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Strong local grading and admission of transported coefficient modules

Statement

Let $V$ be a strongly $A$ -graded Möbius or conformal vertex algebra, $W_{1}, W_{2}$ strongly $\tilde{A}$ -graded generalized modules and $z \neq 0$ . On $(W_{1} \otimes W_{2})^{*}$ use the explicitly transported fields and three Möbius operators. Write $D = L_{P (z)}^{'} (0)$ . Let $λ_{n}$ , $n$ in an arbitrary subset of $R$ , be finite sums of auxiliary-homogeneous generalized $D$ -eigenvectors of weight $n$ , with $(D - n)^{K} λ_{n} = 0$ for one integer $K \geq 1$ . Let $E_{n}$ be the smallest doubly graded subspace containing $λ_{n}$ and stable under all transported modes and the three Möbius operators. Suppose, exactly as in the category-admission clause of the expansion condition, that for each $n$ there is a strongly graded generalized module $X_{n}$ and an injective linear map $e_{n} : E_{n} \to X_{n}$ preserving both gradings and intertwining every transported mode and every Möbius operator. Suppose the smallest such doubly graded stable space $M$ containing all the $λ_{n}$ has finite-dimensional components and is lower truncated in each fixed auxiliary-degree integer weight string. Then $M$ with the transported actions is a strongly graded generalized module, all its generalized weights are real, and $(D - D_{s})^{K} = 0$ on $M$ . In particular its vertex Jacobi identity is an actual finite coefficient identity. No finiteness of the set of weight cosets modulo integers, no CFT-type condition and no global lower bound is required. This lemma does not assert full $P (z)$ -compatibility from the module Jacobi identity; that is a separate condition.

Statement references

Strongly graded generalized module in the HLZ setting

P(z)-compatibility subspace and local grading restriction

Facts & Assumptions

Given: The transported actions, the generalized eigenvectors, the explicit injective mode-intertwining maps, and the local grading bounds of the Statement.

[F1]

The strong grading in Strongly graded generalized module in the HLZ setting means finite-dimensional $(β, h)$ -components and a lower bound in each fixed $(β, h + Z)$ string; it does not require finitely many strings or a bound uniform in auxiliary degree.

[F2]

The transported formulas in P(z)-compatibility subspace and local grading restriction are linear maps on the algebraic dual coefficient by coefficient. Their rational residue derivation gives $[D, v_{m}^{'}] = (r - m - 1) v_{m}^{'}$ for a state of weight $r$ , auxiliary shift $α$ for a state of auxiliary degree $α$ , and $[D, L^{'} (j)] = - j L^{'} (j)$ . These identities also follow on each $E_{n}$ by injecting it into $X_{n}$ and using the corresponding identities there; thus their use here does not assume a separate unproved compatibility-module theorem.

Proof

1.1givenF1F2algebra

On $E_{n}$ , the injective map $e_{n}$ transports the action to its stable image in the actual module $X_{n}$ . Hence lower truncation, vacuum, the three Möbius brackets and commutators, and the full vertex Jacobi identity hold on $E_{n}$ . For Jacobi this can be checked one coefficient at a time: every coefficient is a finite sum because the inner modes truncate and the state products truncate; its image under $e_{n}$ is the identical coefficient in $X_{n}$ , which is zero. Injectivity makes the original coefficient zero. No infinite weight sum is involved in this argument.

2.1givenF2step 1.1induction

Each finite word in transported modes and Möbius operators shifts the generalized weight of its input by an integer and its auxiliary degree by a finite sum in $A$ . By the commutators in [F2], if the input has generalized weight $n$ and nilpotence exponent at most $K$ , the output has generalized weight $n + d$ , $d \in Z$ , and the same exponent bound: for one mode this is $(D - n - r + m + 1)^{K} v_{m}^{'} λ = v_{m}^{'} (D - n)^{K} λ = 0$ , and the Möbius case is identical. Induction on word length proves it for every generated vector. Since $E_{n}$ includes the finite auxiliary-homogeneous components of $λ_{n}$ , these words already span a doubly graded stable subspace. Therefore $M = \sum_{n} E_{n}$ , where an algebraic sum consists only of finite sums of vectors. Its weights are real and the bound $(D - D_{s})^{K} = 0$ holds on all of $M$ . This does not restrict the number of distinct real weight cosets.

3.1givenF1F2step 2.1algebra

Lower truncation of the field action on $M$ follows both from the finite sum in Step 2.1 and from the given local grading restriction. Explicitly, if $λ$ is in component $(β, h)$ and $v$ has degree $(α, r)$ , then $v_{m}^{'} λ$ lies in $(β + α, h + r - m - 1)$ . As $m \to + \infty$ these weights go down the single fixed integer string with auxiliary degree $β + α$ ; that string vanishes sufficiently far below by [F1]. A vector has only finitely many components, so taking the maximum of its finitely many bounds gives lower truncation for every vector. No bound over all auxiliary degrees is needed.

4.1givenF1step 1.1step 2.1step 3.1algebra∎

Vacuum, Jacobi and all Möbius identities on $M$ now follow by linearity from their identities on each $E_{n}$ , since every vector of $M$ is a finite sum from such spaces. Its finite-dimensional components and lower truncation in each weight string are precisely the assumed all-terms grading bounds. Together with the generalized-eigenspace decomposition and auxiliary-degree preservation established above, these are all the strongly graded generalized-module axioms. The conclusion is an actual module construction under the explicit cyclic admission hypothesis, rather than an inference from a rational-current Lie bracket. Full $P (z)$ -compatibility is not among these deductions and must still be proved for the coefficient vectors by the analytic compatibility argument.

Depends on

Dependency tree · 2 levels

Sources

Huang, Lepowsky and Zhang, Logarithmic tensor category theory, VI