The actual ghost BRST Virasoro anomaly

Abstract

Mixed Virasoro anomaly of the cubic ghost charge

The formulas are the classical bc ghost BRST construction, with vacuum cuts b at minus one and c at two. See Kato and Ogawa, Nuclear Physics B 212 (1983), 443–460, and Polchinski, String Theory, volume 1, chapter 4. No novelty of these formulas is claimed. The formal carrier is the original complex finitely supported occupation space.

Theorem 1.1 (R covariance).

$$\forall m\in Int,\forall r\in Int,L\left(m\right)\cdot R\left(r\right)-R\left(r\right)\cdot L\left(m\right)=\mathrm{smul}\left(\mathrm{Int}.\mathrm{cast}\left(-2\cdot m-r\right),R\left(m+r\right)\right)$$

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

Source. Repository-derived.

Commentary.

The actual c Ward identity acts on each pure-c pair. A bijective integer shift aligns both finite sums and their coefficients combine to minus two m minus r.

Theorem 1.2 (Qg odd).

$$\mathrm{parity}\cdot \mathrm{Qg}+\mathrm{Qg}\cdot \mathrm{parity}=0$$

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

Source. Repository-derived.

Commentary.

Actual occupation parity anticommutes with each b and c mode. A product of three odd modes remains odd, and statewise finite supports justify passing parity through the cubic sum.

Definition 1.3 (anomalyCoefficient).

$$\forall r\in Int,\mathrm{anomalyCoefficient}\left(r\right)=\mathrm{div}\left(26,12\right)\cdot \left(\mathrm{Int}.\mathrm{cast}{\left(r\right)}^3-\mathrm{Int}.\mathrm{cast}\left(r\right)\right)$$

Formalization. D5/S3/Quantum/Fermionic/BcGhostAnomaly.anomalyCoefficient (✓ std3).

Source. Repository-derived.

Commentary.

This coefficient uses the ghost central charge minus twenty-six and the commutator direction.

Theorem 1.4 (mixed anomaly).

$$\forall r\in Int,\mathrm{Qg}\cdot L\left(r\right)-L\left(r\right)\cdot \mathrm{Qg}=\mathrm{smul}\left(\mathrm{div}\left(26,12\right)\cdot \left(\mathrm{Int}.\mathrm{cast}{\left(r\right)}^3-\mathrm{Int}.\mathrm{cast}\left(r\right)\right),c\left(r\right)\right)$$

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

Source. Repository-derived.

Commentary.

Graded Jacobi, both cubic Ward identities, finite R covariance and the actual Virasoro theorem make the residual anticommute with all b and c. Multiplication by actual parity puts it in the scalar commutant; oddness forces it to vanish.

This is an algebraic ghost-sector construction. It supplies a dependency for the broader Monster, vertex algebra, conformal field theory and string programme. It does not establish the matter-tensor BRST charge, its cohomology, a no-ghost theorem, string dynamics or AdS/CFT.

References

• Truth anchor: D5/S3/Quantum/Fermionic/BcGhostAnomaly.Qg_odd
• Truth anchor: D5/S3/Quantum/Fermionic/BcGhostAnomaly.R_covariance
• Truth anchor: D5/S3/Quantum/Fermionic/BcGhostAnomaly.anomalyCoefficient
• Truth anchor: D5/S3/Quantum/Fermionic/BcGhostAnomaly.mixed_anomaly
• Dependency: D5/S3/Quantum/Fermionic/BcGhostBRST