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The Actual Ungraded Lattice Vertex Algebra

Abstract

The actual lattice state-field construction is an ungraded vertex algebra.

Let D be any finite-rank ordinary lattice: its Gram matrix G is integral and symmetric with even diagonal. Rank zero is included. No positivity, nondegeneracy or unimodularity is assumed. Charges are Fin(rank(D)) to Z, oscillators are complex multivariate polynomials indexed by Fin(rank(D)) times N, and V is the finite-support charge direct sum of that polynomial algebra. Write B for the original integral bilinear form. Normalized coefficient q means the Laurent coefficient at -q-1. The vacuum is single(0,1), Y is the constructed actual state-field map, T is the charge-sensitive translation, and mu(a,q,b)=(Y(a))_q b.

StateFieldVertexAlgebra records a linear state-field map, vacuum, translation, vacuum field, creation, creativity, translation vacuum, covariance, locality and full integer Borcherds with all three finite supports. actualVertexAlgebra(D) fills this record with the actual Y, single(0,1), charge-sensitive T and the proved concrete axioms for every ordinary D. This constructor imposes no desired algebra laws as additional premises.

Theorem 1.1 (Concrete translation is intrinsic).

Lean statement: D5/S3/VertexAlgebra/LatticeActualVertexAlgebra.translation_as_mode

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/LatticeActualVertexAlgebra.translation_as_mode (✓ 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.

For every a, T(a)=(Y(a))_(-2) vacuum. Evaluate the proved mode -1 covariance on vacuum, use creation and T(vacuum)=0, and obtain the actual minus-two mode. The record and constructor remain ungraded.

Bakalov-Kac, arXiv math/0402315v1, section 4.1, equations (4.12)-(4.16), DOI 10.1142/9789812702562_0001, supplies the lattice field, ordered-product, translation and conformal construction. Equation numbers refer to arXiv v1.

Matsuo-Nagatomo, hep-th/9706118v1, Proposition 1.5.5 and Theorem 5.4.1, supplies residue locality and reconstruction by creative local fields, divided derivatives and nested normal products.

The carrier and formal-series interfaces use pinned Mathlib revision db584cd6d46c92f209a44c0f1c829460d327499d and Lean 4.33.0.

This is algebraic ungraded vertex-algebra mathematics. Finite graded pieces, positivity, PCT, Leech specialization, twisted extensions, the Monster, anomaly, fusion categories, string theory, AdS/CFT and physical completion are not proved here.

References