Highest Weights from the Actual Vacuum Null State
Abstract
The actual norm-four lattice Virasoro modes restrict highest weights to 0, 1/16 and 1/2.
Let D have an integral symmetric Gram matrix with even diagonal, and let a be a charge with B(a,a)=4. For s in C with s squared equal to one, let v be the production norm-four lattice state and put L_m=Y(v)_(m+1) on the actual finite-support lattice carrier.
Theorem 1.1 (Allowed highest weights in the actual carrier).
Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourHighestWeight.highest_weight_values
Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/NormFourHighestWeight.highest_weight_values (✓ std3). ∎
Source. Repository-derived.
Commentary.
For every such D,a,s, every complex h and every nonzero actual state w, if L_0 w=h w and L_m w=0 for every positive integer m, then h=0 or h=1/16 or h=1/2. No positivity, rootlessness, rank, determinant, simplicity or diagonalization assumption is required.
The state-field iterate identity is summed over all nonnegative offsets. Above the descendant’s degree, the left sum vanishes by the shorter descendant’s annihilation bound. Every right offset at least two vanishes by the positive-mode hypothesis on w, leaving exactly offsets zero and one. This proves annihilation for all higher modes and the top-coefficient recurrence together.
The identity L_(-1)=T is proved only on the vacuum and its consumed Virasoro descendants, by comparing the Virasoro and actual translation commutators. It is not imposed on w. The actual mode commutator and derivative-state identity then give the top coefficient of L_(-k)u as (-1)^k(d+(k-1)h)c w when u has degree d and top coefficient c w.
At the literal field coefficient five, the four level-six descendants L_(-2)^3 vacuum, L_(-3)^2 vacuum, L_(-4)L_(-2)vacuum and L_(-6)vacuum act on w by h(h+2)(h+4), 2h(2h+3), h(3h+2) and 5h respectively. Applying this coefficient to the actual null relation with coefficients 64,93,-264,-108 gives 64h(h-1/16)(h-1/2)w=0. Nonvanishing of w and the field property of C give the three asserted values.
The norm-four state and its attribution are those of NormFourVirasoroVector: Shimakura, arXiv:1202.6107, equation (1.3), citing Dong-Mason-Zhu, Theorem 6.3. The actual lattice-field construction follows Bakalov-Kac, arXiv:math/0402315, section 4.1. Imported residue and derivative calculus retain their Carnahan and Matsuo-Nagatomo attribution. This theorem supplies a necessary restriction on actual highest-weight states; it does not assert existence of each weight or classification of representations.