Truncated Polynomial Presentation of the Actual C2 Algebra
Abstract
Truncated Polynomial Presentation of the Actual C2 Algebra.
Theorem 1.1 (Truncated Polynomial Presentation of the Actual C2 Algebra).
Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourVacuumC2Presentation.actual_vacuum_c2_presentation
Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/NormFourVacuumC2Presentation.actual_vacuum_c2_presentation (✓ std3). ∎
Source. Repository-derived.
Commentary.
For every finite integral symmetric even-diagonal lattice D, charge a with B(a,a)=4, and complex s with s squared equal to one, let S be the actual vacuum descendant space, and let C be the inverse image in S of literal C2, whose two input states both belong to S. Equip S/C with the actual coefficient algebra constructed in NormFourVacuumC2Algebra. There exists a complex algebra equivalence from AdjoinRoot(X cubed), equivalently the polynomial algebra modulo the ideal generated by X cubed, to this exact quotient. The class of X maps to the class of the actual norm-four Virasoro axis vector.
Let x denote that axis class. Actual representative multiplication identifies x to the nth power with the class of L_(-2) to the nth power applied to vacuum. The actual level-six vacuum null relation proves x cubed is zero in literal C2. AdjoinRoot.liftAlgHom therefore constructs the map with the specified root image.
The range contains the three actual representatives: vacuum, L_(-2) vacuum and L_(-2) squared vacuum. The existing reduction of every finite negative word into C2 plus their span proves surjectivity on the whole vacuum descendant quotient. No basis or abstract quotient presentation is assumed. The monic cubic gives dimension three of the domain, and the accepted actual low-weight separator theorem gives dimension three of the codomain. Equality of these dimensions and surjectivity prove injectivity.
This is the classical Ising C2 presentation realized on the production lattice modes. Li, Abelianizing vertex algebras supplies the C2 framework; the norm-four vector retains its Dong-Mason-Zhu and Shimakura attribution. The conclusion is an algebra equivalence with the literal actual quotient and the actual axis image. It makes no claim about rationality, Zhu algebra, fusion, analytic CFT or string completion.