LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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.

Current Ward identity for WZW sphere coinvariants

Statement

Let $F_{z}$ be a linear functional in the dual of the fixed-coordinate coinvariant quotient in def-cft-wzw-sphere-current-coinvariants, with the infinity factor equal to the vacuum. For top vectors $v_{j} \in V_{λ_{j}}$ , $x \in g$ , and each $i \leq n$ , $F_{z} (v_{1} \otimes \dots \otimes x_{- 1} v_{i} \otimes \dots \otimes v_{n} \otimes 1) = \sum_{j \neq i} \frac{F_{z} (v_{1} \otimes \dots \otimes x_{0} v_{j} \otimes \dots \otimes v_{n} \otimes 1)}{z_{i} - z_{j}} .$ Here $x_{- 1}$ at the $i$ th slot and $x_{0}$ at the $j$ th slot are the local affine current modes in the fixed coordinates.

Statement references

Fixed-coordinate WZW sphere current coinvariants

Facts & Assumptions

[A1]

Full AC is assumed in Integrable WZW insertions on a punctured sphere for existence of each supplied insertion module and its top. The identity is conditional on those modules.

Given: Distinct $z_{1}, \dots, z_{n}$ , top vectors $v_{j}$ , a common level $k$ , and $F_{z}$ as in the Statement.

[F1]

Fixed-coordinate WZW sphere current coinvariants: a block functional annihilates the summed action of every rational current section.

[F2]

Integrable WZW insertions on a punctured sphere: positive current modes kill each top vector, and the horizontal zero modes give the $g$ -action.

[F3]

Universal affine vacuum module: $x_{m} 1 = 0$ for $m \geq 0$ at infinity.

Proof

1.1A1F1F2F3given

Apply [F1] to the rational current $x / (z - z_{i})$ . At $z_{i}$ , its Laurent expansion is $x t_{i}^{- 1}$ , whose local action on the slot is $x_{- 1}$ . At $z_{j} \neq z_{i}$ , its Taylor expansion is $\frac{x}{z - z_{i}} = \sum_{m \geq 0} (- 1)^{m} (z_{j} - z_{i})^{- m - 1} x t_{j}^{m} .$ The residue-mode convention sends $x t_{j}^{m}$ to $x_{m}$ . On the top vector only the $m = 0$ term survives, by [F2]. At infinity, $\frac{x}{z - z_{i}} = \frac{x t_{\infty}}{1 - z_{i} t_{\infty}} = \sum_{m \geq 0} z_{i}^{m} x t_{\infty}^{m + 1},$ whose modes all have positive index and kill the vacuum by [F3].

2.1step 1.1algebra∎

The annihilation equation from step 1.1 is the left side of the Statement plus $\sum_{j \neq i} (z_{j} - z_{i})^{- 1} F_{z} (\dots, x_{0} v_{j}, \dots)$ , equal to zero. Since $(z_{j} - z_{i})^{- 1} = - (z_{i} - z_{j})^{- 1}$ , moving that sum to the other side gives exactly the displayed identity.

Current Ward identity for WZW sphere coinvariants

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources