DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

Formal distributions and fields on a vector space

Definition

Let $V$ be a complex vector space. A formal distribution in variables $z, w$ with coefficients in a complex vector space $E$ is an arbitrary coefficient family $F (z, w) = \sum_{p, q \in Z} F_{p, q} z^{p} w^{q}, F_{p, q} \in E .$ No convergence or summability is assumed. Multiplication by a polynomial in $z, w$ is defined coefficientwise; each resulting coefficient is a finite sum. The residue ${Res}_{z} F$ is the coefficient of $z^{- 1}$ , retaining the other variables and coefficients. For a one-variable series $f (z) = \sum_{m \in Z} f_{m} z^{m}$ , write ${Res}_{z} f = f_{- 1}$ , in the same support convention as Formal Laurent series $K ((x))$ , their order, derivative, and residue.

The formal delta distribution is $δ (z, w) = \sum_{n \in Z} z^{- n - 1} w^{n} .$ For each integer $r$ , the two binomial expansions of $(z - w)^{r}$ are $ι_{z, w} (z - w)^{r} = \sum_{j \geq 0} (\binom{r}{j}) (- 1)^{j} z^{r - j} w^{j}, ι_{w, z} (z - w)^{r} = \sum_{j \geq 0} (\binom{r}{j}) z^{j} (- w)^{r - j},$ where $(\binom{r}{j}) = r (r - 1) \dots (r - j + 1) / j!$ and $(\binom{r}{0}) = 1$ . The first expansion is in nonnegative powers of $w$ and the second is in nonnegative powers of $z$ ; they are distinct when $r < 0$ .

A formal distribution $A (z) = \sum_{n \in Z} A_{n} z^{- n - 1}$ with $A_{n} \in {End}_{C} (V)$ is a field on $V$ if for every $v \in V$ there is an integer $N (v, A)$ such that $A_{n} v = 0$ for all $n \geq N (v, A)$ . Equivalently, $A (z) v \in V ((z))$ : its powers of $z$ are bounded below, as in Formal Laurent series $K ((x))$ , their order, derivative, and residue, with coefficients in $V$ . Define its modes by $A_{(n)} = A_{n} = {Res}_{z} z^{n} A (z)$ .

Two fields $A (z), B (w)$ are mutually local if some integer $N \geq 0$ satisfies $(z - w)^{N} [A (z), B (w)] = 0$ as a formal distribution, where $[A (z), B (w)] = A (z) B (w) - B (w) A (z)$ . For any integer $n$ , their $n$ th product field is $A (z)_{(n)} B (z) = {Res}_{x} (ι_{x, z} (x - z)^{n} A (x) B (z) - ι_{z, x} (x - z)^{n} B (z) A (x)) .$ These conventions use ordinary, ungraded vector spaces and ordinary commutators; no super sign is implicit.

Remarks

A formal distribution may have infinitely many positive and negative powers. A field has the additional vectorwise lower-truncation condition; the two notions are not interchangeable.
The symbols $ι_{z, w}$ and $ι_{w, z}$ specify formal expansions only. They do not refer to analytic regions or convergence.

Depends on

Formal Laurent series $K((x))$, their order, derivative, and residue

Used by

Dependency tree · two levels

5 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, §§1.3–1.4 (standard reference, not scraped)
Tomoyuki Arakawa and Anne Moreau, Arc spaces and vertex algebras, §§2.1–2.3 (standard reference, not scraped)