The Actual Polynomial Fock C2 Subspace

Abstract

The actual polynomial Fock minus-two span is the higher-variable ideal, and all nonpositive modes have explicit images.

Let F=C[X_0,X_1,…] over the complex numbers. The current mode -j-1 multiplies by X_j, mode zero is zero, and mode j+1 is (j+1) times partial differentiation in X_j, for every natural j. Use the actual state-field map Y formed from the monomial basis and right-nested normal products of divided current derivatives. Write mu(u,n,v)=Y(u)_n v for its normalized integer coefficients.

Define pi:F -> C[x] as the complex algebra homomorphism sending X_0 to x and X_j to zero for every j>=1. Define C2 as the complex linear span of all mu(u,-2,v), with u and v arbitrary in F. Let I be the ideal generated by X_1,X_2,…, regarded as a complex linear subspace. The kernel below is also a complex linear subspace.

Theorem 1.1 (The actual C2 span and every nonpositive coefficient).

$$C_2=I\wedge C_2=\ker \left(pi\right)\wedge \forall u,v\in F,\forall n\in \mathbb{Z},n\le 0\Rightarrow ((n=-1\Rightarrow \mathrm{pi}\left(\mathrm{mu}\left(u,n,v\right)\right)=\mathrm{pi}\left(u\right)\mathrm{pi}\left(v\right))\wedge (n\ne -1\Rightarrow \mathrm{pi}\left(\mathrm{mu}\left(u,n,v\right)\right)=0))$$

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/PolynomialFockC2.actual_c2_modes (✓ std3). ∎

Citation. Tomoyuki Arakawa (2016). Introduction to W-algebras and their Representation Theory. URL: https://arxiv.org/abs/1605.00138v2.

Commentary.

There are no additional hypotheses on either polynomial or on the integer n beyond n<=0. At n=-1 the projected coefficient is pi(u) pi(v); at every other nonpositive mode it is zero. In particular the induced minus-one operation is multiplication and the zero-mode bracket vanishes.

For every finite word, every input polynomial and every nonpositive mode, induction evaluates the projected field. Each of the two original polynomial-valued normal-product summands has finite support by the Laurent order of its actual intermediate field on the fixed input, before pi is moved through either sum. A divided current mode -k-1 creates binomial(j+k,j) X_(j+k), so projection retains only j=k=0. Its mode zero vanishes. In the nonnegative branch, k>=1 leaves a tail mode n-k-1<=-2; induction applies to the actual changed input. The monomial-basis definition of Y extends this calculation to every state.

The actual singleton coefficient is mu(X_j,-2,v)=(j+1) X_(j+1) v for every natural j and every polynomial v. Inverting the nonzero complex scalar j+1 puts every higher-variable multiple in C2 without assuming that C2 is an ideal. Polynomial coefficient support identifies the higher-variable ideal with ker(pi), and the all-state minus-two calculation gives the opposite inclusion.

Arakawa, Section 3.8, equation (3.19), describes the classical limit of universal affine vacuum algebras. Li, Corollary 3.6 and Proposition 3.7, gives the general C2 multiplication and bracket. Matsuo-Nagatomo, Sections 2.1-2.3, supplies the free-boson normalization. These classical results provide context for the specified actual coefficient calculation. No mathematical novelty is asserted. No positive-mode descent or vertex-algebra quotient structure is asserted. Rationality, C2 cofiniteness, charged-module Jacobi, fusion, a Monster realization, string theory and AdS/CFT spacetime dynamics are beyond this result.

References

• Truth anchor: D5/S3/VertexAlgebra/PolynomialFockC2.actual_c2_modes
• Dependency: D5/S3/VertexAlgebra/PolynomialFockJacobi