Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

LatticeSectorIntertwiner

Abstract

LatticeSectorIntertwiner

Construction uses arbitrary ordinary symmetric even integral lattice data, including degenerate forms, zero charges and rank zero. Classification and universality require an explicit two-sided complex inverse Gram matrix, with no positivity or integer unimodularity.

Definition 1.1 (Sector).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorIntertwiner.Sector

Formalization. D5/S3/VertexAlgebra/Sectors/LatticeSectorIntertwiner.Sector (✓ std3).

Citation. Atsushi Matsuo; Kiyokazu Nagatomo (1997). On axioms for a vertex algebra and the locality of quantum fields. URL: https://arxiv.org/abs/hep-th/9706118v1.

Commentary.

Sector on the actual charge-labelled oscillator sectors; the precise parameters and conclusion are the compiled Lean statement.

Definition 1.2 (inclusion).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorIntertwiner.inclusion

Formalization. D5/S3/VertexAlgebra/Sectors/LatticeSectorIntertwiner.inclusion (✓ std3).

Citation. Atsushi Matsuo; Kiyokazu Nagatomo (1997). On axioms for a vertex algebra and the locality of quantum fields. URL: https://arxiv.org/abs/hep-th/9706118v1.

Commentary.

inclusion on the actual charge-labelled oscillator sectors; the precise parameters and conclusion are the compiled Lean statement.

Definition 1.3 (projection).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorIntertwiner.projection

Formalization. D5/S3/VertexAlgebra/Sectors/LatticeSectorIntertwiner.projection (✓ std3).

Citation. Atsushi Matsuo; Kiyokazu Nagatomo (1997). On axioms for a vertex algebra and the locality of quantum fields. URL: https://arxiv.org/abs/hep-th/9706118v1.

Commentary.

projection on the actual charge-labelled oscillator sectors; the precise parameters and conclusion are the compiled Lean statement.

Theorem 1.4 (charge_homogeneity).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorIntertwiner.charge_homogeneity

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorIntertwiner.charge_homogeneity (✓ std3). ∎

Citation. Atsushi Matsuo; Kiyokazu Nagatomo (1997). On axioms for a vertex algebra and the locality of quantum fields. URL: https://arxiv.org/abs/hep-th/9706118v1.

Commentary.

Every actual ordinary field mode on arbitrary original oscillator inputs has output charge equal to the sum of its two input charges. Both normal-product branches and the polynomial linear extension are covered.

Definition 1.5 (canonical).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorIntertwiner.canonical

Formalization. D5/S3/VertexAlgebra/Sectors/LatticeSectorIntertwiner.canonical (✓ std3).

Citation. Atsushi Matsuo; Kiyokazu Nagatomo (1997). On axioms for a vertex algebra and the locality of quantum fields. URL: https://arxiv.org/abs/hep-th/9706118v1.

Commentary.

canonical on the actual charge-labelled oscillator sectors; the precise parameters and conclusion are the compiled Lean statement.

Theorem 1.6 (inclusion_canonical).

Lean statement: D5/S3/VertexAlgebra/Sectors/LatticeSectorIntertwiner.inclusion_canonical

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Sectors/LatticeSectorIntertwiner.inclusion_canonical (✓ std3). ∎

Citation. Atsushi Matsuo; Kiyokazu Nagatomo (1997). On axioms for a vertex algebra and the locality of quantum fields. URL: https://arxiv.org/abs/hep-th/9706118v1.

Commentary.

inclusion_canonical on the actual charge-labelled oscillator sectors; the precise parameters and conclusion are the compiled Lean statement.

Classical sources are Bakalov-Kac, arXiv math/0402315v1, section 4.1, and Matsuo-Nagatomo free-boson reconstruction. Imported residue/Pascal and Sugawara suppliers retain Scott Carnahan and Kalle Kytola source attribution and licenses. No literature novelty is claimed.

The category is the neutral Heisenberg sector category. Full charged lattice-VOA modules, dual-lattice coset fusion, coherent associativity and braiding, pentagon and hexagon, Monster or twisted anomaly, CFT, string theory and AdS/CFT remain additional unproved edges. Rank zero has one charge and a one-dimensional IO space; it is not the zero theory.

References