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Full Integer Borcherds on the Actual Lattice

Abstract

The actual fields satisfy the full integer Borcherds identity with three finite supports.

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.

Define the left kernel as choose(p,j) mu(mu(a,r+j,b),p+q-j,c). The two right kernels use (-1)^j choose(r,j) mu(a,p+r-j,mu(b,q+j,c)) and its (-1)^r weighted reverse mu(b,q+r-j,mu(a,p+j,c)). Actual Hahn truncation separately bounds all three supports.

Integer residue closure gives the p=0 iterate seed. Actual locality gives the high-r zero region. Supported Pascal reindexing proves the discrepancy recurrence at (p+1,q,r), (p,q+1,r) and (p,q,r+1). Positive p induction and nested induction on negative p and distance below the locality boundary give every integer triple.

Theorem 1.1 (All states and all three integer indices).

Lean statement: D5/S3/VertexAlgebra/LatticeAllStateJacobi.borcherds

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/LatticeAllStateJacobi.borcherds (✓ 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,b,c in V and integers p,q,r all three kernels have finite support, and the left sum equals the signed difference of the two right composition sums. There is no Jacobi, support, expansion or locality premise: the concrete construction supplies every one of those obligations.

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.

Finite normal-product and integer residue kernels retain Scott Carnahan attribution: vertexAlg revision 4453e34ec390e82a0c789c731ada8f9a6e86bdea, VertexAlg/VertexBasic/VertexOperator.lean, Apache-2.0. Actual coefficient proofs are re-elaborated on V; no polynomial Fock theorem is transferred between carriers.

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