Rationality of lisse affine W algebras under the orbit-closure hypothesis

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra, let $B$ be its normalized invariant form, and let $k\in \mathbb{C}$. Let $\widehat{\mathfrak{g}}$ be the untwisted affine central extension of $\mathfrak{g}[t,t^{-1}]$ defined using $B$, with central generator $K$. The universal affine vacuum vertex algebra is $V^k(\mathfrak{g})=U(\widehat{\mathfrak{g}}){\otimes }_{U(\mathfrak{g}[t]\oplus \mathbb{C}K)}{\mathbb{C}}_k,$ where $\mathfrak{g}[t]$ acts trivially on ${\mathbb{C}}_k$ and $K$ acts as $k$; let $L_k(\mathfrak{g})$ denote its unique simple graded quotient. For a vertex algebra $V$, define $C_2(V)={\mathrm{span}}_{\mathbb{C}}\{u_{(-2)}v:u,v\in V\},R_V=V\mathrm{/}{C}_2(V),$ with the commutative product on $R_V$ induced by $u\cdot v=u_{(-1)}v$, and set $X_V=\mathrm{Specm}(R_V)$, the set of maximal ideals of $R_V$. For the simple affine quotient $L_k(\mathfrak{g})$, the current states define the surjective algebra homomorphism ${\rho }_k:S(\mathfrak{g})\twoheadrightarrow R_{L_k(\mathfrak{g})},x⟼[x_{-1}\mathbf{1}]\mathrm{.}$ Here $S(\mathfrak{g})$ is the symmetric algebra and the bracket denotes the class in $R_{L_k(\mathfrak{g})}$. Surjectivity follows because $V^k(\mathfrak{g})$ and its quotient $L_k(\mathfrak{g})$ are generated by current states. Identify $\mathrm{Specm}S(\mathfrak{g})$ with ${\mathfrak{g}}^{\ast }$. The induced map on maximal spectra identifies $X_{L_k(\mathfrak{g})}=\mathrm{Specm}(R_{L_k(\mathfrak{g})})$ with the closed subset $V(\ker {\rho }_k)\subseteq {\mathfrak{g}}^{\ast }$. Use $B$ to identify $\mathfrak{g}$ with ${\mathfrak{g}}^{\ast }$ when viewing nilpotent adjoint orbits there.

Let $G$ be the connected adjoint group of $\mathfrak{g}$ and let $f\in \mathfrak{g}$ be nilpotent, meaning $(\mathrm{ad}f{)}^m=0$ for some integer $m\ge 1$. Choose a semisimple grading element inducing a decomposition $\mathfrak{g}={⨁}_{j\in \frac{1}{2}\mathbb{Z}}{\mathfrak{g}}_j$ with $[{\mathfrak{g}}_i,{\mathfrak{g}}_j]\subseteq {\mathfrak{g}}_{i+j}$ and $B({\mathfrak{g}}_i,{\mathfrak{g}}_j)=0$ unless $i+j=0$. Require $f\in {\mathfrak{g}}_{-1}$ and that $\mathrm{ad}f:{\mathfrak{g}}_j\to {\mathfrak{g}}_{j-1}$ is injective for $j\ge \frac{1}{2}$ and surjective for $j\le \frac{1}{2}$. On ${\mathfrak{g}}_{1\mathrm{/}2}$ set ${\omega }_f(u,v)=B(f,[u,v])$, choose a maximal isotropic subspace $\mathrm{\ell }$ for ${\omega }_f$, and put $\mathfrak{m}=\mathrm{\ell }\oplus {⨁}_{j\ge 1}{\mathfrak{g}}_j,\chi =B(f,\cdot ){\mathrm{\mid }}_{\mathfrak{m}}\mathrm{.}$ Choose a basis $u_1,\dots ,u_d$ of $\mathfrak{m}$ with $[u_i,u_j]={\sum }_s{c}_{ij}^s{u}_s$. Use the charged-ghost Fock superalgebra $F(\mathfrak{m})$ generated by odd fields ${\psi }_i(z)={\sum }_{n\in \mathbb{Z}}{\psi }_{i,n}{z}^{-n-1},{\psi }_i^{\ast }(z)={\sum }_{n\in \mathbb{Z}}{\psi }_{i,n}^{\ast }{z}^{-n},$ whose modes satisfy $\{{\psi }_{i,m},{\psi }_{j,n}\}=\{{\psi }_{i,m}^{\ast },{\psi }_{j,n}^{\ast }\}=0$ and $\{{\psi }_{i,m},{\psi }_{j,n}^{\ast }\}={\delta }_{ij}{\delta }_{m+n,0}$; its vacuum is killed by ${\psi }_{i,n}$ for $n\ge 0$ and ${\psi }_{i,n}^{\ast }$ for $n\ge 1$. Give currents degree $0$, ${\psi }_i$ degree $-1$, and ${\psi }_i^{\ast }$ degree $+1$. Write $J^{u_i}(z)={\sum }_n(u_i{)}_n{z}^{-n-1}$ for the affine current of $u_i$. With the charged-ghost normal ordering, the BRST state is $Q(z)={\sum }_i(J^{u_i}(z)-\chi (u_i)){\psi }_i^{\ast }(z)-\frac{1}{2}{\sum }_{i,j,s}{c}_{ij}^s:{\psi }_i^{\ast }(z){\psi }_j^{\ast }(z){\psi }_s(z):\mathrm{.}$ The charged-ghost BRST construction proves $Q_{(0)}^2=0$. Thus $C^k(\mathfrak{g},f)=V^k(\mathfrak{g})\otimes F(\mathfrak{m})$ is a cochain complex with degree-zero cohomology $W^k(\mathfrak{g},f)=H^0(C^k(\mathfrak{g},f),Q_{(0)})$.

Assume $H^0(C^k(\mathfrak{g},f),Q_{(0)})$ is nonzero and a separate conformal-structure construction supplies a specified conformal vector ${\omega }_{\mathrm{D}\mathrm{S}}$; write $Y({\omega }_{\mathrm{D}\mathrm{S}},z)={\sum }_{n\in \mathbb{Z}}L(n)z^{-n-2}$, require $[L(m),L(n)]=(m-n)L(m+n)+(m^3-m){\delta }_{m+n,0}{c}_{\mathrm{D}\mathrm{S}}I\mathrm{/}12$ for some $c_{\mathrm{D}\mathrm{S}}\in \mathbb{C}$, and require $L(-1)=T$. Let $J_k$ be the specified homogeneous maximal proper vertex ideal defining the source's simple quotient, and set $q_k:H^0(C^k(\mathfrak{g},f),Q_{(0)})\twoheadrightarrow W_k(\mathfrak{g},f)=H^0(C^k(\mathfrak{g},f),Q_{(0)})\mathrm{/}{J}_k$. Require that $q_k$ descend ${\omega }_{\mathrm{D}\mathrm{S}}$. Give $W_k$ the induced conformal grading and assume it is of CFT type: $W_k={⨁}_{n\ge 0}(W_k{)}_n$, where $(W_k{)}_n=\{w:L(0)w=nw\}$, each weight space is finite dimensional, and $(W_k{)}_0=\mathbb{C}\mathbf{1}$. This makes lisse mean that $R_{W_k}$ is finite dimensional and uses rationality in the admissible-module convention where every admissible module is an internal direct sum of irreducible admissible submodules with its specified grading inherited by the summands.

Let $\mathcal{O}=G\cdot f$ be the nilpotent adjoint orbit, viewed in ${\mathfrak{g}}^{\ast }$ through $B$, and let $\bar{\mathcal{O}}$ denote its Zariski closure. Suppose $X_{L_k(\mathfrak{g})}=\bar{\mathcal{O}}$ and $W_k(\mathfrak{g},f)$ is lisse. Must $W_k(\mathfrak{g},f)$ be rational?

Good grading and isotropic data for affine W-algebra reduction

Affine W algebra by quantum Drinfeld-Sokolov reduction

Universal affine vertex algebras exist at every complex level

Universal affine vacuum module

C2-cofiniteness

Admissible module for a vertex operator algebra

Rationality in the admissible-module convention

Arakawa and Moreau, §14.2

Arakawa, Creutzig, and Kawasetsu, §1

Status reviewed 2026-10-26: the 2024 Arakawa–Moreau monograph states this formulation as Conjecture 14.2 and records known cases for the Deligne exceptional series at $k=-h^{\vee }\mathrm{/}6-1$. Arakawa–Creutzig–Kawasetsu (2024) proves additional specific rational non-admissible principal-W cases, while separately stating further minimal-W cases as conjectures. These results do not resolve the full formulation above. Targeted searches of the exact conjecture and non-admissible lisse-W rationality literature found no full proof or counterexample by this cutoff; the search was not exhaustive.

The current supplier Affine W algebra by quantum Drinfeld-Sokolov reduction fixes the charged-ghost/Lagrangian BRST complex and the interface $W^k(\mathfrak{g},f)=H^0(C^k(\mathfrak{g},f),Q_{(0)})$ for supplied finite-dimensional complex simple $\mathfrak{g}$, nilpotent $f$, compatible good grading, Lagrangian data and nilpotent character. The displayed charged-ghost square-zero identity is now proved in that definition. Nonzero cohomology is separate from the definition itself. A conformal structure and the chosen simple quotient $W_k(\mathfrak{g},f)$ retain their separate construction hypotheses. The question above assumes its specified construction data and conditions. This item remains a genuine open question and is not a logical supplier.

Alpha status review 2026-11-04: the cited conjecture and the specific 2024 results were rechecked, together with a targeted search for later work. No general solution was identified in that bounded search; this is not an exhaustive claim about the literature. The title describes the orbit-closure hypothesis without imposing a non-admissibility restriction on $k$. The stable item ID retains its historical wording.