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Locality determines the formal OPE coefficients

Statement

Let $A (z), B (w)$ be mutually local fields on a complex vector space $V$ , with locality order $N \geq 0$ . Put $C (z, w) = [A (z), B (w)]$ and, for $0 \leq j < N$ , define $C_{j} (w) = {Res}_{z} ((z - w)^{j} C (z, w)) .$ Then $[A (z), B (w)] = \sum_{j = 0}^{N - 1} C_{j} (w) \frac{\partial_{w}^{j} δ (z, w)}{j!},$ and the coefficients are unique. We use the formal OPE shorthand $A (z) B (w) \sim \sum_{j = 0}^{N - 1} \frac{C_{j} (w)}{(z - w)^{j + 1}},$ to record these coefficients: its precise meaning is the commutator expansion above, equivalently the difference of the prescribed expansions $ι_{z, w} (z - w)^{- j - 1}$ and $ι_{w, z} (z - w)^{- j - 1}$ . No analytic convergence is implied. For all integers $m, n$ the modes satisfy $[A_{(m)}, B_{(n)}] = \sum_{j = 0}^{N - 1} (\binom{m}{j}) (C_{j})_{(m + n - j)} .$ Here the binomial coefficient is the generalized integer binomial from Formal distributions and fields on a vector space. If $N = 0$ , the sums are empty and the commutator is zero.

Facts & Assumptions

Given: The complex vector space, fields $A, B$ , and locality order $N$ .

[F1]

Formal distributions and fields on a vector space: formal distributions are coefficient families; ${Res}_{z}$ extracts the $z^{- 1}$ coefficient; $δ (z, w) = \sum_{r \in Z} z^{- r - 1} w^{r}$ ; and $ι_{z, w}, ι_{w, z}$ are the two specified binomial expansions.

Proof

1.1F1algebra

Set $δ (z, w) = \sum_{r \in Z} z^{- r - 1} w^{r}$ . Comparing coefficients gives $(z - w) δ = 0$ . Differentiating this identity $j$ times in $w$ gives $(z - w) \partial_{w}^{j} δ = j \partial_{w}^{j - 1} δ$ for $j \geq 1$ . Also, direct residue extraction gives ${Res}_{z} ((z - w)^{r} \partial_{w}^{j} δ (z, w) / j!) = 1$ if $r = j$ and $0$ otherwise for $r, j \geq 0$ : repeated use of the differentiated identity reduces $r < j$ to a positive derivative of ${Res}_{z} δ = 1$ , and $r > j$ to a positive power of $(z - w)$ times $δ$ .

2.1givenF1step 1.1

If $N = 0$ , locality says $C = 0$ . If $N = 1$ , writing $C = \sum_{p, q} c_{p, q} z^{- p - 1} w^{- q - 1}$ , the equation $(z - w) C = 0$ gives $c_{p + 1, q} = c_{p, q + 1}$ . Hence $c_{p, q}$ depends only on $p + q$ , so $C = ({Res}_{z} C) δ$ . This is the required expansion for order one.

2.2givenstep 1.1step 2.1algebra

Assume $N \geq 2$ and that the expansion is proved for order $N - 1$ . Since $(z - w)^{N - 1} ((z - w) C) = 0$ , write $(z - w) C = \sum_{j = 0}^{N - 2} d_{j} (w) \partial_{w}^{j} δ / j!$ by induction. The identity in step 1.1 shows that $C - \sum_{j = 0}^{N - 2} d_{j} (w) \frac{\partial_{w}^{j + 1} δ}{(j + 1)!}$ is killed by $z - w$ , hence is a multiple $d_{0}^{'} (w) δ$ by step 2.1. This gives a finite delta-derivative expansion for $C$ . Taking residues after multiplying by $(z - w)^{r}$ and using the residue calculation in step 1.1 identifies its $r$ th coefficient as ${Res}_{z} ((z - w)^{r} C)$ and proves uniqueness.

3.1F1step 2.2algebra

The geometric-series identity $δ = i_{z, w} (z - w)^{- 1} - i_{w, z} (z - w)^{- 1}$ follows by writing the first expansion in nonnegative powers of $w / z$ and the second in nonnegative powers of $z / w$ . Differentiating $j$ times gives $\partial_{w}^{j} δ / j! = i_{z, w} (z - w)^{- j - 1} - i_{w, z} (z - w)^{- j - 1}$ . Substitution in step 2.2 yields the stated singular OPE coefficients and fixes both expansion conventions.

4.1F1step 2.1step 2.2algebra∎

Since ${Res}_{z} (z^{m} δ (z, w)) = w^{m}$ , differentiation gives ${Res}_{z} (z^{m} \partial_{w}^{j} δ / j!) = (\binom{m}{j}) w^{m - j}$ for every integer $m$ . Apply ${Res}_{z} z^{m} {Res}_{w} w^{n}$ to step 2.2. The result is $[A_{(m)}, B_{(n)}] = \sum_{j = 0}^{N - 1} (\binom{m}{j}) (C_{j})_{(m + n - j)}$ . The sum is finite; the $N = 0$ case was handled in step 2.1.

Depends on

Formal distributions and fields on a vector space

Used by

Affine current fields are local with the normalized current OPE Lemma

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Christophe Nozaradan, Introduction to Vertex Algebras, §§1.4 and 3.2 (standard reference, not scraped)
Tomoyuki Arakawa and Anne Moreau, Arc spaces and vertex algebras, §§2.2–2.3 (standard reference, not scraped)