Open Bosonic BRST No Ghost

In the target $M={\mathbb{R}}^{1,25}$, define $X^{\pm }=(X^0\pm X^{25})\mathrm{/}\sqrt{2}$, $p^{\pm }=(p^0\pm p^{25})\mathrm{/}\sqrt{2}$, and ${\alpha }_n^{\pm }=({\alpha }_n^0\pm {\alpha }_n^{25})\mathrm{/}\sqrt{2}$, with transverse indices $i=1,\dots ,24$; then ${\eta }_{+-}=-1$ and $[{\alpha }_m^{+},{\alpha }_n^{-}]=-m{\delta }_{m+n,0}$. Set $k^{\mu }=\sqrt{2{\alpha }^{\mathrm{\prime }}}{p}^{\mu }$ and work in the target momentum chart $p^{+}\mathrm{\ne }0$ (equivalently $k^{+}\mathrm{\ne }0$); this is a change of target basis, not a gauge condition on the BRST complex. Work at $D=26,a=1$, real fixed $k$ with $k^{+}\mathrm{\ne }0$. Let ${\mathcal{C}}_N(k)$ be the relative algebraic subspace generated over $\mathrm{\mid }k⟩\mathrm{\mid }\downarrow ⟩$ by negative-index matter modes and $b_{-n},c_{-n}$, $n>0$, of total oscillator level $N$. Its zero-mode equation is $L_0^{tot}=k^2\mathrm{/}2+N-1=0$. This space is finite dimensional, in all ghost numbers, and the relative differential $d$ is $Q_B$ with its $c_0,b_0$ terms removed. Nilpotence gives $d^2=0$ on this on-shell subspace. Assign degrees $\mathrm{deg}{\alpha }_n^{+}=\mathrm{deg}{c}_n=1$, $\mathrm{deg}{\alpha }_n^{-}=\mathrm{deg}{b}_n=-1$, $\mathrm{deg}{\alpha }_n^i=0$; the relative vacuum has degree zero. At fixed level this degree is bounded above and below. Expanding the quadratic matter Virasoro modes and the cubic ghost term gives $d=d_0+\delta ,d_0=-k^{+}{\sum }_{n\mathrm{\ne }0}{c}_{-n}{\alpha }_n^{-},$ where every term of $\delta$ raises degree by one or two. Explicitly the degree-one terms are $c$ times the transverse quadratic and ${\alpha }^{+}{\alpha }^{-}$ quadratic terms, the transverse momentum term, and $ccb$; the degree-two term is $-k^{-}\sum c_{-n}{\alpha }_n^{+}$. All preserve total level. This includes nonzero transverse momentum; it need not be set to zero. Every sum is locally finite.

The dagger in the existing definition is the antilinear Krein adjoint. The real-momentum charge is Hermitian: the matter term pairs index $n$ with $-n$, the cubic term does so after the CAR reversal, and its zero-mode coefficient is real. In the decomposition $Q_B=c_0{L}_0^{tot}-b_{0}M+d$ all three displayed terms are separately Hermitian; $d^{†}=d$ and $\{d,c_0\}=0$. Therefore for relative $\chi$ and closed $\psi$, $⟨d\chi ,c_0\psi {⟩}_K=⟨\chi ,d{c}_0\psi {⟩}_K=-⟨\chi ,c_{0}d\psi {⟩}_K=0.$ Thus exact vectors are orthogonal to all closed vectors for $h_{rel}$, and the form descends. No BPZ form is substituted here.

Define $K=\frac{1}{k^{+}}{\sum }_{n\mathrm{\ne }0}\frac{{\alpha }_{-n}^{+}{b}_n}{\mathrm{\mid }n\mathrm{\mid }}\mathrm{.}$ The CAR and oscillator relations give $\{d_0,K\}=\mathcal{N}={\sum }_{n>0}[-\frac{{\alpha }_{-n}^{+}{\alpha }_n^{-}+{\alpha }_{-n}^{-}{\alpha }_n^{+}}{n}+c_{-n}{b}_n+b_{-n}{c}_n]\mathrm{.}$ For clarity the two un-normal-ordered contributions are ${\sum }_{n\mathrm{\ne }0}\mathrm{sgn}(n)c_{-n}{b}_n$ and $-{\sum }_{n\mathrm{\ne }0}\mathrm{\mid }n{\mathrm{\mid }}^{-1}{\alpha }_{-n}^{+}{\alpha }_n^{-}$; their constants cancel when the negative indices are moved to down-vacuum normal order. Commuting $\mathcal{N}$ with each of ${\alpha }_{-n}^{+},{\alpha }_{-n}^{-},c_{-n},b_{-n}$ gives that same creator. Its value on the vacuum is zero. Thus its eigenvalue is exactly the total number of these four kinds of creators, a nonnegative integer, and its kernel is precisely the transverse oscillator space ${\mathcal{T}}_N(k)\mathrm{\mid }\downarrow ⟩$. Its ghost number is exactly one. Also $d_0^2=K^2=0$ and $[\mathcal{N},d_0]=[\mathcal{N},K]=0$. Let $\iota :{\mathcal{T}}_N\to {\mathcal{C}}_N$ be inclusion and $\pi$ the projection onto $\mathcal{N}=0$. Set $h=K\mathrm{/}\mathcal{N}$ on positive eigenspaces and $h=0$ on the kernel. Then $d_{0}h+h{d}_0=1-\iota \pi ,h^2=0,\pi h=h\iota =0.$ In particular $H(d_0)={\mathcal{T}}_N$, in ghost number one and in no other ghost number. This is a complete finite-level contraction; $1\mathrm{/}n$ rather than $1\mathrm{/}\mathrm{\mid }n\mathrm{\mid }$ would not give a nonnegative count and is not used.

The operator $h\delta$ strictly raises degree, hence is nilpotent on this finite-level space. Define the finite sums $I=(1+h\delta {)}^{-1}\iota ={\sum }_{r\ge 0}(-h\delta {)}^r\iota ,U=I\pi +dh+h{d}_0\mathrm{.}$ One has $\pi \delta I=0$: $I$ has leading degree zero and strictly positive higher degrees, $\delta$ raises degree, whereas $\pi$ has image only in degree zero. From $I=\iota -h\delta I$, the contraction identity and $d_0\delta +\delta d_0+{\delta }^2=0$ give $(1+h\delta )dI=\iota \pi \delta I=0.$ Thus $dI=0$. Moreover $U=1+(I-\iota )\pi +\delta h$ differs from the identity by a strictly degree-raising operator. Its inverse is a finite geometric sum. Direct multiplication, using $d^2=d_0^2=0$, gives $dU=dh{d}_0=U{d}_0\mathrm{.}$ This is an actual invertible similarity $U^{-1}dU=d_0$ on every on-shell finite-level relative complex, with no term described merely as trivial on physical states. In particular $U\iota =I$ identifies the complete relative cohomology with ${\mathcal{T}}_N$ in ghost number one, all other relative ghost numbers being zero.

The vacuum value is $⟨\downarrow \mathrm{\mid }{c}_0\mathrm{\mid }\downarrow {⟩}_K=⟨0\mathrm{\mid }{c}_{-1}{c}_0{c}_1\mathrm{\mid }0{⟩}_K=1$. In the relative oscillator basis, a nonzero Krein contraction pairs ${\alpha }^{+}$ with ${\alpha }^{-}$ and $c$ with $b$. Dagger preserves the declared degree. Consequently two homogeneous vectors have zero $c_0$ pairing unless their degrees sum to zero. Every $It$ is $t\mathrm{\mid }\downarrow ⟩$ plus strictly positive-degree terms. It follows exactly that $h_{rel}([It],[Is])=⟨t,s{⟩}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}}\mathrm{.}$ The transverse Heisenberg norm is positive definite: a monomial with occupation numbers $r_{n,i}$ has squared norm ${\prod }_{n,i}{n}^{r_{n,i}}{r}_{n,i}!$, and distinct normalized occupation monomials are orthogonal. This proves relative BRST positivity in all levels without claiming that $U$ is unitary on the ambient Krein space.

The preceding finite-level similarity identifies every relative ghost-number-one class with exactly one transverse class, and step 4.1 identifies its form with the positive transverse Fock form. Hence the induced Hermitian form is positive definite in every level. Spatial rotations preserve the metric, oscillator CAR, adjoint, ghost vacuum and $c_0$ insertion, so [F5] transports this result to every nonzero on-shell momentum. No zero-momentum or closed-string sector is added.