Corrected principal BRST induced complexes and minus reduction of affine Vermas

Use odd modes ${\psi }_{\alpha }(n)$, alpha any finite root, with anticommutators ${\delta }_{\alpha +\beta ,0}{\delta }_{m+n,0}$. The ghost vacuum is killed when $n>0$, or $n=0$ and $\alpha >0$. Put deg ${\psi }_{\alpha }=-1$ for $\alpha >0$ and +1 for $\alpha <0$. For the positive nilradical the standard cubic BRST operator is $Q_{st,+}$ and put $Q_{+}^{\mathrm{\prime }}=Q_{st,+}$ $-{\sum }_{\alpha \mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}$ ${\chi }_{+}(x_{\alpha }){\psi }_{-\alpha }(0)$. This is the actual J−chi convention of F1. For the negative nilradical put Q_-=sum_(alpha<0,n) x_alpha(−n)psi_(−alpha)(n) −(1/2)sum_(alpha,beta,gamma<0;k+l+m=0)c^gamma_(alpha,beta):psi_(−alpha)(k)psi_(−beta)(l)psi_gamma(m): +sum_(alpha<0)chi_-(x_alpha)psi_(−alpha)(0). Here ${\chi }_{-}(x)=(e,x)$; its sign in the coefficient homology is exactly the displayed plus sign. Give M tensor F minus cohomological degree opposite to the above ghost degree. The actual differential squares to zero: current cocycles vanish on a nilradical, cubic single contractions cancel by Jacobi, double ghost contractions are traces of strictly root-height-lowering adjoint operators and vanish, and the remaining character bracket is ${\chi }_{-}([n-,n-])=0$. The same calculation with positive height verifies Q'_+ squared zero. Each fixed vector sees finitely many modes by current restrictedness and ghost annihilation.

Write $b_{\alpha }={\psi }_{\alpha }$ and $c^{\alpha }={\psi }_{-\alpha }$ for $\alpha >0$. Define corrected fields Jhat_a=J_a−sum_(beta,gamma>0)c^gamma_(a,beta):c^beta b_gamma:. For a in b− these close with the c fields. Their current bracket is the b− bracket at level k+hvee: the double ghost contraction is ${\mathrm{Tr}}_{n+}$ of the two quotient-adjoint matrices, equal to hvee(a,b). For Cartan a,b it is ${\sum }_{\alpha >0}\alpha (a)\alpha (b)=h^{\vee }(a,b)$; all other b− weight pairs give zero on both sides. Single contractions give the Lie bracket and the coadjoint action on c. Expanding $Q_{st,+}$ gives d c=−(1/2)[c,c] and d Jhat_a=sum_(alpha>0,b in b−;r+s=n)c^b_(alpha,a):c^alpha(s)Jhat_b(r): − n k_a psi_a(n) on mode n, where the last term is zero for Cartan a and for a negative root k_a=k−sum_(beta,gamma<0)c^gamma_(a,beta)c^(−gamma)_(-a,−beta). The actual negative character part gives [−chi_+,Jhat_a(n)]=−sum_(beta>0)([f,a],x_beta)psi_(−beta)(n+1); for its old-loop shifted version replace n+1 by n. These identities follow by moving the one b through each ghost pair; the single current contraction gives the displayed bracket, its derivative gives −n k, and the double ghost contraction gives the finite trace correction in $k_a$. They fix the actual complex, including the derivative term, rather than merely its symbol.

Let C'' be generated by the corrected b− current states ${\widehat{J}}_a(-1)1$ and the c states ${\psi }_{-\alpha }(0)1$. The displayed formulas show it is a subcomplex of the universal plus complex. Its creator-mode PBW map into the relative quotient from F1 is triangular and has identity leading symbol: replacing a corrected current by its original current plus a ghost bilinear lowers affine-current word length, and the killed antighost/gauge creators disappear. Thus at each finite positive improved-weight bound it is a vector-space isomorphism onto that relative quotient. Its restriction is a chain map, so C'' is a concrete relative model of $W^k$. C'' has only nonnegative cohomological degrees; hence $W^k=\ker$ $Q_{+}$ in C'' degree0 is an actual embedded vertex subalgebra, with no degree-zero boundary quotient inside C''.

For the actual minus complex of $M_{aff}(\lambda )$, let $a=n-[t^{-1}]$ include mode0. Its antighost creators are $b_x(n)={\psi }_x(n)$, x in n−, $n\le 0$. Set ${\theta }_x(n)=[Q_{-},b_x(n)]={\widehat{J}}_x(n)+{\chi }_{-}(x){\delta }_{n,0}$, with the corrected negative-nilradical currents obtained by the same CAR calculation as step 1.2. They satisfy [theta_x(m),theta_y(n)]=theta_[x,y](m+n), [theta_x(m),b_y(n)]=b_[x,y](m+n), [b_x,b_y]=0, and d b_x=theta_x, d theta_x=0. There is no central term on the nilradical and ${\chi }_{-}([x,y])=0$. Thus these are the ACTUAL operators of the Lie-superalgebra cone on a. The remaining corrected b+ negative-loop currents and positive-root ghost negative-loop creators generate a cyclic subspace $T_{\lambda }$ on $v_{\lambda }$ tensor1. Its top is killed by positive-root corrected zero modes and positive ghost zero modes, and $Q_{-}$ kills it. The mirrored identities of step 1.2 make $T_{\lambda }$ a subcomplex. Ordering cone operators first gives a vector-space isomorphism U(cone a) tensor T_lambda -> C_-(M_aff(lambda)). Indeed corrected-current replacement is triangular in affine-current length; at mode0, ${\theta }_x(0)={\widehat{J}}_x(0)+{\chi }_{-}(x)$ is an invertible polynomial translation of the free finite Verma creators. There are no other PBW relations. Since Q is a derivation on the cone operators and preserves $T_{\lambda }$, this map identifies the ACTUAL tensor differential.

Define the transpose $t$ on the finite Lie algebra to fix the Cartan and exchange opposite root spaces, with $e^t=f$ and $f^t=e$. Such a Chevalley anti-involution is obtained by scaling the simple-root exchanges to match the nonzero simple-root coefficients of the pinned opposite principal elements; their products on each pair are one, so the Cartan and Serre relations are preserved. Choose dual opposite-root bases with $x_{\alpha }^t=x_{-\alpha }$. Define $\mathrm{♯}$ on old-loop modes by $x(n)\mapsto -x^t(n)$, $h(n)\mapsto -h(n)$, ${\psi }_{\alpha }(n)\mapsto -{\psi }_{-\alpha }(n)$, and $K\mapsto K$. The invariant form and CAR are preserved, so this is an involution of the current/Clifford algebra, preserving old-loop degree. Substitution in the linear and cubic terms of 1.1 gives $\mathrm{♯}(Q_{+}^{\mathrm{\prime }})=Q_{-}$: the two minus signs in each current/ghost term cancel; the transpose reverses the Lie bracket in the cubic term; and $-{\chi }_{+}(x_{\alpha }){\psi }_{-\alpha }(0)$ becomes $+{\chi }_{-}(x_{-\alpha }){\psi }_{\alpha }(0)$ since $e^t=f$. Cohomological degree is also preserved when the minus grading is reversed as in 1.1. Thus $\mathrm{♯}\circ \tau$ is the actual minus action of the embedded W algebra, and its modes commute with $Q_{-}$.

Give a corrected root current ${\widehat{J}}_{-\alpha }$ height-filtration degree ht(alpha), a Cartan current degree0, and $c^{\alpha }$ degree ht(alpha)−1; derivatives have degree0 for this filtration. The standard differential lowers this degree by1, while the character differential preserves it, as the two bracket formulas show: the bracket-root heights add, and every c factor subtracts1. In each finite improved-weight piece the filtration has finitely many nonnegative values. Principal sl2 strings give $b-=g^f$ plus a homogeneous complement I, with [f,I]=n−. Choose a basis $I_{\alpha }$ with $[f,I_{\alpha }]$ the negative of the vector dual to $x_{\alpha }$, so the actual negative character differential pairs with coefficient+1. The character differential pairs ${\widehat{J}}_{I_{\alpha }}(-n)$ with ${\psi }_{-\alpha }(-n+1)$, $n\ge 1$, and kills the ${\widehat{J}}_{P_i}$ for a basis $P_i$ of $g^f$. PBW symmetrization is a chain isomorphism for this linear odd derivation, so its contraction is the polynomial/exterior Koszul contraction sum I_alpha partial_(psi_alpha)/N, where N counts paired variables. It leaves exactly the polynomial negative modes of $P_i$. Finite filtration lifting therefore produces closed fields $W_i$ with leading $P_i$ and weights $m_i+1$, and gives gr $W^k=C$[$P_i$ and their derivatives]. In particular ordered normally ordered derivatives of $W_i$ form a free state PBW basis.

Define K(mu) as the actual C'' module generated by a top vector $u_{\mu }$ on which all positive old-loop modes vanish, ${\widehat{J}}_{-\alpha }(0)$ and ${\psi }_{-\alpha }(0)$ vanish, and ${\widehat{J}}_h(0)$ acts by mu(h). Its creator space is the PBW space of ${\widehat{J}}_{b-}(-n)$, ${\psi }_{-\alpha }(-n)$, $n\ge 1$. The b− current/odd-n− brackets verified in step 1.2 define this induced current representation, and their local fields extend by normal products and derivatives to an actual C'' module: locality follows by clearing the finitely many pairwise poles before taking nth-product residues; the same residue identity is module Jacobi. Define its differential by Q'_+$(a{u}_{\mu })=[Q_{+}^{\mathrm{\prime }},a]u_{\mu }$. The top relations are preserved by the formulas in step 1.2 (zero modes and every c(0) kill the top), so this is well defined and square zero. It preserves old loop energy. Each energy piece is finite dimensional because every creator has strictly positive old energy.

The cone factor of step 2.2 has an exact explicit contraction. PBW supersymmetrization intertwines the odd linear derivation db=theta, dtheta=0, because it is a Lie derivation; hence its vector complex is the polynomial/exterior complex with differential sum theta_i partial_(b_i). On a monomial of positive total paired degree N, use sum b_i partial_(theta_i)/N; on constants use zero. Its two composites give identity minus constant extraction. Each word involves finitely many generators, including the mode0 ones, so this is algebraic. Tensoring with $T_{\lambda }$ contracts the full minus complex onto the ACTUAL subcomplex $T_{\lambda }$, not merely its leading symbols. Old loop energy is preserved because $b_x(n)$ and ${\theta }_x(n)$ have the same energy, including the zero-mode translation. The inclusion of $T_{\lambda }$ is W-equivariant: the tau/sharp images of embedded W lie in its corrected b+ algebra. Therefore the resulting cohomology identification is an actual W-module identification. No unbounded perturbation series, character comparison, or assumed acyclicity is used.

Write ${\widehat{J}}_a^{+}$ for 1.2 and ${\widehat{J}}_a^{-}$ for its negative-nilradical counterpart in 2.2. Normal ordering gives $\mathrm{♯}({\widehat{J}}_{-\alpha }^{+}(n))=-{\widehat{J}}_{\alpha }^{-}(n)$ for $\alpha >0$, and $\mathrm{♯}({\widehat{J}}_h^{+}(n))=-{\widehat{J}}_h^{-}(n)-2\rho (h){\delta }_{n,0}$. Here is the scalar calculation, which cannot be discarded. In $E_{\alpha }(n)={\sum }_m:{\psi }_{-\alpha }(m){\psi }_{\alpha }(n-m):$, the first ghost is a creator at $m=0$. After applying $\mathrm{♯}$, its order differs from the order for a positive first ghost precisely at $m=0$; CAR contributes ${\delta }_{n,0}$. The remaining reordered bilinear is $-E_{\alpha }(n)$. Multiplication by the coefficient $\alpha (h)$ and summation give $-{\sum }_{\alpha >0}\alpha (h){\delta }_{n,0}=-2\rho (h){\delta }_{n,0}$. For a root current the analogous trace is zero because its matrix shifts root weight, so it has no diagonal entries. This proves the displayed identities, including the zero-mode terms, by the same finite matrix substitution in the corrected-current definition.

On K(mu), use the height filtration of step 3.2. Its leading character differential pairs ${\widehat{J}}_{I_{\alpha }}(-n)$ with ${\psi }_{-\alpha }(-n)$, $n\ge 1$, and leaves the negative $P_i$ modes. PBW symmetrization and the same explicit Koszul contraction compute it. Every fixed old-energy piece has finitely many height values, so successive lifting proves $H^i(K(\mu ))=0$ for i≠0 and identifies its $H^0$ leading space with the polynomial ring in $P_i(-n)$, $n\ge 1$. The W action is the actual tau-shifted action from the embedded algebra of step 2.1, and its negative $W_i$ mode has leading multiplication by $P_i(-n)$. Thus ordered negative $W_i$ words on the retained top are linearly independent and span $H^0(K(\mu ))$.

Put $\mu =-\bar{\lambda }-2\rho$. Regard $T_{\lambda }$ as a module over the old corrected ${\mathfrak{b}}_{-}$ current/ghost algebra by letting a mode $a$ act as $\mathrm{♯}(a)$. Its generator $v_{\lambda }\otimes 1$ satisfies exactly the defining top relations of $K(\mu )$ in 3.3. Positive loop modes kill it; the transformed negative-root and negative-ghost zero modes are positive-root zero modes and kill it; and 4.1 gives the Cartan eigenvalue $-\bar{\lambda }(h)-2\rho (h)$. For the last assertion the mirrored corrected Cartan has eigenvalue $\bar{\lambda }(h)$ on this vector: all its normally ordered ghost bilinears kill the ghost vacuum. Consequently the assignment $a_1\cdots a_r{u}_{\mu }\mapsto \mathrm{♯}(a_1)\cdots \mathrm{♯}(a_r)(v_{\lambda }\otimes 1)$ is a well-defined module map $\mathrm{\Phi }:K(\mu )\to T_{\lambda }$. The creator identities in 4.1 exchange its negative-root currents with the positive-root currents of $T_{\lambda }$, send Cartan creators to their negatives, and exchange opposite ghost creators. Their scalar corrections occur only at mode zero. The two PBW bases of 2.2 and 3.3 therefore map bijectively, so $\mathrm{\Phi }$ is an isomorphism. It preserves old-loop degree, with both generators assigned relative degree zero.

Its actual top character is ${\gamma }_{\mu }$ under F3. Take the finite lowest Verma with lowest weight mu+2rho, tensored with Lambda(n+), in homological degrees reversed. PBW makes its coefficient module free of rank one over U(n+). For $m\otimes x_1\wedge \cdots \wedge x_r$, define the character-twisted homology differential by the coefficient terms ${\sum }_j(-1{)}^{j+1}(x_j+{\chi }_{+}(x_j))m\otimes x_1\wedge \cdots \widehat{x_j}\cdots \wedge x_r$ and bracket terms ${\sum }_{j<l}(-1{)}^{j+l+1}m\otimes [x_j,x_l]\wedge x_1\wedge \cdots \widehat{x_j}\cdots \widehat{x_l}\cdots \wedge x_r$. Its square is zero by the module commutator identity, ${\chi }_{+}([x,y])=0$, and Jacobi. Filter by coefficient PBW length plus exterior degree. Its leading complex is the polynomial/exterior Koszul complex with differential sum $x_i$ ${\mathrm{contraction}}_i$; the bracket and character terms lower filtration. On positive total degree N its contraction is sum ${\mathrm{\Lambda }}_i$ ${\mathrm{\partial }}_{x_i}\mathrm{/}N$. This proves that its graded homology is C in degree zero; lifting primitives lowers the nonnegative filtration and terminates, proving the same for the actual complex. Its $H_0$ is evaluation at $x=-{\chi }_{+}(x)$, and the constant word maps to 1. The finite ghost vacuum line is killed by every negative corrected root and every negative ghost zero mode; its corrected Cartan is (mu+2rho)(h)−sum_(alpha>0)alpha(h)=mu(h). Its inclusion is consequently a cochain map and induces the nonzero isomorphism onto $H_0$. These are exactly the corrected-Borel top relations of 3.3, so evaluating a closed finite BRST representative on this line agrees with evaluating it on K(mu)'s top. The lowest Verma's central character is ${\gamma }_{\mu }$ because w0(mu+2rho)+rho=w0(mu+rho) and F4 gives Weyl invariance. The natural map F3 therefore gives the stated top character, including its Cartan translation. This finite free-character calculation is supplied here rather than inferred from a cohomology theorem for a different coefficient module.

This is an isomorphism of the actual complexes and their W actions. Both differentials kill the generators. On each creator word, the supercommutator rule and $\mathrm{♯}(Q_{+}^{\mathrm{\prime }})=Q_{-}$ give $Q_{-}\mathrm{\Phi }(a{u}_{\mu })=\mathrm{\Phi }([Q_{+}^{\mathrm{\prime }},a]u_{\mu })$. Thus the differentials intertwine on a PBW basis, including derivative and lower-order terms. For a W mode $u$, its action on $K(\mu )$ is through $\tau (u)$; its action on $T_{\lambda }$ is through $\mathrm{♯}\tau (u)$. The module property of $\mathrm{\Phi }$ intertwines these actions coefficientwise. Every formula acts on finite words in restricted modules, so this verification uses no completed vector or infinite correction to a generator. Together with the W-equivariant inclusion in 3.4, it gives $H_{-}^i(M_{aff}(\lambda ))\cong H^i(K(-\bar{\lambda }-2\rho ))$ as graded W modules, with the specified top retained.

The induced module $M_W({\gamma }_{\mu })$ from F5 maps to $H^0(K(\mu ))$. Ordered negative $W_i$ words span any highest module generated by that top: replace composite-state fields by their finite normal-product mode formulas, move positive modes right until they kill the top, evaluate degree-zero modes there, and order negative modes. In the filtration of step 3.2 every commutator correction lowers degree, so induction first on that degree and then on inversions terminates at each fixed source/target grade. Its associated leading monomials are the polynomial ones of step 4.2. Therefore the map is surjective and injective, since those monomials' images are independent. It preserves the highest vector and grading. This proves the asserted K identification without assuming a PBW theorem for arbitrary W modules.

Apply 4.2–6.2 to the complex in 6.1. For $\mu =-\bar{\lambda }-2\rho$, the Harish–Chandra arguments satisfy $w_0(\mu +\rho )=-w_0\bar{\lambda }+\rho$ because $w_0\rho =-\rho$. Weyl invariance in F4 therefore gives ${\gamma }_{\mu }={\gamma }_{-w_0\bar{\lambda }}$ on every central element, with no restriction on $\lambda$. Hence $H_{-}^i(M_{aff}(\lambda ))=0$ for $i\mathrm{\ne }0$ and $H_{-}^0(M_{aff}(\lambda ))\cong M_W({\gamma }_{-w_0\bar{\lambda }})$. The intermediate complex is $K(-\bar{\lambda }-2\rho )$; it need not be identified with $K(-w_0\bar{\lambda })$ as a corrected-Borel module. Their degree-zero W cohomologies are identified by the same universal highest module and normalized top. The chain map 5.1 and the PBW isomorphism 6.2 preserve relative mode degree, so the asserted identification preserves the specified grading and top. The independent calculation 4.2–6.2 already proves the K clause for every $\mu$. No external ordinary-Verma reduction theorem is a premise.