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Bosonic BRST Complex

Physical scope

Open or single-chiral closed-string BRST oscillator complex at fixed target momentum on the algebraic finite-particle domain; the Krein form is indefinite off physical cohomology.

Definition

At fixed target momentum p, choose chiral zero mode $k^{μ} = α_{0}^{μ}$ by $k = \sqrt{2 α^{'}} p$ for the open string and $k = \sqrt{α^{'} / 2} p$ for each closed-string chiral half. On the algebraic tensor product of the corresponding matter oscillator Fock space and finite-particle bc ghost Fock space, with $gh (c_{n}) = + 1$ and $gh (b_{n}) = - 1$ , put $∣ ↓ ⟩ = c_{1} ∣ 0 ⟩$ . In the BRST cubic mode sum only, normal order relative to this relative vacuum: $b_{n}$ and $c_{n}$ annihilate it for $n > 0$ , while $b_{0} ∣ ↓ ⟩ = 0$ and $c_{0}$ is the separate zero mode. Define $Q_{B} = \sum_{n} c_{- n} (L_{n}^{m} - a δ_{n, 0}) - \frac{1}{2} \sum_{m, n} (m - n) : c_{- m} c_{- n} b_{m + n} :$ . On the unital complex oscillator mode superalgebra, before fixing the zero-mode eigenvalue, define the linear graded BPZ anti-involution by $bpz (α_{n}^{μ}) = (- 1)^{n + 1} α_{- n}^{μ}, bpz (b_{n}) = (- 1)^{n} b_{- n}, bpz (c_{n}) = (- 1)^{n + 1} c_{- n}$ , and $bpz (1) = 1$ , $bpz (A B) = (- 1)^{∣ A ∣ ∣ B ∣} bpz (B) bpz (A)$ for parity-homogeneous mode polynomials $A, B$ . Here parity $∣ A ∣ \in {0, 1}$ is the number of ghost generators modulo two in each homogeneous monomial; matter generators and complex scalars have parity zero. Extend by complex linearity. Since $bpz (α_{0}^{μ}) = - α_{0}^{μ}$ , this map exchanges the zero-mode specializations k and -k; it is not an endomorphism of a nonzero fixed-momentum fibre or the Hermitian physical form. Separately define an antilinear Krein adjoint dagger by $(α_{n}^{μ})^{†} = α_{- n}^{μ}, b_{n}^{†} = b_{- n}, c_{n}^{†} = c_{- n}$ , with scalar fixed-momentum matter vacuum pairing $⟨ 0; k ∣ 0; k ⟩_{K} = 1$ and ghost normalization $⟨ 0 ∣ c_{- 1} c_{0} c_{1} ∣ 0 ⟩_{K} = 1$ . The resulting matter-plus-ghost ambient Hermitian Krein form is indefinite. A delta normalization between distinct generalized momentum eigenstates is not a scalar form on this fixed-momentum fibre and is not used here. Descent to cohomology and positivity require separate results and are not asserted by this definition. Then $gh (Q_{B}) = + 1$ and $L_{0}^{t o t} = {Q_{B}, b_{0}}$ . When $Q_{B}$ is nilpotent, define $C_{r e l}^{q} (p) = {ψ : gh (ψ) = q, b_{0} ψ = 0, L_{0}^{t o t} ψ = 0}$ and $H_{r e l}^{q} (p) = \ker (Q_{B} ∣ C_{r e l}^{q}) / im (Q_{B} ∣ C_{r e l}^{q - 1})$ ; on $q = 1$ first define the scalar chain form $h_{r e l} (ψ, ϕ) = ⟨ ψ ∣ c_{0} ∣ ϕ ⟩_{K}$ on $C_{r e l}^{1} (p)$ , with $c_{0}^{†} = c_{0}$ . For cycles psi,phi the notation $h_{r e l} ([ψ], [ϕ])$ denotes the candidate induced form only when independence of cycle representatives has been proved. With the relative vacuum assigned ghost number one, define the total oscillator level operator $N_{t o t} = N_{m} + \sum_{n > 0} n (b_{- n} c_{n} + c_{- n} b_{n})$ , where $N_{m} = \sum_{n > 0} α_{- n} \cdot α_{n}$ . Its eigenvalue on $∣ k ⟩ ∣ ↓ ⟩$ is zero, and every negative-index matter or ghost creator of index -n adds n. For each nonnegative integer N let $F_{N} (p)$ be the span of all such monomials of total level N over $∣ k ⟩ ∣ ↓ ⟩$ , with no $c_{0}$ factor. Define $C_{r e l}^{q} (p, N) = C_{r e l}^{q} (p) \cap F_{N} (p)$ and $H_{r e l}^{q} (p, N)$ as its $Q_{B}$ cohomology when $Q_{B}$ is nilpotent. On $F_{N} (p), L_{0}^{t o t} = k^{2} / 2 + N - a$ . Thus $C_{r e l} (p, N) = F_{N} (p)$ when this scalar vanishes and is zero otherwise; N labels total matter-plus-ghost oscillator level, not matter level alone. The full algebraic relative complex is the direct sum of these total-level complexes.

Definition references

David Tong, Lectures on String Theory, arXiv:0908.0333v3, full 218-page lecture notes, §§5.4–5.4.3 (PDF file pp.127–131; printed pp.120–124); BRST context, not the full relative-complex convention.

Kazuyuki Furuuchi and Nobuyoshi Ohta, On the No-ghost Theorem in String Theory, arXiv:hep-th/0607105v3, pp. 2–4; $Q_{B}$ decomposition, on-shell condition, and relative $b_{0}$ restriction.

Sebastian Uhlmann, String Field Theory: Methods And Solutions (2003), Chapter III.2, printed p.38 (PDF p.48), equations (III.20)–(III.23); linear BPZ and graded mode-product reversal.

Facts & Assumptions

[F1]

Bosonic Matter Oscillator Modes defines the dimensionless modes with $[α_{m}^{μ}, α_{n}^{ν}] = m η^{μ ν} δ_{m + n, 0}$ and $L_{n}^{m} = \frac{1}{2} \sum_{r} : α_{n - r} \cdot α_{r} :$ , normally ordered on the algebraic finite-particle Fock domain at fixed momentum.

[F2]

Bosonic Bc Ghost System fixes the ghost sector

[F3]

Bosonic Covariant State Space defines the dimensionless normal-ordering intercept parameter.

The momentum vector $p$ is a fixed element of $R^{1, D - 1}$ , and the two displayed values of $k^{μ} = α_{0}^{μ}$ are the open and single-chiral closed normalizations. The matter Fock space is the algebraic span of finite products of negative-index matter modes; the ghost Fock space is the algebraic span of finite products of creation modes from the SL(2)-invariant vacuum. Matter normal ordering is relative to the momentum vacuum. The BRST cubic mode sum is normal ordered relative to $∣ ↓ ⟩ = c_{1} ∣ 0 ⟩$ , while the underlying bc fields and ghost stress tensor retain their SL(2)-vacuum convention. Only the conjugate pair $c_{1}, b_{- 1}$ changes creator/annihilator roles between the two vacua: $c_{1}$ annihilates $∣ ↓ ⟩$ and $b_{- 1}$ creates, whereas the SL(2) roles are reversed. In the cubic sum a contraction of this pair requires $(m, n) = (- 1, 0)$ or $(0, - 1)$ . Since ${b_{- 1}, c_{1}} = 1$ , $- b_{- 1} c_{0} c_{1} = c_{1} c_{0} b_{- 1} + c_{0}$ ; thus the first ordered triple changes by $c_{0}$ with prefactor $1 / 2$ , and the second changes by $- c_{0}$ with prefactor $- 1 / 2$ . All other triples have unchanged ordering or zero contraction. Consequently $G_{↓} - G_{S L (2)} = c_{0}$ , so the displayed physical intercept a corresponds to the SL(2)-ordered charge $C + G_{S L (2)} - (a - 1) c_{0}$ . All mode sums act locally finitely on every finite-particle vector. The BPZ map uses graded product reversal, whereas $†$ uses ordinary reversal $(A B)^{†} = B^{†} A^{†}$ and is antilinear. In particular $bpz ({b_{m}, c_{n}}) = (- 1)^{m + n} {b_{- m}, c_{- n}} = δ_{m + n, 0}$ , so the odd reversal sign preserves the ghost CAR, including ${b_{0}, c_{0}} = 1$ . For matter modes, ordinary reversal gives $bpz ([α_{m}^{μ}, α_{n}^{ν}]) = (- 1)^{m + n} [α_{- n}^{ν}, α_{- m}^{μ}] = m η^{μ ν} δ_{m + n, 0}$ ; zero ghost anticommutators and mixed commutators remain zero. Applying BPZ twice fixes each generator and each product, so it is a well-defined graded anti-involution of the stated abstract algebra. BPZ and the Hermitian Krein adjoint are not interchangeable.

Verification

Given: Use the BRST operator in the Definition on the algebraic tensor-product domain, the matter, ghost, and intercept conventions of [F1]–[F3], and assume it is nilpotent when defining cohomology.

1.1givenF1F2F3algebra

In the anticommutator ${Q_{B}, b_{0}}$ , the linear matter term contributes $L_{0}^{m} - a$ because ${c_{- n}, b_{0}} = δ_{n, 0}$ . Anticommuting $b_{0}$ through the cubic term selects $m = 0$ or $n = 0$ , since ${b_{0}, c_{- m}} = δ_{m 0}$ and ${b_{0}, c_{- n}} = δ_{n 0}$ . The remaining relative-vacuum normally ordered sum is $\sum_{n > 0} n (b_{- n} c_{n} + c_{- n} b_{n})$ . Their sum is exactly the total zero mode $L_{0}^{t o t}$ .

2.1givenF1F2step 1.1algebra

The oscillator commutators and ghost CAR give $[N_{t o t}, α_{- n}^{μ}] = n α_{- n}^{μ}$ , $[N_{t o t}, b_{- n}] = n b_{- n}$ and $[N_{t o t}, c_{- n}] = n c_{- n}$ for $n > 0$ , with zero eigenvalue on the relative vacuum. Anticommuting the charge with $b_{0}$ in the relative-vacuum ordering therefore gives $L_{0}^{t o t} = k^{2} / 2 + N_{t o t} - a$ . Every monomial of $Q_{B}$ has net mode index zero, so $[N_{t o t}, Q_{B}] = 0$ . At fixed $N$ only indices $1 \leq n \leq N$ occur; matter occupations are bounded by $N / n$ and each odd ghost creator occurs at most once. Thus $F_{N}$ is finite dimensional across all ghost numbers, including the one-dimensional $N = 0$ vacuum case. Fixed matter-level spaces alone need not be preserved: $c_{- n} L_{n}^{m}$ transfers $n$ units from matter to ghosts.

3.1givenstep 1.1F1F2algebra

If $Q_{B}^{2} = 0$ , then $[Q_{B}, L_{0}^{t o t}] = [Q_{B}, {Q_{B}, b_{0}}] = 0$ . For $ψ$ with $b_{0} ψ = L_{0}^{t o t} ψ = 0$ , one has $b_{0} Q_{B} ψ = L_{0}^{t o t} ψ - Q_{B} b_{0} ψ = 0$ and $L_{0}^{t o t} Q_{B} ψ = Q_{B} L_{0}^{t o t} ψ = 0$ . Thus the relative subspaces and, by step 2.1, their fixed total-level subspaces are preserved. The displayed quotients are cohomologies when nilpotence holds.

4.1givenF1F2∎

The declared ghost-number-one sesquilinear form is a candidate physical form only after it descends to cohomology; this definition alone asserts neither that descent nor positivity.

Bosonic BRST Complex

Definition

Definition references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources