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Closed Fully Relative Kunneth Sector

Physical scope

Only the total ghost-number-two (1,1) summand of the fully-relative closed complex at nonzero momentum and fixed chiral levels; not the full semi-relative or absolute closed cohomology.

Statement

For $D = 26$ , $a = 1$ , fixed nonzero target momentum p and nonnegative finite total chiral oscillator levels N and tilde-N satisfying $α^{'} p^{2} / 4 + N - 1 = α^{'} p^{2} / 4 + \tilde{N} - 1 = 0$ , the fully-relative closed complex $C_{c l, r e l} (p; N, \tilde{N})$ of def-st-closed-fully-relative-brst-complex has the canonical Künneth decomposition $H_{c l, r e l}^{2} (p; N, \tilde{N}) = ⨁_{q_{L} + q_{R} = 2} H_{r e l, L}^{q_{L}} (p, N) \otimes H_{r e l, R}^{q_{R}} (p, \tilde{N})$ . Its (1,1) summand has the positive Hermitian form $h_{r e l, L} \otimes h_{r e l, R}$ and, in a nonzero- $p^{+}$ chart, is isometrically identified with the tensor product of the transverse matter Fock spaces of weighted levels N and tilde-N. Here the cohomology labels are total matter-plus-ghost levels; under the isomorphism these labels become the weighted transverse matter levels N and tilde-N. A dressed BRST cycle is constrained by total level, rather than by matter level alone. The two mass-shell equations imply $N = \tilde{N}$ , so this physical tensor sector is level matched. This theorem concerns the fully-relative (1,1) tensor sector and makes no identification with the full semi-relative or absolute closed-string cohomology.

Statement references

Kazuyuki Furuuchi and Nobuyoshi Ohta, On the No-ghost Theorem in String Theory, arXiv:hep-th/0607105v3, Single-chiral source context only. No source theorem for the full closed sector is claimed; the finite-level Künneth argument is supplied locally.

Facts & Assumptions

[F1]

Closed Fully Relative BRST Complex defines the graded left/right tensor complex and the product Hermitian form.

[F2]

Open Bosonic BRST No Ghost proves single-chiral relative ghost-number-one positivity at every finite level and nonzero on-shell effective momentum, with the declared Hermitian $c_{0}$ form.

[F3]

Closed Bosonic Lightcone Spectrum defines the weighted left/right levels $N = \sum_{n > 0, i} α_{- n}^{i} α_{n}^{i}$ and $\tilde{N} = \sum_{n > 0, i} {\tilde{α}}_{- n}^{i} {\tilde{α}}_{n}^{i}$ , the intercept equations $L_{0}^{m} - a = {\tilde{L}}_{0}^{m} - \tilde{a} = 0$ , and their mass-shell consequences with $k_{L} = k_{R} = \sqrt{α^{'} / 2} p$ .

[A1]

Fix the total-level bidegree $(N, \tilde{N})$ in [F1]. Its chiral factors are the actual finite-dimensional relative subcomplexes $C_{r e l, L} (p, N)$ and $C_{r e l, R} (p, \tilde{N})$ , not fixed matter-level spaces.

[A2]

For each closed chiral factor use the open theorem's effective momentum $p_{e f f} = p / 2$ , since $\sqrt{2 α^{'}} p_{e f f} = \sqrt{α^{'} / 2} p$ . This is nonzero exactly when $p \neq 0$ .

Given: Assume $D = 26$ , intercept one, fixed nonzero on-shell closed momentum, finite total chiral levels $N, \tilde{N}$ , and [F2] with its actual nonzero-momentum and relative-complex hypotheses.

Proof

1.1givenF1A1algebra

For finite-dimensional chain complexes over a field, split each degree as boundaries plus chosen cohomology representatives plus a complement mapped isomorphically by the differential onto the next boundaries. Tensoring these decompositions shows that every summand containing a boundary/complement pair is contractible, while the cohomology representatives tensor to the surviving homology. Thus $H^{2} (C_{L} \otimes C_{R}) = ⨁_{q_{L} + q_{R} = 2} H^{q_{L}} (C_{L}) \otimes H^{q_{R}} (C_{R})$ and the $(1, 1)$ summand is canonically $H_{r e l, L}^{1} (p, N) \otimes H_{r e l, R}^{1} (p, \tilde{N})$ . The canonical map sends $[x] \otimes [y]$ to $[x \otimes y]$ ; the auxiliary splittings prove bijectivity without making the map choice dependent.

2.1givenF2F3A2step 1.1algebra

The two mass-shell equations in [F3] are $α^{'} p^{2} / 4 + N - 1 = 0$ and $α^{'} p^{2} / 4 + \tilde{N} - 1 = 0$ . Subtracting gives $N = \tilde{N}$ . The conversion in [A2] matches each chiral zero mode to the nonzero open-string parameter, so the finite-total-level construction in [F2], steps 1.1–4.1, applies to both factors. Its contraction has kernel the transverse matter Fock space $T_{N}$ tensored with $∣ ↓ ⟩$ , and its finite degree-raising correction preserves total level. Its $c_{0}$ pairing equals the transverse positive form exactly. Thus it gives isometric identifications $H_{r e l, L}^{1} (p, N) ≅ T_{N}$ and $H_{r e l, R}^{1} (p, \tilde{N}) ≅ T_{\tilde{N}}$ . The labels become matter/transverse weighted levels through these isomorphisms; a generic chain vector can mix matter levels. Transport from a nonzero- $p^{+}$ chart is the rotation transport included in [F2].

3.1step 1.1step 2.1F1algebra∎

Choose orthonormal bases $e_{i}$ and $f_{j}$ for the two positive finite-dimensional factor forms. For a nonzero tensor $v = \sum_{i, j} z_{i j} e_{i} \otimes f_{j}$ , the product form gives $⟨ v, v ⟩ = \sum_{i, j} ∣ z_{i j} ∣^{2} > 0$ . Hence the $(1, 1)$ summand has positive product form and is level matched. The result concerns only the fully-relative $(1, 1)$ summand, not semi-relative or absolute closed cohomology.

Depends on

Dependency tree · 2 levels

Sources

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