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First-level oscillator pairing and screening at $p = 2$

Example

Take $p = 2$ in the generic oscillator spaces, with weight $wt (f e^{t α}) = \deg f + 2 t^{2} - t$ for homogeneous $f$ . Use $L_{t} (n)$ for the mode of index $n$ on $F_{t}$ and $Q_{t}$ for the screening coefficient with input charge $t$ . Then

A vector $u \in F_{t}$ is called singular here if $u \neq 0$ , $L_{t} (0) u = λ u$ for some $λ \in C$ , and $L_{t} (n) u = 0$ for every $n > 0$ .

$L_{0} (- 1) e^{0} = 0, L_{1 / 2} (- 1) e^{α / 2} = \frac{1}{2} x_{1} e^{α / 2}, Q_{- 1 / 2} e^{- α / 2} = x_{1} e^{α / 2} .$

The reflected-charge pairing satisfies

$⟨ x_{1} e^{0}, x_{1} e^{α / 2} ⟩ = - 4.$

Both $e^{0}$ and $e^{α / 2}$ have oscillator degree and weight zero; $e^{- α / 2}$ has oscillator degree zero and weight one. The vectors $x_{1} e^{0}$ and $x_{1} e^{α / 2}$ have oscillator degree and weight one. The screened vector $x_{1} e^{α / 2}$ is singular with $L_{1 / 2} (0)$ -eigenvalue one, and every $L_{1 / 2} (n)$ with $n > 0$ kills it. The vector $x_{1} e^{0}$ is not singular, since $L_{1} x_{1} e^{0} = - 2 e^{0}$ .

Example references

Rank-one Feigin–Fuchs oscillator spaces and long-screening coefficients

Background-charge oscillator modes and integral long screenings

Reflected oscillator pairing and transpose of the long screening

Adamović–Milas, Section 1

Arakawa–Moreau, Sections 2.4–2.5

Facts & Assumptions

Given: Integer $p = 2$ , the polynomial $P = C [x_{1}, x_{2}, \dots]$ , the labelled oscillator spaces, the explicit operators $L_{n}$ , and the long-screening coefficient $Q_{t} : F_{t} \to F_{t + 1}$ . All vectors in this example have polynomial degree at most one.

[F1]

Rank-one Feigin–Fuchs oscillator spaces and long-screening coefficients defines $α_{- m} = x_{m}$ , $α_{m} = 4 m \partial_{x_{m}}$ , $α_{0} = 4 t$ , the normal-ordered formula for $L_{n}$ , and the substitution/creation formula for $Q$ .

[F2]

Background-charge oscillator modes and integral long screenings proves that these operators are a Virasoro representation and that the degree-zero operator is $\sum_{m \geq 1} m x_{m} \partial_{x_{m}} + 2 t^{2} - t$ .

[F3]

Reflected oscillator pairing and transpose of the long screening defines the pairing of reflected charges $t$ and $1 / 2 - t$ , with monomial diagonal entries $\prod_{m} (- 4 m)^{r_{m}} r_{m}!$ .

Verification

1.1givenF1algebra

In [F1]'s normally ordered formula for $L_{0} (- 1)$ and $L_{1 / 2} (- 1)$ , the only terms surviving on a vacuum have indices $0, - 1$ and $- 1, 0$ . Their sum is $2 α_{0} α_{- 1} / 8 = t x_{1}$ , and the improvement term is zero because its factor is $(- 1) + 1 = 0$ . Thus $L_{t} (- 1) e^{t α} = t x_{1} e^{t α}$ , which gives both stated values at $t = 0, 1 / 2$ . For $t = - 1 / 2$ , the screening field on the vacuum is $E_{1, - 1 / 2} (z) e^{- α / 2} = z^{- 2} \exp (\sum_{m \geq 1} \frac{x_{m}}{m} z^{m}) e^{α / 2} .$ Its coefficient of $z^{- 1}$ is $x_{1} e^{α / 2}$ because the creation exponential begins $1 + x_{1} z + O (z^{2})$ . This proves the screening equation directly.

2.1F1F2F3step 1.1algebra

At $p = 2$ , the vacuum weights $2 t^{2} - t$ at $t = 0, 1 / 2, - 1 / 2$ are respectively $0, 0, 1$ . Multiplication by $x_{1}$ adds one to the oscillator degree and to the weight by [F2], proving every claimed degree and weight. Charges $0$ and $1 / 2$ are reflected partners. Their degree-one monomials therefore pair by [F3] to $(- 4 \cdot 1)^{1} 1! = - 4$ , as asserted. Both monomials are nonzero because they are basis elements of their labelled polynomial copies.

3.1F1F2step 2.1algebra∎

Directly commute the quadratic and improvement terms of [F1] with $x_{1} = α_{- 1}$ . For $n > 0$ , the quadratic term gives $α_{n - 1}$ , while the improvement contributes $- n (n + 1) δ_{n, 1} I$ . Hence $[L_{t} (n), x_{1}] = α_{n - 1} - 2 δ_{n, 1} I .$ Positive modes annihilate each vacuum directly in the normal-ordered formula. On $x_{1} e^{α / 2}$ , the case $n = 1$ therefore gives $(4 (1 / 2) - 2) e^{α / 2} = 0$ , and every $n \geq 2$ gives $α_{n - 1} e^{α / 2} = 0$ because that oscillator is a derivative of a constant polynomial. Its weight-one eigenvalue was computed in step 2.1, so it is the stated nonzero singular vector. For $x_{1} e^{0}$ , the same $n = 1$ computation gives $- 2 e^{0}$ , proving its nonsingularity. All assertions follow by these finite polynomial computations.

Remarks

This example computes the first oscillator level at one value of $p$ . It verifies the stated screening and reflected-pairing conventions. It does not prove an all-grade character formula, completeness of Fock submodules, or any triplet endpoint or logarithmic-extension theorem.

First-level oscillator pairing and screening at $p = 2$

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 1 level

Sources

First-level oscillator pairing and screening at p=2p=2

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 1 level

Sources

First-level oscillator pairing and screening at $p = 2$