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Polynomial Fock Zero-Mode Spectrum

Abstract

The concrete polynomial Fock zero mode has finite energy fibers and exact finite-dimensional eigenspaces.

The operator is the pointwise finite Sugawara L_0 on the complex polynomial Fock space from Polynomial Fock Sugawara Support. A monomial has energy equal to the sum of its exponents weighted by variable index plus one. The result concerns nonnegative integer eigenvalues; it does not construct state fields or a conformal character.

Theorem 1.1 (Each weighted-monomial energy fiber is finite).

Lean statement: D5/S3/VertexAlgebra/PolynomialFockLZeroSpectrum.energyFiber_finite

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

Source. Repository-derived.

Commentary.

At energy N, every occupied variable index is below N and each exponent is at most N. Thus exponent vectors of energy N embed in a finite product of finite intervals, including N = 0.

Theorem 1.2 (The zero-mode eigenspace has the weighted-monomial basis).

Lean statement: D5/S3/VertexAlgebra/PolynomialFockLZeroSpectrum.lZero_spectrum

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

Source. Repository-derived.

Acknowledgement. Yanjun Chu and Zongzhu Lin (2018). Moduli spaces of conformal structures on Heisenberg vertex algebras. URL: https://arxiv.org/abs/1812.11378v1.

Commentary.

For every natural N, the kernel of L_0 minus N times the identity is precisely the span of monomials of energy N, and its complex dimension is the number of those monomials. The proof uses the concrete Sugawara-current commutator and coefficient uniqueness; it does not establish the full Virasoro commutator.

References