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Off-shell gauge complexes, BRST doublets and anomaly obstructions — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-generatedVerification: AI-generatedverified 2026-10-22 (gpt-6.1-sol)Open item page →

Gauge-fixing doublets cannot create a new polynomial class

Example

In the local Abelian BRST algebra, take the nonminimal pair $\overset{ˉ}{c}, B$ with $\overset{ˉ}{c}$ odd, ghost number $- 1$ , $B$ even, ghost number $0$ , and $s \overset{ˉ}{c} = B$ , $s B = 0$ . The polynomial $B^{2}$ is $s$ -closed, has doublet degree two, and is exact: $B^{2} = s (\overset{ˉ}{c} B) = s (\frac{h B^{2}}{2}) .$ This finite polynomial calculation illustrates removal of a doublet class. It does not establish positivity of any physical Hilbert-space quotient.

Example references

Abelian local jet BRST differential

Nilpotence and contraction of nonminimal doublets

Facts & Assumptions

Given: The nonminimal BRST pair and the doublet homotopy.

[F1]

Abelian local jet BRST differential: $\overset{ˉ}{c}$ is odd with ghost number $- 1$ , $B$ is even with ghost number $0$ , and $s \overset{ˉ}{c} = B$ , $s B = 0$ .

[F2]

Nilpotence and contraction of nonminimal doublets: for this pair, $h = \overset{ˉ}{c} \partial_{L} / \partial B$ , $N = \overset{ˉ}{c} \partial_{L} / \partial \overset{ˉ}{c} + B \partial_{L} / \partial B$ , and $s h + h s = N$ .

Verification

Given: [F1]–[F2].

1.1givenF1algebra

Since $B$ is even, the graded Leibniz rule gives $s (B^{2}) = (s B) B + B (s B) = 0$ , so $B^{2}$ is $s$ -closed.

1.2F1algebra

Because $\overset{ˉ}{c}$ is odd, $s (\overset{ˉ}{c} B) = (s \overset{ˉ}{c}) B - \overset{ˉ}{c} (s B) = B^{2}$ . Thus $B^{2}$ is exact with primitive $\overset{ˉ}{c} B$ , whose ghost number is $- 1$ .

2.1F2step 1.2algebra∎

The operator $N$ counts the two factors of $B$ , so $N (B^{2}) = 2 B^{2}$ , while $h B^{2} = 2 \overset{ˉ}{c} B$ . Therefore $s (h B^{2} / 2) = B^{2}$ , agreeing with the general contraction of [F2]. This is an algebraic polynomial identity and makes no statement about a positive norm or a state space.