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Higher Zhu action on the Omega spaces of an arbitrary weak module

Statement

Let V be a vertex operator algebra with finite-dimensional nonnegative integral state grades, allowing V0 larger than C1, and let M be any weak left V-module. For an integer $N \geq 0$ define $Ω_{N} (M) = {w \in M : a_{(m)} w = 0 whenever a is homogeneous and m \geq w t (a) + N} .$ For homogeneous a of weight r put $o (a) = a_{(r - 1)}$ , extending linearly in a. Every o(a) preserves $Ω_{N} (M)$ , every element of $O_{N} (V)$ acts by zero, and $o (a *_{N} b) w = o (a) o (b) w, o (1) w = w .$ When V is of CFT type, the linked associative-quotient theorem consequently makes $Ω_{N} (M)$ a unital left $A_{N} (V) - m o d u l e$ . The preservation, $O_{N} - a n n i h i l a t i o n$ and multiplicativity identities themselves hold in the full nonnegative scope just stated. No grading, local finiteness or semisimplicity of L(0) on M is assumed. This is the forward action; no converse rationality criterion is asserted.

Facts & Assumptions

Given: The full non-super conformal state-field setup and weak module in the Statement. Here the broader term “vertex operator algebra” means the following data, without the CFT-type condition $V_{0} = C 1$ .

[F1]

$V$ is a complex vector space with vacuum $1 \in V_{0}$ , translation $T$ , and state-field map $Y (a, z) = \sum_{j \in Z} a_{(j)} z^{- j - 1}$ satisfying the lower-truncation, vacuum/creation, translation-covariance, and locality axioms of Vertex algebra in the state–field convention. It has a conformal vector $ω \in V_{2}$ with $Y (ω, z) = \sum_{j \in Z} L (j) z^{- j - 2}$ and central charge $c \in C$ , where $[L (m), L (n)] = (m - n) L (m + n) + \frac{c}{12} (m^{3} - m) δ_{m + n, 0} {id}_{V}$ for all $m, n \in Z$ , and $L (- 1) = T$ . The operator $L (0)$ is diagonalizable and $V = ⨁_{r \geq 0} V_{r}$ , where $V_{r} = {a : L (0) a = r a}$ is finite dimensional. No condition $V_{0} = C 1$ is imposed. A weak left module $M$ is a complex vector space with a lower-truncated field map $Y_{M} (a, z) = \sum_{m \in Z} a_{(m)}^{M} z^{- m - 1}$ , vacuum action, and the full module Jacobi identity of Module for an ordinary vertex algebra; it has no grading or $L (0)$ -semisimplicity assumption. For homogeneous $a \in V_{r}$ , write $w t (a) = r$ .

[F2]

Taking the two residues of the module Jacobi identity in [F1] gives, for all integers $m, n$ , $[a_{(m)}^{M}, b_{(n)}^{M}] = \sum_{j \geq 0} (\binom{m}{j}) (a_{(j)} b)_{(m + n - j)}^{M}, (a_{(m)} b)_{(n)}^{M} = \sum_{t \geq 0} (- 1)^{t} (\binom{m}{t}) (a_{(m - t)}^{M} b_{(n + t)}^{M} - (- 1)^{m} b_{(m + n - t)}^{M} a_{(t)}^{M}) .$ The first sum is finite by state truncation; on each fixed vector the two terms of the iterate sum are separately finite by rightmost-mode truncation. For negative integer upper indices, $(\binom{m}{t}) = m (m - 1) \dots (m - t + 1) / t!$ . Translation covariance and the conformal-vector residues give $(T a)_{(m)}^{M} = - m a_{(m - 1)}^{M}$ , $[L_{M} (0), a_{(m)}^{M}] = (r - m - 1) a_{(m)}^{M}$ , and $w t (a_{(j)} b) = r + w t (b) - j - 1$ when the state product is nonzero. These are consequences of the stated Jacobi axioms; they do not require $V_{0} = C 1$ .

[F3]

For homogeneous $a \in V_{r}$ and $b \in V$ , and each $N \geq 0$ , the formulas used here are $a \circ_{N} b = \sum_{j = 0}^{r + N} (\binom{r + N}{j}) a_{(j - 2 N - 2)} b, a *_{N} b = \sum_{p = 0}^{N} (- 1)^{p} (\binom{N + p}{N}) \sum_{j = 0}^{r + N} (\binom{r + N}{j}) a_{(j - N - p - 1)} b,$ $D a = (T + L (0)) a = T a + r a, O_{N} (V) = {span}_{C} {a \circ_{N} b, D a : a, b \in V},$ where the circle expression is extended linearly over homogeneous first inputs. This is the full nonnegative-grade convention used locally; in the CFT-type case it agrees with Higher Zhu quotients and candidate products, and Associative structure of the higher Zhu quotients separately proves associativity and descent of the quotient product.

Proof

1.1givenF2algebra

Let a have weight r and b have weight s. For $m \geq s + N$ , commuting $b_{m}$ past $a_{r - 1}$ gives terms $(b_{j} a)_{m + r - 1 - j}$ with coefficient binom(m,j). Their state weight is s+r-j-1 and their mode index is at least this weight plus N. Each kills w in $Ω_{N} (M)$ , as does $a_{r - 1} b_{m}$ w. The commutator sum is finite by state truncation. Thus o(a) preserves $Ω_{N} (M)$ . Vacuum acts identically by the module vacuum identity.

1.2F1F2algebra

The zero mode of a circle generator is $\sum_{j = 0}^{r + N} (\binom{r + N}{j}) (a_{(j - 2 N - 2)} b)_{(r + s + 2 N - j)} .$ Apply the iterate identity. In every first term the b index is $r + s + 2 N - j + t \geq s + N$ , so that term kills w. In the second terms the b index is the fixed r+s-2-t. Values $t \geq r + N$ kill w through $a_{t}$ . For each remaining $t < r + N$ , the summed coefficient is a scalar multiple of $\sum_{j = 0}^{r + N} (- 1)^{j} (\binom{r + N}{j}) (\binom{j - 2 N - 2}{t}) = 0,$ the finite difference of order r+N of a polynomial in j of degree t. Thus every circle generator acts by zero. All sums can be exchanged here: j ranges over a finite set; in the first iterate sum $b_{r + s + 2 N - j + t} w$ is zero for all large t, and in the second $a_{t}$ w is zero for all large t. Also $o (T a + r a) = - r a_{r - 1} + r a_{r - 1} = 0$ . Therefore $o (O_{N} (V)) = 0$ . This includes $r + N = 0$ , where the second surviving t range is empty.

2.1F1F2step 1.2algebra

Expand $o (a *_{N} b)$ using F1 and the iterate identity. For a fixed outer star index p, the second iterate terms again have b index r+s-2-t independent of j. They vanish for $t \geq r + N$ , and for $t < r + N$ their sum over j is the finite difference of the degree-t polynomial binom(j-N-p-1,t), hence zero. In the first terms put the a index $k = j - N - p - 1 - t$ . Its b index is r+s-2-k. The Omega cutoff on b eliminates $k \leq r - N - 2$ . The inequalities $j \leq r + N$ and $t \geq 0$ give $k \leq r - 1$ . Thus the only possible terms are $a_{(r - 1 - a)} b_{(s - 1 + a)} w, 0 \leq a \leq N .$ Write $l = r + N - j$ . Then $t = a - p - l$ , so their coefficient is exactly $C_{a} = \sum_{p = 0}^{a} (- 1)^{p} (\binom{N + p}{N}) \sum_{l = 0}^{a - p} (\binom{r + N}{l}) (- 1)^{a - p - l} (\binom{r - p - l - 1}{a - p - l}) .$ This expression accounts for every surviving first term; it uses no cancellation of unknown infinite sums.

3.1F1step 1.1step 1.2step 2.1algebra∎

The inner sum is the coefficient of $x^{a - p}$ in $(1 - x)^{r - p - 1} {(1 + \frac{x}{1 - x})}^{r + N} = (1 - x)^{- N - p - 1} .$ It therefore equals binom(N+a,a-p). Consequently $C_{a} = \sum_{p = 0}^{a} (- 1)^{p} (\binom{N + p}{N}) (\binom{N + a}{a - p}) = (\binom{N + a}{a}) \sum_{p = 0}^{a} (- 1)^{p} (\binom{a}{p}) = {\begin{cases} 1 & a = 0, \\ 0 & a > 0. \end{cases}$ Only o(a)o(b)w remains, proving multiplicativity. Step 1.1 and step 1.2 give the claimed full-scope identities. In the CFT-type case F1 then gives the claimed unital $A_{N} (V) - a c t i o n$ . No existence/induction theorem or nonresonance of conformal-weight differences was used.

Depends on

Dependency tree · 2 levels

Sources

Dong, Li and Mason, Vertex operator algebras and associative algebras