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The Full Cyclic Vacuum State Commutant

Abstract

The literal norm-four generator annihilator equals the commutant of every actual state in the existing cyclic vacuum span.

Let D be the production finite-rank even integral lattice data, a any charge and s any complex scalar. The ambient space is the full production Carrier(D), with its actual vacuum and actual state-field map Y. Put e=NormFourVirasoroVector.vector(D,a,s) and L_m=Y(e)(m+1). Use exactly NormFourVacuumIrreducible.vacuumDescendants(D,a,s), denoted W: the complex linear span of all finite words L(-(k+1)) on the actual vacuum, where each k is a natural number. The empty word is the vacuum; words may vanish or be linearly dependent. A span element is a finite linear combination, while the collection of word lengths and indices is unrestricted.

Definition 1.1 (Every nonnegative coefficient of every actual cyclic state).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourFullCommutant.fullStateCommutant

Formalization. D5/S3/VertexAlgebra/Lattice/NormFourFullCommutant.fullStateCommutant (✓ std3).

Source. Repository-derived.

Commentary.

fullStateCommutant(D,a,s) is the submodule of all v in the full ambient carrier such that, for every u in this exact W and every integer j with j at least zero, Y(u)_j v=0. The coefficient convention is the Laurent exponent -j-1. The definition quantifies over all states of the existing span, all nonnegative integer coefficients and the whole ambient carrier.

Theorem 1.2 (The literal generator annihilator is exactly the full cyclic state commutant).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourFullCommutant.space_eq_full_state_commutant

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

Source. Repository-derived.

Commentary.

For every production D, every charge a and every complex s, NormFourCommutant.space(D,a,s)=fullStateCommutant(D,a,s). Thus Y(e)_j v=0 for all integers j at least zero holds if and only if Y(u)_j v=0 for every u in the supplied cyclic span and every integer j at least zero. No norm-four equation, sign equation, positivity, inverse Gram matrix or Virasoro relation is needed for this equality.

Fix v and let B_v be the submodule of states u whose nonnegative Y(u) coefficients annihilate v. Linearity of the actual state-field map makes B_v a submodule. The actual vacuum field is the identity field, whose only nonzero coefficient is at -1, so vacuum belongs to B_v. For e,b in B_v, the actual all-state iterate formula, for every integer r and every integer n at least zero, expresses Y(e_r b)n v as the first finite family minus (-1)^r times the second finite family. For each natural i, the first term is (-1)^i binomial(r,i)e(r-i)(b_(n+i)v), and the second is (-1)^i binomial(r,i)b_(r+n-i)(e_i v). The first is zero because n+i is nonnegative; the second is zero because i is nonnegative. Every term in both complete families vanishes, including all tails; no finite cutoff is substituted.

The existing recursive word k::w is L_(-(k+1))word(w)=e_(-k)word(w). Induction on that exact word, using the iterate result with r=-k, places every word in B_v whenever v is in the generator annihilator. Its submodule property then places the entire supplied span W in B_v. Conversely, actual creation gives e=Y(e)(-1)vacuum=L(-2)vacuum=word([1]), so e belongs to W and the full-state condition includes the original generator condition.

The existing negative-word span and its definition are owned by NormFourVacuumIrreducible; its irreducibility theorem is not reproved or needed for this equality. The two finite iterate families are supplied by LatticeAllStateReconstruction.stateField_iterate in LatticeAllStateLocality, using the actual lattice fields and the StateFieldResidueReconstruction calculus. The lattice realization follows Bakalov-Kac, math/0402315, section 4.1. The norm-four state retains the Shimakura, arXiv:1202.6107, (1.3), and Dong-Mason-Zhu, Theorem 6.3 attribution. Imported residue and normal-product sources retain their Carnahan, Matsuo-Nagatomo and Kytola attribution and license notices. This is a repository derivation for these actual objects, with no novelty claim for the classical generator criterion for commutants. It identifies the commutant of the exact cyclic vacuum state span; it does not identify that carrier with an independently constructed Ising VOA. No simplicity, rationality, fusion, Baby Monster, Monster or full physical programme completion is asserted.

References