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A commutative differential algebra gives a vertex algebra

Example

Let $V = C [u]$ with its usual polynomial addition and multiplication, vacuum $1 = 1$ , and derivation $T = d / d u$ . Define $Y (a, z) b = (e^{z T} a) b = (\sum_{r \geq 0} \frac{z^{r} T^{r} a}{r!}) b .$ Then this state–field map makes $V$ a vertex algebra. The series is finite for each $a$ , so it is also a useful example where formal lower truncation is immediate.

Facts & Assumptions

Given: $V = C [u]$ , $T = d / d u$ , and the state-field convention of Vertex algebra in the state–field convention.

[A1]

$T$ is the coefficientwise formal derivative on polynomials and obeys $T (a b) = (T a) b + a (T b)$ ; the multiplication on $V$ is commutative with unit $1$ .

[F1]

Vertex algebra in the state–field convention requires lower truncation, $Y (1, z) = id$ , $T 1 = 0$ , $Y (a, z) 1 = e^{z T} a$ , translation covariance, and locality.

Proof

1.1A1F1given

If $a = 0$ , then $e^{z T} a = 0$ ; otherwise let $d = \deg a$ , so $T^{r} a = 0$ for $r > d$ and $e^{z T} a$ is a polynomial in $z$ . Hence $Y (a, z) b$ has finite support in nonnegative powers for every $b$ , and its coefficients define a field. Direct substitution gives $Y (1, z) b = b$ and $Y (a, z) 1 = e^{z T} a$ . Also $T 1 = 0$ .

2.1A1F1step 1.1algebra

For any $b \in V$ , the derivation law in [A1] gives $[T, Y (a, z)] b = T ((e^{z T} a) b) - (e^{z T} a) T b = (T e^{z T} a) b = \partial_{z} Y (a, z) b$ , so translation covariance holds.

3.1A1F1algebra∎

Each $Y (a, z)$ acts by multiplication by $e^{z T} a$ . Multiplication operators commute because $V$ is commutative, so $[Y (a, z), Y (c, w)] = 0$ for every $a, c \in V$ . Thus locality holds with exponent $N = 0$ . All axioms of [F1] are verified.

Depends on

Vertex algebra in the state–field convention

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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, Example 4.2.6 (standard reference, not scraped)
Tomoyuki Arakawa and Anne Moreau, Arc spaces and vertex algebras, §2.7 (standard reference, not scraped)