Classifying Relaxed Highest-Weight Modules for Admissible-Level Bershadsky–Polyakov Algebras

Abstract

The Bershadsky–Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to $\mathfrak {sl}_{3}$ and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications (Arakawa in Commun Math Phys 323:627–633, 2013, Adamović and Kontrec in Classification of irreducible modules for Bershadsky–Polyakov algebra at certain levels). In particular, we prove that the simple Bershadsky–Polyakov algebras with admissible nonintegral $\mathsf {k}$ are always rational in category $\mathscr {O}$, whilst they always admit nonsemisimple relaxed highest-weight modules unless $\mathsf {k}+\frac{3}{2} \in \mathbb {Z}_{\geqslant 0}$.

Appendix A: Proof of Theorem 4.8

In this appendix, we adopt the notation of Sect. 4.1 and assume throughout that $\lambda _0 \notin \mathbb {Z}_{\geqslant 0}$ so that $H^0(\mathcal {L}_{\lambda }) \ne 0$ (and that the level $\mathsf {k}$ is as in Assumption 1). With these assumptions, the aim is to prove the following assertion:

$$\begin{aligned} \mathsf {I}^{\mathsf {k}}\cdot \mathcal {L}_{\lambda } \ne 0 \quad \Rightarrow \quad H^0(\mathsf {I}^{\mathsf {k}}) \cdot H^0(\mathcal {L}_{\lambda }) \ne 0. \end{aligned}$$

(A.1)

$H^0({-})$ is a cohomological functor which involves tensoring with a ghost vertex operator superalgebra whose vacuum element will be denoted by ${|}{0}{\rangle }$. With this, we shall prove (A.1) by exhibiting elements $\chi \in \mathsf {I}^{\mathsf {k}}$ and $v \in \mathcal {L}_{\lambda }$ for which $\chi \otimes {|}{0}{\rangle }$ and $v\otimes {|}{0}{\rangle }$ are (degree-0) closed elements of the appropriate BRST complexes and the (clearly closed) element $\chi _n v\otimes {|}{0}{\rangle }$ is not exact, for some $n \in \mathbb {Z}$. Using brackets to denote cohomology classes, $[\chi _n v\otimes {|}{0}{\rangle }]$ then gives a nonzero element of $H^0(\mathsf {I}^{\mathsf {k}}) \cdot H^0(\mathcal {L}_{\lambda })$:

$$\begin{aligned} {[}\chi \otimes {|}{0}{\rangle }] \cdot [v \otimes {|}{0}{\rangle }] \equiv [\chi \otimes {|}{0}{\rangle }](z) [v \otimes {|}{0}{\rangle }] = [\chi (z) v \otimes {|}{0}{\rangle }] \ne 0. \end{aligned}$$

(A.2)

As noted at the end of Sect. 4.1, this amounts to a proof of Theorem 4.8. To prove (A.1) however, we need to delve a little deeper into the details of minimal quantum hamiltonian reduction for .

1.1 A.1. Minimal quantum hamiltonian reduction

Recall from [6] that the minimal quantum hamiltonian reduction functor $H^0({-})$ computes the cohomology of the tensor product of a given -module with certain ghost vertex operator superalgebras. Specifically, we need a fermionic ghost system $\mathsf {F}^{\alpha }$ for each positive root $\alpha \in \Delta _+$ of $\mathfrak {sl}_{3}$ and one bosonic ghost system $\mathsf {B}$ corresponding to the two simple roots $\alpha _{1}$ and $\alpha _{2}$. Denoting the fermionic ghosts by $b^{\alpha }$ and $c^{\alpha }$, $\alpha \in \Delta _+$, and the bosonic ghosts by $\beta$ and $\gamma$, we take the defining operator product expansions to be

(A.3)

understanding that the remaining operator product expansions between ghost generating fields are regular. The tensor product of these ghost vertex operator superalgebras will be denoted by $\mathsf {G}= \mathsf {F}^{\alpha _{1}} \otimes \mathsf {F}^{\alpha _{2}} \otimes \mathsf {F}^{\theta } \otimes \mathsf {B}$, for convenience.

We fix a basis of $\mathfrak {sl}_{3}$ for the computations to follow. Let $E_{ij}$ denote the $3 \times 3$ matrix with 1 in the (i, j)-th position and zeroes elsewhere. Then, we set

$$\begin{aligned} e^{\theta } = E_{13}, \quad \begin{aligned} e^{\alpha _{1}}&= E_{12},&h^{\alpha _{1}}&= E_{11}-E_{22},&f^{\alpha _{1}}&= E_{21}, \\ e^{\alpha _{2}}&= E_{23},&h^{\alpha _{2}}&= E_{22}-E_{33},&f^{\alpha _{2}}&= E_{32}, \end{aligned} \quad f^{\theta } = E_{31}. \end{aligned}$$

(A.4)

Here, $\theta = \alpha _{1} + \alpha _{2}$ is the highest root of $\mathfrak {sl}_{3}$ and we shall also set $h^{\theta } = h^{\alpha _{1}} + h^{\alpha _{2}} = E_{11}-E_{33}$.

To define $H^0(\mathcal {M})$ for a -module $\mathcal {M}$, one first grades $\mathcal {M} \otimes \mathsf {G}$ by the fermionic ghost number, that is by the total number of c-modes minus the total number of b-modes. Equivalently, the ghost number is the eigenvalue of the zero mode of the field $\sum _{\alpha \in \Delta _+} \mathopen {:} b^{\alpha }(z) c^{\alpha }(z) \mathclose {:}$. Next, one introduces [5, 6] the following fermionic field of ghost number 1:

(A.5)

A straightforward computation verifies that $d(z) d(w) \sim 0$. We then form a differential complex by requiring that d(z) is homogeneous of conformal weight 1 and equipping $\mathcal {M} \otimes \mathsf {G}$ with the differential $d = d_0$ (which obviously squares to 0).

With (A.5), this requirement on d(z) requires that the conformal weight of $c^{\theta }$ is also 1, whilst that of $e^{\theta }$ is 0. The latter may be achieved by adding $\frac{1}{2} \partial h^{\theta }$ to the standard Sugawara energy-momentum tensor $T^{\text {Sug.}}$ of . When this is done, homogeneity and (A.5) now fix the conformal weight $\widetilde{\Delta }$ of all the generating fields as in Table 1. The energy-momentum tensor of is thus

$$\begin{aligned} \begin{aligned}&\widetilde{L} = T^{\text {Sug.}} + \frac{1}{2} \partial h^{\theta } + \sum _{\alpha \in \Delta _+} T^{\mathsf {F}^{\alpha }} + T^{\mathsf {B}}, \\&\quad \text {where} \quad T^{\mathsf {F}^{\alpha _{i}}} = \frac{1}{2} \mathopen {:} \partial b^{\alpha _{i}} c^{\alpha _{i}} + \partial c^{\alpha _{i}} b^{\alpha _{i}} \mathclose {:},\\&\quad T^{\mathsf {F}^{\theta }} = \mathopen {:} \partial b^{\theta } c^{\theta } \mathclose {:} \quad \text {and} \quad T^{\mathsf {B}} = \frac{1}{2} \mathopen {:} \partial \gamma \beta - \partial \beta \gamma \mathclose {:}. \end{aligned} \end{aligned}$$

(A.6)

The central charge matches that of $\mathsf {BP}^{\mathsf {k}}$, see (2.2):

$$\begin{aligned} \frac{8\mathsf {k}}{\mathsf {k}+3} - 6\mathsf {k}+ 1 + 1 - 2 - 1 = -\frac{(2\mathsf {k}+3)(3\mathsf {k}+1)}{\mathsf {k}+3}. \end{aligned}$$

(A.7)

Table 1 The ghost numbers $\#$, charges $\widetilde{j}$ and conformal weights $\widetilde{\Delta }$ of the generating fields of the vertex operator superalgebra

As the notation suggests, $\widetilde{L}$ is closed and its image in cohomology (that is, in ) is L. Note that the “symmetric” deformation of adding $\frac{1}{2} \partial h^{\theta }$ to $T^{\text {Sug.}}$ ensures this result. There are other deformations consistent with d being a differential—they correspond to adding a multiple of $\partial J$ to L. Speaking of which, the element

$$\begin{aligned} \widetilde{J} = \frac{1}{3} (h^{\alpha _{1}} - h^{\alpha _{2}}) + \mathopen {:} b^{\alpha _{1}} c^{\alpha _{1}} \mathclose {:} - \mathopen {:} b^{\alpha _{2}} c^{\alpha _{2}} \mathclose {:} - \mathopen {:} \beta \gamma \mathclose {:} \end{aligned}$$

(A.8)

is likewise closed and its image in cohomology is J [6]. We give the charge ($\widetilde{J}_0$-eigenvalue) of the generating fields of in Table 1 for completeness. We also note that

$$\begin{aligned} \begin{aligned} \widetilde{G}^+&= f^{\alpha _{2}} + \mathopen {:} h^{\alpha _{2}} \beta \mathclose {:} - \mathopen {:} b^{\alpha _{1}} c^{\theta } \mathclose {:} - \mathopen {:} b^{\alpha _{1}} c^{\alpha _{1}} \beta \mathclose {:} + 2 \mathopen {:} b^{\alpha _{2}} c^{\alpha _{2}} \beta \mathclose {:} + \mathopen {:} b^{\theta } c^{\theta } \beta \mathclose {:} + \mathopen {:} \beta \beta \gamma \mathclose {:} + (\mathsf {k}+1) \partial \beta \\ \text {and} \quad \widetilde{G}^-&= f^{\alpha _{1}} - \mathopen {:} h^{\alpha _{1}} \gamma \mathclose {:} + \mathopen {:} b^{\alpha _{2}} c^{\theta } \mathclose {:} - 2 \mathopen {:} b^{\alpha _{1}} c^{\alpha _{1}} \gamma \mathclose {:} + \mathopen {:} b^{\alpha _{2}} c^{\alpha _{2}} \gamma \mathclose {:} - \mathopen {:} b^{\theta } c^{\theta } \gamma \mathclose {:} + \mathopen {:} \gamma \gamma \beta \mathclose {:} - (\mathsf {k}+1) \partial \gamma \end{aligned} \end{aligned}$$

(A.9)

are both closed. Their images in cohomology are $G^+$ and $G^-$, respectively [8].

We remark that deforming the energy-momentum tensor of means that we now have two distinct mode conventions for affine fields. Our convention will be that mode indices with respect to the deformed conformal weight will be denoted with parentheses. Thus, for an affine generator a with deformed conformal weight $\widetilde{\Delta }$ as in Table 1, we shall write

$$\begin{aligned} a(z) = \sum _{n \in \mathbb {Z}} a_n z^{-n-1} = \sum _{n \in \mathbb {Z}- \widetilde{\Delta }} a_{(n)} z^{-n-\widetilde{\Delta }}. \end{aligned}$$

(A.10)

We shall not bother to so distinguish mode indices for ghost fields: their expansions will always be taken with respect to the conformal weights in Table 1.

1.2 A.2. The proof

We start with a well known fundamental result for the highest-weight vector v of $\mathcal {L}_{\lambda }$, recalling that we are assuming throughout that $\lambda _0 \notin \mathbb {Z}_{\geqslant 0}$ and that $\mathsf {k}$ satisfies Assumption 1. Let ${|}{0}{\rangle }$ denote the vacuum vector of $\mathsf {G}$. By [9, Lem. 4.6.1 and Prop. 4.7.1], we have the following lemma.

Lemma A.1

For all $n \in \mathbb {Z}_{\geqslant 0}$, $(e^{\theta }_{-1})^n v \otimes {|}{0}{\rangle }$ is closed and inexact. In particular, $[v \otimes {|}{0}{\rangle }] \ne 0$.

We next consider the maximal ideal $\mathsf {I}^{\mathsf {k}}$ of .

Lemma A.2

$\mathsf {I}^{\mathsf {k}}$ is generated by a single singular vector $\chi$ whose $\mathfrak {sl}_{3}$-weight and conformal weight are $(\mathsf {u}-2) \theta$ and $(\mathsf {u}-2) \mathsf {v}$, respectively. Moreover, $\chi \otimes {|}{0}{\rangle }$ is closed.

Proof

This follows easily from [38, Cor. 1], which says that the maximal submodule of a Verma module whose highest weight is admissible is generated by singular vectors of known weight. In our case, the highest weight is $\mathsf {k}\omega _{0}$ (which is admissible because $\mathsf {k}$ is) and the only generating singular vector that is nonzero in the quotient of this Verma module has weight $w_{} \cdot (\mathsf {k}\omega _{0})$, where $w_{}$ is the Weyl reflection corresponding to the root $-\theta + \mathsf {v}\delta$. Here, $\delta$ denotes the standard imaginary root of $\widehat{\mathfrak {sl}}_{3}$. This singular vector is $\chi$ and its $\mathfrak {sl}_{3}$- and conformal weights are now easily computed. The fact that $\chi \otimes {|}{0}{\rangle }$ is closed follows from $\chi$ being a highest-weight vector. $\square$

In fact, $\chi \otimes {|}{0}{\rangle }$ is also inexact, though we will not need to a priori establish this fact for what follows.

We remark that a nice conceptual proof of [38, Cor. 1] starts from the celebrated fact that the submodule structure of a Verma module only depends on the corresponding integral Weyl group [64]. This structure is therefore the same for all admissible highest-weight $\widehat{\mathfrak {sl}}_{3}$-modules, irrespective of their level. In particular, this structure matches that of a Verma module whose simple quotient is integrable, integrability being equivalent to admissibility for simple highest-weight modules with $\mathsf {v}=1$. However, the fact that the maximal submodule is generated by singular vectors is well known in the integrable case, see [65] or [66].

Suppose now that $\chi (z) v = 0$. Because $\chi$ generates $\mathsf {I}^{\mathsf {k}}$, it follows that $\mathsf {I}^{\mathsf {k}}\cdot v = 0$. Since v generates $\mathcal {L}_{\lambda }$, as a -module, and $\mathsf {I}^{\mathsf {k}}$ is a two-sided ideal of , we get $\mathsf {I}^{\mathsf {k}}\cdot \mathcal {L}_{\lambda } = 0$. Thus, the hypothesis of (A.1), that $\mathcal {L}_{\lambda }$ is not an -module, requires that $\chi _n v \ne 0$ for some $n \in \mathbb {Z}$. As $\chi$ has $\mathfrak {sl}_{3}$-weight $(\mathsf {u}-2) \theta$, our knowledge of the weights of $\mathcal {L}_{\lambda }$ lets us refine this requirement to $\chi _{-(\mathsf {u}-2)-i} v \ne 0$ for some $i \in \mathbb {Z}_{\geqslant 0}$. There is therefore a minimal $N \in \mathbb {Z}_{\geqslant 0}$ such that $\chi _{-(\mathsf {u}-2)-N} v \ne 0$.

As $\mathcal {L}_{\lambda }$ is simple, there therefore exists a Poincaré–Birkhoff–Witt monomial such that

$$\begin{aligned} U \chi _{-(\mathsf {u}-2)-N} v = v \end{aligned}$$

(A.11)

(rescaling $\chi$ if necessary). We choose an ordering for U so that

$$\begin{aligned} f^{\alpha }_{n\leqslant 0}< h^{\alpha }_{n<0}< e^{\alpha }_{n<0}< f^{\alpha }_{n>0}< h^{\alpha }_{n>0} < e^{\alpha }_{n\geqslant 0} \end{aligned}$$

(A.12)

(obviously we may omit the $h^{\alpha }_0$ and K). This means, for example, that the $f^{\alpha }_n$ with $n\leqslant 0$ are ordered to the left while the $e^{\alpha }_n$ with $n\geqslant 0$ are ordered to the right. For $n\geqslant 0$, we have $e^{\alpha }_0 \chi = 0$ and $e^{\alpha }_n v = 0$, hence

$$\begin{aligned} e^{\alpha }_n \chi _{-(\mathsf {u}-2)-N} v = (e^{\alpha }_0 \chi )_{-(\mathsf {u}-2)-N+n} v = 0. \end{aligned}$$

(A.13)

We may therefore assume that U contains no $e^{\alpha }_n$-modes with $n\geqslant 0$. Similarly,

$$\begin{aligned} h^{\alpha }_n \chi _{-(\mathsf {u}-2)-N} v = (\mathsf {u}-2) \theta (h^{\alpha }) \chi _{-(\mathsf {u}-2)-(N-n)} v = 0 \end{aligned}$$

(A.14)

for $n>0$, by the minimality of N. Thus, we may assume that U contains no $h^{\alpha }_n$-modes with $n>0$ either. Finally, v is not in the image of any $f^{\alpha }_n$, with $n\leqslant 0$, $h^{\alpha }_n$, with $n<0$, or $e^{\alpha }_n$, with $n<0$. All these modes may therefore also be excluded from U. We conclude that U may be taken to be a monomial in the modes $f^{\alpha }_n$ with $n>0$.

Given a partition $\xi = [\xi _1 \geqslant \xi _2 \geqslant \cdots ]$, let $\ell (\xi )$ denote its length and ${|}{\xi }{|}$ denote its weight. We write $f^{\alpha }_{\xi } = f^{\alpha }_{\xi _1} f^{\alpha }_{\xi _2} \cdots$. Then, there exist partitions $\xi$, $\pi$ and $\rho$ such that $U = f^{\theta }_{\xi } f^{\alpha _{2}}_{\pi } f^{\alpha _{1}}_{\rho }$ and so

$$\begin{aligned} f^{\theta }_{\xi } f^{\alpha _{2}}_{\pi } f^{\alpha _{1}}_{\rho } \chi _{-(\mathsf {u}-2)-N} v = v. \end{aligned}$$

(A.15)

Moreover, considering $\mathfrak {sl}_{3}$- and conformal weights gives

$$\begin{aligned} \ell (\pi ) = \ell (\rho ), \quad \ell (\xi ) + \ell (\pi ) = \mathsf {u}-2 \quad \text {and} \quad {|}{\xi }{|} + {|}{\pi }{|} + {|}{\rho }{|} = \mathsf {u}-2+N. \end{aligned}$$

(A.16)

Lemma A.3

Let F(z), $F \in \mathfrak {sl}_{3}$, be an affine field and let $U_0$ be a monomial in the negative root vectors $f^{\alpha }_0$ of $\widehat{\mathfrak {sl}}_{3}$. Then, the modes of the field $(U_0 \chi )(w)$ satisfy

(A.17)

Proof

Observe that $U_0 \chi$ is annihilated by the $F_m$ with $m>0$. Consequently, the assertion follows easily from the operator product expansion

$$\begin{aligned} F(z) (U_0 \chi )(w) \sim \frac{(F_0 U_0 \chi )(w)}{z-w}. \end{aligned}$$

(A.18)

$\square$

We apply Lemma A.3 to the left-hand side of (A.15), noting that the $f^{}$-modes all annihilate v. The result is

(A.19)

using (A.16). This looks complicated, but it allows us to determine the partitions $\xi$, $\pi$ and $\rho$.

Lemma A.4

If any of the parts of $\xi$, $\pi$ or $\rho$ are greater than 1, then $f^{\theta }_{\xi } f^{\alpha _{2}}_{\pi } f^{\alpha _{1}}_{\rho } \chi _{-(\mathsf {u}-2)-N} v = 0$.

Proof

Suppose that $\xi$ has a part $\xi _i > 1$ (the argument is identical if $\pi$ or $\rho$ has a part greater than 1). Then, we can form a new partition $\xi '$ from $\xi$ by subtracting 1 from $\xi _i$ and reordering parts if necessary. Note that $\ell (\xi ') = \ell (\xi )$ and ${|}{\xi '}{|} = {|}{\xi }{|} - 1$. Then, Lemma A.3 and N being minimal give

(A.20)

But, this is the right-hand side of (A.19). $\quad \square$

Combining (A.15), which is manifestly nonzero, with Lemma A.4 now forces all parts of $\xi$, $\pi$ and $\rho$ to be 1. As partition lengths and weights are now equal, the relations of (A.16) are easily solved to give ${|}{\xi }{|} = \mathsf {u}-2-N$ and ${|}{\pi }{|} = {|}{\rho }{|} = N$. In particular, (A.15) now becomes

$$\begin{aligned} (f^{\theta }_1)^{\mathsf {u}-2-N} (f^{\alpha _{2}}_1)^N (f^{\alpha _{1}}_1)^N \chi _{-(\mathsf {u}-2)-N} v = v. \end{aligned}$$

(A.21)

By rescaling $\chi$ again, if necessary, we arrive at following key result.

Proposition A.5

If N is the minimal integer such that $\chi _{-(\mathsf {u}-2)-N} v \ne 0$, then

$$\begin{aligned} (f^{\alpha _{2}}_1)^N (f^{\alpha _{1}}_1)^N \chi _{-(\mathsf {u}-2)-N} v = (e^{\theta }_{-1})^{\mathsf {u}-2-N} v. \end{aligned}$$

(A.22)

The idea now is to use the fact that the right-hand side of (A.22) is inexact when tensored with ${|}{0}{\rangle }$ (Lemma A.1) to prove that the same is true for $\chi _{-(\mathsf {u}-2)-N} v$. For this, we need to replace the action of $f^{\alpha _{2}}_1$ and $f^{\alpha _{1}}_1$ with elements that preserve exactness, for example any closed elements.

Lemma A.6

For all $i,j \in \mathbb {Z}_{\geqslant 0}$, we have

(A.23)

Proof

We start with (A.9), which gives

$$\begin{aligned} \widetilde{G}^-_{(1/2)} = f^{\alpha _{1}}_{(1/2)} - \sum _{m \in \mathbb {Z}} h^{\alpha _{1}}_{(m)} \gamma _{-m+1/2} + \cdots = f^{\alpha _{1}}_1 - \sum _{m \in \mathbb {Z}} h^{\alpha _{1}}_m \gamma _{-m+1/2} + \cdots , \end{aligned}$$

(A.24)

where the $\cdots$ stands for pure ghost terms. As these ghost terms annihilate ${|}{0}{\rangle }$, we have

(A.25)

for any $j \in \mathbb {Z}_{\geqslant 0}$. Now, $m\geqslant 1$ implies that $h^{\alpha _{2}}_m v = 0$, hence that

(A.26)

The first commutator on the right-hand side is a sum of terms, each obtained from $(f^{\alpha _{1}}_1)^j$ by replacing one of the $f^{\alpha _{1}}_1$ by $-2 f^{\alpha _{1}}_{m+1}$. However, each of these terms is 0 by Lemma A.4. On the other hand, the second commutator is proportional to $\chi _{-(\mathsf {u}-2)-(N-m)}$, so it annihilates v by minimality of N. We therefore obtain

(A.27)

from which we conclude inductively that , for all $i\in \mathbb {Z}_{\geqslant 0}$.

To deduce (A.23), we now repeat the argument by acting with $\widetilde{G}^+_{(1/2)}$ on $(f^{\alpha _{2}}_1)^i (f^{\alpha _{1}}_1)^j \chi _{-(\mathsf {u}-2)-N} v \otimes {|}{0}{\rangle }$. There are no essential differences between this case and that described above, so we omit the details.

Corollary A.7

$\chi _{-(\mathsf {u}-2)-N} v \otimes {|}{0}{\rangle }$ is closed and inexact.

Proof

We have already seen that $\chi _{-(\mathsf {u}-2)-N} v \otimes {|}{0}{\rangle }$ is closed. Suppose therefore that it is exact. As , since $\widetilde{G}^{\pm }$ is closed, it now follows from Proposition A.5 and Lemma A.6 that

(A.28)

is also exact. But, this contradicts Lemma A.1. $\square$

This corollary completes the proof of Theorem 4.8.

We conclude with a few remarks about this proof. First, proving that quantum hamiltonian reduction indeed realises all simple highest-weight modules is obviously desirable and has been studied in several settings. However, Arakawa’s general results [9, 54] in this direction for universal minimal and regular W-algebras do not immediately imply the desired results for their simple quotients. Indeed, the cases where this completeness result for simple W-algebras is known seem to be cases in which the simple quotient is rational and $C_2$-cofinite, see for example [1, 54, 67]. Our proof, applying as it does to the nonrational and non-$C_2$-cofinite simple Bershadsky–Polyakov algebras, is therefore quite novel and seems to be very different from the rational proofs in the literature.

Second, this proof relies on certain key facts that might be regarded as special to the Bershadsky–Polyakov algebras. In particular, we use the explicit realisation (A.9) of the charged generators of $\mathsf {BP}^{\mathsf {k}}$. However, the pure ghost terms played no role in the proof, so it may be possible to generalise this part of the argument to other minimal, or perhaps even subregular, W-algebras. On the other hand, the proof also exploits the fact that the maximal ideal of is generated by a single singular vector, which does not normally hold when generalising to nonadmissible levels. It is therefore not clear that this proof can be adapted for the nonadmissible case, but it would of course be interesting to try.

Alternatively, it may be that one can prove more general completeness results of this type by further developing the inverse quantum hamiltonian reduction methods introduced in [18, 33] and extended to the Bershadsky–Polyakov algebras in [31]. These methods have the advantage of building up the representation theory iteratively from that of the so-called exceptional W-algebras [67], in particular from the regular ones. This may then lead to uniform methods for all W-algebras, at least when the level is admissible and (sufficiently) nondegenerate. We hope to have the opportunity to report on this promising direction in the future.