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Original bc conformal state and cubic-current zero mode

Abstract

Conformal and cubic-current states of the original occupation ghost fields.

These are the classical algebraic bc ghost formulas in the original finite-occupation Fock representation, with b vacuum cut at minus one and c cut at two. The conventions follow Kato and Ogawa, Nuclear Physics B 212 (1983), 443–460, and Polchinski, String Theory, volume 1, chapter 4. The proofs use the existing original all-state Y and signed integer iterate; no abstract replacement state-field map or transported charge is introduced.

Definition 1.1 (The explicit conformal state).

Lean statement: D5/S3/Quantum/Fermionic/BcGhostConformalBRSTState.Omega

Formalization. D5/S3/Quantum/Fermionic/BcGhostConformalBRSTState.Omega (✓ std3).

Source. Repository-derived.

Commentary.

Omega is minus twice ket{0,3} plus ket{1,2}, equivalently minus twice b(-2)c(0)vacuum minus b(-3)c(1)vacuum. The sorted-occupation signs and original mode shifts are retained.

Definition 1.2 (The explicit raw odd cubic-current state).

Lean statement: D5/S3/Quantum/Fermionic/BcGhostConformalBRSTState.Jg

Formalization. D5/S3/Quantum/Fermionic/BcGhostConformalBRSTState.Jg (✓ std3).

Source. Repository-derived.

Commentary.

Jg is ket{0,1,3}, literally b(-2)c(1)c(0)vacuum. There is no improvement term.

Theorem 1.3 (Every integer conformal mode is the original Virasoro mode).

Lean statement: D5/S3/Quantum/Fermionic/BcGhostConformalBRSTState.conformal_modes

Proof. Machine-checked in Lean as D5/S3/Quantum/Fermionic/BcGhostConformalBRSTState.conformal_modes (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every integer m, the (m+1)-st mode of Y(Omega) equals the original BcGhostVirasoro.L(m). The two negative-residue normal products are reindexed on their actual statewise finite integer half-lines. Their coefficients combine to the literal weight m+q in the original normal-ordered bilinear sum.

Theorem 1.4 (Actual odd parity of the cubic state).

Lean statement: D5/S3/Quantum/Fermionic/BcGhostConformalBRSTState.current_odd

Proof. Machine-checked in Lean as D5/S3/Quantum/Fermionic/BcGhostConformalBRSTState.current_odd (✓ std3). ∎

Source. Repository-derived.

Commentary.

Parity sends the sorted three-occupation state Jg to minus Jg. This supplies the odd sign in the zero-residue identity and is consumed in the current comparison and signed derivation.

Theorem 1.5 (The current zero mode is the entire original cubic charge).

Lean statement: D5/S3/Quantum/Fermionic/BcGhostConformalBRSTState.current_zero_mode

Proof. Machine-checked in Lean as D5/S3/Quantum/Fermionic/BcGhostConformalBRSTState.current_zero_mode (✓ std3). ∎

Source. Repository-derived.

Commentary.

The zero mode Y(Jg)[[0]] equals the complete original locally finite sum over integer pairs (p,q), with coefficient one half times (p-q) and the original normal ordering of c(p)c(q)b(-p-q). Its anticommutators on all b and c modes follow from actual generator contractions and the zero-residue iterate. The difference from Qg anticommutes with the generators; composing with parity gives an ordinary commutant, to which scalar_commutant legitimately applies. Vacuum vanishing removes the scalar, and parity involution removes the composition.

Theorem 1.6 (Signed derivation on every actual homogeneous-state field mode).

Lean statement: D5/S3/Quantum/Fermionic/BcGhostConformalBRSTState.Qg_signed_derivation

Proof. Machine-checked in Lean as D5/S3/Quantum/Fermionic/BcGhostConformalBRSTState.Qg_signed_derivation (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every original Fock state a, every natural parity exponent p with parity(a)=(-1)^p a, and every integer n, Qg Y(a)[[n]] minus (-1)^p Y(a)[[n]] Qg equals Y(Qg a)[[n]]. This is an operator equality on the full original finite-occupation carrier. The signed integer iterate at residue zero collapses to its i=0 terms; the actual current equality and odd parity identify the operator with Qg. No ambient square-zero, rank-26 or matter-plus-ghost premise is used.

The pure ghost charge is anomalous: no nilpotence is asserted. These results do not establish a total coupled BRST system, BRST cohomology, a no-ghost theorem, Monster moonshine, string dynamics or AdS/CFT. Native verification, independent review and official admission are distinct statuses.

References