Parabolic BRST restriction through the simple rank-one affine vacuum

A finite perturbation contraction, with actual formulas. Let (C,d0) be a cochain complex, and let i:H→C, p:C→H, h:C→C[-1] satisfy $pi=1,d0i=pd0=0,d0h+hd0=1-ip$ and $h^2=hi=ph=0$. Let δ have degree1, (d0+δ)²=0. Assume δh and hδ strictly raise a finite filtration on each relevant homogeneous block. Define finite sums A=(1+δh)^(-1)δ=δ(1+hδ)^(-1), dH=pAi, I=i-hAi, P=p-pAh, Hc=h-hAh.

Then $dI=IdH,Pd=dHP,PI=1$ and $dHc+Hcd=1-IP$, where $d=d0+\delta$. The inverse sums terminate on each block, so no analytic completion or infinite cancellation is used.

Here is an algebraic check sufficient to reconstruct every identity. The two defining inverse equations imply A=δ-δhA=δ-Ahδ. Substitution of these into (d0+δ)²=0 gives d0 A + A d0 + A i p A=0.

For example, expand dI-IdH, replace d0h by 1-ip-hd0, cancel δi against the leading term Ai using A=δ-δhA, and use the displayed identity on i. The remaining terms are $h(d0A+AipA)i=0$. The P identity is its reversed calculation. $PI=1$ follows from $ph=hi=h^2=0$. Expanding dHc+Hcd with the same substitutions gives 1-(i-hAi)(p-pAh). Squaring $dH=pAi$ and using the displayed identity, together with $pd0=d0i=0$, gives $d{H}^2=0$. Thus the perturbed homology is exactly the homology of (H,dH); it need not equal H.

The last qualification is essential for the Wakimoto problem. Oscillator/Koszul contraction can remove contractible variables, but it does not prove that the transferred differential vanishes or that its remaining complex is exact. Those are separate coefficient calculations.

Explicit oscillator contraction. On a polynomial-superalgebra $C[x_a,y_b]\otimes \mathrm{\Lambda }({\xi }_a,{\eta }_b)$, with x,y even and ξ,η odd, put d0=Σ_a ξ_a ∂/∂x_a + Σ_b y_b ∂/∂η_b, κ=Σ_a x_a ∂/∂ξ_a + Σ_b η_b ∂/∂y_b.

Odd Leibniz expansion gives d0κ+κd0=N, where N counts the total number of x,y,ξ,η factors. Both cross-index anticommutators vanish. Hence on positive total degree h=κ/N contracts the complex, while on constants $h=0$; p is constant extraction and i is inclusion. The side conditions above hold, and d0h+hd0=1-ip. Every monomial involves only finitely many variables, so the formulas are algebraic even for countably many oscillators.

For an affine Fock/ghost block one must additionally prove that the intended perturbation raises a filtration which is bounded on the fixed loop-energy AND horizontal-weight block. Loop energy alone does not bound zero-mode coordinate degree: a beta-gamma zero-mode coordinate can have energy zero. The horizontal weight is therefore essential. Nor does finite-dimensionality of each block alone prove strict triangularity of the perturbation. The actual current/screening differential must be checked term by term before applying the finite inverse above.

Nilpotent-loop semiregular contraction. For a finite-dimensional positively root-graded nilpotent Lie algebra m, let $Lm=m[t,t^{-1}]$, split into m[t] and $m[t^{-1}]t^{-1}$. Its semiregular space is the ordered PBW space $U(m[t^{-1}]t^{-1})\otimes U(m[t]{)}^{\ast }$, with graded dual on the second factor; the action is the regular/co-regular action obtained by moving a generator through the PBW order. The loop algebra has no central cocycle here, since the invariant form vanishes on m. The semi-infinite ghost complex uses that same polarization.

For abelian m this complex is precisely the oscillator complex above. A negative-mode regular coordinate y has its contraction ghost η and differential y∂η. A nonnegative-mode co-regular coordinate x has its exterior ghost ξ and differential ξ∂x. The constant survives in degree zero and every other monomial is contracted by κ/N. The nonnegative zero-mode coordinates must be included; their horizontal root weights make their degree bounded on a fixed principal-weight block even though their loop energy is zero.

For general m, use a central series with root-graded central last term z. PBW orders z modes first and factors the underlying semiregular space and ghost space into the z oscillator part and the quotient part. The z action is exactly free on its negative-mode U(z) factor and cofree on its nonnegative-mode dual factor: moving a z generator introduces no commutator. Consequently the z differential is exactly the abelian d0, not merely its leading symbol.

Filter the whole BRST complex by quotient ghost degree, with central ghost degree as the other index. The Chevalley differential has components of bidegree (0,1), (1,0), and (2,-1): respectively the central differential, the quotient differential/action, and the bracket of two quotient lifts with value in z. This follows directly by sorting the three possible bracket/action terms in the BRST formula. The (0,1) component is the explicit central oscillator differential above. Its cohomology is the quotient semiregular space in the single central cohomological degree zero. On these central constants, the (1,0) component is exactly the quotient BRST differential: the commutator of two lifted quotient actions differs from the quotient action by a central action, which is zero on central cohomology. Thus the first cohomology page is the quotient complex in row zero. After taking its differential, every later differential of bidegree (r,1-r), $r\ge 2$, has zero source or zero target. This is an actual bounded filtration argument, not an Euler-characteristic argument. The possible normal-order scalar is absent: a product of adjoint actions on the positively root-graded nilpotent algebra has trace zero, because each factor strictly raises finite root height. Repeat along the central series. At each fixed principal weight, all creation variables have negative principal weight, only finitely many modes can occur, and their total number is bounded by that weight's height. Exterior factors obey the same bound. Thus the filtration is bounded on every block, all perturbation inverses are finite there, and central-series induction proves that the semiregular BRST cohomology is C in degree zero and zero otherwise. Tensoring with a spectator module preserves this contraction.

CAR Fock spaces and the precise correction. For a finite Lie-algebra basis $x_a$ use odd generators $b_a(n),c^a(n),n$ integral, with anticommutators $\{b_a(m),c^b(n)\}={\delta }_{a,b}{\delta }_{m+n,0}$ and all other anticommutators zero. Wedge multiplication and contraction construct their Fock space faithfully. Give c degree1 and b degree-1. The standard loop vacuum is killed by $b_a(n),n\ge 0$, and $c^a(n),n>0$. Its fields $b(z)=\sum b(n)z^{-n-1}$ and $c(z)=\sum c(n)z^{-n}$ have OPE $b_a(z)c^b(w){\delta }_{a,b}\mathrm{/}(z-w)$, by the CAR mode sum. Normal-ordered derivatives span their creative fields; the same finite-pole residue proof as for local affine fields, with the Koszul signs, gives the super vertex algebra and its module Jacobi identity. No ghost construction is assumed merely from its name.

For induction one uses instead the principal polarization. At loop mode0 every negative finite root and every Cartan direction interchanges its creator c with its annihilator b; positive finite roots retain the Chevalley convention. Thus the b creators represent the nonpositive principal subalgebra, including the Cartan, while c creators represent the dual of the strictly positive principal subalgebra. The extra Cartan flip contributes no root-weight trace because ad H is zero on the Cartan. This Fock space is a module for the same ghost fields, not a new creative vacuum vertex algebra. Set its principal vacuum cohomological degree to0; c and b retain degrees1 and-1, fixing the semi-infinite degree normalization. The finite change of normal order in the cubic BRST expression contributes the linear term $\gamma (H)c^H(0)$, with gamma=-sum_(negative finite roots) $\alpha =2{\rho }_g$. Explicitly each reordered pair uses $cb=1-bc$; the two antisymmetric structure-constant terms combine to minus that root's Cartan weight. The opposite parabolic contains ALL negative finite roots, so its correction is exactly2rho_g, also when the Levi is proper. On nonzero loop modes, the normal-order anomaly is the invariant form Tr_pminus(ad a ad b). The zero-mode linear term is thus fixed by an explicit CAR calculation, not chosen as an unsupported extra hypothesis.

The BRST operator is the actual mode sum Q=sum_(a,n) c^a(-n) J_a(n) -(1/2)sum_(a,b,c,m,n) f_ab^c :c^a(-m)c^b(-n)b_c(m+n): +sum_H 2rho_g(H)c^H(0).

Each fixed vector sees finitely many modes: a sufficiently positive current kills it and the complementary sufficiently positive ghost mode annihilates its finite exterior word. In principal polarization the finite zero-mode correction above is included. In nilpotent positive-loop restriction no negative finite roots are flipped and no Cartan correction occurs. For that restriction the BRST operator is the zero mode of a genuine weight-one ghost/current state, hence a derivation of the tensor super vertex algebra. Its square and its module action are checked by the mode calculation below.

Currents and the actual anomaly. Let ${\mathfrak{p}}_{-}=l+{\mathfrak{m}}_{-}$ be the finite opposite parabolic, with $l$=its center plus simple factors $l_i$. Gauge its centerless loop algebra $q={\mathfrak{p}}_{-}[t,t^{-1}]$. Use charged ghosts for a finite basis of pminus and the principal polarization (including the finite zero-mode root signs). The BRST operator is the zero mode of sum_a c^a J_a - (1/2)sum_(a,b,c) f_ab^c :c^a c^b b_c:

with the linear Cartan ghost correction gamma(H)=2rho_g(H). On finite Levi coroots this equals 2rho_l; its difference is a character of the parabolic, so the outside-Cartan choice causes no bracket anomaly. The correction cancels precisely the finite zero-mode coboundary caused by the principal polarization.

Let $S_g^{opp}$ be the opposite-principal-polarization semiregular bimodule constructed by the same restricted-coinduction argument as the tilting bimodule, with gamma=-2rhohat. Fix its left central level k. Its right central level is k+2hvee_g, since left $K=\text{right }K-2h_g^{\vee }$. Regard minus the right action as a left q action. Tensor with a module M in the Levi highest-weight category (finite principal-graded pieces and a finite union of upper cones) for the direct sum of affine Levi factors, extending mminus by zero on M. The Wakimoto modules and the Hom spaces used below have exactly these properties. The resulting q-current cocycle is -(k+2hvee_g)(a,b)_g + k_i(a,b)_li

on the simple Levi factor i, and has the analogous central-Cartan form. This is a projective representation of the centerless loop parabolic; no inconsistent identification of its independent Levi centers is made.

The double ghost contraction contributes Tr_pminus(ad a ad b). On $l_i$, write $c_i=2\mathrm{/}({\theta }_i,{\theta }_i{)}_g$, so $(a,b{)}_g=c_i(a,b{)}_{li}$. The decomposition $g={\mathfrak{m}}_{+}+l+{\mathfrak{m}}_{-}$ and duality of mplus,mminus give Tr_pminus(ad a ad b) =2hvee_i(a,b)_li+Tr_mminus(ad a ad b) =(c_i hvee_g+hvee_i)(a,b)_li.

On the Levi center it is hvee_g(a,b)_g. Wick expansion of Q squared has only the current-cocycle plus this ghost-cocycle term: the cubic terms cancel by Jacobi, and single-contraction bracket terms cancel between the linear and cubic summands. The stated Cartan correction removes the zero-mode coboundary. Therefore Q squared is zero exactly at k_i=c_i(k+hvee_g)-hvee_i, k_center=k+hvee_g.

This proves the required shift, including its sign, in the actual induction complex. It is not unshifted induction from $g[t^q]$. The left g action on $S_g^{opp}$ commutes with all right-current and ghost operations, hence descends to a genuine affine g action on its cohomology.

The actual restriction currents. Write $\mathfrak{m}={\mathfrak{m}}_i^{+}$ and choose a basis $u_b$ of $\mathfrak{m}$. For $a\in \mathfrak{l}$, write $[a,u_b]={\sum }_c{f}_{ab}^c{u}_c$. In the STANDARD-loop ghost complex of steps 13.1 and 16.1, define ${\tilde{J}}_a(z)=J_a(z)-{\sum }_{b,c}{f}_{ab}^c:c^b(z)b_c(z):\mathrm{.}$ The ghost part acts by $c^d\mapsto -{\sum }_b{f}_{ab}^d{c}^b$ and $b_d\mapsto {\sum }_c{f}_{ad}^c{b}_c$. Consequently its commutator with the action term of $Q$ cancels the commutator of $J_a$ with that term; its commutator with the cubic term is zero by the derivation identity for $\mathrm{ad}a{\mathrm{\mid }}_{\mathfrak{m}}$. Thus $[Q,{\tilde{J}}_a(n)]=0$ for every $n$. Single contractions give ${\tilde{J}}_{[a,a^{\mathrm{\prime }}]}$; double contractions give the central form $k(a,a^{\mathrm{\prime }}{)}_{\mathfrak{g}}+{\mathrm{Tr}}_{\mathfrak{m}}(\mathrm{ad}a\mathrm{ad}{a}^{\mathrm{\prime }})$. Step 21.1 evaluates this as $k_i(a,a^{\mathrm{\prime }}{)}_{{\mathfrak{s}\mathfrak{l}}_2}$ on the simple Levi and $(k+h^{\vee })(a,a^{\mathrm{\prime }}{)}_{\mathfrak{g}}$ on its center. These are actual cochain operators, natural in every restricted current module. On the vacuum tensor ghost vertex algebra they are creative fields and define the universal Levi vertex map $\mathrm{\Psi }$ into $Q$-closed states by the current reconstruction in F4.

The induced module and an explicit cohomological reduction. Define $F(M)=H^{\mathrm{\infty }\mathrm{/}2+0}(q,S_g^{\mathrm{o}\mathrm{p}\mathrm{p}}\otimes M)$ with the action and cocycle just specified. The following PBW reduction proves its vanishing in other degrees and exactness, rather than assuming an induction theorem.

Comparison with the actual simple vacuum. Put $\lambda =k{\mathrm{\Lambda }}_0$ and let $(C^{\bullet },d_C)$ be the actual Wakimoto resolution in F2, with its identity-label summand $W_{\mathfrak{g}}(\lambda )\subset C^0$. The canonical current-module map $r_{\mathfrak{g}}:V^k(\mathfrak{g})\to W_{\mathfrak{g}}(\lambda )$ takes the vacuum to its normalized top vector $v_W$. To check the vacuum relations in general rank, each finite-simple vector $f_j{v}_{\lambda }$ is killed by its one-root polarization map (F2, the one-root four-term sequence), which factors into a sufficiently large translation comparison (F2, the actual semiregular current-linear comparisons). Thus the canonical map from $M(\lambda )$ factors through the actual finite-simple vacuum quotient, exactly as in the named actual vacuum singular-line interface of F2. Every other Weyl label has highest weight strictly below $\lambda$. Hence $d_C{v}_W=0$, and current linearity gives $d_C{r}_{\mathfrak{g}}=0$. Also $Q(v_W\otimes 1_{\mathrm{g}\mathrm{h}})=0$, since $r_{\mathfrak{g}}$ obeys the vacuum current relations. The map $r_{\mathfrak{g}}\otimes 1$ therefore maps the universal-vacuum BRST complex into the total complex of $C^{\bullet }\otimes F_{\mathrm{g}\mathrm{h}}$. It intertwines each corrected-current mode in step 23.2, since it intertwines the current modes and is the identity on ghosts. No vertex-algebra structure on the entire resolution is asserted.

Order the root coordinates first outside the parabolic, then in q. The two-sided PBW models of $S_g^{opp}$ separate its q regular/co-regular coordinates. Transfer the diagonal M action to the spectator factor by the coproduct maps f tensor u tensor m ↦ sum_(u) f tensor u_(1) tensor u_(2)m

and its graded-dual transpose. Their inverses use the antipode. For one generator X the identities Delta(X)=X tensor1+1 tensor X and S(X)=-X verify that the old diagonal action becomes the action on the semiregular q factor alone. Multiplication proves the assertion for a word. The two maps commute with the differential and the remaining outside actions. Their central terms agree exactly because the preceding cocycle calculation made the combined q level minus its ghost trace. The Cartan zero-mode constraint is -right H + M_H + 2rho_g(H)=0.

The full Cartan action retains the dual adjoint term. In the right PBW model of F1 it is $H(f\otimes b)=-(f\circ \mathrm{ad}H)\otimes b+f\otimes (H-2\widehat{\rho }(H))b$. The constraint in step 26.1 applies on the surviving q augmentation, after the q ghost Cartan action is included. It cancels the scalar shift; it does not remove the weights of outside coordinates. If $\xi$ is a homogeneous word in the outside regular and dual coordinates, the surviving action is $H(\xi \otimes m)={\mathrm{w}\mathrm{t}}_{\mathrm{o}\mathrm{u}\mathrm{t}}(\xi )(H)\xi \otimes m+\xi \otimes Hm$, where a regular root coordinate has its root weight and a dual coordinate has the negative of its root weight. In particular the outside augmentation has weight zero, so its top finite weight is the original M weight. This is the weight comparison for the induced Wakimoto top.

Acyclicity for the loop parabolic, without nilpotence. Put $\mathfrak{b}={⨁}_{j\le 0}{\mathfrak{q}}_j$ and $\mathfrak{n}={⨁}_{j>0}{\mathfrak{q}}_j$ in the principal grading. Both are subalgebras; $\mathfrak{q}=\mathfrak{b}\oplus \mathfrak{n}$ as vector spaces and ${\mathfrak{q}}_0$ is its finite Cartan. The q coefficient factor separated in steps 25.1–26.1 has the two semiregular PBW models: it is free over $U(\mathfrak{b})$ and cofree over $U(\mathfrak{n})$, with the normal-order character already included. Cofree here means the restricted ${\mathrm{Hom}}_{\mathbb{C}}(U(\mathfrak{n}),V)$ with action $(x\phi )(u)=\phi (ux)$ and algebraic multiplicity space V; its graded pieces involve only finitely many positive-degree words. The outside coordinates and M are spectators after the Hopf change of variables. The current cocycle restricts to zero on each of $\mathfrak{b}$ and $\mathfrak{n}$, since it pairs only opposite nonzero principal degrees; its mixed term is canceled by the ghost anomaly of step 22.1. Thus the ordinary cochain and chain calculations for each half apply to this BRST complex.

Write S for this q coefficient factor, retaining spectator multiplicities. The ghost space in the polarization of step 14.1 is $\mathrm{\Lambda }(\mathfrak{b})\otimes \mathrm{\Lambda }({\mathfrak{n}}^{\ast })$, with b ghosts of degree $-1$ and n ghosts of degree $+1$. There are two filtrations of its normal-ordered BRST complex. Filtering first by the number of b ghosts gives the first cohomology page $E_1^{p,q}=H^q(\mathfrak{n},{\mathrm{\Lambda }}^{-p}(\mathfrak{q}\mathrm{/}\mathfrak{n})\otimes S)$, for $p\le 0$, $q\ge 0$. Filtering in the other direction gives $E_1^{\mathrm{\prime }}{}^{p,q}=H_{-q}(\mathfrak{b},{\mathrm{\Lambda }}^p((\mathfrak{q}\mathrm{/}\mathfrak{b}{)}^{\ast })\otimes S\otimes {\mathbb{C}}_{\gamma })$, for $p\ge 0$, $q\le 0$, with the character $\gamma (H)=2{\rho }_{\mathfrak{g}}(H)$ on b and zero on its negative-root directions. These formulas follow by sorting the action and cubic terms in step 15.1: terms preserving the first ghost count are the ordinary n cochain differential with its quotient adjoint action, and terms preserving the second count are the ordinary b chain differential with the stated character. The remaining mixed terms change the filtration. Closure of each half under brackets ensures compatibility; the canceled mixed anomaly ensures the square is zero.

Tensoring a cofree n module with the exterior coefficient module in the first page remains cofree: the coproduct and antipode maps of step 26.1 move its diagonal action to the cofree factor. The coproduct sums on a homogeneous vector are finite by its finite ghost word and bounded principal support. Its ordinary n cohomology therefore vanishes in degrees $q>0$. Explicitly apply restricted Hom to the augmented free bar resolution of the trivial n module; insertion of the unit contracts its underlying vector-space resolution, and tensor-Hom adjunction gives the cofree cochain contraction. The same argument on the second page, using the free b factor, gives $H_j(\mathfrak{b},-)=0$ for $j>0$: tensor the bar resolution with the free coefficient module and use the unit-insertion contraction. Antisymmetrization gives the displayed Chevalley complexes; its comparison is obtained by the ordered PBW filtration, whose associated complex is the polynomial Koszul resolution, and every correction lowers finite enveloping word length. This establishes the same vanishing for their action/bracket differentials. The first filtration thus places total cohomology in degrees at most zero, and the second places it in degrees at least zero. All nonzero cohomological degrees vanish. No central-series argument for q is used.

In degree zero these same filtrations identify cohomology with the image of the n invariant subspace in the b coinvariants with the character of step 28.1. In the cofree PBW model n invariants are the augmentation functional tensored with the remaining regular words. In the free PBW model passing to b coinvariants evaluates those regular words at their character. Their composite is precisely constant extraction on the q coordinates, leaving every outside coordinate and M. The scalar Cartan correction cancels as in step 27.1. This identifies the surviving vector space and its natural projection, not just the concentration degree. The identifications are natural in M, and the surviving outside and Levi actions are obtained from their commuting current operators.

The two filtrations converge in the algebraic complex at issue. On a fixed total principal depth all non-Cartan creation variables, including both ghost types, have strictly negative principal degree. Their heights and numbers are bounded by that depth and the upper bound of the M grading; only finitely many root modes are then available. Cartan zero ghosts form a finite exterior algebra, so they add a finite bound on ghost numbers. Cartan coefficient polynomials can have arbitrarily large word degree globally, but each vector is a finite polynomial, and each ordinary bar or PBW comparison above uses only finitely many of its words. Hence both ghost filtrations are bounded on the fixed-depth complex, even though its Cartan coefficient space need not be finite dimensional. Restricted duals in the positive half involve finite-degree words only. There is no unproved finite perturbation inverse for a nonnilpotent q, and no infinite sum at a fixed weight.

What remains after the q contraction is precisely the polarized semiregular space of the outside nilpotent loop algebra Lmplus, tensor M. Consequently F(M) as an Lmplus-module = US(Lmplus) tensor M, H^(infinity/2+r)(q,S_g^opp tensor M)=0 for r!=0.

To compare with $L(\lambda )$ without assuming an augmentation of the two-sided complex, use its good truncation $T^p=C^p$ for $p<0$, $T^0=\ker (d_C:C^0\to C^1)$, and $T^p=0$ for $p>0$. There are actual quasi-isomorphisms $C\leftarrow T\to H^0(C)=L(\lambda )$, the second normalized to send $[v_W]$ to the simple vacuum. Tensoring with the ghosts and taking the BRST total complexes preserves these quasi-isomorphisms. Indeed at a fixed full Cartan weight and loop depth only finitely many Weyl labels and creation monomials occur, by the finite-weight property in F2 and the bounds in steps 8.1 and 31.1; the zero-mode exterior space is finite, and the Cartan contraction is pointwise finite. The resulting finite block filtrations prove convergence of both spectral-sequence orders. All maps commute with the corrected-current modes. Thus this zigzag identifies the class of every corrected-current mode polynomial on $v_W\otimes 1_{\mathrm{g}\mathrm{h}}$ with the same polynomial on the actual simple vacuum in $L(\lambda )\otimes F_{\mathrm{g}\mathrm{h}}$.

This is natural in M and proves exactness of F. Applying Lmplus restriction contracts that outside semiregular factor by the central-series oscillator homotopy already written out. The surviving corrected Levi currents act on M itself, with the levels above. Hence restriction composed with F is naturally the identity. In particular F is faithful on Hom spaces; a nonzero Levi map induces a nonzero g map whose restriction is the original map.

For $M=W_l({\nu }_l)$, this actual induced module is $W_g(\nu )$. This is a PBW/action identification, not a character comparison: its outside root creators and dual creators are exactly the missing roots in the full Wakimoto polarization. At each fixed depth choose the finite antidominant translation that has already dualized all roots relevant to that depth. The ordered normal commutators in the induction quotient and in the finite-polarization module are identical; the Cartan equality just proved gives the same top nu, and the cocycle calculation gives the same current central terms. Thus the basis identification intertwines every current mode. This proves both F(W_l(nu_l))=W_g(nu), H^(infinity/2+r)(Lmplus,W_g(nu))=0 for r!=0, H^(infinity/2+0)(Lmplus,W_g(nu))=W_l(nu_l).

Here nu_l has the same finite Cartan weight as nu and the shifted levels $k_i$ and $k_{center}$.

In the other order, steps 33.1–33.1 identify the $Q$-cohomology of each Wakimoto term with its actual Levi Wakimoto module in degree zero, and with zero in other degrees. The contraction is normalized by constant extraction on the outside semiregular coordinates; it sends $[v_W\otimes 1_{\mathrm{g}\mathrm{h}}]$ to the Levi top vector $v_{W,\mathfrak{l}}$. The surviving operators are exactly the Levi current modes, by step 33.1 and the explicit formula in step 23.2. Therefore this spectral sequence identifies total cohomology with the cohomology of the complex $R(C^{\bullet })$ of restricted Wakimoto modules, and identifies the class of a corrected-current polynomial with that same current polynomial on $v_{W,\mathfrak{l}}$. This identifies a specified state, rather than only a Hom space or an abstract isomorphism of cohomology vector spaces.

The actual singular state under the edge. Reduce $k_i+2=c_{i}p\mathrm{/}q$ to $p_i\mathrm{/}{q}_i$, where $g_i=\gcd (c_i,q)$, $p_i=c_{i}p\mathrm{/}{g}_i$ and $q_i=q\mathrm{/}{g}_i$. The rank-one extra integral root is ${\beta }_i=-{\alpha }_i+q_i\delta$; it is a semi-infinite cover of the identity of cohomological degree $-1$. Let $j_i:M(s_{{\beta }_i}\cdot k_i{\mathrm{\Lambda }}_0)\to M(k_i{\mathrm{\Lambda }}_0)$ be the actual ordinary singular embedding and $q_i^{\mathrm{v}\mathrm{a}\mathrm{c}}:M(k_i{\mathrm{\Lambda }}_0)\twoheadrightarrow V^{k_i}({\mathfrak{s}\mathfrak{l}}_2)$ the finite-simple vacuum quotient. By F3 and F4, $A_i=q_i^{\mathrm{v}\mathrm{a}\mathrm{c}}{j}_i(v_i)$ is the actual nonzero generator of the maximal rank-one vacuum ideal. Choose its scalar by this equality. The repaired actual-current/singular-line interface of F2 gives the commuting equality $d_i{r}_{i,\mathrm{s}\mathrm{r}\mathrm{c}}(v_i)=r_{i,\mathrm{v}\mathrm{a}\mathrm{c}}{j}_i(v_i)={\bar{r}}_i(A_i)\mathrm{.}$ Here $r_{i,\mathrm{s}\mathrm{r}\mathrm{c}}$ is the actual finite-polarization comparison into the source Wakimoto module, $r_{i,\mathrm{v}\mathrm{a}\mathrm{c}}={\bar{r}}_i{q}_i^{\mathrm{v}\mathrm{a}\mathrm{c}}$ is the canonical CURRENT-module comparison into $W_{{\mathfrak{s}\mathfrak{l}}_2}(k_i{\mathrm{\Lambda }}_0)$, and $d_i$ is the normalized rank-one Wakimoto edge. The excluded-root half-determinant test proves this particular vector nonzero. The equality concerns the actual vacuum-quotient singular state; it is not the scalar null polynomial on a horizontal highest line. No injectivity assertion about the whole map ${\bar{r}}_i$ is needed.

Tensor this edge with the spectator Heisenberg top and induce it by $F$. Step 33.1 gives $R\circ F=1$, so the induced map is nonzero and restricts to $d_i$. The actual global cover Hom space is one-dimensional by F2's restricted-adjunction cover calculation, so this induced map is a nonzero scalar multiple of the component of $d_C$ from the $s_{{\beta }_i}$ label to the identity label. Therefore the IDENTITY COMPONENT of $R(d_C)$ on the source highest vector is a nonzero scalar multiple of ${\bar{r}}_i(A_i)\otimes 1_{\mathrm{H}\mathrm{e}\mathrm{i}\mathrm{s}}$. This component equality alone does not prove that the isolated identity-summand vector belongs to the image of the FULL differential.

The other components of this column vanish. The restricted differential commutes with every spectator Heisenberg zero mode, by the natural corrected currents in step 23.2. Each restricted Wakimoto summand $W_{\mathfrak{l}}((w\cdot \lambda {)}_{\mathfrak{l}})$ has a SINGLE spectator zero-mode charge, namely the restriction of the finite part of $w\cdot \lambda$ to $\mathfrak{z}(\mathfrak{l})$: neither the rank-one root modes nor the Heisenberg creation modes change it. A global cover component from $w$ to $w^{\mathrm{\prime }}=s_{\gamma }w$ changes the highest weight by $m\gamma$, where $\gamma$ is a real integral root and $m=⟨w(\lambda +\widehat{\rho }),{\gamma }^{\vee }⟩\mathrm{\ne }0$ by regularity. If its restricted map is nonzero, equality of spectator charges forces ${\gamma }_{\mathrm{f}\mathrm{i}\mathrm{n}}{\mathrm{\mid }}_{\mathfrak{z}(\mathfrak{l})}=0$. The annihilator of $\mathfrak{z}(\mathfrak{l})$ in the finite root span is exactly $\mathbb{C}{\alpha }_i$, so the reduced finite root system forces ${\gamma }_{\mathrm{f}\mathrm{i}\mathrm{n}}=\pm {\alpha }_i$. The integral roots with these finite parts have exactly the rank-one period $q_i$ in step 37.1: evaluating the affine coroot pairing gives the condition $q\mid c_{i}n$ for a root $\pm {\alpha }_i+n\delta$, equivalently $q_i\mid n$. Hence $s_{\gamma }$ lies in the rank-one integral affine Weyl subgroup generated by $s_{{\alpha }_i}$ and $s_{{\beta }_i}$. Starting with $w=s_{{\beta }_i}$, every nonzero restricted outgoing component therefore remains in that subgroup. In affine rank one the semi-infinite lengths are the even integers on translations and the odd integers on translated finite reflections, with a unique label at each integer; in particular its only degree-zero label is the identity. Thus EVERY other degree-zero component on this source summand is zero. Combining this with step 38.1 and rescaling the source highest vector gives an actual finite $y_i\in R(C^{-1})$ satisfying $R(d_C)y_i={\iota }_1({\bar{r}}_i(A_i)\otimes 1_{\mathrm{H}\mathrm{e}\mathrm{i}\mathrm{s}})\mathrm{.}$ Here ${\iota }_1$ is the inclusion of the identity summand into the full degree-zero row. This proves a full-column boundary equation, rather than inferring one from a single differential component.

From the row boundary to a boundary in the vacuum BRST algebra. Write the finite vacuum state $A_i$ as a finite linear combination of negative Levi-current words on $1$. Applying the corrected-current words to the actual simple vacuum defines precisely $\mathrm{\Psi }(A_i)$ in $A=L(\lambda )\otimes F_{\mathrm{g}\mathrm{h}}$. Steps 24.2, 32.2 and 36.1 identify its $Q$-cohomology class with the class of ${\bar{r}}_i(A_i)\otimes 1_{\mathrm{H}\mathrm{e}\mathrm{i}\mathrm{s}}$ in $H^0(R(C^{\bullet }))$: current linearity, vacuum normalization, and the ghost identity fix this identification on each word. That vector is a genuine row boundary by the full-column equation in step 39.1. Therefore $[\mathrm{\Psi }(A_i)]=0$ in $H^0(A,Q)$. Equivalently there is a finite vector $b_i\in A$ with $Q{b}_i=\mathrm{\Psi }(A_i)$. Finiteness follows either from the definition of algebraic cohomology or from the finite-block total-complex comparisons above. A boundary equation in a Wakimoto module alone would not establish this conclusion; the current-equivariant comparison with the actual simple vacuum in steps 24.2, 32.2 and 36.1 is essential.

The map from the universal Levi vertex algebra to $H^0(A,Q)$ consequently kills its actual maximal rank-one vacuum generator. The kernel of a vertex-algebra homomorphism is a vertex ideal; F3 identifies the ideal generated by $A_i$ with the maximal rank-one vacuum ideal. Thus this homomorphism factors through $L_{{\widehat{\mathfrak{s}\mathfrak{l}}}_2}(k_i{\mathrm{\Lambda }}_0)$. This is an actual current/state factorization, before its field homotopy is applied to arbitrary modules below.

Its vacuum is nonzero: in $L(\lambda )\otimes F_{\mathrm{g}\mathrm{h}}$ the lowest-loop-depth complex is the ordinary finite Chevalley complex of $\mathfrak{m}$ with coefficients $\mathbb{C}$. It has no cochains in negative cohomological degree, and its degree-zero vacuum spans $H^0(\mathfrak{m},\mathbb{C})=\mathbb{C}$. Since $Q$ preserves loop depth, that vacuum cannot become a boundary at another depth. The induced vertex homomorphism from the simple rank-one vacuum is therefore nonzero and hence injective. The commuting spectator Heisenberg currents have the level already computed in step 23.2.

The boundary acts by an actual field homotopy on every weak module. Use the STANDARD-loop nilradical restriction ghost vertex algebra from steps13.1/16.1, not the noncreative principal Fock vacuum used for induction. Its tensor product $A=V\otimes F_{ghost}$ contains the odd BRST state q; $Q=q_0$ is its differential. For any weak V-module $M,C(M)=M\otimes F_{ghost}$ is a weak A-module. The zero-mode sum defining $Q_M$ is pointwise finite: sufficiently positive currents kill a fixed coefficient vector, while the complementary positive ghost modes kill its finite exterior word. The same fixed-level current/CAR anomaly calculation proving $Q^2=0$ gives $Q_M^2=0$; no positive-energy or highest-weight restriction on M is used here.

Super module Jacobi at the zero mode gives the coefficient identities $[Q_M,Y_{C(M)}(a,z){]}_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{r}}=Y_{C(M)}(Qa,z),[q_0,a_n{]}_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{r}}=(q_{0}a{)}_n\mathrm{.}$ There are no higher-binomial terms because the current mode is zero. Thus closed states act by cochain maps, and the cohomology is an actual weak module over H(A,Q).

Let Ψ be the corrected universal Levi current vertex map into Q-closed states of A constructed above. Step 40.1 establishes that the actual vacuum singular state $\mathrm{\Psi }(A_i)$, not merely its highest-vector scalar polynomial, represents zero in H(A,Q). By the definition of algebraic cohomology there is $b_i$ in A with $Q{b}_i=\mathrm{\Psi }(A_i)$. It is an actual finite vector: the comparison and contractions in steps 24.2, 32.2 and 36.1 take place in the fixed homogeneous block containing $\mathrm{\Psi }(A_i)$. No multiplicative identification of the whole Wakimoto resolution with A is needed for this conclusion; the compatible current comparison in steps 24.2, 32.2 and 36.1 identifies this precise state and its zero cohomology class.

On C(M) the field of this generator has the explicit nullhomotopy $Y_{C(M)}(\mathrm{\Psi }(A_i),z)=Q_M{Y}_{C(M)}(b_i,z)-(-1{)}^{\mathrm{\mid }{b}_i\mathrm{\mid }}{Y}_{C(M)}(b_i,z)Q_M\mathrm{.}$ Every mode therefore annihilates cohomology. The annihilator of a weak vertex-module representation is a vertex ideal: its iterate identity expresses every mode of $u_m$ a as locally finite sums of compositions with a-modes, and translation preserves a zero field. Hence the entire ideal generated by $A_i$ annihilates $H(C(M),Q_M)$. By F3 this is precisely the maximal rank-one vacuum ideal, so the action factors through the SIMPLE rank-one quotient for every weak M. This field argument does not infer arbitrary-module annihilation from a null polynomial on one highest line.

The construction is also linear over any commuting coefficient algebra, including $D=C[t]\mathrm{/}(t^2)$. In particular a genuine generalized affine highest-plane extension is covered without imposing Cartan semisimplicity: all current/ghost operators commute with D, and the displayed field homotopy is D-linear. This is the operator interface used by generalized highest planes in the central-reduction argument.

For a positively loop-graded highest affine module, the lowest-loop-depth complex is its coefficient top tensor exterior(mplus^*). Its degree convention has no negative-cohomological cochains there. For a simple L(lambda) the coefficient top is $L_g(lambdabar)$, giving exactly the lowest-depth complex asserted in the Statement. This last identification needs a lowest depth; the all-weak-module factorization in step 46.1 does not.

and its differential is exactly the ordinary finite Chevalley differential: all nonzero loop modes raise the depth, and the zero modes give the displayed action/bracket formula. Its r-th cohomology is $H^r({\mathfrak{m}}_{+},L_g(lambdabar))$, with no negative r at that depth. In degree0 the highest vector is a nonzero invariant and cannot be a boundary. It yields an actual rank-one highest module, so Admissible affine sl2 vacuum null polynomial applies with its stated rank-one level parameters. More generally each finite-Levi highest vector in these lowest-depth Chevalley groups yields such a rank-one highest subquotient.