LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Affine current fields are local with the normalized current OPE

Statement

Let $V^{k} (g)$ be the vacuum module of Universal affine vacuum module, with $g$ finite-dimensional complex simple and $k \in C$ . Then for every $x \in g$ , $J^{x} (z) = \sum_{n \in Z} x_{n} z^{- n - 1}$ is a field, is creative with $J^{x} (z) 1 = e^{z T} (x_{- 1} 1)$ , and is translation-covariant. For $x, y \in g$ , $[J^{x} (z), J^{y} (w)] = J^{[x, y]} (w) δ (z, w) + k B (x, y) \partial_{w} δ (z, w),$ so the current fields are pairwise local of order at most $2$ , with formal OPE $J^{x} (z) J^{y} (w) \sim \frac{J^{[x, y]} (w)}{z - w} + \frac{k B (x, y) id}{(z - w)^{2}} .$

Facts & Assumptions

Given: The simple Lie algebra, normalized invariant form, level $k$ , and induced vacuum module.

[F1]

Universal affine vacuum module: the central generator acts as $k$ and $g [t]$ kills $1$ .

[F2]

The universal affine vacuum module is restricted: all sufficiently large positive modes of a fixed current kill each fixed vector of $V^{k} (g)$ .

[F3]

Mode translation descends to the affine vacuum module: $T 1 = 0$ and $[T, x_{n}] = - n x_{n - 1}$ .

[F4]

Untwisted affine central extension: $[x_{m}, y_{n}] = [x, y]_{m + n} + m δ_{m, - n} B (x, y) K$ .

[F5]

Locality determines the formal OPE coefficients: $δ (z, w) = \sum_{m \in Z} z^{- m - 1} w^{m}$ , $(z - w) δ = 0$ , and $(z - w)^{2} \partial_{w} δ = 0$ ; its locality expansion identifies the formal OPE coefficients.

Proof

1.1F1F2F3algebra

By [F2], for each vector $v$ the modes $x_{n} v$ vanish for sufficiently large $n$ , so $J^{x} (z)$ is a field. By [F1], $x_{n} 1 = 0$ for $n \geq 0$ , whence $J^{x} (z) 1 = \sum_{r \geq 0} x_{- r - 1} 1 z^{r}$ . From [F3], induction gives $T^{r} x_{- 1} 1 = r! x_{- r - 1} 1$ : the induction step is $T x_{- r - 1} 1 = (r + 1) x_{- r - 2} 1$ . Thus the displayed series is $e^{z T} (x_{- 1} 1)$ . The same relation gives $[T, J^{x} (z)] = \sum_{n} (- n) x_{n - 1} z^{- n - 1} = \partial_{z} J^{x} (z)$ after the index change $m = n - 1$ .

2.1F1F4F5step 1.1algebra

By [F4] and the scalar action of $K$ in [F1], $[J^{x} (z), J^{y} (w)] = \sum_{m, n} [x_{m}, y_{n}] z^{- m - 1} w^{- n - 1}$ . For the loop term set $r = m + n$ ; it becomes $J^{[x, y]} (w) \sum_{m} z^{- m - 1} w^{m} = J^{[x, y]} (w) δ (z, w)$ . The central term is $k B (x, y) \sum_{m} m z^{- m - 1} w^{m - 1} = k B (x, y) \partial_{w} δ (z, w)$ . Both reindexings are coefficient identities of formal distributions; each coefficient in the two-variable series is a finite sum.

3.1F5step 2.1algebra∎

Multiplying the commutator in step 2.1 by $(z - w)^{2}$ gives zero: $(z - w) δ = 0$ and differentiating it gives $(z - w) \partial_{w} δ = δ$ , hence $(z - w)^{2} \partial_{w} δ = 0$ . This proves locality. The coefficient formula in [F5] gives the two singular coefficients $J^{[x, y]}$ and $k B (x, y) id$ and no higher ones, yielding the displayed OPE.

Depends on

Used by

Dependency tree · two levels

14 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

Tomoyuki Arakawa and Anne Moreau, Arc spaces and vertex algebras, §§2.2–2.3 and 3.1 (standard reference, not scraped)
Christophe Nozaradan, Introduction to Vertex Algebras, §§1.4 and 3.2 (standard reference, not scraped)