Finite mode windows for a nonnegative C2-cofinite VOA

Fix the total operator degree d, and induct on total state weight $p\ge 0$, then on word length l. Replacing a homogeneous state by a representative plus C2-products lowers p in every iterate term, since $\mathrm{wt}(a_{-2}b)=\mathrm{wt}(a)+\mathrm{wt}(b)+1$ while the two replacement states have total wt(a)+wt(b). The replacement is finite on its fixed tail by the mode truncations in F1 and F2. A vacuum mode either vanishes or removes a letter, lowering length. Nonvacuum weight-zero representatives remain in A; they are NOT silently discarded. At $p=0$ there are no C2 replacement terms, since those would have positive weight, and the same length and tuple induction below applies.

Sort adjacent indices by the commutator. Each commutator state has smaller total state weight by j+1, so the error is already reduced by the p induction. After sorting, a last index $\ge N$ kills the cyclic vector. For fixed p,l,d the surviving sorted tuples have fixed sum p-l-d and every index $\le N-1$, hence every index is also at least p-l-d-(l-1)(N-1). There are only finitely many tuples, with a finite lexicographic order.

At a repeated adjacent index i use $(a_{-1}b{)}_{(2i+1)}={\sum }_{t\ge 0}(a_{(-1-t)}{b}_{(2i+1+t)}+b_{(2i-t)}{a}_t)$. For $i<0$ the first sum contains exactly one desired repeated pair at $t=-i-1$; for $i\ge 0$ the second contains exactly one reversed pair at $t=i$. Commute that reversed pair, whose error has smaller p. Every other pair has unequal indices summing to 2i, so after sorting it has a smaller first index and a lexicographically smaller full tuple, or dies by the N cutoff. The collapsed state $a_{-1}b$ has the same p and shorter word length. Its C2 replacement lowers p as before. Induction on this finite tuple set, after the outer p and length inductions, therefore removes all equal indices with coefficient ONE. The argument applies at $p=0$ as well: commutator states of negative weight are zero, and the collapsed weight-zero representative is handled by shorter length.

These reductions cover every finite p and l, not a fixed truncated filtration only. In a resulting strict word, $i_j\le N-1-(l-j)$, while $\mathrm{wt}(a_j)\ge 0$. Consequently $d={\sum }_j(\mathrm{wt}(a_j)-i_j-1)\ge l(l-1)\mathrm{/}2-Nl$. This quadratic is bounded below and tends to positive infinity. At fixed d it bounds l. For each such l there are finitely many states and their fixed total weight fixes the index sum; step 2.1 bounds all indices. Hence the full unbounded-filtration degree span is finite dimensional and is zero below the quadratic minimum.

For $k\ge 2$ let $C_k(V)=\mathrm{span}\{a_{-k}b\}$. Apply the spanning result to V generated by its vacuum. Nonnegative modes kill the vacuum, so all surviving indices are negative. A word whose smallest index is -m with $m\ge k$ belongs to $C_k$, because $(T^{(m-k)}a{)}_{(-k)}=k(k+1)\mathrm{.}\mathrm{.}\mathrm{.}(m-1)a_{(-m)}$. Otherwise all distinct indices lie in the finite interval -k+1,...,-1; finitely many such words remain. Thus $C_k(V)$ is homogeneous, has finite codimension in V, and contains all sufficiently high state-weight spaces. Modulo $O_N(V)$, the circle identity reduces $a_{(-2N-2)}b$ to $-{\sum }_{j\ge 1}\mathrm{binom}(\mathrm{wt}(a)+N,j)a_{(j-2N-2)}b$. Each term has smaller state weight. Induction on nonnegative state weight therefore reduces $V\mathrm{/}{O}_N$ to finitely many low grades. This proves its finite dimension, including nonvacuum weight-zero states. No ideal property of $O_N$ is used.

Let $F$ be the free complex vector space on the set ${\bigcup }_{\mathrm{\ell }\ge 0}(V\times \mathbb{Z}{)}^{\mathrm{\ell }}$ of finite mode words, including the empty word, and let $P_{a,m}$ prepend the letter $(a,m)$. Let $\mathcal{K}$ consist of all subspaces $K\subseteq F$ stable under every $P_{a,m}$ for which the induced operators on $F\mathrm{/}K$ are linear in $a$ and satisfy the weak-module axioms. This is a set, as a subset of the power set of $F$; it involves no basis or isomorphism-representative choices. For every cyclic weak module generated by $w$, evaluation of a word on $w$ gives a surjection $F\to Vw$ intertwining all prepend operators. Its kernel belongs to $\mathcal{K}$, and the induced isomorphism identifies $Vw$ with that quotient. For fixed $N\ge 0$, map $a\in V$ to the family of its zero-mode operators in ${\prod }_{K\in \mathcal{K}}{\mathrm{End}}_{\mathbb{C}}({\mathrm{\Omega }}_N(F\mathrm{/}K))$. By [F4] these operators preserve the indicated spaces, kill $O_N(V)$, take the vacuum to the identity family, and multiply according to $a{\ast }_{N}b$. The image $Z_N$ is therefore an associative unital algebra of dimension at most $\dim V\mathrm{/}{O}_N(V)$, hence finite. Every $w\in {\mathrm{\Omega }}_N(M)$ belongs to its cyclic weak submodule, and $w\in {\mathrm{\Omega }}_N(Vw)$, so this family detects exactly the state zero-mode identities on all weak ${\mathrm{\Omega }}_N$ spaces. Thus the action on every ${\mathrm{\Omega }}_N(M)$ factors through $Z_N$. In particular a polynomial annihilating $o(\omega )$ in this finite-dimensional algebra annihilates $L_M(0)$ on every weak ${\mathrm{\Omega }}_N(M)$, uniformly in $M$.