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Polynomial Fock Mode Commutators

Abstract

The concrete polynomial Fock modes satisfy the Heisenberg and Sugawara-current relations.

The modes and pointwise finite Sugawara operators are those of Polynomial Fock Sugawara Support. The generic Sugawara proof of Kytola supplies a reference for the commutator calculation; the concrete Heisenberg premise is established here from polynomial differentiation.

Theorem 1.1 (Every pair of integer modes has the Heisenberg commutator).

Lean statement: D5/S3/VertexAlgebra/PolynomialFockSugawaraCommutators.mode_heisenberg

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

Source. Repository-derived.

Acknowledgement. Kalle Kytölä (2025). VirasoroProject, Sugawara.lean. URL: https://github.com/kkytola/VirasoroProject/blob/5ff4245383b2cdd4eea7a0524bc1274c32041eb4/VirasoroProject/Sugawara.lean.

Commentary.

For all integer m and n, mode m times mode n minus the reverse product is m times the identity when m+n=0, and zero otherwise. The mixed sign case differentiates a variable multiplier; the two same sign cases commute.

Theorem 1.2 (The Sugawara operator acts on every mode with weight one).

Lean statement: D5/S3/VertexAlgebra/PolynomialFockSugawaraCommutators.L_mode_commutator

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

Source. Repository-derived.

Acknowledgement. Kalle Kytölä (2025). VirasoroProject, Sugawara.lean. URL: https://github.com/kkytola/VirasoroProject/blob/5ff4245383b2cdd4eea7a0524bc1274c32041eb4/VirasoroProject/Sugawara.lean.

Commentary.

For all integer m and r, L m times mode r minus the reverse product is minus r times mode (m+r). Commuting through the pointwise finite sum leaves two singleton contributions. This result does not assert the L-L commutator or its central term.

References