bibkey: chulin2018heisenberg authors: Yanjun Chu and Zongzhu Lin year: 2018 title: Moduli spaces of conformal structures on Heisenberg vertex algebras doi: null url: https://arxiv.org/abs/1812.11378v1 claim: Section 3.1 gives the Heisenberg vacuum representation and conformal vectors; Section 3.2 states their central charges and common conformal grading. strata_touched: [] license: citation-only triage: anchor

Heisenberg vertex algebra

Chu and Lin, Section 3.1, define the affine Heisenberg bracket, the charge-zero vacuum representation, and the field expansion $Y(h,z)={\sum }_{n\in \mathbb{Z}}h(n)z^{-n-1}$ over $\mathbb{C}$. These are the literature anchors for the normalized mode convention and Fock construction.

The rank-one polynomial formulas over $\mathbb{Q}$, $a_{k+1}=(k+1)\partial /\partial X_k$, $a_{-(k+1)}=X_k\cdot (-)$, and $a_0=0$ are an algebraic specialization of the complex vacuum representation. The all-integer commutator relation and second-order formal locality are standard consequences of these formulas; changing the coefficient field or rewriting locality as a coefficient shift does not make them new mathematical results. A future Lean construction of the pointwise Laurent field would be a formalization contribution, not a claim that Chu and Lin supply its Mathlib proof.

For the rank-one complex vacuum module, Section 3.1 identifies the grading-preserving conformal vectors as ${\omega }_t={\omega }_0+th(-2)1$. Section 3.2 states that their central charges are $1-12t^2$ and that the underlying vertex algebra and conformal grading are the same. Thus the unshifted graded dimension series is independent of $t$. The character including the vacuum-energy factor $q^{-c/24}$ is a different observable and does depend on $t$.

Verified locator

• arXiv:1812.11378v1, Sections 3.1-3.2: https://arxiv.org/abs/1812.11378v1
• The arXiv metadata and Sections 3.1-3.2 source HTML were inspected on 28 September 2026. The source uses $\mathbb{C}$; the rational polynomial realization above is a specialization.