Natural principal VOA Zhu and actual-ideal BRST comparison

Fix the conformal-shift sign explicitly. Put $\xi =h[-1]1$ and ${\omega }_{\text{new}}={\omega }_{\text{old}}+(1\mathrm{/}2)T$ xi, so $H_{\text{new}}=H_{\text{old}}-{\xi }_0\mathrm{/}2$. Define $T_{\mathrm{\Delta }}=\exp ({\sum }_{n\ge 1}(-1{)}^n{\xi }_n\mathrm{/}(2n))$ and $\mathrm{\Delta }(s)=s^{-{\xi }_0\mathrm{/}2}$ $\exp ({\sum }_{n\ge 1}(-1{)}^n{\xi }_n{s}^{-n}\mathrm{/}(2n))$. Positive xi modes commute and lower old energy, so these operators and their inverses are locally finite. From the mode commutator identity, the coefficient of ${\xi }_j$ in conjugation by log $T_{\mathrm{\Delta }}$ is -log(1+z)/2 for $j=0$ and $(-1{)}^j(1+z{)}^{-j}\mathrm{/}(2j)$ for $j\ge 1$: expand the binomial coefficients binom(n,j)/n. Exponentiating the commuting operators gives $T_{\mathrm{\Delta }}$ $Y(a,z)=Y(\mathrm{\Delta }(1+z)a,z)$ $T_{\mathrm{\Delta }}$. If ${\xi }_0$ $a=2j$ a and $\mathrm{\Delta }(s)a=\sum$ $s^{-j-n}$ $a_n$, then $w{t}_{\text{new}}(a_n)=w{t}_{\text{old}}(a)-j-n$. Multiplying the circle or either star residue by $(1+z{)}^{w{t}_{\text{old}}(a)}$, or its right-product exponent one less, proves exactly that $T_{\mathrm{\Delta }}$ takes old circles/products to new circles/products. It induces $A_{\text{old}}(M)$ to $A_{\text{new}}(M)$ and its inverse for every module here. It preserves the actual ideal because N is stable under every ${\xi }_n$. On an affine current x its map is $[x{]}_{\text{new}}-(k\mathrm{/}2)B(h,x)$; roots are unchanged. The test x[-2]+x[-1] for a root of grade j becomes x[-2]+(1-j)x[-1], fixing the sign rather than leaving an unspecified spectral-flow translation.

Compute the actual Zhu complex before its cohomology. For the charged ghosts of $n_j$ the improved weights are $\mathrm{wt}(b)=1-j$ and $\mathrm{wt}(c)=j$. Their circle/translation relations eliminate derivative ghost modes. In physical mode notation the zero-energy pair is b[-j],c[j]; declare b[m] with $m>-j$ and c[n] with $n>j$ to annihilate the top, let b[-j],c[j] act in the finite regular Clifford representation, and generate the module by the remaining, strictly energy-raising exterior modes. The relation $\{b[m],c[n]\}={\delta }_{m+n,0}$ makes this induction consistent and gives the current ghost fields with the stated shifted Hamiltonian. Their star anticommutator is $b\ast c+c\ast b=1$: the two contraction coefficients sum to $\mathrm{wt}(b)+\mathrm{wt}(c)=1$, and the remaining normally ordered sum is zero. The other anticommutators vanish. Ordered ghost monomials therefore give exactly Cl(n direct-sum n*). There are no further relations: the shifted zero-energy ghost modes act faithfully on that finite faithful Clifford top, whose loop Fock induction was explicitly constructed above. Mixed coefficient/ghost fields have zero cross OPE. Expanding $(1+z{)}^{a+b}=(1+z{)}^a(1+z{)}^b$ in their circle residues, and using the factor circle and translation relations, reduces mixed words to the tensor products of their two Zhu classes; conversely the two factor maps commute and recover every mixed normally ordered word. These inverse reductions prove $A(C(M))=A_{\text{new}}(M)$ tensor Cl, including bimodule actions. This calculation uses only integer Hamiltonian residue identities; positivity of the full unreduced ghost/current Hamiltonian is not asserted. After step 1.1 this is $A_{\text{old}}(M)$ tensor Cl.

The BRST state Q has improved weight one, so its zero-mode derivation induces the supercommutator by its Zhu class. Its class under step 2.1 is the actual finite $Q=\sum (u_i-{\chi }_i)c^i$ minus one half sum $f_{ij}^s$ $c^i$ $c^j$ $b_s$. The affine shift in step 1.1 does not change any constrained positive-root current. In converting ghost normal products to stars, the possible linear correction is a contraction times $f_{ij}^j$ or $f_{ij}^i$; these are zero because positive-root n is nilpotent and has zero adjoint traces. Thus the differential is precisely the finite BRST differential of F4, with coefficient A(M) and the same actual ideal. If a cohomology state is changed by a BRST boundary, its finite Zhu class changes by a finite boundary. A circle of closed representatives maps to a circle in the original complex. Hence there is a well-defined natural map ${\eta }_M:A(H^{0}C(M))$ to Phi(A(M)); for the adjoint quotient it respects products. No injection or surjection has been assumed to construct it.

For finite affine generation, A(M) is an HC bimodule with a good PBW filtration: the current-word reductions in F2 and the leading circle relations give a surjection $R_M=M\mathrm{/}{C}_2(M)$ to $g{r}_{PBW}$ A(M); finitely many current-module generators make $R_M$ finite over S(g), hence the target is finite as well. Reweight by h/2 to obtain the good Kazhdan filtration K of F4. Step 1.1 is triangular in old energy and preserves Cartan weight, so it identifies this actual K with the state-weight filtration for the shifted Zhu space; no changed ideal is hidden in the reweighting. On the W side the analogous leading circle/star argument gives the surjection $R_{H^{0}C(M)}$ to $g{r}_K$ $A(H^{0}C(M))$. By F3, the first ring/module also surjects onto classical reduction of $R_M$ via its induced Li degree zero. By F4, classical reduction of the surjection $R_M$ to $g{r}_K$ A(M) is surjective and its target is $g{r}_K$ Phi(A(M)). Taking closed representatives shows that these maps form a commutative square: either route takes the same leading state, first modulo its C2 terms and then modulo the finite moment/ghost differential. Therefore $g{r}_K$ ${\eta }_M$ is surjective. Its target has the good bounded-below reduced filtration supplied by F4. Finite leading-symbol lifting proves ${\eta }_M$ surjective.

For the universal vacuum, the same composite is an isomorphism. Indeed $R_W=O(S)$ by F3, $R_{V^k}=S(g)$, and $R_{V^k}$ to $g{r}_{PBW}$ $A(V^k)$ is the identity under F2. Finite HC strictness and F5 identify $g{r}_K$ Phi(U) with O(S) by the same actual invariant restriction. Consequently the composite $R_W$ to $g{r}_K$ A(W) to $g{r}_K$ Phi(U) is the identity on O(S), under the closed-symbol identification, not merely a comparison of dimensions. The first map is surjective; their composite being an isomorphism makes both maps isomorphisms. The filtrations on A(W) and $\mathrm{\Phi }(U)=Z$ are bounded below, by positive W grading and the positive invariant degrees. Finite-degree lifting and the leading-symbol injectivity argument give the actual natural isomorphism ${\eta }_{univ}:A(W^k)$ to Z. In particular this isomorphism is the map of step 3.1, not an abstract center identification with untracked lower terms.

The surjectivity in step 4.1 extends to any loop-bounded M in the present class by filtered unions of finitely generated current submodules. Circle spaces, actual BRST degree-zero cohomology, and the finite Clifford complexes commute with this union on finite representatives. The same is true for the HC functor in F4 on such unions: both its quotient and its finite Chevalley cochain model have finite witnessing equations. In particular ${\eta }_N$ is surjective for the actual vacuum ideal, without assuming that this ideal is finitely generated. Right exactness in F3 gives $H^0$ C(N) to $W^k$ to $H^0$ C(V) to zero. Applying the right exact circle-space functor of F1 yields the top exact row $A(H^{0}C(N))$ to $A(W^k)$ to A(H) to zero. The actual affine Zhu row is A(N) to U to U/J to zero; F4 makes its finite reductions the bottom exact row Phi(A(N)) to Phi(U) to Phi(U/J) to zero. Naturality of the closed-representative map gives their commuting vertical maps ${\eta }_N$, ${\eta }_{univ}$ and ${\eta }_V$. Since ${\eta }_N$ is onto and ${\eta }_{univ}$ is an isomorphism, the two images in the middle terms agree under ${\eta }_{univ}$, so their cokernels are isomorphic. This proves ${\eta }_V:A(H)$ to Phi(U/J). No left injection of $H^0$ C(N) was needed; negative quantum cohomology vanishing was not smuggled into this argument.

Finally right-n coinvariants of U/J are exactly Y/JY. F4 identifies their invariant space with Phi(U/J), and F5 identifies JY with Y times the actual central ideal Wh(JY). Thus $\mathrm{Wh}(Y\mathrm{/}JY)=Z\mathrm{/}\mathrm{Wh}(JY)$, with the normalizer-product algebra structure, not only as a vector space. Since all maps arose from multiplication of closed states, the ${\eta }_V$ just obtained is the claimed algebra isomorphism and agrees with ${\eta }_{univ}$ on the universal source. All occurrences of J are its actual image from A(N); no radical or highest-weight scalar polynomial was substituted. This proves the Statement.