Regularity of positive-integral-level affine VOAs

Let $W\subseteq M$ be the sum of all simple ordinary submodules. Taking ${\mathrm{Res}}_{z_0}$ in module Jacobi and then the coefficient residues in $z_1,z_2$ gives the mode commutator formula $[u_{(m)}^M,a_{(n)}^M]={\sum }_{j\ge 0}\left(\genfrac{}{}{0px}{}{m}{j}\right)(u_{(j)}a{)}_{(m+n-j)}^M$. For homogeneous $a$ of conformal weight $r$, take $u=\omega$ and $m=1$: because ${\omega }_{(0)}a=Ta$ and ${\omega }_{(1)}a=ra$, this gives $[L_M(0),a_{(n)}^M]=(Ta{)}_{(n+1)}^M+r{a}_{(n)}^M$. Creation gives $Ta=a_{(-2)}\mathbf{1}$. The module Jacobi iterate with $(a,\mathbf{1})$ and $Y_M(\mathbf{1},z)=I$ gives $Y_M(Ta,z)={\mathrm{Res}}_x({\iota }_{x,z}-{\iota }_{z,x})(x-z{)}^{-2}{Y}_M(a,x)={\mathrm{\partial }}_z{Y}_M(a,z)$ by the formal delta/residue identity in [F4]. Hence $(Ta{)}_{(n+1)}^M=-(n+1)a_{(n)}^M$, so every homogeneous state mode shifts $L_0$ eigenvalues by the integer $r-n-1$. For each simple ordinary submodule $S={⨁}_{h\in \mathbb{C}}{S}_h$, the sum ${⨁}_{h\in \xi }{S}_h$ over any coset $\xi \in \mathbb{C}\mathrm{/}\mathbb{Z}$ is a submodule, so simplicity leaves exactly one nonzero coset. Its eigenvalues have real parts bounded below by [F4], so the corresponding integer-indexed subset has a least element $h_S$ and the spectrum is contained in $h_S+{\mathbb{Z}}_{\ge 0}$. By [F1], only finitely many isomorphism classes of simple ordinary modules occur at this level, and isomorphic modules have the same $L_0$ spectrum. Hence the spectrum of $W$ lies in finitely many such cosets. Every vector of $W$ is a finite sum of vectors from ordinary modules, each a finite sum of $L_0$ eigenvectors; therefore $L_0$ is locally finite on $W$, and $W$ is the direct sum of its generalized eigenspaces. Choosing one lowest representative for each distinct coset shows that $W$ satisfies [F3] and belongs to VOA category $\mathcal{O}$.

Suppose $M\mathrm{/}W\mathrm{\ne }0$. It is a weak module, so [F2] supplies a simple ordinary submodule $S\subseteq M\mathrm{/}W$. Let $M_1$ be its preimage in $M$, giving $0\to W\to M_1\to S\to 0$. For any $m\in M_1$, its image in $S$ is a finite sum of $L_0$ eigenvectors; choose a monic polynomial $p$ annihilating that image. Then $p(L_0)m=w\in W$. Since $L_0$ is locally finite on $W$, the span $W_m=\mathrm{span}\{L_0^{q}w:q\ge 0\}$ is finite-dimensional and invariant. If $d=\mathrm{deg}p$, the finite-dimensional space $W_m+\mathrm{span}\{m,L_{0}m,\dots ,L_0^{d-1}m\}$ is $L_0$-invariant by the monic recurrence $p(L_0)m=w$ (for $d=0$, $m=w\in W$). Thus $L_0$ is locally finite on $M_1$. Every vector therefore lies in a finite-dimensional invariant subspace, and [F5] decomposes each such subspace into generalized eigenspaces; these give the algebraic generalized-eigenspace decomposition of $M_1$. Every generalized eigenvalue of $M_1$ lies in the support of $W$ or $S$: a generalized eigenvector either maps to a nonzero generalized eigenvector in $S$ with the same eigenvalue or lies in $W$. Their supports occupy finitely many integer cosets bounded below, so $M_1$ is in VOA category $\mathcal{O}$ by [F3].

The induced grading in [F3] makes $M_1$ admissible. By [F1], $M_1$ is a direct sum of simple ordinary submodules. Each is also a simple ordinary submodule of $M$ and hence is contained in $W$ by its definition. This forces $M_1=W$, contradicting $M_1\mathrm{/}W=S\mathrm{\ne }0$. Therefore $M=W$, so $M$ is a sum of simple ordinary submodules.

By step 3.1, $M=W$, and step 1.1 puts $W$ in VOA category $\mathcal{O}$. Its induced grading in [F3] makes $M$ admissible. With the Axiom of Choice [A1] required by [F1], the admissible-module decomposition in [F1] expresses $M$ as an algebraic direct sum of simple ordinary modules. By Regular conformal vertex algebra, this is regularity.