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Higher Zhu quotients and candidate products

Definition

Let $V = ⨁_{r \geq 0} V_{r}$ be a conformal vertex algebra of CFT type. For each integer $n \geq 0$ and homogeneous $u, v \in V$ , define $u \circ_{n} v = {Res}_{z} Y (u, z) v \frac{(1 + z)^{w t (u) + n}}{z^{2 n + 2}}, D u = L (- 1) u + L (0) u, O_{n} (V) = {span}_{C} {u \circ_{n} v, D u : u, v \in V} .$ Define the candidate operation on representatives $u *_{n} v = \sum_{m = 0}^{n} (- 1)^{m} (\binom{m + n}{n}) {Res}_{z} Y (u, z) v \frac{(1 + z)^{w t (u) + n}}{z^{n + m + 1}} .$ Extend $\circ_{n}$ and $*_{n}$ uniquely to bilinear maps $V \times V \to V$ by summing over the finite homogeneous decompositions of both inputs; $D = T + L (0)$ is linear. The span defining $O_{n} (V)$ uses these extensions. Write $A_{n} (V) = V / O_{n} (V)$ for the vector-space quotient; the operation $*_{n}$ descends to an associative unital algebra only under the separate theorem cited below. At $n = 0$ the extra span $D (V)$ is redundant and the formulas reduce to the Zhu circle subspace and candidate product.

Statement references

Associative structure of the higher Zhu quotients

Facts & Assumptions

Given: $V$ is a conformal vertex algebra of CFT type in the non-super state–field convention.

[F1]

Conformal vertex algebra of CFT type: $T = L (- 1)$ , homogeneous weights are nonnegative integers, and $L (0) u = w t (u) u$ for homogeneous $u$ .

[F2]

Vertex algebra in the state–field convention: $Y (u, z) v = \sum_{j \in Z} u_{(j)} v z^{- j - 1}$ and the vacuum/creation axiom is $Y (u, z) 1 = e^{z T} u$ .

Proof

1.1givenF1F2algebra

For fixed $n \geq 0$ and homogeneous $u$ of weight $r$ , the exponent $r + n$ is a nonnegative integer. The two residue formulas are therefore finite sums: $u \circ_{n} v = \sum_{j = 0}^{r + n} (\binom{r + n}{j}) u_{(j - 2 n - 2)} v, u *_{n} v = \sum_{m = 0}^{n} \sum_{j = 0}^{r + n} (- 1)^{m} (\binom{m + n}{n}) (\binom{r + n}{j}) u_{(j - n - m - 1)} v .$ Thus every displayed vector lies in $V$ , and $O_{n} (V)$ is a vector subspace.

2.1givenF1F2step 1.1algebra∎

For homogeneous $u$ , the creation axiom and the finite binomial expansion give $u \circ_{0} 1 = {Res}_{z} e^{z T} u (1 + z)^{w t (u)} z^{- 2} = T u + w t (u) u = L (- 1) u + L (0) u = D u .$ Hence $D (V) \subseteq span {u \circ_{0} v}$ and $O_{0} (V)$ equals the ordinary Zhu circle subspace. The $n = 0$ candidate operation is the ordinary Zhu star operation.

Proof status

This definition fixes the higher Zhu quotient as in DLM §2, including the additional $D (V)$ summand needed for the algebra theorem. It defines a vector quotient and a candidate operation only; associativity and descent are not implicit.

Depends on

Dependency tree · 2 levels

Sources

Chongying Dong, Haisheng Li, and Geoffrey Mason, Vertex operator algebras and associative algebras