BRST vanishing and PBW theorem for affine W algebras

Use $b_a={\psi }_a$ and $c^a={\psi }_a^{\ast }$. Their singular contractions are $b_a(z)c^b(w)\sim {\delta }_a^b\mathrm{/}(z-w)$ and $c^b(z)b_a(w)\sim {\delta }_a^b\mathrm{/}(z-w)$, with zero b–b and c–c contractions. These follow directly by splitting the displayed Clifford modes into the given annihilation and creation parts. Currents commute with the ghost modes. Let $j_a=J^{u_a}-{\chi }_a$ and $f_{ab}^s$ be the structure constants. Since $(m\mathrm{\mid }m)=0$ and $f_{ab}^s{\chi }_s=0$, their OPE is $j_a(z)j_b(w)\sim f_{ab}^s{j}_s(w)\mathrm{/}(z-w)$, with no double pole. The current is $Q=Q_1+Q_2$ with $Q_1=:j_a{c}^a:$ and $Q_2=-\frac{1}{2}{f}_{ab}^s:c^a{c}^b{b}_s:$. Normal ordering may be defined by moving the annihilation modes to the right, with a minus for each swap of two odd modes. Expanding a product of two such ordered monomials and commuting annihilation modes through the opposite monomial gives precisely the sum over cross-contractions; this follows by induction on the number of modes moved, since the only nonzero cross-commutators are the displayed scalar contractions or the current bracket. Thus the following is a finite Wick calculation justified from the declared mode relations.

Assume the chosen good grading is compatible with (e,h,f): e has grading1, h grading0, f grading−1. Put $m=\mathrm{\ell }⨁{⨁}_{j\ge 1}{g}_j$, $\chi =(f\mathrm{\mid }-)$, $Y=f+m\perp$, and $S=f+g^e$. By the actual good-grading proof, ℓ is Lagrangian in the nondegenerate skew space $g_{1\mathrm{/}2}$, m is positively graded and nilpotent, and $\chi ([m,m])=0$. Nilpotence follows directly because brackets strictly increase the positive grading and g has a largest grade. Thus exp(ad a) is a finite polynomial for $a\in m$; its group law on m is the finite Baker–Campbell–Hausdorff polynomial, denoted M. This uses only the finite-dimensional nilpotent Lie algebra, without an algebraic-group existence theorem.

The single-pole $Q_1{Q}_1$ contribution is $f_{ab}^s:j_s{c}^a{c}^b:\mathrm{/}(z-w)$. In the sum $Q_1{Q}_2+Q_2{Q}_1$, contracting the one b against either c of $Q_1$ gives its negative, −$f_{ab}^s:j_s{c}^a{c}^b:\mathrm{/}(z-w)$. There are no double contractions in these mixed terms because the j current has no ghost contraction. If J rather than j is used to perform the calculation, the uncancelled character term is $f_{ab}^s{\chi }_s:c^a{c}^b:\mathrm{/}(z-w)$, identically zero. The affine double-contraction term is $k(u_a\mathrm{\mid }{u}_b)c^a(z)c^b(w)\mathrm{/}(z-w{)}^2$, also identically zero.

For $Q_2{Q}_2$, a single contraction leaves three c's and one b. Anticommuting the three c's to a common order antisymmetrizes its coefficient in a,b,c; that coefficient is a scalar multiple of $f_{ab}^r{f}_{rc}^s+f_{bc}^r{f}_{ra}^s+f_{ca}^r{f}_{rb}^s$, which is zero by Lie Jacobi. This is zero as a tensor, so any overall Wick sign or common nonzero normalization does not affect its vanishing. The only possible double contractions use the sole b of each factor against a c in the other. The remaining coefficient is a scalar multiple of $f_{ar}^s{f}_{bs}^r={\mathrm{tr}}_m(\mathrm{ad}{u}_a\mathrm{ad}{u}_b)$, multiplying $c^a(z)c^b(w)\mathrm{/}(z-w{)}^2$. This tensor vanishes identically: choose a homogeneous basis of the positive-graded m; $\mathrm{ad}(u_a)$ and $\mathrm{ad}(u_b)$ strictly raise degree whenever the u's are homogeneous, so their product has zero diagonal and trace. Bilinearity gives the result for every basis, including the arbitrary one in the original definition. There cannot be three contractions because each monomial contains only one b and all c–c contractions are zero. Consequently there are no other poles.

The linear map $m⨁g^e\to m\perp$, (a,s)↦[a,f]+s, is an isomorphism. Here is the grading calculation. Decompose g into finite-dimensional sl2 strings, taking homogeneous highest vectors; the commuting grading element minus h/2 preserves each multiplicity space, so such homogeneous strings exist by diagonalization there. Goodness forces each string's highest grade to be nonnegative: if the highest grade were at most−1/2, surjectivity of f from the next grade to that grade would fail. Likewise its lowest grade is at most0 by injectivity at grades at least1/2. In the part of each string of grade at least0, every vector except its highest vector is the f-image of the vector one step above, which has grade at least1. Consequently $g_{\ge 0}=[f,g_{\ge 1}]⨁g^e$. The remaining part of m⊥ is the annihilator of ℓ inside $g_{-1\mathrm{/}2}$. This equals [f,ℓ]: orthogonality follows from $([f,u]\mathrm{\mid }v)=(f\mathrm{\mid }[u,v])=0$ on ℓ, and dimensions agree because $f:g_{1\mathrm{/}2}\to g_{-1\mathrm{/}2}$ is an isomorphism and ℓ has half dimension. Thus $m\perp =[f,m]⨁g^e$, with f injective on m. This proves the asserted linear isomorphism, including half-integer grades.

Thus the entire singular part of Q(z)Q(w) is zero. In particular $Q_{(0)}Q=0$. The super mode commutator, obtained by taking the two residues of signed Jacobi, gives $[Q_{(0)},Q_{(0)}{]}_{+}=(Q_{(0)}Q{)}_{(0)}=0$. Over C this is $2Q_{(0)}^2=0$, hence $d^2=0$. This proves the exact missing condition for the actual charged-ghost construction, for every level k and all the supplied good-grading/isotropic data. It proves neither nonzero cohomology nor its PBW identification.

The polynomial map $\mathrm{\Phi }:M\times S\to Y$, $(\exp a,f+s)\mapsto \exp (\mathrm{ad}a)(f+s)$, is well defined: Y is M-stable because χ is a character, so the restriction to m of exp(ad a)(f+s) stays χ. Give a homogeneous $a_j\in m_j$ coordinate weight j, and a homogeneous $s_j\in (g^e{)}_j$ coordinate weight j+1. The output's grade-j component has weight j+1. Every weight is positive: m has positive grades, $g^e$ has nonnegative grades, and m⊥ has grades at least−1/2. The map is homogeneous for these weights: a term $[a_{j_1},\dots ,[a_{j_r},f]\dots ]$ has output weight $j_1+\cdots +j_r$; a term with $s_j$ has output weight $j_1+\cdots +j_r+j+1$. Its linear part is the isomorphism just proved. After composing with that linear inverse, Φ has the form $x_v\mapsto x_v+P_v$, where $P_v$ has weight v and ordinary polynomial degree at least2. Every variable in such a monomial has strictly smaller positive weight than v. Order the finite set of variable weights increasingly and invert recursively: at the smallest weight $P_v=0$; at each later weight all variables in $P_v$ have already been recovered. This gives a polynomial inverse. Hence Φ is an affine isomorphism, not merely an étale or formal neighborhood assertion.

The action of M on Y corresponds under Φ to left multiplication on its first factor and the trivial action on S. This follows directly from $\exp (\mathrm{ad}a)\exp (\mathrm{ad}b)=\exp (\mathrm{ad}\mathrm{BCH}(a,b))$. Taking N-jets gives an isomorphism $Y_N\cong M_N\times S_N$ for every $N\ge 0$. This is an actual polynomial statement: substitute a(t),s(t) modulo $t^{N+1}$ in Φ and its polynomial inverse. The group $M_N$ has Lie algebra $m[t]\mathrm{/}{t}^{N+1}m[t]$, is polynomially isomorphic by exponential coordinates to the affine space $m^{N+1}$, and acts by left multiplication on its $M_N$ factor.

Write $b_{a,n}={\psi }_{a,n}$ and $c_n^a={\psi }_{a,n}^{\ast }$ in the actual carrier's convention. The creation modes are $b_{a,n}$, $n<0$, and $c_n^a$, $n\le 0$. Give each c-creation count1 and each b-creation count1, separately. Affine currents have both counts0. Normal ordering expresses every vector uniquely as a finite sum of affine PBW words times exterior ghost-creation monomials. On this vector space the BRST residue splits as $d=d_{-}+d_{+}$, with bidegrees $(\mathrm{\Delta }c,\mathrm{\Delta }b)=(0,-1)$ and (1,0), respectively. Indeed the current term is ${\sum }_{a,m}{J}_{a,m}{c}_{-m}^a$; $m<0$ uses an annihilating c and lowers b, while $m\ge 0$ uses a creating c. The character term is $-{\sum }_a{\chi }_a{c}_1^a$, lowering b, NOT a c-zero-mode creation. In the cubic term two annihilating c's and a creating b give $\mathrm{\Delta }b=-1$; two creating c's and an annihilating b give $\mathrm{\Delta }c=1$; the mixed case belongs to one of these two degrees according to whether its b is creating or annihilating. These exhaust the signs and modes. Square-zero already proved for the actual Q implies $d_{-}^2=d_{+}^2=d_{-}{d}_{+}+d_{+}{d}_{-}=0$ by their distinct bidegrees.

For $x\in g_j$ set the Kazhdan degree of its current state to 1−j. Set ${\mathrm{deg}}_K{b}_a=1-j_a$ and ${\mathrm{deg}}_K{c}^a=j_a$ when $u_a\in m_{j_a}$; a derivative has the SAME Kazhdan degree as its state. Extend additively over PBW monomials and use the increasing filtration. Current and ghost commutator corrections lower this degree, because a current bracket has degree one less than the sum of its two inputs, as does a b–c contraction. The central current term lowers it further. The exact BRST differential preserves the filtration: its current-adjoint and cubic terms have the same degree as their inputs, its level-dependent ∂c term is one degree lower, and its character term has degree0 because ${\chi }_a$ is zero unless $j_a=1$.

Define the relative classical complex at jet level N as the ordinary Chevalley–Eilenberg cochain complex $C_N=C[Y_N]\otimes \mathrm{\Lambda }((m[t]\mathrm{/}{t}^{N+1}{)}^{\ast })$ for this action. Explicitly, on a function F its differential is ${\sum }_{a,r}{c}^{a,r}(u_a{t}^r)F$, and on the odd coordinate $c^{a,n}$ it is $-(1\mathrm{/}2){\sum }_{b,c;r+s=n}{f}_{bc}^a{c}^{b,r}{c}^{c,s}$. These formulas fix all signs and the exact finite complex. By $Y_N\cong M_N\times S_N$, it is the tensor product of $C[S_N]$ with the algebraic de Rham complex of $M_N$: the invariant vector fields form a frame, their brackets are $f_{bc}^a$, and the Maurer–Cartan equation for its dual frame is precisely the displayed ghost differential. In polynomial exponential coordinates, the latter de Rham complex is contracted to constants by the following explicit homotopy. For a form whose coefficient has homogeneous polynomial degree p and whose exterior degree is q, use ${\iota }_E\mathrm{/}(p+q)$, where $E=\sum x_i{\mathrm{\partial }}_{x_i}$, whenever $p+q>0$, and zero on constants. The identity $d{\iota }_E+{\iota }_{E}d=(p+q)$ proves the contraction. Therefore $H^i(C_N)=0$ for i≠0 and $H^0(C_N)=C[S_N]$. No external de Rham vanishing theorem is used.

Put $n=m[t^{-1}]t^{-1}$. The $d_{-}$ complex is the ordinary Lie homology complex of n on $V^k(g)\otimes \mathrm{\Lambda }(c_{\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}})$, with the b-creations as its exterior chain variables. The coefficient action is the diagonal action of the negative current modes on $V^k$ and the induced coadjoint action on the c-creations, shifted by the character $\lambda (u_a{t}^{-r})={\delta }_{r,1}{\chi }_a$. The first current/character terms give the coefficient-action part of the chain differential. The two-annihilating-c cubic terms give its Lie bracket part. The mixed cubic terms with annihilating b give the coadjoint coefficient action. This verifies all components from the displayed mode split, including the shifted mode $r=1$. The shift λ is a character because a bracket of two negative powers has exponent at most−2. There is no current anomaly on n, since the invariant form vanishes on m×m and the ghost trace form vanishes by positive grading, as checked in the preceding calculation.

The complexes inject as N increases: the $degree\le N$ jet coefficients and their ghosts have differentials involving only $indices\le N$, by the formulas above. Their algebraic union is the relative classical jet BRST complex $C_{\mathrm{\infty }}$; each polynomial cochain contains only finitely many jet coordinates and ghosts. A cycle, boundary and any witnessing primitive therefore lie at a finite stage. Cohomology consequently commutes with this union, giving $H^i(C_{\mathrm{\infty }})={\delta }_{i,0}C[J_{\mathrm{\infty }}S]$, as differential commutative algebras. The derivation is the jet translation; Φ and its inverse are substituted coefficientwise and commute with this translation, so the degree-zero identification respects it.

There is also a direct local comparison for the full classical complex; this does not prove its quantum analogue. At jet stage N, use the independent linear moment-coordinate functions $y_{a,n}={\xi }_{u_a,n}-{\delta }_{n,0}{\chi }_a$ on $g_N$, for $0\le n\le N$. Extend these to affine linear coordinates on $g_N$. The classical BRST algebra is $C[g_N]\otimes \mathrm{\Lambda }(b_{a,n})\otimes \mathrm{\Lambda }(c^{a,n})$, where b has degree−1 and c degree+1. Its differential is the sum of the Koszul differential $d_K{b}_{a,n}=y_{a,n}$, $d_{K}y=0$, $d_{K}c=0$, and the Chevalley–Eilenberg differential for $m[t]\mathrm{/}{t}^{N+1}$. On b the latter is the same equivariant linear action as on its moment y. In coordinates this is $(u_a{t}^r)y_{b,n}=y_{[u_a,u_b],n-r}$ for $n\ge r$ and zero otherwise, and the identical formula for b. The character contributes no error since $\chi ([m,m])=0$. Thus the CE differential preserves the Koszul differential, so $d_K$ and $d_{C}E$ anticommute, and the total differential is square zero.

Affine PBW, ordered with the n-negative currents first, makes $V^k$ a free left U(n)-module $U(n)\otimes Z$, where Z is spanned by ordered negative-current words from a fixed complement of m in g. Tensoring with the ghost exterior algebra preserves freeness of the diagonal coefficient action. Here is the explicit untwisting: if T is the c-ghost representation, the map $u\otimes v\mapsto \sum u_{(1)}\otimes u_{(2)}v$ identifies the regular action on the first factor with the diagonal action; its inverse replaces the second coproduct factor by its antipode. This is checked on a primitive generator x, for which $\mathrm{\Delta }x=x\otimes 1+1\otimes x$, then on products. The coproduct is finite on every word. The character shift is the algebra automorphism x↦x−λ(x), valid because $\lambda ([n,n])=0$. Thus the coefficient module is free for the exact shifted action, not only for its associated graded.

The Koszul complex is explicitly contracted onto $C[Y_N]$. For each monomial whose total y-polynomial degree plus b-exterior degree is $p>0$, use $h_K=(\sum b_{a,n}\mathrm{\partial }\mathrm{/}\mathrm{\partial }{y}_{a,n})\mathrm{/}p$; on $p=0$ use zero. Then $d_K{h}_K+h_K{d}_K=Id-e{v}_{y=b=0}$. This proves its sole cohomology is $C[Y_N]$ in degree0. Filter the finite-stage total complex by its c-exterior degree. There are finitely many c and b generators, so this filtration is bounded in every cohomological degree; elementary successive lifting of cocycles therefore gives the spectral sequence without a convergence assumption. Its first page is exactly the relative CE complex $C_N$ already computed, and its next cohomology is $C[S_N]$ in degree0 only. Hence the full classical complex has $H^i={\delta }_{i,0}C[S_N]$, and the natural projection to the relative complex induces that isomorphism. Taking the algebraic union over N, with finite-support cochains as above, proves the same for the full classical arc BRST complex. This is an explicit Koszul contraction plus affine de Rham contraction; neither is an imported cohomology theorem.

For completeness its regular-module Lie chain contraction is local and finite. Identify U(n) with its ordered PBW monomials and filter $U(n)\otimes \mathrm{\Lambda }(n)$ by the sum p of polynomial PBW length and exterior length. The leading differential is $\delta =\sum x_i\mathrm{\partial }\mathrm{/}\mathrm{\partial }{b}_i$ on $\mathrm{Sym}(n)\otimes \mathrm{\Lambda }(n)$; the remainder R decreases p, since each commutator and each chain-bracket replaces two letters by one, and the character term replaces a letter by a scalar. On $p>0$ the leading contraction is $h=\sum b_i\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}_i$ divided by p; on $p=0$ it is zero. Thus $\delta h+h\delta =1-i\pi$ on the constant subspace. Correct it by the finite series $h^{\mathrm{\prime }}=h(1+Rh{)}^{-1}=h{\sum }_{r\ge 0}(-Rh{)}^r$, with the associated corrected projection ${\pi }^{\mathrm{\prime }}=\pi (1+Rh{)}^{-1}$. The sum terminates on every word because Rh lowers p. Substituting the finite geometric-series identities and $(\delta +R{)}^2=0$ gives $(\delta +R)h^{\mathrm{\prime }}+h^{\mathrm{\prime }}(\delta +R)=1-i{\pi }^{\mathrm{\prime }}$. This proves zero homology outside chain degree0 and identifies degree-zero homology with the coinvariants. It applies to countably many generators since every polynomial word and every differential term has finite support.

In the original variables, the coefficient action of $u_a{t}^{-r}$ is the mode of ${\theta }_a=d{b}_a$. Hence the coinvariant quotient is exactly $C^{-}$ obtained by quotienting the full BRST space by the span of ${\theta }_{a,-r}v$ and $b_{a,-r}v$, $r\ge 1$. The quotient is a complex since $db=\theta$ and $d\theta =0$. This is a quotient of complexes of vector spaces; no vertex-ideal assertion is made for $C^{-}$.

The preceding contraction also justifies convergence and filtration strictness. Use the old energy with current state weight1, b-state weight1 and c-state weight0, derivatives adding1. Its vacuum module has finite-dimensional subspaces of $energy\le P$: every positive-energy creation has energy at least1, while there are only dim(m) exterior c-zero modes. Both $d_{-}$ and $d_{+}$ preserve this finite-dimensional subspace; the character term lowers energy by1. The PBW Koszul homotopy replaces a current mode by its corresponding b-mode of the same old energy; commutator corrections preserve energy and the character corrections lower it. The diagonal untwisting preserves old energy, since a negative current of loop order r raises the c-ghost energy by r. Consequently the full contraction preserves $energy\le P$. On that finite space c-count and b-count are bounded, so perturbing by $d_{+}$ uses only finitely many terms of the same homotopy formula: each $d_{+}{h}^{\mathrm{\prime }}$ raises c-count. The induced differential on the $b=0$ coinvariants is the actual quotient differential, with no higher terms, because h' raises b-count and $d_{+}$ preserves it, so every correction containing h' projects to zero. This proves that the natural full-to-relative projection is a quasi-isomorphism on each energy bound, hence on their algebraic union.

PBW triangularity identifies $C^{-}$ as a vector space with ordered negative modes of a complement g/m and the c-creations. One can see this directly by replacing each m-current mode with θ plus its explicit quadratic ghost term and character, and moving θ,b modes to the left; the leading affine PBW term is unchanged, so this is an invertible triangular basis change. In the quotient θ and b disappear. All remaining current-state Kazhdan degrees are positive, since g/m has grading at most1/2, so $1-j\ge 1\mathrm{/}2$. All c-state degrees $j_a$ are positive too.

The contraction and comparison above preserve this filtration as well. Replacing a negative m-current by the corresponding b has the same Kazhdan degree; commutator and bracket corrections lower it, and the nonzero character corrections have input and output degree0. In the Hopf untwisting, the coadjoint action of an m-current of grade j shifts c-ghost Kazhdan degree by −j, whereas that current's degree is 1−j. Every nontrivial coproduct correction therefore lowers the total Kazhdan degree by at least one. Both the untwisting and its antipode inverse are filtered, and the corrected Koszul homotopy is filtered. Perturbation by $d_{+}$ remains filtered since $d_{+}$ is filtered. Thus the cohomology isomorphism between the full complex and $C^{-}$ identifies their induced Kazhdan filtrations strictly; this is established by the filtered contraction, not inferred from an unfiltered quasi-isomorphism.

Define the new energy on these generators by the same numbers 1−j and $j_a$, but let every derivative add1. The differential preserves this new energy: this is checked on db, dc and dJ from the actual BRST state; for the level-dependent derivative term, $(u_a\mathrm{\mid }x)\mathrm{\ne }0$ requires $j_a+j_x=0$, so $1+j_a=1-j_x$. Each new-energy subspace of $C^{-}$ is finite dimensional, since there are finitely many positive-weight generators, finitely many derivative levels below a bound, bounded word length and the usual finite PBW count. At a fixed new energy Δ the Kazhdan filtration has only finitely many values between0 andΔ. This supplies an actual finite filtered complex, rather than an unbounded formal spectral-sequence claim.

The associated graded of the relative quotient is the classical relative complex $C[J_{\mathrm{\infty }}Y]\otimes \mathrm{\Lambda }(c_{\mathrm{j}\mathrm{e}\mathrm{t}\mathrm{s}})$ with the explicit CE differential written in the preceding calculation. The m-current constraints become ${\xi }_{u_a,n}={\delta }_{n,0}{\chi }_a$ after b and θ are removed. Noncommutative reorderings and the k-dependent derivative term are lower Kazhdan degree, so disappear; the surviving current-adjoint and ghost-quadratic terms are exactly the classical CE action and its bracket differential. This also identifies the multiplication and jet derivation on its degree-zero cohomology. The preceding classical calculation establishes $H^i={\delta }_{i,0}C[J_{\mathrm{\infty }}{S}_f]$ by the polynomial $M\times S_f$ isomorphism and explicit de Rham Euler contraction, with the full classical Koszul comparison separately verified.

On each finite new-energy subcomplex, compute cohomology by its finite Kazhdan filtration. Its associated-graded cohomology is concentrated in cohomological degree0. Successively lifting a cocycle down the finite filtration therefore proves vanishing in every nonzero cohomological degree; in degree0 the same finite lifting identifies its associated-graded cohomology with that first-page algebra. This is elementary finite filtered linear algebra: when a leading term is a graded boundary, subtract its lifted primitive and decrease the filtration degree; only finitely many decreases are possible. The same procedure proves strictness. Taking the algebraic direct sum of the new-energy pieces gives $H^i(C^{-})=0$ for i≠0 and $g{r}_K{H}^0(C^{-})=C[J_{\mathrm{\infty }}{S}_f]$. The filtered full-to-relative comparison in the preceding calculation transfers this to the original charged-ghost complex. Its degree-zero cohomology is the actual universal $W^k(g,f)$, whose differential vertex-algebra multiplication is inherited because $Q_0$ is a signed derivation. The comparison of associated-graded cohomology with the classical full complex is multiplicative: lift two cocycles in the full complex, take their normal product, and take its leading symbol; it is the ordinary product of their classical symbols, with lower commutators already removed. The translation derivation is preserved at every step. Hence the resulting isomorphism is one of differential commutative algebras, exactly the original Statement.