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The Heisenberg Jordan Fock family is grading-restricted; it is logarithmic exactly when λ is nonzero

Statement

Let $E = C e_{0} \oplus C e_{1}$ , define $N e_{0} = 0$ , $N e_{1} = e_{0}$ , and let $A_{λ} = λ I + N$ for $λ \in C$ . The oscillator module $F_{λ} = F (E, A_{λ})$ constructed in Heisenberg normal-ordered fields give modules with an arbitrary zero-mode action is a module over the conformal CFT-type rank-one Heisenberg VOA of central charge $1$ . Its conformal modes are

$L_{F} (n) = \frac{1}{2} \sum_{k \in Z} : a_{k} a_{n - k} :, L_{F} (0) = D \otimes I_{E} + \frac{1}{2} I_{P} \otimes A_{λ}^{2} = D \otimes I_{E} + \frac{λ^{2}}{2} I + λ I_{P} \otimes N,$

where $D ∣_{P_{d}} = d I$ . The degree spaces $P_{d} \otimes E$ give an admissible grading and are exactly the generalized weight spaces of weights $d + λ^{2} / 2$ , of dimension $2 \dim P_{d}$ . Thus $F_{λ}$ is grading-restricted generalized. It is logarithmic exactly when $λ \neq 0$ ; at $λ = 0$ it is ordinary despite its nonsemisimple current zero mode. In particular $F_{1}$ has $a_{0} = I + N$ and $L_{F} (0) ∣_{E} = \frac{1}{2} I + N$ .

Facts & Assumptions

Given: The rank-one polynomial VOA, the explicit two-dimensional $E, N, A_{λ}$ and its normal-ordered Fock action.

[F1]

Heisenberg normal-ordered fields give modules with an arbitrary zero-mode action proves the full module Jacobi identity, lower truncation and the shift $v_{(k)}^{F} (P_{d} \otimes E) \subset P_{d + w - k - 1} \otimes E$ for a state of weight $w$ .

[F2]

The rank-one Heisenberg vacuum space is a conformal VOA of central charge one proves $ω = x_{1}^{2} / 2$ , central charge $1$ , the normal-square conformal field and finite polynomial weight spaces $P_{d}$ .

[F3]

Admissible module for a vertex operator algebra fixes the nonnegative integer grading and mode shift. Grading-restricted generalized module requires finite generalized weight spaces and a lower bound on real parts.

[F4]

Logarithmic generalized module means grading-restricted generalized with nonsemisimple $L_{F} (0)$ ; Ordinary module means that $L_{F} (0)$ is semisimple instead.

Proof

1.1givenF1F2algebra

Apply [F1] with the displayed $E, A_{λ}$ . This already gives a genuine module with all three-variable Jacobi coefficients verified, not merely a representation of the current-mode Lie algebra. Formula [F2] and the action definition give $Y_{F} (ω, z) = \frac{1}{2} : a^{F} (z)^{2} :$ , so its modes are the displayed quadratic sums, finite on each polynomial vector by the same normal-order bounds as [F1]. At $n = 0$ this gives $L_{F} (0) = \sum_{r \geq 1} a_{- r} a_{r} + a_{0}^{2} / 2 = D \otimes I + A_{λ}^{2} / 2$ . Since $N^{2} = 0$ , $A_{λ}^{2} / 2 = (λ^{2} / 2) I + λ N$ .

2.1givenF1F2step 1.1algebra

The conformal modes also satisfy the central-charge-one Virasoro relations directly. The oscillator commutator gives $[L_{F} (n), a_{m}] = - m a_{n + m}$ . Thus $C_{m, n} = [L_{F} (m), L_{F} (n)] - (m - n) L_{F} (m + n)$ commutes with every multiplication and derivative mode and is determined by its action on the common positive-mode kernel $E$ . Its degree is $- m - n$ ; an operator commuting with all positive modes must send $E$ into $E$ , so $C_{m, n} ∣_{E} = 0$ if $m + n \neq 0$ . For $m > 0$ , $L_{F} (m) E = 0$ and $L_{F} (- m) e = x_{m} A_{λ} e + \frac{1}{2} \sum_{r = 1}^{m - 1} x_{r} x_{m - r} e$ . Applying $L_{F} (m)$ gives $m A_{λ}^{2} e + \frac{1}{2} \sum_{r = 1}^{m - 1} r (m - r) e$ . Subtracting $2 m L_{F} (0) e = m A_{λ}^{2} e$ leaves $(m^{3} - m) e / 12$ . Antisymmetry and the $m = 0$ case complete the Virasoro relation; commuting with negative multiplication modes extends it from $E$ to every polynomial vector.

2.2givenF1F2F3step 1.1algebra

The degree shift in [F1] is precisely [F3]'s admissible shift, so $F_{λ} = ⨁_{d \geq 0} P_{d} \otimes E$ is admissible. On its degree- $d$ subspace, step 1.1 gives $(L_{F} (0) - d - λ^{2} / 2)^{2} = 0$ . Distinct degrees have distinct eigenvalues. If $h \neq d + λ^{2} / 2$ , then $(d + λ^{2} / 2 - h) I + λ N$ is invertible on that degree, with inverse $c^{- 1} I - λ c^{- 2} N$ where $c = d + λ^{2} / 2 - h$ . Since vectors have only finitely many degree components, the generalized eigenspaces are exactly the stated degree spaces. They have finite dimension $2 \dim P_{d}$ by [F2], and their real parts are bounded below by $Re (λ^{2} / 2)$ . This verifies grading restriction in [F3].

3.1givenF4step 1.1step 2.2algebra∎

If $λ \neq 0$ , on the lowest space $E$ the vector $e_{1}$ satisfies $(L_{F} (0) - λ^{2} / 2) e_{1} = λ e_{0} \neq 0$ and its second iterate is zero. Thus $L_{F} (0)$ is not semisimple, and the module is logarithmic by [F4]. If $λ = 0$ , step 1.1 gives $L_{F} (0) = D$ , diagonal on the displayed grading, so the module is ordinary by [F4] and not logarithmic. At $λ = 1$ the explicit formulas become $a_{0} = I + N$ and $L_{F} (0) ∣_{E} = I / 2 + N$ . No unproved module-twist, reconstruction-for-modules, classification, or rationality theorem is used.

Depends on

Used by

An explicit logarithmic conformal Heisenberg module Example

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Sources

Tomoyuki Arakawa and Anne Moreau, Arc Spaces and Vertex Algebras (standard reference, not scraped)
Yi-Zhi Huang, James Lepowsky and Lin Zhang, Logarithmic Tensor Category Theory for Generalized Modules for a Conformal Vertex Algebra, I (standard reference, not scraped)