The exact matter anomaly after shifted finite cancellation
Abstract
The exact matter anomaly after shifted finite cancellation
Assume only that H is a two-sided inverse of the complex lattice Gram matrix. The original matter Virasoro theorem and original ghost weight-minus-one Ward identity determine the two commutators. The construction remains on the algebraic tensor product and uses no completion or tensor commutant assumption.
Definition 1.1 (The shifted original-operator action family).
Lean statement: D5/S3/VertexAlgebra/Lattice/LatticeMatterGhostCovariance.paired
Formalization. D5/S3/VertexAlgebra/Lattice/LatticeMatterGhostCovariance.paired (✓ std3).
Source. Repository-derived.
Commentary.
For every D,H and integers r,k, paired D H r k is the tensor map sugawaraMode D H k tensor BcGhostFock.c (r-k). Its statewise support is finite. Translating the finite captures by m maps to m+r supplies the actual noncentral cancellation.
Theorem 1.2 (Original-operator covariance with the rank anomaly).
Lean statement: D5/S3/VertexAlgebra/Lattice/LatticeMatterGhostCovariance.covariance
Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/LatticeMatterGhostCovariance.covariance (✓ std3). ∎
Source. Repository-derived.
Commentary.
For every D,H with H G=G H=1 and every integer r, [T,M_r+G_r]=-(rank(D)/12)(r^3-r) C_r. For a simple tensor choose U=U(v), V=U(sugawaraMode r v), d=max(0,-r), J=[r-A,U] and I=[min(-A-d,-r),max(V,U+d)]. I captures x, M_r x and G_r x, contains J and J-r, and contains the central index -r. A finite bijection m maps to m+r converts the matter noncentral coefficient into k-2r; it cancels the ghost coefficient 2r-k. At m=-r the central cubic is the negative of r^3-r. The statement neither constructs the total BRST charge nor proves nilpotence. The cubic ghost supplier, its composition with this matter term and later physical interpretation remain separate obligations.