B=V^G$, and $P_G=r^{-1}\sum_{g\in G}g$ for the Reynolds projection. We use the mode convention $Y(a,z)=\sum_{m\in\mathbb Z}a_mz^{-m-1}$ and write $T$ for translation. For homogeneous $a\in V_h$, write $\wt a=h$. Normally ordered products are associated to the right. In this section, assume that $R_V$ is finite dimensional. \subsection{Strong generators}\label{subsec:generators} A subset $S\subset V$ \emph{strongly generates} $V$ if $V$ is spanned by the monomials \begin{equation}\label{eq:monomials} W=u^1_{-n_1}\cdots u^{\ell}_{-n_\ell}\1, \qquad \ell\ge0,\quad u^i\in S,\quad n_i\ge1. \end{equation} The empty product is the vacuum. The elements of $S$ are called \emph{strong generators}. If $S$ can be chosen finite, $V$ is \emph{strongly finitely generated}. A subspace strongly generates $V$ if its elements do. For homogeneous positive-weight $u^i$, the monomial \eqref{eq:monomials} has \emph{length} $\ell$ and weight \begin{equation}\label{eq:monomialweight} \wt W=\sum_i\wt u^i+\sum_i(n_i-1). \end{equation} For a nonempty monomial, its \emph{height} is $\max_i n_i$; in particular, $\operatorname{ht}(W)\le\wt W$. The identity $(T^{n-1}u)_{-1}=(n-1)!u_{-n}$ expresses these monomials as normally ordered products of derivatives of the generators, up to nonzero scalar factors; see \cite[Section~3]{LianLinshaw}. Write $R_{V,+}=\bigoplus_{w>0}(R_V)_w$. Let $s$ be a graded linear section of the quotient map on $R_{V,+}$. Taking invariants in the space of linear maps gives the equivariant section \[ s_G(x)=\frac1r\sum_{g\in G}g\,s(g^{-1}x),\qquad U=s_G(R_{V,+})\subset V_{>0}. \] Thus $U$ is a finite-dimensional graded $G$-stable space of lifts. Let \[ C_1(V)=TV+\Span\{a_{-1}b:a,b\in V_{>0}\}. \] The identities $a_{-2}b=(Ta)_{-1}b$ and $T\1=0$ give $C_2(V)\subseteq C_1(V)$. Hence $V_{>0}=U+C_1(V)$, and induction on weight shows that $U$ strongly generates $V$; see \cite[Theorem 4.10]{Li}. In particular, each $V_w$ is finite dimensional: after choosing a finite homogeneous basis of $U$, both the length and the sum of the derivative orders of a monomial of weight $w$ are bounded. \subsection{The relative quotient}\label{subsec:relative} For a vertex algebra $W$ and a $W$-module $M$, write \[ R_W(M)=M/\Span\{w_{-2}m:w\in W,\ m\in M\}. \] In particular, $R_W(W)=R_W$. We regard $V$ as a module over $B=V^G$ and denote $R_B(V)=R_{V^G}(V)$ by $Q$ for brevity. Thus \begin{equation}\label{eq:relative} D_B=\Span\{b_{-2}v:b\in B,\ v\in V\},\qquad Q=V/D_B,\qquad q:V\longrightarrow Q. \end{equation} Both $D_B$ and $Q$ are graded. Translation gives $(T^{m-2}b)_{-2}=(m-1)!b_{-m}$, so \begin{equation}\label{eq:negative} q(b_{-m}v)=0\qquad(b\in B,\ m\ge2). \end{equation} Applying $P_G$ to $\sum_jb^j_{-2}v_j\in D_B\cap B$ replaces each $v_j$ by $P_Gv_j\in B$. Hence \begin{equation}\label{eq:exact} D_B\cap B=C_2(B),\qquad Q^G\simeq R_B. \end{equation} The second identification uses exactness of finite-group invariants. It is therefore enough to show that $Q$ is finite dimensional. \subsection{The length estimate}\label{subsec:length} \begin{lemma}\label{lem:orbit} Let $A$ be a commutative $G$-algebra and $J$ a $G$-stable ideal. Then \[ J^r\subseteq A J^G, \] where $A J^G$ is the ideal generated by the invariant elements of $J$. \end{lemma} \begin{proof} For $x\in J$, the nonleading coefficients of $\prod_{g\in G}(t-gx)$ lie in $J^G$. Evaluating at $t=x$ therefore gives $x^r\in A J^G$. Replace $x$ by $t_1x_1+\cdots+t_rx_r$ and compare the coefficient of $t_1\cdots t_r$. It is $r!x_1\cdots x_r$, and $r!\ne0$. \end{proof} The positive-weight ideal $R_{V,+}$ of the finite-dimensional, connected graded algebra $R_V$ is nilpotent. Choose $\nu\ge1$ with \begin{equation}\label{eq:nu} R_{V,+}^{\nu}=0. \end{equation} We filter by the sum of the weights of the generating states, with no contribution from the mode indices. For $p\ge0$, define \begin{equation}\label{eq:goodfiltration} F_pV=\Span\left\{ u^1_{-n_1}\cdots u^{\ell}_{-n_\ell}\1: \sum_i\wt u^i\le p\right\}. \end{equation} Here $\ell\ge0$, the $u^i\in U$ are homogeneous, and $n_i\ge1$. Set $F_{-1}V=0$, so $F_0V=\C\1$. The sum $\sum_i\wt u^i$ differs from both the length $\ell$ and the weight $\sum_i\wt u^i+\sum_i(n_i-1)$ of the monomial. To check that translation preserves this filtration, use $T\1=0$ and $[T,u_{-n}]=n u_{-n-1}$: \[ T\bigl(u^1_{-n_1}\cdots u^\ell_{-n_\ell}\1\bigr) =\sum_{j=1}^{\ell}n_j\, u^1_{-n_1}\cdots u^j_{-n_j-1}\cdots u^\ell_{-n_\ell}\1. \] Each summand uses the same generating states, so $\sum_i\wt u^i$ is unchanged. Hence $TF_pV\subseteq F_pV$, although translation raises the weight in $V$ by one. For example, $T^ju\in F_hV\cap V_{h+j}$ for $u\in U\cap V_h$. The length bound below concerns the number $\ell$ of factors. By \cite[Theorem 4.11]{Li}, this is a good increasing filtration: \[ (F_pV)_mF_qV\subseteq \begin{cases} F_{p+q}V,&m<0,\\ F_{p+q-1}V,&m\ge0. \end{cases} \] Thus $A=\gr_FV$ is a commutative differential algebra. For homogeneous $u\in U$, let $\bar u$ be its class in $F_{\wt u}V/F_{\wt u-1}V$. The elements $T^j\bar u$, $j\ge0$, generate $A$, where $T$ denotes the induced derivation. Each $F_pV$ is graded by the original weight grading and is $G$-stable. The two degrees are related by \begin{equation}\label{eq:filtrationdrop} V_w\subseteq F_wV,\qquad C_2(V)\cap V_w\subseteq F_{w-1}V. \end{equation} The first inclusion follows from \eqref{eq:monomialweight}. For $a\in V_h$ and $b\in V_k$, the filtration properties give $a_{-2}b=(Ta)_{-1}b\in F_{h+k}V$. This vector lies in $V_{h+k+1}$, proving the second inclusion. \begin{lemma}[Length bound]\label{lem:length} Every class in $Q$ is represented by a linear combination of monomials \eqref{eq:monomials} in $U$ of length at most \[ L=r\nu-1. \] \end{lemma} \begin{proof} Let $\mathfrak a\subset A$ be the ideal generated by the symbols of elements of $U$. We first show \begin{equation}\label{eq:idealnil} \mathfrak a^{\nu}=0. \end{equation} Take $\nu$ homogeneous elements of $U$, of total weight $w$. Their normally ordered product lies in $C_2(V)$ by \eqref{eq:nu}, so its symbol in filtration degree $w$ vanishes by \eqref{eq:filtrationdrop}. This proves \eqref{eq:idealnil}. Consider the commutative algebra \[ \bar A=A/A\,T(A^G),\qquad J=(\bar A)_{>0}, \] where the subscript denotes positive filtration degree. We claim \begin{equation}\label{eq:invnil} (J^G)^{\nu}=0. \end{equation} Taking invariants gives \[ \bar A^G\simeq A^G/\bigl(A^G T(A^G)\bigr), \] since projecting the coefficients in $\sum_j a_jT(f_j)$, $f_j\in A^G$, onto invariants shows that $(A\,T(A^G))^G=A^G T(A^G)$. Every positive-degree invariant is obtained by applying $P_G$ to a sum of monomials in $T^j\bar u$. The identity \[ P_G((Ta)b)=T P_G(ab)-P_G(a(Tb)) \] gives, by iteration, \[ P_G((T^k\bar u)b)\equiv(-1)^kP_G(\bar u\,T^kb) \pmod{A\,T(A^G)}. \] Since $U$ is $G$-stable, the right side belongs to $\mathfrak a\cap A^G$. Thus every element of $J^G$ has a representative in $\mathfrak a\cap A^G$. Products of $\nu$ such representatives vanish by \eqref{eq:idealnil}, proving \eqref{eq:invnil}. Lemma~\ref{lem:orbit} now gives \[ J^{r\nu}\subseteq\bar A(J^G)^{\nu}=0. \] Thus $\bar A$ is spanned by monomials in $T^j\bar u$ of length at most $r\nu-1$. Give $D_B$ and $Q$ the induced filtrations. Taking invariants gives $\gr_FB=A^G$. If $b\in B\cap F_pV$, $v\in F_sV$, and $\sigma_j$ denotes the class in $F_jV/F_{j-1}V$, then \[ \sigma_s(v)\,T\sigma_p(b) =\sigma_{p+s}(b_{-2}v)\in\gr_FD_B. \] Consequently \begin{equation}\label{eq:gradedquotient} A\,T(A^G)\subseteq\gr_FD_B, \qquad \bar A\twoheadrightarrow\gr_FQ \end{equation} as graded vector spaces. Lift a homogeneous spanning expression in $\bar A$ to normally ordered products of derivatives of elements of $U$. Subtracting these lifts from a representative in $Q$ lowers its filtration degree. Induction on filtration degree proves the length bound; the lifts may be chosen to preserve the original weight. \end{proof} It remains to bound the heights of these monomials. We do this by bounding the weights in $Q$ of coefficients in products of a fixed number of fields. \section{The differential equations}\label{sec:fields} \subsection{ \texorpdfstring{$n$}{n}-point functions} Fix $n\ge1$ and finite-dimensional homogeneous $G$-stable subspaces $E_i\subset V_{h_i}$, $1\le i\le n$, and put $E=\bigotimes_{i=1}^nE_i$. For the products $Y(a_1,z_1)\cdots Y(a_n,z_n)\1$, set \begin{equation}\label{eq:Delta} \Delta(z)=\prod_{i0$, there is a homogeneous $w\in V_k$, $k>0$, such that \[ u_{h+k-1}w\ne0\quad\text{in }\C\1. \] \end{lemma} \begin{proof} Let $\pi_0:V\to V_0=\C\1$ be the graded projection, and put \[ J=\{v\in V:\pi_0Y(a,z)v=0\text{ for all }a\in V\}. \] The grading shows that $J$ is graded. Let $a,b,v$ be homogeneous with $v\in J$, and set $N=\wt a+\wt b$. Every coefficient of $\pi_0Y(Y(a,x)b,t)v$ vanishes. The two-field case of \eqref{eq:productiterate}, and the fact that $\pi_0$ commutes with $\tau_{\{1,2\}}$, give \[ \tau_{\{1,2\}}\bigl(\pi_0 (z_1-z_2)^N Y(a,z_1)Y(b,z_2)v\bigr)=0. \] The expression inside the map belongs to $\mathcal L_{V_0}(z_1,z_2)$. Injectivity of the map makes it zero. The scalar factor $(z_1-z_2)^N$ is invertible in the original ordered space $V_0((z_1))((z_2))$, so $\pi_0Y(a,z_1)Y(b,z_2)v=0$ there. Taking coefficients shows that $b_mv\in J$ for all $b,m$. Also $\pi_0T=0$ and translation covariance give \[ \pi_0Y(a,z)Tv=-\partial_z\pi_0Y(a,z)v=0. \] Thus $J$ is stable under all left modes and $T$, hence is an ideal by skew symmetry. It does not contain $\1$, and simplicity gives $J=0$. Thus $a_{h+k-1}u\in\C^\times\1$ for some homogeneous $a\in V_k$. Necessarily $k>0$, since $Y(\1,z)u=u$. Skew symmetry $Y(u,z)a=e^{zT}Y(a,-z)u$ gives \[ u_{h+k-1}a=(-1)^{h+k}a_{h+k-1}u. \] The terms involving positive powers of $T$ have zero vacuum component. Take $w=a$. \end{proof} Assume $r>1$. For each $g\ne1$, choose a homogeneous $a_g$ with $ga_g\ne a_g$, and set $u_g=(1-g)a_g$. By Lemma~\ref{lem:detect}, we can choose and rescale a homogeneous $w_g$ so that \begin{equation}\label{eq:pairunit} (u_g)_{h_g+k_g-1}w_g=\1, \qquad h_g=\wt u_g,\quad k_g=\wt w_g. \end{equation} All higher modes vanish by the grading. Let \[ U_g=\Span\{ha_g:h\in G\},\qquad W_g=\Span\{hw_g:h\in G\}. \] These are finite-dimensional homogeneous $G$-spaces, and $a_g,u_g\in U_g$. Fix finite-dimensional homogeneous $G$-stable subspaces $E_i\subset V_{h_i}$, $1\le i\le k$, and put $E=\bigotimes_iE_i$. We adjoin $2r-1$ copies of each pair $(u_g,w_g)$ by setting \begin{align*} \widetilde E&=E\otimes \bigotimes_{g\ne1}(U_g\otimes W_g)^{\otimes(2r-1)},\\ \xi&=\bigotimes_{g\ne1}(u_g\otimes w_g)^{\otimes(2r-1)}. \end{align*} Write the pairs in $\xi$ as $(u_t,w_t)$, $1\le t\le N$, where \begin{equation}\label{eq:N} N=(r-1)(2r-1). \end{equation} \subsection{The polynomial relation module} We encode the differential equations of Lemma~\ref{lem:PDE} as relations over a polynomial ring. The tensor product $\widetilde E$ has $n=k+2N$ factors; let $d_I$ be the corresponding orders. Put $R=\C[p_1,\ldots,p_n]$ and $p_I=\sum_{i\in I}p_i$. Define \begin{equation}\label{eq:M} M=\frac{R\otimes\widetilde E} {\displaystyle\sum_{\varnothing\ne I\subseteq[n]} R\,p_I^{d_I}P_I\widetilde E}. \end{equation} The variable $p_i$ represents differentiation in the variable of the $i$th field. Let $\mathcal N$ be the $R$-submodule of $M$ generated by the image of $E\otimes\xi$. It is finitely generated because $E$ is finite dimensional. \begin{lemma}[Support at the origin]\label{lem:support} For some integer $q_0\ge1$, \begin{equation}\label{eq:origin} (p_1,\ldots,p_n)^{q_0}\mathcal N=0. \end{equation} \end{lemma} \begin{proof} For $\lambda=(\lambda_1,\ldots,\lambda_n)\in\C^n$, write $\m_\lambda=(p_1-\lambda_1,\ldots,p_n-\lambda_n)$ for the corresponding maximal ideal of $R$. We show that the localization $\mathcal N_{\m_\lambda}$ vanishes whenever $\lambda\ne0$. Put $S=\{i:\lambda_i\ne0\}$. Suppose first that $|S|\ge2r-1$. Choose a line through the origin which contains none of the nonzero $\lambda_i$. One of its open half-planes contains at least $r$ of these numbers; let $J$ be the corresponding set of $r$ indices. Every nonempty subset sum of these numbers is nonzero, because its projection onto a common real direction is strictly positive. Hence $p_I$ is a unit in $R_{\m_\lambda}$ for every nonempty $I\subseteq J$. The relations in \eqref{eq:M} kill $P_I\widetilde E$ in $M_{\m_\lambda}$ for all such $I$. Equation \eqref{eq:invariantsubsets}, with the remaining tensor factors fixed, shows that these images span $\widetilde E$. Thus $M_{\m_\lambda}=0$. Now suppose $1\le|S|\le2r-2$, and choose $j\in S$. For each $g\ne1$, one of the $2r-1$ copies of $u_g$ in $\xi$ has an index $i_g\notin S$. The indices $i_g$ are distinct. Apply \eqref{eq:difference} to the factor indexed by $j$ and these copies of $u_g=(1-g)a_g$, leaving the remaining factors fixed. By linearity, for every $e\in E$ this gives \[ e\otimes\xi\in \sum_{I\subseteq S^c}P_{\{j\}\cup I}\widetilde E. \] For each such $I$, $p_{\{j\}\cup I}(\lambda)=\lambda_j\ne0$. The defining relations therefore kill all the indicated images in $M_{\m_\lambda}$. The generators of $\mathcal N$ vanish there, so $\mathcal N_{\m_\lambda}=0$. Since $\mathcal N$ is finitely generated, its support is the zero set of its annihilator. The preceding localizations and the Nullstellensatz imply \[ (p_1,\ldots,p_n)\subseteq\sqrt{\Ann_R\mathcal N}; \] see \cite[Tags 00L2 and 00FV]{Stacks}. Choose $e_i\ge1$ with $p_i^{e_i}\mathcal N=0$. Every monomial of total degree at least $q_0=1+\sum_i(e_i-1)$ is divisible by one of these powers, which proves \eqref{eq:origin}. If $\mathcal N=0$, take $q_0=1$. \end{proof} \subsection{Proof of the weight bound} \begin{proof}[Proof of Proposition~\ref{prop:cutoff}] By linearity, it suffices to take $a_i$ in homogeneous components $E_i=X\cap V_{h_i}$. We fix $k$ and $h_1,\ldots,h_k$ and prove a bound independent of the $a_i$ and the mode indices. Since $k\le s$ and $X$ has finitely many weights, the maximum of these bounds and $0$ gives $H(X,s)$. If $r=1$, every singleton projection is the identity. The equations in Lemma~\ref{lem:PDE} bound the degree in each variable of $H_Q$, so \eqref{eq:degreeweight} gives the result after division by $\Delta$ in the ordered Laurent expansion. Assume $r>1$ and form $\widetilde E$ as above. Give the original fields variables $z_1,\ldots,z_k$ and the $t$th auxiliary pair variables $x_t,y_t$. Put $H_t=\wt u_t$ and $K_t=\wt w_t$. For auxiliary states $b_t,c_t$ in their respective tensor factors, list all $n=k+2N$ states and variables in tensor order: \[ \begin{aligned} (A_1,\ldots,A_n)&=(a_1,\ldots,a_k,b_1,c_1,\ldots,b_N,c_N),\\ (\zeta_1,\ldots,\zeta_n)&=(z_1,\ldots,z_k,x_1,y_1,\ldots,x_N,y_N). \end{aligned} \] Let $\eta_i$ be the weight of the $i$th tensor factor and write $\widetilde e=A_1\otimes\cdots\otimes A_n$. Define \begin{equation}\label{eq:tildeDelta} \widetilde\Delta(z,x,y) =\prod_{1\le iH:=\max\{0,\widetilde h+q_0-1-\widetilde D\}. \end{equation} It remains to transfer this vanishing to the original product. Equation \eqref{eq:pairunit} and the grading give \[ d^{H_t+K_t}Y(u_t,d)w_t=\1+dA_t(d),\qquad A_t(d)\in V[[d]]. \] Apply Lemma~\ref{lem:iterate} to this auxiliary pair, with common coordinate $y_t$ and relative coordinate $d=x_t-y_t$. The full factor $\widetilde\Delta$ remains present. The internal series for the pair is $\1+dA_t(d)$, so the constant coefficient in $d$ of its iterated field is $Y(\1,y_t)=\id$. The cross factors are polynomials in $d$ with nonnegative powers, and their constant coefficients are their values at $x_t=y_t$. Thus taking $[d^0]$ removes the pair and evaluates these factors on that diagonal. On the left this is the ordinary polynomial substitution $x_t=y_t$, since $F$ is already a polynomial. Apply this argument to the auxiliary pairs in turn. Previously extracted scalar factors remain polynomials with nonnegative powers of each new relative coordinate. We obtain \begin{equation}\label{eq:collision} F(z,y,y)=C(z,y)H_Q(z), \end{equation} where $H_Q$ is defined by $E_1,\ldots,E_k$ and \begin{equation}\label{eq:C} C(z,y)=\prod_{i=1}^k\prod_{t=1}^N (z_i-y_t)^{2h_i+H_t+K_t} \prod_{1\le tH$, project \eqref{eq:collision} to $Q_w$. Equation \eqref{eq:weightbound} gives $C(z,y)(H_Q)_w(z)=0$. Multiplication by the nonzero scalar polynomial $C$ is injective on vector-valued polynomials, so $(H_Q)_w=0$. Finally, $\Delta$ is invertible in the original ordered space $Q((z_1))\cdots((z_k))$. Dividing by it shows that every coefficient of weight $w$ in the original product vanishes. \end{proof} \section{Cofiniteness and its consequences}\label{sec:finish} \begin{proof}[Proof of Theorem~\ref{thm:main}] Let $U$ be the finite-dimensional graded $G$-stable strong generating space of Section~\ref{subsec:generators}, and choose $\nu$ with $R_{V,+}^{\nu}=0$. Lemma~\ref{lem:length} spans $Q$ by monomials of length at most $L=r\nu-1$. Apply Proposition~\ref{prop:cutoff} with $X=U$ and $s=\max\{1,L\}$. With $H=H(U,s)$, we obtain $Q_w=0$ for $w>H$. Each $V_w$ is finite dimensional, and hence so is $Q=\bigoplus_{w\le H}Q_w$. Equation \eqref{eq:exact} now gives $\dim B/C_2(B)<\infty$. More explicitly, for a homogeneous basis $u^1,\ldots,u^d$ of $U$, \[ Q=\Span\left\{q(\1),\ q(u^{i_1}_{-n_1}\cdots u^{i_\ell}_{-n_\ell}\1): 1\le\ell\le L,\ 1\le i_j\le d,\ 1\le n_j\le H\right\}. \] The height bound follows from \eqref{eq:monomialweight}. \end{proof} Recall that a vertex operator algebra is \emph{strongly rational} if it is simple, of CFT type, self-contragredient, rational, and $C_2$-cofinite. Here rationality means complete reducibility of admissible modules. \begin{samepage} \begin{corollary}\label{cor:rational} Let $V$ be a strongly rational vertex operator algebra and $G$ a finite group of automorphisms. Then $V^G$ is strongly rational. \end{corollary} \begin{proof} Theorem~\ref{thm:main} gives $C_2$-cofiniteness of $V^G$. McRae's theorem \cite[Corollary 4.23]{McRae} then gives strong rationality. \end{proof} \end{samepage} \begin{proof}[Proof of Theorem~\ref{thm:sfg}] Choose a finite-dimensional graded $G$-stable strong generating space $X\subset V_{>0}$ and use the filtration \eqref{eq:goodfiltration}, with $X$ in place of $U$: each generating state contributes its weight, independently of its mode index. Put $S=\Sym(\bigoplus_{j\ge0}X^{(j)})$, where $X^{(j)}$ is a copy of $X$ representing $T^jX$. The equivariant surjection $S\twoheadrightarrow\gr_FV$ induces \[ S^G\twoheadrightarrow(\gr_FV)^G=\gr_FB \] by exactness of taking $G$-invariants. Each polynomial involves only finitely many copies $X^{(j)}$. Applying Noether's degree bound \cite{Noether} to their direct sum shows that $S^G$ is generated by the elements $P_G(m)$, where $m$ ranges over monomials with at most $r$ factors. Choose a finite homogeneous basis $\mathcal X$ of $X$. Lift these invariant elements by applying $P_G$ to the corresponding right-associated normally ordered products of derivatives of elements of $\mathcal X$. Induction on filtration degree expresses every state of $B$ as a linear combination of normally ordered products of these lifts. Their classes therefore generate $R_B$ as an algebra. By Proposition~\ref{prop:cutoff} with $s=r$, every lift of weight greater than $H_X=H(X,r)$ belongs to $D_B\cap B=C_2(B)$. Only finitely many lifts have weight at most $H_X$: each uses at most $r$ elements of the finite basis $\mathcal X$, and their derivative orders are bounded. Their positive-weight members span $B_{>0}/C_1(B)$: products of two positive-weight states and $C_2(B)$ lie in $C_1(B)$. Weight induction, or \cite[Theorem 4.10]{Li}, now gives finite strong generation. \end{proof} \begin{remark} In Theorems~\ref{thm:main} and~\ref{thm:sfg}, simplicity can be replaced by the following condition: for each $g\in G$ acting nontrivially on $V$, there are homogeneous $a_g,w_g$ such that $u_g=(1-g)a_g\ne0$ and \[ (u_g)_{\wt u_g+\wt w_g-1}w_g\in\C^\times\1. \] These pairs replace those supplied by Lemma~\ref{lem:detect}. \end{remark} \appendix \setcounter{equation}{0} \renewcommand{\theequation}{A.\arabic{equation}} \section{Proof of product--iterate associativity}\label{app:associativity} We verify the identity used in \eqref{eq:productiterate} directly from locality. Let $a_i\in V_{h_i}$, $1\le i\le m$, and put \[ \Delta_m(z)=\prod_{i