RESEARCH PAPER / NOTE

Introduction to Vertex Operator Algebras

Chenqi Meng

Peerpedia 261022.405c1131e2fdCorresponding author: ChenqiMengLatest acceptance: 30 Oct 2026, 14:29 UTC

License · CC0 1.0 · Peerpedia collaboration terms

Abstract

Vertex operator algebra is a rigorous formulation of 2D chiral conformal field theory. This note reviews (will do!) its definition, construction, and properties. It in particular emphasizes the field (mathematics)-like properties of a VOA. A VOA can be viewed as a quantized field. We will eventually (hopefully) arrive at the quantum Galois theory of VOA.

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Remove the VOSA chapter now published independently as Vertex Operator Superalgebras and the Majorana Fermion (https://peerpedia.dev/papers… Current

Accepted 30 Oct 2026, 14:29 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Use wedge. And move properties out from definition

Accepted 30 Oct 2026, 11:29 UTC · Contribution by ChenqiMeng

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Add the super Jacobi identity to the VSA exercises with explicit parity and expansion conventions, and derive the field supercommutator by …

Accepted 30 Oct 2026, 10:03 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Replace the chiral correlation-functions outline with a chapter on vertex operator superalgebras and update the roadmap. Define parity, sup…

Accepted 30 Oct 2026, 09:57 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Add Virasoro minimal models as the second worked example and remove the lattice-VOA outline from the basic examples. Construct the universa…

Accepted 30 Oct 2026, 09:14 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Add the rank-one Heisenberg VOA as the first worked example, following the notation of the local Heisenberg note. Construct the oscillator …

Accepted 30 Oct 2026, 03:50 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Generalize the descendant-field section to arbitrary vertex-algebra products and move it after Jacobi in the Vertex algebras chapter. Prove…

Accepted 30 Oct 2026, 03:13 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Add fields of Virasoro descendants to the VOA chapter. Prove the iterate formula with explicit Jacobi residues, define normal ordering usin…

Accepted 30 Oct 2026, 03:06 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Extend the weight-one discussion with the canonical invariant symmetric bilinear form and affine current algebra, including short proofs fr…

Accepted 29 Oct 2026, 14:43 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Add the Commutants and cosets chapter. Define the commutant through regular field action, prove equivalence with commuting fields and closu…

Accepted 29 Oct 2026, 09:08 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Clarify the ideal lattice and the largest proper ideal for CFT-type VOAs. Expand the existing proposition and proof to show that the sum of…

Accepted 29 Oct 2026, 08:33 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Remove the proof for orbifold simplicity.

Accepted 28 Oct 2026, 14:28 UTC · Contribution by ChenqiMeng

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Removed redundant announcement

Accepted 28 Oct 2026, 14:17 UTC · Contribution by ChenqiMeng

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Add the G-orbifolds chapter after Basic structure of VOAs. Define fixed-point VOAs, record their inherited conformal structure, and develop…

Accepted 28 Oct 2026, 14:04 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Revise the constructions chapter as Basic structure of VOAs. Use field notation for definitions, leave routine verifications as exercises, …

Accepted 28 Oct 2026, 13:21 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Add a constructions chapter with definitions and proofs for vertex subalgebras, VOA subalgebras, homomorphisms, automorphisms, extensions, …

Accepted 28 Oct 2026, 09:26 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Simplified the title for the chapter.

Accepted 28 Oct 2026, 09:04 UTC · Contribution by ChenqiMeng

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Rename the vertex algebra chapter heading to Vertex algebras and define the operator commutator at first use. Simplify the weight-one Lie a…

Accepted 28 Oct 2026, 09:02 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Replace the schematic stress-tensor OPE with exact conformal-vector products and Laurent series, including regular terms. Show the calculat…

Accepted 28 Oct 2026, 08:41 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Explain the three special Virasoro modes as global conformal transformations: translations, dilations, and special conformal transformation…

Accepted 28 Oct 2026, 08:33 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Expand the Virasoro covariance proof to show how the left-hand residues give the commutator, how the delta kernel produces the binomial exp…

Accepted 28 Oct 2026, 08:26 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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Add six VOA properties with compact proofs: vacuum and conformal weights, Virasoro covariance, mode weights and dilations, the stress-tenso…

Accepted 28 Oct 2026, 08:10 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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A bug

Accepted 28 Oct 2026, 08:03 UTC · Contribution by ChenqiMeng

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Added VOA definition

Accepted 28 Oct 2026, 08:02 UTC · Contribution by ChenqiMeng

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Convert this test article to BananaMark syntax: labels, environment titles, math blocks, diagrams and YAML bibliography; update the demonst…

Accepted 28 Oct 2026, 06:01 UTC · Contribution by ChenqiMeng · via CLI · AI-assisted

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ChenqiMeng · 27 accepted changes