bibkey: matsuo1997freeboson authors: Atsushi Matsuo; Kiyokazu Nagatomo year: 1997 title: A Note on Free Bosonic Vertex Algebra and its Conformal Vectors doi: null url: https://arxiv.org/abs/hep-th/9704060v1 claim: Free bosonic divided derivatives have explicit contractions, and the charged polynomial Heisenberg Fock representation is irreducible. strata_touched: [] license: citation-only triage: anchor

Divided derivatives in the polynomial Fock representation

Sections 2.1–2.3 of arXiv hep-th/9704060v1 discuss free bosonic fields, the polynomial Fock realization, vacuum and state fields. Printed page 19 gives the contraction of the divided derivatives of orders a and b as

$$(-1{)}^a\frac{(a+b+1)!}{a!b!}(y-z{)}^{-a-b-2}.$$

Printed page 20 realizes positive modes as n times differentiation with respect to x_n, negative modes as multiplication by x_{-n}, and mode zero as charge. For charge zero and x_{j+1}=X_j, these are the modes used in the complex polynomial Fock modules. Printed pages 21–22 discuss the vacuum and the state-field correspondence. The divided derivative is 1/a! times the ordinary derivative; its creation series has coefficient binomial(j+a,a) X_{j+a}. The contraction constant is (-1)^a (b+1) binomial(a+b+1,a).

The complement index domains in printed Theorem 2.1 and the x_i indexing in Section 2.3 appear inconsistent. These displays are not copied as formal statements. The source supplies classical background and the normalization, rather than an exact statement of the all-integer polynomial output formula or Lean proof terms. The actual formula requires supported evaluation of the nested normal products and identification with an independently defined weighted power-series convolution.

This paper is distinct from hep-th/9706118v1, On axioms for a vertex algebra and the locality of quantum fields. Chu–Lin, arXiv 1812.11378v1 Section 3.1, supplies Heisenberg normalization background; Tong’s String Theory Section 4.3.3 supplies pairing background with contraction -alpha’/2. Neither is an exact source for the labelled coefficient formula in the unit-normalized realization. These formal algebraic calculations do not assert analytic convergence, module fusion, a Monster realization or spacetime dynamics.

Locator

• Matsuo–Nagatomo, arXiv hep-th/9704060v1, Sections 2.1–2.3, printed pages 19–22: https://arxiv.org/abs/hep-th/9704060v1

Charged Heisenberg irreducibility

Section 2.2, printed page 20, defines the action on $F_r=\mathbb{C}[x_1,x_2,\dots ]$ by ${\alpha }_n=n{\partial }_{x_n}$ for $n>0$, ${\alpha }_0=r\mathrm{id}$, and multiplication by $x_{-n}$ for $n<0$. Printed page 21 states: “This is an irreducible representation of A and is called the Fock representation of charge r.” It also states that $F_r$ is generated by the unit $1$ as an $A$-module.

Under $x_{j+1}=X_j$, this is precisely the actual complex polynomial mode action with positive mode $j+1$ equal to $(j+1){\partial }_{X_j}$ and negative mode $-j-1$ equal to multiplication by $X_j$. Irreducibility means that every complex linear subspace invariant under every integer mode is zero or the whole polynomial space. The charge $r$ in this paragraph is independent of the background-charge parameter used to choose a conformal vector. The Heisenberg assertion does not assert Virasoro irreducibility or supply a proof of the all-state charged vertex-algebra module Jacobi identity, intertwiner or fusion laws.

Verified locator

• Exact frontmatter URL: https://arxiv.org/abs/hep-th/9704060v1
• Versioned PDF: https://arxiv.org/pdf/hep-th/9704060v1
• PDF SHA-256: b15b4c89a2ee0a8c38019a29270e5ab5cd6000590bd394e57fb8b2de4aaa16c1.
• Sections 2.2–2.3, printed pages 20–22: charged polynomial modes, the vacuum and state-field background. The correspondence is $x_{j+1}=X_j$ on the existing $\mathrm{MvPolynomial}\mathbb{N}\mathbb{C}$. For every complex charge $\lambda$, mode zero is $\lambda \mathrm{id}$, negative mode $-j-1$ multiplies by $X_j$, and positive mode $j+1$ is $(j+1){\partial }_{X_j}$.

The exact positional charged expansion in question 2147.7 is a proposed implementation question, not a solved formula, a verbatim source theorem or a mathematical novelty claim. It asks for arbitrary complex charge, arbitrary polynomial states $u,v$ and every integer mode, using the actual right-nested divided-derivative normal products and an independently defined finite powerset of positions. Repeated positions stay distinct, and the coefficient suppliers are the existing zero-charge subword fields on the same arbitrary $v$, not only the vacuum.

The existing Verified locator for https://arxiv.org/abs/hep-th/9706118v1 is reused. Sections 1.2–1.4, printed pages 5–8, supply normalized modes, divided derivatives, statewise truncation and the ordered minus-one residue split. Original summands and each individual transformed normal-product summand need finite support before rearrangement. These classical references supply conventions and background only; they do not supply the charged module proof or settle module Jacobi, fusion, a Monster realization, string theory or AdS/CFT.