DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-05 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Closed Fully Relative BRST Complex

Physical scope

The fixed-level fully-relative left/right tensor complex of the critical flat closed bosonic string; it is distinct from semi-relative and absolute complexes.

Definition

At fixed target momentum $p$ , take left and right copies of the single-chiral relative complex of def-st-bosonic-brst-complex with $k_{L} = k_{R} = \sqrt{α^{'} / 2} p$ , and order the graded tensor product left factor before right factor. Define $Q_{c l} = Q_{L} \otimes 1 + (- 1)^{{g h}_{L}} 1 \otimes Q_{R}$ , and impose $b_{0}^{L} = b_{0}^{R} = 0$ and $L_{0}^{t o t, L} = L_{0}^{t o t, R} = 0$ . The fully-relative complex has total ghost number ${g h}_{L} + {g h}_{R}$ . For chiral chain states $ψ_{L}, ϕ_{L}$ in $C_{r e l, L}^{1} (p)$ and $ψ_{R}, ϕ_{R}$ in $C_{r e l, R}^{1} (p)$ , define $h_{c l} (ψ_{L} \otimes ψ_{R}, ϕ_{L} \otimes ϕ_{R}) = ⟨ ψ_{L} ∣ c_{0}^{L} ∣ ϕ_{L} ⟩_{K} ⟨ ψ_{R} ∣ c_{0}^{R} ∣ ϕ_{R} ⟩_{K}$ and extend sesquilinearly to their algebraic tensor product. These are chain forms, not evaluations on unnamed cohomology classes. If both chiral forms descend to $H_{r e l}^{1}$ , their product defines the candidate induced form on $H_{r e l, L}^{1} \otimes H_{r e l, R}^{1}$ by the same formula on cycle representatives. The displayed product fixes its sign convention; any combined zero-mode representation must agree with it, with $c_{0}^{L}$ preceding $c_{0}^{R}$ . For nonnegative integers $N$ and $\tilde{N}$ define $C_{c l, r e l}^{r} (p; N, \tilde{N}) = ⨁_{q_{L} + q_{R} = r} C_{r e l, L}^{q_{L}} (p, N) \otimes C_{r e l, R}^{q_{R}} (p, \tilde{N})$ , with each chiral level the total matter-plus-ghost level defined in the supplier. Define $H_{c l, r e l}^{r} (p; N, \tilde{N})$ as the cohomology of this tensor complex. The chiral zero-mode equations are $α^{'} p^{2} / 4 + N - 1 = 0$ and $α^{'} p^{2} / 4 + \tilde{N} - 1 = 0$ at $D = 26, a = 1$ ; off either equation the corresponding relative factor is zero. This is a fully-relative tensor complex, not the full closed semi-relative BRST complex.

Definition references

Kazuyuki Furuuchi and Nobuyoshi Ohta, On the No-ghost Theorem in String Theory, arXiv:hep-th/0607105v3, pp. 2–5, single-chiral relative BRST source conventions only; the left/right tensor complex is defined locally.

Facts & Assumptions

[F1]

Bosonic BRST Complex defines each chiral factor.

[F2]

Bosonic BRST Nilpotence proves square zero for the open-string oscillator charge at $D = 26, a = 1$ . Step 1.1 transfers that result to each closed chiral factor by an explicit fixed-zero-mode identification; no equality of the open and closed target momenta is assumed.

The tensor product is the algebraic graded tensor product, left factor first. The sign $(- 1)^{g h_{L}}$ is the Koszul sign required for an odd differential on a graded tensor product. The zero-mode constraints are imposed separately in each factor. The product form is the ordinary tensor-product Hermitian form with the left zero-mode insertion preceding the right one if a representative formula is used. This is a definition of the fully-relative complex only; no identification with the full semi-relative or absolute closed-string complex is part of it.

Verification

Given: The critical parameters $D = 26, a = 1$ , the fixed target momentum $p$ , the chiral oscillator/ghost data and relative constraints from [F1], and independent left/right factors on their graded algebraic tensor product.

1.1givenF1F2algebra

Put $p_{o p} = p / 2$ . The open zero mode at this momentum is $\sqrt{2 α^{'}} p_{o p} = \sqrt{α^{'} / 2} p$ , exactly the zero mode in either closed chiral factor. Identify the open matter and ghost vacuum at $p_{o p}$ with the chosen chiral vacuum at $p$ and send each finite creation-mode monomial to the monomial with the same mode indices and target indices. By F1, the oscillator commutators, ghost CAR, ghost grading, vacuum annihilators and relative-vacuum normal ordering are identical; this is a linear isomorphism intertwining every oscillator and ghost generator. Hence it intertwines every $L_{n}^{m}$ , the same intercept term and the full locally finite cubic BRST sum. It also intertwines $b_{0}$ , $L_{0}^{t o t} = {Q_{B}, b_{0}}$ and the total-level operator. Thus the open charge of F2 and each closed chiral charge are conjugate under this explicit identification. F2 at $D = 26, a = 1$ gives $Q_{L}^{2} = Q_{R}^{2} = 0$ at every $p$ , with the relative subspaces and fixed levels preserved as in F1. This is an algebraic change of momentum label, not an identification of the full open and closed physical theories.

2.1givenF1F2step 1.1algebra

For a homogeneous left state of ghost number $q$ , the graded tensor differential is $Q_{c l} = Q_{L} \otimes 1 + (- 1)^{q} 1 \otimes Q_{R}$ . Squaring gives $Q_{L}^{2} \otimes 1 + 1 \otimes Q_{R}^{2}$ plus two mixed terms with signs $(- 1)^{q + 1}$ and $(- 1)^{q}$ ; those cancel. Hence $Q_{c l}^{2} = 0$ .

3.1givenF1F2step 1.1step 2.1algebra∎

Each chiral differential preserves its own relative subspace by the conditional identity in [F1]'s verification, so their signed sum preserves both $b_{0}^{L} = b_{0}^{R} = 0$ and both $L_{0}^{t o t} = 0$ constraints. Each fixed total-level factor is finite dimensional and preserved by its differential by [F1], so the displayed bidegree is an actual finite-dimensional tensor subcomplex. The scalar chain forms in the Definition are sesquilinear, so their tensor product is well-defined on the chain (1,1) subspace. If a cycle representative is changed by a chiral boundary, the product changes by zero whenever that chiral form pairs boundaries with cycles as zero; this proves the stated conditional descent to the tensor product of chiral cohomologies. Descent and positivity are not consequences of the tensor-complex definition alone; the displayed product fixes the zero-mode sign convention.

Depends on

Dependency tree · 2 levels

Sources

Kazuyuki Furuuchi and Nobuyoshi Ohta, On the No-ghost Theorem in String Theory, arXiv:hep-th/0607105v3