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The Actual Norm-Four Virasoro State

Abstract

A norm-four lattice state has all self-products and every integer Virasoro commutator at central charge one half.

Let D be a finite-rank lattice with integral symmetric Gram matrix and even diagonal. Charges are maps from Fin(rank(D)) to Z, and the oscillator algebra is the complex multivariate polynomial ring on Fin(rank(D)) times N. The carrier V is its finite-support charge direct sum. Write B for the integral bilinear form, vacuum=single(0,1), T for the charge-sensitive translation, and Y for the constructed lattice state-field map. The normalized coefficient q is the Laurent coefficient at -q-1, and mu(u,q,w)=Y(u)_q(w).

For a charge a with B(a,a)=4 and a complex scalar s with s squared equal to one, put A=sum_i a_i X(i,0), Q=single(0,A squared), E=single(a,1)+single(-a,1), and v=(1/16)Q+(s/4)E. Thus both signs are included. Define L_m=Y(v)_(m+1). The statement uses no positivity, rootlessness, unimodularity, nondegeneracy, ambient conformal grading or additional rank assumption.

Theorem 1.1 (All actual singular self-products and the unbounded tail).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourVirasoroVector.self_products

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/NormFourVirasoroVector.self_products (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every such D,a,s, the five conclusions hold simultaneously: mu(v,0,v)=T(v), mu(v,1,v)=2v, mu(v,2,v)=0, mu(v,3,v)=(1/4)vacuum, and mu(v,n,v)=0 for every integer n at least four. The proof evaluates the neutral, same-charge, opposite-charge and both mixed products in the actual carrier. The norm-four cocycle phase is one. Opposite charges cancel odd oscillator terms, and negative creation-series indices prove the entire tail.

Theorem 1.2 (Every integer mode satisfies the central-charge-one-half relation).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourVirasoroVector.virasoro_commutator

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/NormFourVirasoroVector.virasoro_commutator (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every integer m,n, L_m L_n-L_n L_m=(m-n)L_(m+n) plus ((m cubed-m)/24) times the identity when m+n=0, and zero central term otherwise. The supported actual mode-commutator formula and the self-products reduce the sum to indices zero through three. The integer binomial polynomial includes negative m+1; no nonnegative upper-index restriction is imposed.

The norm-four formula is classical: Shimakura, arXiv:1202.6107, page 4, equation (1.3), attributes its Ising-vector assertion to Dong-Mason-Zhu, Theorem 6.3. Here only the self-product and mode-action conclusions above are established. The lattice-field construction follows Bakalov-Kac, arXiv:math/0402315, section 4.1. Imported normal-product and Sugawara sources retain Scott Carnahan and Kalle Kytola Apache-2.0 attribution.

These algebraic conclusions do not establish the simple minimal-model quotient, simplicity, an ambient conformal vector, a Virasoro frame, Monster moonshine, string theory or AdS/CFT.

References