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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Accepted

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

ChenqiMeng · via CLI · AI-assisted · 30 Oct 2026, 14:28 UTC · 261030.5a7f58f4e09c

Accepted

Use wedge. And move properties out from definition

ChenqiMeng · 30 Oct 2026, 11:29 UTC · 261030.6b9c1cee3c0f

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 30 Oct 2026, 10:02 UTC · 261030.3c6c6609ead5

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 30 Oct 2026, 09:48 UTC · 261030.03344a2883a9

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 30 Oct 2026, 08:45 UTC · 261030.e3e919f838b4

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 30 Oct 2026, 03:49 UTC · 261030.67fe6430233d

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 30 Oct 2026, 03:12 UTC · 261030.4b5a5ae729fb

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 30 Oct 2026, 03:05 UTC · 261030.7dcceb9ba6e2

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 29 Oct 2026, 14:41 UTC · 261029.ffe122dca3bc

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 29 Oct 2026, 09:08 UTC · 261029.9c52d363e410

Accepted

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…

ChenqiMeng · via CLI · AI-assisted · 29 Oct 2026, 08:31 UTC · 261029.50daa62cd5c3

Accepted

Remove the proof for orbifold simplicity.

ChenqiMeng · 28 Oct 2026, 14:28 UTC · 261028.39988ee71ae2

Accepted

Removed redundant announcement

ChenqiMeng · 28 Oct 2026, 14:08 UTC · 261028.928ba4647b57

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 28 Oct 2026, 13:57 UTC · 261028.c5dfb1a65ef0

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 28 Oct 2026, 13:19 UTC · 261028.ccad06b5be94

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 28 Oct 2026, 09:25 UTC · 261028.4d9a88200081

Accepted

Simplified the title for the chapter.

ChenqiMeng · 28 Oct 2026, 09:03 UTC · 261028.9f3591393185

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 28 Oct 2026, 09:01 UTC · 261028.d91af2ea7ee6

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 28 Oct 2026, 08:40 UTC · 261028.4a5afd9e6ab7

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 28 Oct 2026, 08:32 UTC · 261028.8fc42c1c620f

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 28 Oct 2026, 08:25 UTC · 261028.b0222d54e9a8

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 28 Oct 2026, 08:09 UTC · 261028.ddca2d933f06

Accepted

A bug

ChenqiMeng · 28 Oct 2026, 08:03 UTC · 261028.17178c02565b

Accepted

Added VOA definition

ChenqiMeng · 28 Oct 2026, 08:02 UTC · 261028.96b81e26267e

Accepted

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

ChenqiMeng · via CLI · AI-assisted · 28 Oct 2026, 06:01 UTC · 261028.16a57477b500