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Irreducibility of the Actual Cyclic Vacuum Representation

Abstract

The actual cyclic norm-four lattice Virasoro vacuum representation is irreducible.

Let D have an integral symmetric Gram matrix with even diagonal, let a be a charge with B(a,a)=4, and let s in C satisfy s squared equal to one. Write L_m for the actual mode Y(v)_(m+1) of the norm-four lattice state v on the finite-support carrier.

Definition 1.1 (The cyclic vacuum space).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourVacuumIrreducible.vacuumDescendants

Formalization. D5/S3/VertexAlgebra/Lattice/NormFourVacuumIrreducible.vacuumDescendants (✓ std3).

Source. Repository-derived.

Commentary.

For every D,a,s, vacuumDescendants is the complex linear span of all finite words L_(-(k_1+1)) … L_(-(k_r+1)) vacuum, with all k_i natural. The empty word is the actual vacuum. These word vectors may vanish or be linearly dependent.

Definition 1.2 (Every integer mode restricts to the cyclic space).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourVacuumIrreducible.restrictedMode

Formalization. D5/S3/VertexAlgebra/Lattice/NormFourVacuumIrreducible.restrictedMode (✓ std3).

Source. Repository-derived.

Commentary.

For every such D,a,s and every integer m, restrictedMode is the restriction of the actual L_m to vacuumDescendants. Closure follows by induction on negative words using the actual Virasoro commutator, including its central term, and creativity on the vacuum.

Theorem 1.3 (The invariant-subspace dichotomy).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourVacuumIrreducible.vacuum_descendants_irreducible

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

Source. Repository-derived.

Commentary.

For every such D,a,s and every complex submodule U of vacuumDescendants, if restrictedMode m maps every element of U into U for every integer m, then U is zero or the whole cyclic space. There is no positivity, rootlessness, PBW basis, eigenbasis or grading assumption on U.

A negative word has natural grade equal to the sum of its positive indices. The actual L_0 acts on it by that grade, and grade zero is exactly the vacuum line. Span induction gives a finite homogeneous expansion of every cyclic vector. For a finite support S and n in S, the polynomial product of L_0-j over j in S different from n kills every other component. Its value on the n component is a nonzero scalar, so every L_0-invariant subspace contains each component.

A nonzero invariant subspace therefore has a nonzero vector of least natural grade n. Every positive actual mode kills this vector: otherwise the commutator would produce a nonzero eigenvector of smaller natural grade. The highest-weight restriction gives n equal to 0, 1/16 or 1/2. Since n is natural, it equals zero. The subspace contains the vacuum line and hence every negative word, proving cyclic irreducibility.

The norm-four state retains its Dong-Mason-Zhu and Shimakura attribution; the lattice construction and residue calculus retain their Bakalov-Kac, Carnahan and Matsuo-Nagatomo attribution. This conclusion concerns the cyclic Virasoro representation. Bundled VOA simplicity, fusion, anomalies, Monster moonshine and string or AdS-CFT realization require further results.

References