Defining Relations for Minimal Unitary Quantum Affine W-Algebras

Abstract

We prove that any unitary highest weight module over a universal minimal quantum affine W-algebra at non-critical level descends to its simple quotient. We find the defining relations of the unitary simple minimal quantum affine W-algebras and the list of all their irreducible positive energy modules. We also classify all irreducible highest weight modules for the simple affine vertex algebras in the cases when the associated simple minimal W-algebra is unitary.

1 Introduction

Vertex algebras have been playing an increasingly important role in quantum physics (see e.g. [BML15, BMR19, CKLW, Ch19, Beem20] and references therein). Some of the most relevant to physics among them are unitary vertex algebras.

The problem of unitarity of highest weight representations of infinite-dimensional Lie superalgebras has been a hot topic in Mathematics and Physics in the 1980s and 1990s. Unitary highest weight representations have been classified for many infinite-dimensional Lie algebras and superalgebras, including affine Lie algebras [K], Virasoro algebra [FQS85-1, GKO85], $N{=}1,2,3$ and 4 superconformal algebras [FQS85-2, FQS85-1, FQS86, KR, KW85, KT85, BFK86, GKO86, DPYZ86, ET87, ET88-1, ET88-2, KS89, M90, FST, Ad01, CLRW]. All these Lie (super)algebras arise naturally in the context of a special class of vertex algebras, called quantum affine W-algebras, which are obtained by quantum Hamiltonian reduction from affine vertex algebras.

Quantum affine W-algebras $W_k({\mathfrak {g}},x,f),$ are simple vertex algebras constructed in [KRW03, KW04], starting from a datum $({\mathfrak {g}},x,f)$ and $k\in {\mathbb {C}}$. Here ${\mathfrak {g}}={\mathfrak {g}}_{{\bar{0}}}\oplus {\mathfrak {g}}_{{\bar{1}}}$ is a basic Lie superalgebra, i.e. ${\mathfrak {g}}$ is simple, its even part ${\mathfrak {g}}_{{\bar{0}}}$ is a reductive Lie algebra and ${\mathfrak {g}}$ carries an even invariant non-degenerate supersymmetric bilinear form (.|.), x is an ad–diagonalizable element of ${\mathfrak {g}}_{{\bar{0}}}$ with eigenvalues in $\tfrac{1}{2}{\mathbb {Z}}$, $f\in {\mathfrak {g}}_{{\bar{0}}}$ is such that $[x,f]=-f$ and the eigenvalues of $ad\,x$ on the centralizer ${\mathfrak {g}}^f$ of f in ${\mathfrak {g}}$ are non-positive, and k is non-critical, i.e. $k\ne -h^\vee$, where $h^\vee$ is the dual Coxeter number of ${\mathfrak {g}}$. Recall that $W_k({\mathfrak {g}},x,f)$ is the unique simple quotient of the universal W-algebra, denoted by $W^k({\mathfrak {g}},x,f),$ which is freely strongly generated by elements labeled by a basis of the centralizer of f in ${\mathfrak {g}}$ [KW04].

In the paper [KMP23] we focused on minimal data $({\mathfrak {g}},x,f)$, i.e. the cases when f is an even minimal nilpotent element. In this case x and f are contained in a (unique) subalgebra ${\mathfrak {s}}$, isomorphic to sl(2). The Virasoro, Neveu–Schwarz and $N=2$ algebras cover all minimal W-algebras (associated with ${\mathfrak {g}}=sl(2), spo(2|1),sl(2|1)$ respectively), such that the 0-eigenspace ${\mathfrak {g}}_0$ of $ad\,x$ is abelian: their unitarity, and that of their irreducible highest weight modules, has been studied in the papers quoted above. In [KMP23] we dealt with the cases when ${\mathfrak {g}}_0$ is not abelian, and we found all non-critical $k\in {\mathbb {C}}$ for which the simple minimal W-algebras $W_k({\mathfrak {g}},x,f)$, denoted henceforth by $W_k^{\min }({\mathfrak {g}})$, are unitary, along with the (partly conjectural) classification of unitary highest weight modules over their universal covers $W^k_{\min }({\mathfrak {g}})$. We call this set of values of k the unitarity range. Recall from [AKMPP18] that a level k is said to be collapsing if $W_k^{\min }({\mathfrak {g}})$ is the simple affine vertex algebra attached to the centralizer ${\mathfrak {g}}^\natural$ in ${\mathfrak {g}}_0$ of ${\mathfrak {s}}$. If k is collapsing, then the unitarity of $W_k^{\min }({\mathfrak {g}})$ reduces to the unitarity of an affine vertex algebras, which is well-understood. Requiring that the unitary range contains non-collapsing levels imposes severe restrictions on the Lie superalgebra ${\mathfrak {g}}$, namely ${\mathfrak {g}}$ must be one of the algebras listed in Table 1. It turns out that the unitarity range (described explicitly in Table 1) is precisely the set of levels for which the affine vertex subalgebra of $W_k^{\min }({\mathfrak {g}})$ generated by ${\mathfrak {g}}^\natural$ is integrable when viewed as a $\widehat{{\mathfrak {g}}}^\natural$-module.

The analysis of the unitarity of $W_k^{\min }({\mathfrak {g}})$ developed in [KMP23] led to the more general problem of classifying unitary highest weight modules over $W_k^{\min }({\mathfrak {g}})$; these results, regarding modules in the Neveu–Schwarz sector, are summarized in Sect. 2.

In particular, we gave a mathematically rigorous proofs of the classification for the $N=3$ and $N=4$ superconformal algebras, due to Miki [M90] and Eguchi-Taormina [ET87, ET88-1, ET88-2], respectively.

It was conjectured in [KMP23] that actually any unitary highest weight $W^k_{\min }({\mathfrak {g}})$-module descends to $W_k^{\min }({\mathfrak {g}})$. In the present paper we prove this conjecture (Theorem 5.1 and Corollary 5.2). The proof is based on the classification of irreducible highest weight modules over the simple affine vertex algebras $V_k({\mathfrak {g}})$ for these k (Theorem 3.1). We also find generators for the maximal ideal $I^k$ of $W^k_{\min }({\mathfrak {g}})$, so that $W_k^{\min }({\mathfrak {g}})=W^k_{\min }({\mathfrak {g}})/I^k$ (Theorem 4.1). Finally, for all non-critical $k\in {\mathbb {C}}$ in the unitarity range, we classify all irreducible highest weight modules over $W_k^{\min }({\mathfrak {g}})$ (Theorem 6.1), using the Zhu algebra method. It follows from this classification that all these vertex algebras have infinitely many irreducible modules, and therefore are not rational, unless k is a collapsing level. We also prove that for $W_k^{\min }({\mathfrak {g}})$ all irreducible positive energy modules are highest weight modules (Theorem 6.9), whereas this statement does not hold for $V_k({\mathfrak {g}})$ (Remark  3.3). In the final section, we show examples of positive energy modules over $V_k({\mathfrak {g}})$ which are not highest weight and we outline an approach to more general representations of $W_k^{\min }({\mathfrak {g}})$, showing the existence of non-positive energy modules.

In the present paper we keep notation and terminology of [KMP23]; in particular, ${\mathbb {N}}$ and $\mathbb {Z}_+$ denote the sets of positive and non-negative integer numbers, respectively. The base field is ${\mathbb {C}}$.

2 Setup

2.1 Minimal W-algebras

Let ${\mathfrak {g}}={\mathfrak {g}}_{{\bar{0}}}\oplus {\mathfrak {g}}_{{\bar{1}}}$ be a simple finite-dimensional Lie superalgebra over ${\mathbb {C}}$ with a reductive even part ${\mathfrak {g}}_{{\bar{0}}}$ and a non-degenerate even supersymmetric invariant bilinear form (.|.). They are classified in [K77]. Let ${\mathfrak {s}}=\{e,x,f\}\subset {\mathfrak {g}}_{{\bar{0}}}$ be a minimal sl(2)-triple, i.e. $[x,e]=e, [x,f]=-f, [e,f]=x$ and the $ad\,x$ gradation of ${\mathfrak {g}}$ has the form

$$\begin{aligned} {\mathfrak {g}}=\bigoplus _{j\in \tfrac{1}{2}\mathbb {Z}}{\mathfrak {g}}_j,\quad \text {where }{\mathfrak {g}}_j=0\text { for } |j|>1, {\mathfrak {g}}_1={\mathbb {C}}e,\,{\mathfrak {g}}_{-1}={\mathbb {C}}f. \end{aligned}$$

(2.1)

Denote by ${\mathfrak {g}}^\natural$ the centralizer of ${\mathfrak {s}}$ in ${\mathfrak {g}}_0$. Let $W^k_{\min }({\mathfrak {g}})$ be the associated to ${\mathfrak {s}}$ and $k\in {\mathbb {C}}$ universal quantum affine W-algebra [KW04]. This vertex algebra is strongly generated by fields $J^{\{a\}},\,a\in {\mathfrak {g}}^{\natural },G^{\{v\}},\,v\in {\mathfrak {g}}_{-1/2}, L$ of respective conformal weight 1, 3/2, 2 and $\lambda$-brackets explicitly given in [KW04]. We normalize the bilinear form (.|.) by the condition $(x|x)=\tfrac{1}{2}$, and denote by $h^\vee$ the corresponding dual Coxeter number. We shall assume throughout the paper that $k\ne -h^\vee$, i.e. k is non-critical. Then the vertex algebra $W^k_{\min }({\mathfrak {g}})$ has a unique maximal ideal $I^k$, and we let $W_k^{\min }({\mathfrak {g}})=W^k_{\min }({\mathfrak {g}})/I^k$ be the corresponding simple vertex algebra.

2.2 Unitary conformal vertex algebras

Let V be a conformal vertex algebra with conformal vector $L=\sum _{n\in {\mathbb {Z}}} L_nz^{-n-2}$. Let $\phi$ be a conjugate linear involution of V. A Hermitian form $H(\,. \,\,, \,.\, )$ on V is called $\phi$–invariant if, for all $a\in V$, one has [KMP22]

$$\begin{aligned} H(v,Y(a,z)u)=H(Y(A(z)a,z^{-1})v,u),\quad u,v\in V. \end{aligned}$$

(2.2)

Here the linear map $A(z):V\rightarrow V((z))$ is defined by

$$\begin{aligned} A(z)=e^{zL_1}z^{-2L_0}g, \end{aligned}$$

(2.3)

where

$$\begin{aligned} g(a)=(-1)^{L_0+2L_0^2}\phi (a),\quad a\in V. \end{aligned}$$

(2.4)

We say that V is unitary if there is a conjugate linear involution $\phi$ of V and a $\phi$-invariant Hermitian form on V which is positive definite.

2.3 Unitary minimal W-algebras and unitary highest weight modules

In classifying the unitary minimal W-algebras $W_k^{\min }({\mathfrak {g}})$, one first considers the levels k when $W_k^{\min }({\mathfrak {g}})$ is either ${\mathbb {C}}$ or an affine vertex algebra. Such levels are called collapsing levels.

The triples $({\mathfrak {g}},{\mathfrak {s}},k)$ such that $W_k^{\min }({\mathfrak {g}})={\mathbb {C}}$ are described in Proposition 3.4 of [AKMPP18]. Corollary 7.12 of [KMP22] provides the list of triples $({\mathfrak {g}},{\mathfrak {s}},k)$ such that $W_k^{\min }({\mathfrak {g}})$ is an affine unitary vertex algebra.

Turning to the non collapsing levels, it is proven in [KMP22, Proposition 7.9] that, if $W^{\min }_k({\mathfrak {g}})$ is unitary and k is not a collapsing level, then the parity of ${\mathfrak {g}}$ is compatible with the $ad\,x$–gradation, i.e. the parity of the whole subspace ${\mathfrak {g}}_j$ is $2j\mod 2$. It follows from [KRW03, KW04] that for each basic simple Lie superalgebra ${\mathfrak {g}}$ there is at most one minimal datum $({\mathfrak {g}},{\mathfrak {s}})$ that is compatible with parity, and the complete list of the ${\mathfrak {g}}$ which admit such a datum is as follows:

$$\begin{aligned} \begin{aligned} sl(2|m)\text { for }m\ge 3,\quad psl(2|2),\quad spo(2|m)\text { for }m\ge 0,\\ osp(4|m)\text { for }m>2\text { even},\quad D(2,1;a),\quad F(4),\quad G(3). \end{aligned} \end{aligned}$$

(2.5)

Remark 2.1

Recall from [K77] that in the case of D(2, 1; a),  the parameter a ranges over ${\mathbb {C}}\setminus \{0,-1\}$ (if $a=0,-1$ the superalgebra is not simple); moreover the symmetric group $\mathcal S_3$ generated by the tranformations $a\mapsto -1-a, a\mapsto 1/a$ induces Lie superalgebra isomorphisms.

The even part ${\mathfrak {g}}_{{\bar{0}}}$ of ${\mathfrak {g}}$ in these cases is ${\mathfrak {g}}^\natural \oplus \mathfrak s$ with ${\mathfrak {g}}^\natural$ a reductive Lie algebra. In Propositions 7.1. and 7.2 of [KMP23] it is proven that the conjugate linear involutions of $W_k^{\min }({\mathfrak {g}})$ that fix the Virasoro vector L are in one-to-one correspondence with the conjugate linear involutions $\phi$ of ${\mathfrak {g}}$ that fix pointwise the triple $\{e,x,f\}$. It is easy to see that, in order to have $W_k^{\min }({\mathfrak {g}})$ unitary, $\phi _{|{\mathfrak {g}}^\natural }$ must be the conjugation corresponding to a compact real form. Such a conjugate linear involution is called an almost compact involution and Proposition 3.2 of [KMP23] shows that an almost compact involution $\phi _{ac}$ exists in all the cases listed in (2.5). Since the cases ${\mathfrak {g}}=spo(2|m),\,m=0,1,$ and 2, correspond to the well understood cases of the simple Virasoro, Neveu–Schwarz and $N=2$ vertex algebras, we exclude these ${\mathfrak {g}}$ from consideration. For the other cases, Theorem 1.4 of [KMP23] lists all the pairs $({\mathfrak {g}},k)$ for which the $\phi _{ac}$-invariant hermitian form on $W_k^{\min }({\mathfrak {g}})$ is positive definite, making $W_k^{\min }({\mathfrak {g}})$ a unitary vertex algebra. We say that k belongs to the unitary range if k is not collapsing and $W_k^{\min }({\mathfrak {g}})$ is a unitary vertex algebra.

We have [KMP23]

$$\begin{aligned} {\mathfrak {g}}^\natural =\bigoplus _{i\in S}{\mathfrak {g}}^\natural _i\text { and } {\mathfrak {g}}_0={\mathfrak {g}}^\natural \oplus {\mathbb {C}}x,\end{aligned}$$

(2.6)

where ${\mathfrak {g}}^\natural _i$ are simple summands of ${\mathfrak {g}}^\natural$, and either $S=\{1\}$ or $S=\{1,2\}$; the latter happens only for $D(2,1;\tfrac{m}{n})$. Let ${\mathfrak {h}}^\natural$ be a Cartan subalgebra of ${\mathfrak {g}}^\natural$, then ${\mathfrak {h}}={\mathfrak {h}}^\natural +{\mathbb {C}}x$ is a Cartan subalgebra of ${\mathfrak {g}}_0$ and of ${\mathfrak {g}}$. Let $\theta \in {\mathfrak {h}}^*$ be the root of e, so that $(\theta |\theta )=2$, and denote by $\theta _i\in ({\mathfrak {h}}^\natural )^*$ the highest root of ${\mathfrak {g}}^\natural _i$. Denote by $M_i(k)$ the levels of the affine Lie algebras ${\widehat{{\mathfrak {g}}}}_i^\natural$ in $W^k_{\min }({\mathfrak {g}})$. Then we have for $i\in S$ [KMP23]

$$\begin{aligned} M_i(k)&= \frac{2k}{(\theta _i|\theta _i)}+\chi _i, \end{aligned}$$

(2.7)

$$\begin{aligned} \chi _i&=-\xi (\theta _i^\vee ), \end{aligned}$$

(2.8)

where $\xi \in ({\mathfrak {h}}^\natural )^*$ is the highest weight of the ${\mathfrak {g}}^\natural$-module ${\mathfrak {g}}_{-1/2}$ (this module is irreducible, except for ${\mathfrak {g}}=psl(2|2)$, when its two irreducible components turn out to be isomorphic). In [KMP23] it is shown that $\chi _i=-2$ when ${\mathfrak {g}}= spo(2|3)$, and $\chi _i=-1$ otherwise.

In Table 1 we describe the unitary range in all cases, along with the subalgebras ${\mathfrak {g}}^\natural$, the numbers $M_i(k)$, and the dual Coxeter numbers $h^\vee$.

Table 1 Unitarity range, ${\mathfrak {g}}^\natural , M_i(k)$, and $h^\vee$

We also make a specific choice for a set of simple roots $\Pi$ of ${\mathfrak {g}}$ in each case. In Table 2 we list our choice of $\Pi =\{\alpha _1,\ldots \}$ of ${\mathfrak {g}}$, ordered from left to right, as well as the invariant bilinear form (.|.) on ${\mathfrak {h}}^*$, and the root $\theta$.

Table 2 Simple roots $\Pi$, invariant form (.|.), and the highest root $\theta$ of ${\mathfrak {g}}$

Let $\Delta ^\natural$ be the set of roots of $({\mathfrak {g}}^\natural ,{\mathfrak {h}}^\natural )$. We made our choice of $\Pi$ so that $\Pi ^\natural =\Pi \cap \Delta ^\natural$ is a set of simple roots for ${\mathfrak {g}}^\natural$. Write ${\mathfrak {g}}^\natural ={\mathfrak {n}}^\natural _-\oplus {\mathfrak {h}}^\natural \oplus {\mathfrak {n}}^\natural _+$ for the triangular decomposition of ${\mathfrak {g}}^\natural$ corresponding to choosing $\Pi ^\natural$ as a set of simple roots. Note that $\alpha _1$ is an isotropic root, and that we have

$$\begin{aligned} (\alpha _1)_{|{\mathfrak {h}}^\natural }=-\xi . \end{aligned}$$

(2.9)

We parametrize the highest weight modules for $W^k_{\min }({\mathfrak {g}})$ following Section 7 of [KW04]. For $\nu \in ({\mathfrak {h}}^\natural )^*$ and $\ell _0\in {\mathbb {C}}$, let $L^W(\nu ,\ell _0)$ denote the irreducible highest weight $W^k_{\min }({\mathfrak {g}})$–module with highest weight $(\nu ,\ell _0)$ and highest weight vector $v_{\nu ,\ell _0}$. This means that one has

$$\begin{aligned}&J^{\{h\}}_0v_{\nu ,\ell _0}=\nu (h)v_{\nu , \ell _0} \text { for } h\in {\mathfrak {h}}^\natural ,\quad L_0v_{\nu ,\ell _0}=\ell _0v_{\nu ,\ell _0},\\&J^{\{u\}}_nv_{\nu ,\ell _0}=G^{\{v\}}_nv_{\nu ,\ell _0}=L_nv_{\nu ,\ell _0}=0\text { for }n>0,\,u\in {\mathfrak {g}}^\natural , \ v\in {\mathfrak {g}}_{-1/2}, \\&J^{\{u\}}_0v_{\nu ,\ell _0}=0\text { for } u\in {\mathfrak {n}}^\natural _{+}. \end{aligned}$$

Let $P^+\subset ({\mathfrak {h}}^\natural )^*$ be the set of dominant integral weights for ${\mathfrak {g}}^\natural$ and let

$$\begin{aligned} P^{+}_k=\left\{ \nu \in P^+\mid \nu (\theta ^\vee _i)\le M_i(k)\text { for all }i\ge 1\right\} . \end{aligned}$$

(2.10)

Definition 2.2

An element $\nu \in P^+_k$ is called an extremal weight if $\nu +\xi$ doesn’t lie in $P^+_k$. Equivalently, $\nu$ is extremal if

$$\begin{aligned} \nu (\theta _i^\vee )>M_i(k)+\chi _i\text { for some }i\in S. \end{aligned}$$

(2.11)

For $\nu \in P^+$, introduce the following number [KMP23]

$$\begin{aligned} A(k,\nu )=\frac{(\nu |\nu +2\rho ^\natural )}{2(k+h^\vee )}+\frac{(\xi |\nu )}{k+h^\vee }((\xi |\nu )-k-1), \end{aligned}$$

(2.12)

where $2\rho ^\natural$ is the sum of positive roots of ${\mathfrak {g}}^\natural$. The main result of [KMP23] is the following theorem.

Theorem 2.3

Let $L^W(\nu ,\ell _0)$ be an irreducible highest weight $W^k_{\min }({\mathfrak {g}})$–module for ${\mathfrak {g}}= psl(2|2),$ spo(2|m) with $m\ge 3, \, D(2,1;a),\, F(4)$ or G(3).

• (1) This module can be unitary only if the following conditions hold:

  • (a) $M_i(k)$ are non-negative integers,

  • (b) $\nu (\theta _i^\vee ) \le M_i(k)$ for all i,

  • (c)
    $$\begin{aligned} \ell _0\ge A(k,\nu ), \end{aligned}$$

    (2.13)

    and equality holds in (2.13) if $\nu (\theta ^\vee _i)>M_i(k)+\chi _i$ for $i=1$ or 2.

• (2) This module is unitary if the following conditions hold:

  • (a) $M_i(k)+\chi _i\in {\mathbb {Z}}_+$ for all i,

  • (b) $\nu (\theta ^\vee _i) \le M_i(k)+\chi _i$ for all i (i.e. $\nu$ is not extremal),

  • (c) inequality (2.13) holds.

Applying this theorem to $L^W(0,0)=W_k^{\min }({\mathfrak {g}})$ one recovers the unitarity range displayed in Table 1.

2.4 A technical result

Let $\Delta$ be the set of roots of ${\mathfrak {g}}$ and let ${\widehat{\Pi }}=\{\alpha _0=\delta -\theta ,\alpha _1,\ldots \}$ be a set of simple roots of the affine Lie superalgebra $\widehat{{\mathfrak {g}}}$. For an isotropic root $\beta \in {\widehat{\Pi }}$, we denote by $r_{\beta }({\widehat{\Pi }})$ the set of simple roots in the set ${\widehat{\Delta }}$ of roots of $\widehat{{\mathfrak {g}}}$ obtained by the corresponding odd reflection. We denote by $x_\alpha$ a root vector of $\widehat{{\mathfrak {g}}}$, attached to $\alpha \in {\widehat{\Delta }}$, and by w. the shifted action of $\widehat{W}$: $w.\lambda = w(\lambda +{\widehat{\rho }})-{\widehat{\rho }}$.

Lemma 2.4

[KMP23, Lemma 11.3] Let $\widehat{\Pi }'$ be a set of simple roots for ${\widehat{\Delta }}$. Let M be a $\widehat{{\mathfrak {g}}}$–module and assume that $m\in M$ is a singular vector with respect to ${\widehat{\Pi }}'$. If $\alpha \in {\widehat{\Pi }}'$ is an isotropic root and $x_{-\alpha }m\ne 0$, then $x_{-\alpha }m$ is a singular vector with respect to $r_{\alpha }({\widehat{\Pi }}')$.

3 Classification of Irreducible Highest Weight Representations of $V_k({\mathfrak {g}})$ in the Unitary Range

Denote by $V^k({\mathfrak {g}})$ and $V_k({\mathfrak {g}})$ the universal and simple affine vertex algebras of level k associated to ${\mathfrak {g}}$. Denote by $L(\lambda )$ the irreducible highest weight $V^k({\mathfrak {g}})$-module of highest weight $\lambda$. For $h\in {\mathbb {C}}$ and $\nu \in ({\mathfrak {h}}^\natural )^*$, set

$$\begin{aligned} \widehat{\nu }_h=k\Lambda _0+h\theta +\nu . \end{aligned}$$

(3.1)

Note that every highest weight module for $V^k({\mathfrak {g}})$ has highest weight ${\widehat{\nu }}_h$ for some $\nu \in ({\mathfrak {h}}^\natural )^*$ and $h\in {\mathbb {C}}$.

Theorem 3.1

Let k be in the unitary range. Then, up to isomorphism, the irreducible highest weight $V_k({\mathfrak {g}})$-modules are as follows:

• (1) $L({\widehat{\nu }}_h)$ with $\nu \in P^+_k$ non-extremal and $h\in {\mathbb {C}}$ arbitrary;

• (2) $L({\widehat{\nu }}_{h})$ with $\nu$ extremal and h from the set $E_{k,\nu }=\{(\xi |\nu ),k+1-(\xi |\nu )\}$.

Lemma 3.2

Let k be in the unitary range. Then $V_k({\mathfrak {g}})$ is integrable for $\widehat{{\mathfrak {g}}}^\natural$.

Proof of Lemma 3.2

It is enough to check that $(x_{\theta _i})_{(-1)}^N\textbf{1}=0$ for some $N\in {\mathbb {N}}$ and all i. Using Table 2 one easily checks that $\theta -\alpha _1$ is a root of ${\mathfrak {g}}$. Since $(x_{\alpha _1})_{(0)}\textbf{1}=0$, $\textbf{1}$ is a singular vector in $V^k({\mathfrak {g}})$ also for $r_{\alpha _1}({\widehat{\Pi }})$. Since $(x_{\theta -\alpha _1})_{(-1)}\textbf{1}$ is nonzero in $V^k({\mathfrak {g}})$, it follows from Lemma 2.4 that $(x_{\theta -\alpha _1})_{(-1)}\textbf{1}$ is a singular vector in $V^k({\mathfrak {g}})$ for ${\widehat{\Pi }}'=r_{\alpha _0+\alpha _1}r_{\alpha _1}({\widehat{\Pi }})$. Note that, since $k\ne 0$,

$$\begin{aligned}(x_{-\theta +\alpha _1})_{(1)} (x_{\theta -\alpha _1})_{(-1)}\textbf{1}&{=} [(x_{-\theta +\alpha _1})_{(1)}, (x_{\theta -\alpha _1})_{(-1)}]\textbf{1}{=}((h_{\theta -\alpha _1})_{(0)} {+}k(x_{-\theta +\alpha _1}|x_{\theta -\alpha _1}))\textbf{1}\\ {}&=k(x_{-\theta +\alpha _1}|x_{\theta -\alpha _1})\textbf{1}= \gamma \textbf{1},\quad \gamma \ne 0. \end{aligned}$$

In particular, $(x_{\theta -\alpha _1})_{(-1)}\textbf{1}$ generates $V^k({\mathfrak {g}})$.

Let $\Lambda '=k\Lambda _0-\alpha _0-\alpha _1=k\Lambda _0-\delta +\theta -\alpha _1$ be the weight of $(x_{\theta -\alpha _1})_{(-1)}\textbf{1}$. By a case-wise verification one shows that the roots $\eta _i:=\delta -\theta _i$ are in ${\widehat{\Pi }}'$ and, by (2.7) and (2.8),

$$\begin{aligned} (\Lambda '|\eta _i^\vee )=\frac{2}{(\theta _i|\theta _i)}k+(\alpha _1|\theta _i^\vee )=M_i(k)-\chi _i-(\xi |\theta _i^\vee ) =M_i(k). \end{aligned}$$

Hence the vector $(x_{\theta _i})_{(-1)}^{M_i(k)+1}(x_{\theta -\alpha _1})_{(-1)}\textbf{1}$ is singular in $V^k({\mathfrak {g}})$ for ${\widehat{\Pi }}'$. Since the vector $(x_{\theta -\alpha _1})_{(-1)}\textbf{1}$ generates $V^k({\mathfrak {g}})$, it follows that $V_k({\mathfrak {g}})$ is ${\widehat{{\mathfrak {g}}}}^\natural$–integrable. $\square$

Proof of Theorem 3.1

In [GK15, Sections 4 and 6] the characters of highest weight $\widehat{{\mathfrak {g}}}$-modules with highest weight $k\Lambda _0$ have been computed using only their integrability with respect to $\widehat{{\mathfrak {g}}}^\natural$, which implies that such modules are irreducible. It was deduced from this in [GS18, Theorem 5.3.1] that, if $V_k({\mathfrak {g}})$ is integrable as a $\widehat{{\mathfrak {g}}}^\natural$-module, then the $V^k({\mathfrak {g}})$–modules, which are integrable over $\widehat{{\mathfrak {g}}}^\natural$, descend to $V_k({\mathfrak {g}})$.

By Lemma 3.2, we are left with proving that the modules listed in (1), (2) are $\widehat{{\mathfrak {g}}}^\natural$–integrable. Let v be a highest weight vector for $L({\widehat{\nu }}_h)$.

Case (1) $\nu$ is not extremal, $h\in {\mathbb {C}}$. If $(\widehat{\nu }_{h}|\alpha _1)\ne 0$ and $({\widehat{\nu }}_{ h}-\alpha _1|\alpha _0+\alpha _1)\ne 0$, then $(x_{\theta -\alpha _1})_{(-1)}(x_{-\alpha _1})_{(0)}v$ is a singular vector with respect to the set of simple roots $r_{\alpha _0+\alpha _1}r_{\alpha _1}({\widehat{\Pi }})$. Moreover, by (2.7), (2.8), and (2.9) we have for $i\in S$

$$\begin{aligned} m_i:= ({\widehat{\nu }}_{ h}-2\alpha _1-\alpha _0|\eta _i^\vee )&=M_i(k)-\chi _i+2(\alpha _1|\theta _i^\vee )-(\nu |\theta _i^\vee )\nonumber \\&=M_i(k)-\chi _i-2(\xi |\theta _i^\vee )-(\nu |\theta _i^\vee )\nonumber \\&=M_i(k)+\chi _i-(\nu |\theta _i^\vee )\in \mathbb {Z}_+ \end{aligned}$$

(3.2)

by (2.11). Since $\eta _i\in r_{\alpha _0+\alpha _1}r_{\alpha _1}({\widehat{\Pi }})$, we see that, in $L({\widehat{\nu }}_{ h})$,

$$\begin{aligned} (x_{\theta _i})_{(-1)}^{m_i+1}(x_{\theta -\alpha _1})_{(-1)}(x_{-\alpha _1})_{(0)}v=0, \end{aligned}$$

hence $L({\widehat{\nu }}_{ h})$ is $\widehat{{\mathfrak {g}}}^\natural$–integrable.

If $({\widehat{\nu }}_{ h}|\alpha _1)= 0$, since $({\widehat{\nu }}_{ h}|\alpha _0+\alpha _1)=({\widehat{\nu }}_{ h}|\alpha _0)\ne 0$, we have that $(x_{\theta -\alpha _1})_{(-1)}v$ is a singular vector for $r_{\alpha _0+\alpha _1}r_{\alpha _1}({\widehat{\Pi }})$. Moreover,

$$\begin{aligned} ({\widehat{\nu }}_{ h}-\alpha _1-\alpha _0|\eta _i^\vee )=M_i(k)-(\nu |\theta _i^\vee )\in \mathbb {Z}_+ \end{aligned}$$

(3.3)

since $\nu \in P^+_k$. We can therefore conclude as above that $L({\widehat{\nu }}_{ h})$ is $\widehat{{\mathfrak {g}}}^\natural$–integrable.

Finally, if $({\widehat{\nu }}_{ h}|\alpha _1)\ne 0$ and $({\widehat{\nu }}_{ h}-\alpha _1|\alpha _0+\alpha _1)= 0$, then $(x_{-\alpha _1})_{(0)}v$ is a singular vector for $r_{\alpha _0+\alpha _1}r_{\alpha _1}({\widehat{\Pi }})$. Moreover,

$$\begin{aligned} ({\widehat{\nu }}_{ h}-\alpha _1|\eta _i^\vee )=M_i(k)-(\nu |\theta _i^\vee )\in \mathbb {Z}_+, \end{aligned}$$

(3.4)

and we conclude as in the previous case.

Case (2) $\nu$ is extremal, $h\in E_{k,\nu }$. If $h=(\xi |\nu )$, then $({\widehat{\nu }}_{ h}|\alpha _1)=0$. We can then conclude as in the non-extremal case. If $h=k+1-(\xi |\nu )$, we may assume that $({\widehat{\nu }}_{h}|\alpha _1)\ne 0$. Then, since $(\theta |\alpha _1)=1$, we have

$$\begin{aligned} ({\widehat{\nu }}_{ h}-\alpha _1|\alpha _0+\alpha _1)&=(k\Lambda _0+(k+1-(\xi |\nu ))\theta +\nu -\alpha _1|\alpha _0+\alpha _1)\\&=k-2(k+1-(\xi |\nu ))+1+(k+1-(\xi |\nu ))+(\nu |\alpha _1)=0. \end{aligned}$$

We can therefore conclude as in the non-extremal case.

We are left with proving that any irreducible highest weight module for $V_k({\mathfrak {g}})$ is of the form (1) or (2). Let $L({\widehat{\nu }}_h)$ be an irreducible highest weight $V_k({\mathfrak {g}})$-module. We prove that, necessarily, $\nu \in P^+_k$. Indeed, the action of ${\mathfrak {g}}^\natural$ on a highest weight vector v should be locally finite, so that $\nu \in P^+$. If $h\notin E_{k,\nu }$ then $({\widehat{\nu }}_{h}|\alpha _1)\ne 0$ and $(\widehat{\nu }_{ h}-\alpha _1|\alpha _0+\alpha _1)\ne 0$. It follows that $(x_{\theta -\alpha _1})_{(-1)}(x_{-\alpha _1})_{(0)}v$ is a highest weight vector with respect to the set of simple roots $r_{\alpha _0+\alpha _1}r_{\alpha _1}({\widehat{\Pi }})$ of highest weight $\Lambda '={\widehat{\nu }}_h-\alpha _0-2\alpha _1$. If $L({\widehat{\nu }}_h)$ is integrable with respect to $\widehat{{\mathfrak {g}}}^\natural$, the computation done in (3.2) shows that $m_i$ should be a non-negative integer, hence

$$\begin{aligned} (\nu |\theta _i^\vee )\le M_i(k)+\chi _i\le M_i(k). \end{aligned}$$

It follows that $\nu \in P^+_k$ and it is not extremal, i.e. $L({\widehat{\nu }}_h)$ is of type (1).

If $h=(\xi |\nu )$ (resp. $h=k+1-(\xi |\nu )$), then $({\widehat{\nu }}_{ h}|\alpha _1)= 0$ (resp. $({\widehat{\nu }}_{ h}|\alpha _0+\alpha _1)= 0$ ) and as in (3.3) (resp. (3.4)) we get that $\nu \in P^+_k$. In particular, $L({\widehat{\nu }}_h)$ is of type (1) if $\nu$ is not extremal and of type (2) if $\nu$ is extremal. $\square$

Remark 3.3

All modules listed in Theorem 3.1 are of positive energy (the definition is recalled in Sect. 6), but there might exist positive energy $V_k({\mathfrak {g}})$–modules outside of this list. In § 7.1 we present arguments for this claim in the case ${\mathfrak {g}}= spo(2 \vert 3)$, $k =- m/4$, and $m\ge 4$ even.

One of the referees pointed out that in [GS18, 5.6.4] the authors give an example of an irreducible positive energy module which is not a highest weight module. Similar examples, with ${\mathfrak {g}}$ as in Table 1 and k in the unitarity range, are given in [GS18, 5.6.6] (note that in [GS18] a normalization of the bilinear form different from ours is used).

Let us explain what happens in our situation. If M is a positive energy module for a vertex algebra V, we set $M_{top}=Zhu(M)$, where $Zhu( - )$ is the Zhu functor between positive energy modules and modules for the Zhu algebra A(V). We start with the $V_k({\mathfrak {g}})$–module $L({\widehat{\nu }}_h)$ with $\nu \in P^+_k$ non-extremal and $h\in {\mathbb {C}}$ arbitrary, from Theorem 3.1. Then $L({\widehat{\nu }}_h)_{top}$ contains an irreducible ${\mathfrak {g}}_{\bar{0}}$–submodule $E= U({\mathfrak {g}}_{{\bar{0}}}). v$, where v is the highest weight vector of $L({\widehat{\nu }}_h)$. Next we consider the Kac module $E_1$, which is a suitable quotient of $Ind_{{\mathfrak {g}}_{{\bar{0}}}} ^{{\mathfrak {g}}} E$. As mentioned in [GS18], if all $M_i(k)$ are large enough, we get that $E_1$ is a module for Zhu’s algebra $A(V_k({\mathfrak {g}}))$. Using Zhu’s functor we conclude the following:

• there exists a positive energy $V_k({\mathfrak {g}})$–module $\widetilde{L}({\widehat{\nu }}_h)$ such that ${\widetilde{L}}({\widehat{\nu }}_h)_{top} = E_1$.

• the module ${\widetilde{L}}({\widehat{\nu }}_h)$ is indecomposable and has an irreducible subquotient isomorphic to $L({\widehat{\nu }}_h)$.

Hence non-highest weight positive energy modules for $V_k({\mathfrak {g}})$ exist. But still these modules are weight modules with finite-dimensional weight spaces.

Let us mention one important consequence. Take $h \in {{\mathbb {C}}}$ such that E is finite-dimensional, irreducible module and assume that $M_i(k)$ are large enough. Then $E_1$ is an indecomposable finite-dimensional module for Zhu’s algebra $A(V_k({\mathfrak {g}}))$, and therefore the category $KL_k$ [AMP22] is not semisimple. In the case ${\mathfrak {g}}= spo(2 \vert 3)$, one can show that the category $KL_{-m/4}({\mathfrak {g}})$ is not semisimple for $m \ge 4$.

4 Explicit Description of the Maximal Ideal of $W^k_{\min }({\mathfrak {g}})$

Let, as before, $\Pi =\{\alpha _1,\ldots \}$ be the set of simple roots for ${\mathfrak {g}}$ given in Table 2, and $I^k$ be the maximal ideal of $W^k_{\min }({\mathfrak {g}})$. Also denote by $J^k$ the maximal ideal of $V^k({\mathfrak {g}})$. Set

$$\begin{aligned} v_i= (J^{\{x_{\theta _i}\}}_{(-1)})^{M_i(k)+1}\textbf{1},\quad i \in S. \end{aligned}$$

(4.1)

If an irreducible highest weight $W^k_{\min }({\mathfrak {g}})$–module $L^W(\nu ,\ell _0)$ is unitary, then, restricted to the affine subalgebra $V^{\beta _k}({\mathfrak {g}}^\natural )$ (see [KMP23, (7.4), (7.5)] for the definition of this subalgebra), it is unitary, hence a direct sum of irreducible integrable highest weight $\widehat{\mathfrak {g}}^\natural$–modules of levels $M_i(k),\ i\in S$. But it is well-known [FZ92, KWang92] that all these modules descend to the simple affine vertex algebra $V_{\beta _k}({\mathfrak {g}}^\natural )$, and are annihilated by the elements $v_i$. In particular, applying this argument to $W_k^{\min }({\mathfrak {g}})=L^W(0,0)$, we deduce that

$$\begin{aligned} v_i\in I^k. \end{aligned}$$

(4.2)

Theorem 4.1

Let k be in the unitary range. The maximal ideal $I^k$ is generated by the singular vector

$$\begin{aligned} {{\widetilde{v}}}_1= (J^{\{x_{\theta _1}\}}_{(-1)} )^{M_1(k)-1} G^{\{x_{-\alpha _1}\}}_{(-1)}\textbf{1}\end{aligned}$$

(4.3)

if ${\mathfrak {g}}= spo(2|3)$ and by the singular vectors $v_i,\ i\in S$ (cf. (4.1)) in the other cases.

We split the proof according to whether ${\mathfrak {g}}\ne spo(2|3)$ or ${\mathfrak {g}}= spo(2|3)$.

4.1 ${\mathfrak {g}}\ne spo(2|3)$

In this case $\chi _i=-1,\, i\in S$.

Lemma 4.2

Set $\eta _i=\delta -\theta _i,\, i\in S$. One has

$$\begin{aligned}&\dim V^k({\mathfrak {g}})_{k\Lambda _0-j\eta _i}=1\text { for all }j\ge 0, \end{aligned}$$

(4.4)

$$\begin{aligned}&\dim V_k({\mathfrak {g}})_{k\Lambda _0-j\eta _i}=1 \text { if and only if } j\le s, \end{aligned}$$

(4.5)

where

$$\begin{aligned} s=(k\Lambda _0|\eta _i^\vee ). \end{aligned}$$

(4.6)

In particular, for each $M\in {\mathbb {Z}}_{\ge 0}$, the vector $(x_{-\theta +\alpha _1})_{(1)}(x_{\theta _i})_{(-1)}^{M}(x_{\theta -\alpha _1})_{(-1)}\textbf{1}$ is a multiple of $(x_{\theta _i})_{(-1)}^{M}\textbf{1}$.

Proof

Any basic Lie superalgebra ${\mathfrak {g}}$ different form spo(2|3) admits a set of simple roots $\Pi ''$ with the property that $\theta _i$ is the maximal root, hence the root $\eta _i=\delta -\theta _i$ lies in ${\widehat{\Pi }}''$. Since $V^k({\mathfrak {g}})$ is the vacuum module, $x_\alpha \textbf{1}=0$ for any root $\alpha \in \Delta$, in particular $x_\alpha \textbf{1}=0$ for any $\alpha \in \Pi ''$. Since $(x_{-\theta _i})_{(1)}\textbf{1}=0$, $\textbf{1}$ is a singular vector also for ${\widehat{\Pi }}''$. Since $\eta _i$ is simple in ${\widehat{\Pi }}''$, it is clear that $V^k({\mathfrak {g}})_{k\Lambda _0-j\eta _i}\subset {\mathbb {C}}(x_{\theta _i})_{(-1)}^j\textbf{1}$, and indeed $\dim V^k({\mathfrak {g}})_{k\Lambda _0-j\eta _i}=1$, since $(x_{\theta _i})_{(-1)}^j\textbf{1}\ne 0$ in $V^k({\mathfrak {g}})$. This proves (4.4). By Lemma 3.2, $V_k({\mathfrak {g}})$ is integrable for $\widehat{{\mathfrak {g}}}^\natural$, in particular, $(x_{\theta _i})_{(-1)}^{j}\textbf{1}=0$ if and only if $j>(k\Lambda _0|\eta _i^\vee )$, hence (4.5) holds. The final claim follows from (4.4) by comparing weights. $\square$

Introduce the following vectors in $V^k({\mathfrak {g}})$, where $i\in S$:

$$\begin{aligned} w_i&=(x_{\theta _i})_{(-1)}^{M_i(k)+1}(x_{\theta -\alpha _1})_{(-1)}\textbf{1}, \end{aligned}$$

(4.7)

$$\begin{aligned} s_i&=(x_{\alpha _1})_{(0)}w_i, \end{aligned}$$

(4.8)

$$\begin{aligned} u_i&=(x_{-\theta +\alpha _1})_{(1)}w_i. \end{aligned}$$

(4.9)

Proposition 4.3

The vectors $s_i$ are singular in the universal affine vertex algebra $V^k({\mathfrak {g}})$ and generate $J^k$.

Proof

Recall that $\alpha _0=\delta -\theta$. Since $(x_{\alpha _1})_{(0)}\textbf{1}=0$, $\textbf{1}$ is singular also for $r_{\alpha _1}({\widehat{\Pi }})$. Since $(x_{\theta -\alpha _1})_{(-1)}\textbf{1}$ is nonzero, it follows from Lemma 2.4 that $(x_{\theta -\alpha _1})_{(-1)}\textbf{1}$ is singular in $V^k({\mathfrak {g}})$ for ${\widehat{\Pi }}'=r_{\alpha _0+\alpha _1}r_{\alpha _1}({\widehat{\Pi }})$. Let $\Lambda '=k\Lambda _0-\delta +\theta -\alpha _1$ be the weight of $(x_{\theta -\alpha _1})_{(-1)}\textbf{1}$.

Since $\delta -\theta _i$ is in ${\widehat{\Pi }}'$ and, by (2.7) and (2.8),

$$\begin{aligned} (\Lambda '|(\delta -\theta _i)^\vee )=\frac{2}{(\theta _i|\theta _i)}k +(\alpha _1|\theta _i^\vee )=M_i(k)-\chi _i-(\xi |\theta _i^\vee ) =M_i(k), \end{aligned}$$

we see that $w_i$ is singular for ${\widehat{\Pi }}'$. Since $V^k({\mathfrak {g}})/\langle w_i\rangle$ is integrable with respect to $\widehat{{\mathfrak {g}}}^\natural$, it is irreducible, because the computation of its character formula in [GK15] did not use irreducibility, but only integrability. Hence the vectors $w_i$ generate the maximal proper ideal of $V^k({\mathfrak {g}})$.

The weight of $u_i$ is $k\Lambda _0 -s'\eta _i$, where

$$\begin{aligned} s':=1+(\Lambda '|\eta _i^\vee )=1+(k \Lambda _0|\eta _i^\vee )-(\alpha _0+\alpha _1|\eta _i^\vee )=1+s+(\alpha _1,\theta _i^\vee ), \end{aligned}$$

and s is defined in (4.6). Since $(\alpha _1|\theta _i^\vee )\le -1$, we have $s'\le (k \Lambda _0|\eta _i^\vee )$, so $(J^k)_{k\Lambda _0-s'\eta _i} =0$ by Lemma 4.2 (2). Since $w_i\in J^k$, we have $u_i\in J^k$, so $u_i=0$.

We now observe that the fact that $u_i=0$ implies that $w_i$ is a singular vector for $r_{-\alpha _0-\alpha _1}({\widehat{\Pi }}')$ $=r_{\alpha _1}({\widehat{\Pi }})$. Indeed, if $\beta \ne \alpha _0+\alpha _1$ is a simple root for $r_{-\alpha _0-\alpha _1}({\widehat{\Pi }}'),$ then it is a positive root for ${\widehat{\Pi }}'$, hence, since $w_i$ is singular for ${\widehat{\Pi }}'$, $(x_\beta )_{(0)}(x_{\theta _i})_{(-1)}^{M_i(k)+1}(x_{\theta -\alpha _1})_{(-1)}\textbf{1}=0$. Observe that, since $\alpha _0=\delta -\theta$, $u_i=0$ simply means that

$$\begin{aligned} (x_{-\theta +\alpha _1})_{(1)}w_i=0. \end{aligned}$$

Having shown that $w_i$ are singular for $r_{\alpha _1}({\widehat{\Pi }})$, it follows that the $s_i$ are either zero or singular vectors for ${\widehat{\Pi }}$.

Let $\Lambda '''_i=k\Lambda _0-(M_i(k)+1) (\delta -\theta _i)-\delta +\theta -\alpha _1$ be the weight of $w_i$. Since $k\le 0$ (see Table 1), we have

$$\begin{aligned} (\Lambda '''_i|\alpha _1)=((M_i(k)+1)\theta _i|\alpha _1)+1=-k+1>0, \end{aligned}$$

so,

$$\begin{aligned} (x_{-\alpha _1})_{(0)}s_i=c(-k+1)w_i-(x_{\alpha _1})_{(0)}(x_{-\alpha _1})_{(0)}w_i, \text { where }c\in {\mathbb {C}}. \end{aligned}$$

Since $w_i$ are singular vectors for $r_{\alpha _1}({\widehat{\Pi }})$, we see that

$$\begin{aligned} (x_{-\alpha _1})_{(0)}w_i=0. \end{aligned}$$

It follows that $(x_{-\alpha _1})_{(0)}s_i$ are nonzero multiples of $w_i$, hence the $s_i$ generate the maximal proper ideal of $V^k({\mathfrak {g}})$, since the $w_i$ do. $\square$

Proof of Theorem 4.1for ${\mathfrak {g}}\ne spo(2|3)$.

Since $L(k\Lambda _0) = V^k({\mathfrak {g}})/J^k$ is irreducible, $H_0$ is exact on category ${\mathcal {O}}$ and $H_0(L(k\Lambda _0))\ne 0$, by [Ar05], we have that the $W^k_{\min }({\mathfrak {g}})$-module $H_0(V^k({\mathfrak {g}}))/H_0(J^k)$ irreducible, so $I^k=H_0(J^k)$. Assume first that $|S|=1$. From Proposition 4.3, we deduce that $I^k=H_0(J^k)$ is a highest weight module. Due to [KRW03, (2.19)] its highest weight is

$$\begin{aligned} ((M_1(k)+1)\theta _1,M_1(k)+1). \end{aligned}$$

(4.10)

By (4.2), $v_1$ lies in $I^k$; moreover, it has weight (4.10), hence it is a highest weight vector. In particular it is singular and generates $I^k$. When $|S|=2$, ${\mathfrak {g}}$ is of type $D(2,1; \tfrac{m}{n})$ and $k=\tfrac{mn}{m+n}q,\,q\in {\mathbb {N}}$. $H_0(J^k)$ is the sum of two highest weight modules of weights $(mq\theta _1,mq),\,(nq\theta _2,nq)$. By Remark 2.1, we can assume $m>n$; then $v_2$ is singular, by the above comparing weight argument. We should prove that $v_1$ is not in the submodule generated by $v_2$. Otherwise, we can reach $v_1$ applying to $v_2$ a combination of operators $L_m, m<0,\, J^{\{u\}}_{r}, u\in {\mathfrak {g}}^\natural , r\le 0, G^{\{v\}}_s,\,v\in {\mathfrak {g}}_{-1/2}, s<0$. We can clearly assume that the v’s appearing are root vectors; let $\beta _v$ be the corresponding root, so that $\eta _v:=(\beta _v)_{|{\mathfrak {h}}^\natural }$ is the corresponding weight. Let $\Pi =\{\alpha _1,\alpha _2,\alpha _3\}$ with $\alpha _1$ odd. Then $\eta _v\in \pm \{\tfrac{\alpha _2-\alpha _3}{2},\tfrac{\alpha _2+\alpha _3}{2}\}$. Define the weight of $G^{\{v\}}_s$ to be the pair $(\eta _v,-s)$; likewise, if u is a root vector of ${\mathfrak {g}}^\natural$, and $\eta _u$ is the corresponding root, define the weight of $J^{\{u\}}_{r}$ as $(\eta _u, -r)$. Finally, declare that the weight of $L_n$ is $(0,-n)$. Let $\Delta _W$ be the set of weights. Note that any element of $\Delta _W$ is a positive integral linear combination of $\Pi _W=\{(-\alpha _2,0),(-\alpha _3,0),(\tfrac{\alpha _2+\alpha _3}{2},\tfrac{1}{2})\}$. It follows that

$$\begin{aligned} (mq\alpha _2,mq) - (nq\alpha _3,nq) =a_1(-\alpha _2,0)+a_2(-\alpha _3,0)+a_3(\tfrac{\alpha _2+\alpha _3}{2},\tfrac{1}{2}),\quad a_i\in \mathbb {Z}_+, \end{aligned}$$

so that $a_1=-nq, a_2=mq, a_3=2(m-n)q$. Since $a_1$ is negative, we have the required contradiction.

4.2 ${\mathfrak {g}}=spo(2|3)$

Introduce the following vector

$$\begin{aligned} r_1=(x_{\alpha _1})_{(0)}u_1= (x_{\alpha _1})_{(0)}(x_{-\theta +\alpha _1})_{(1)}w_1. \end{aligned}$$

(4.11)

Proposition 4.4

The vector $r_1$ is singular in the universal affine vertex algebra $V^k({\mathfrak {g}})$ and generates $J^k$.

Proof

As in the proof of Proposition 4.3, the vector $w_1$ generates the maximal proper ideal of $V^k({\mathfrak {g}})$. Let $\mu$ be weight of $w_1$. An explicit calculation shows that $(\mu |\alpha _0+\alpha _1)=-k-\tfrac{1}{2}\ne 0$, hence $u_1\ne 0$, and it is singular for $r_{\alpha _1}{\widehat{\Pi }}$. So $r_1$ is either 0 or singular for ${\widehat{\Pi }}$. The first possibility does not occur, since

$$\begin{aligned} (k\Lambda _0-(M_1(k)+1)(\delta -\theta _1)|\alpha _1)=\tfrac{1}{2}(M_1(k)+1)\ne 0. \end{aligned}$$

The claim follows. $\square$

Proof of Theorem 4.1for ${\mathfrak {g}}= spo(2|3)$.

Arguing as in the previous subsection, by Proposition 4.4 it follows that the maximal ideal in $W^k_{\min }({\mathfrak {g}})$ generated by a singular vector v of weight $(2 (m-2) \omega _1,m-3/2)$, where $\omega _1$ is the fundamental weight for $\mathfrak {sl}_2$ and $m=M_1(k)+2$. We observe that also the weight of ${{\widetilde{v}}}_1$ (cf. (4.3)) is $(2 (m-2) \omega _1,m-3/2)$. Moreover, using (4.2) and the relations

$$\begin{aligned} (G^{\{x_{-\alpha _1}\}})_{(0)}{{\widetilde{v}}}_1= c_1v_1,\quad (G^{\{x_{-\xi }\}})_{(1)} v_1 = c_2 {{\widetilde{v}}}_1, \text { where }c_1,c_2\in {\mathbb {C}}\setminus \{0\}, \end{aligned}$$

we see that ${{\widetilde{v}}}_1 \in I^k$, hence it is a multiple of v, so it is singular and generates $I^k$. $\square$

5 Descending from $W^k_{\min }({\mathfrak {g}})$ to $W_k^{\min }({\mathfrak {g}})$

Let $H_0$ be the quantum Hamiltonian reduction functor from the category ${\mathcal {O}}_k$ of $\widehat{{\mathfrak {g}}}$-modules of level k to the category of $W^k_{\min }({\mathfrak {g}})$-modules [KW04]. By [Ar05], it is exact. As in (3.1), for $\nu \in P^+_k$ and $h\in {\mathbb {C}}$ let ${\widehat{\nu }}_h=k\Lambda _0+h\theta +\nu$. By [Ar05, KW04], $H_0(L({\widehat{\nu }}_h))=0$ if $\widehat{\nu }_h(\alpha _0^\vee )=k-2\,h\in {\mathbb {Z}}_{\ge 0}$, and $H_0(L(\widehat{\nu }_h)=L^W(\nu ,\ell _0(h))$ if $\widehat{\nu }_h(\alpha _0^\vee )=k-2\,h\notin {\mathbb {Z}}_{\ge 0},$ where

$$\begin{aligned} \ell _0(h)=\frac{({\widehat{\nu }}_h|{\widehat{\nu }}_h+2\widehat{\rho })}{2(k+h^\vee )}-h, \end{aligned}$$

(5.1)

and $L^W(\nu ,\ell _0)$ is the irreducible highest weight $W^k_{\min }({\mathfrak {g}})$-module with highest weight $(\nu ,\ell _0)$ [KMP23, KW04].

Theorem 5.1

Let k be in the unitary range. Then all irreducible highest weight $W^k_{\min }({\mathfrak {g}})$–modules $L^W(\nu ,\ell _0)$ with $\ell _0\in {\mathbb {C}}$ when $\nu \in P^+_k$ is not extremal, and $\ell _0=A(k,\nu )$ otherwise, descend to $W_k^{\min }({\mathfrak {g}})$.

Proof

If $k-2h\in \mathbb {Z}_{\ge 0}$, then, by [KMP23, Lemma 11.8], for $h'=k+1-h$ we have $\ell _0:=\ell _0(h)=\ell _0(h')$. Since $k -2\,h'\notin \mathbb {Z}_{+}$ we conclude that $H_0(L(\widehat{\nu }_{h'}))=L^W(\nu ,\ell _0)$.

So

$$\begin{aligned} \text {for each }\ell _0\text { there is }{{\tilde{h}}}\in {\mathbb {C}}\text { such that } L^W(\nu ,\ell _0)=H_0(L({\widehat{\nu }}_{{{\tilde{h}}}})). \end{aligned}$$

(5.2)

By Theorem 3.1, if $\nu$ is not extremal, then $L(\widehat{\nu }_{{{\tilde{h}}}})$ is a $V_k({\mathfrak {g}})$–module, hence $L^W(\nu ,\ell _0({{\tilde{h}}}))$ is a $W_k^{\min }({\mathfrak {g}})$-module. Note that $h\in E_{k,\nu }$ if and only if $\ell _0(h)=A(k,\nu )$. It follows from Theorem 3.1 that, if $\nu$ is extremal, then $L(\widehat{\nu }_{{{\tilde{h}}}})$ is a $V_k({\mathfrak {g}})$–module, hence $L^W(\nu ,A(k,\nu ))$ is a $W_k^{\min }({\mathfrak {g}})$-module. $\square$

A simple application of Theorem 5.1 is the proof of Conjecture 4 in [KMP23].

Corollary 5.2

Any unitary $W^k_{\min }({\mathfrak {g}})$–module $L^W(\nu ,\ell _0)$ descends to $W_k^{\min }({\mathfrak {g}})$.

Proof

By Theorem 1.3 (1) of [KMP23] (see Theorem 2.3 (1)), the unitary representations of $W^k_{\min }({\mathfrak {g}})$ occur as representations listed in Theorem 5.1. $\square$

Remark 5.3

Conjecture 2 from [KMP23] that all $W_k^{\min }({\mathfrak {g}})$-modules $L^W(\nu , A(k, \nu ))$ for extremal $\nu$ are unitary is still an open problem, except for ${\mathfrak {g}}=spo(2|3)$ and psl(2|2). We can prove this statement also when ${\mathfrak {g}}=spo(2|n)$ and $k=-1$.

6 Classification of Irreducible Highest Weight Representations of $W_k^{\min }({\mathfrak {g}})$ in the Unitarity Range

The main result of this section is the following theorem.

Theorem 6.1

Let k be in the unitary range. The modules appearing in Theorem 5.1 form the complete list of inequivalent irreducible highest weight representations of $W_k^{\min }({\mathfrak {g}})$.

Remark 6.2

Combining (5.2) and Theorem 6.1, we have proven that the irreducible highest weight modules of $W_k^{\min }({\mathfrak {g}})$ are precisely the non-zero images of the irreducible $V_k({\mathfrak {g}})$ under Hamiltonian reduction.

We need to recall some well-known facts about Zhu algebras in the super case [KWang92].

Let $V=\bigoplus \limits _{n\in \tfrac{1}{2}\mathbb {Z}}V_n$ be a conformal vertex algebra, graded by the eigenspaces of $L_0$, with the parity $p(V_n)\equiv 2n\mod 2$. For $a\in V_n$ we write $\deg a =n$. Define bilinear maps $*:V\times V\rightarrow V,$ $\circ :V\times V \rightarrow V$, setting

$$\begin{aligned} a*b&={\left\{ \begin{array}{ll}Res_z\left( Y(a,z)\frac{(z+1)^{\deg a}}{z}b\right) \quad &{}\text {if }a,b\in V_{{\overline{0}}}, \\ 0 \quad &{}\text {if }a\text { or }b\in V_{{\overline{1}}}. \end{array}\right. }\\ a\circ b&={\left\{ \begin{array}{ll}Res_z\left( Y(a,z)\frac{(z+1)^{\deg a}}{z^2}b\right) \quad &{}\text {for }a\in V_{{\overline{0}}}, \\ Res_z\left( Y(a,z)\frac{(z+1)^{\deg a-\tfrac{1}{2}}}{z^2}b\right) \quad &{}\text {for }a\in V_{{\overline{1}}}. \\ \end{array}\right. } \end{aligned}$$

Denote by $O(V)\subset V$ the ${\mathbb {C}}$-span of elements of the form $a\circ b$, and by Zhu(V) the quotient space V/O(V). Then Zhu(V) is an associative algebra. Recall from [FZ92, (3.1.12)] or [KWang92, Lemma 1.1.(3)] that if [a] denotes the class of an element $a\in V$ in Zhu(V), then for all $a,b\in V_{{\overline{0}}}$ the following relations hold:

$$\begin{aligned}{}[a_{(-1)}\textbf{1}]*[b]=[(a_{(-1)}+a_{(0)})b],\quad [b]*[a_{(-1)}\textbf{1}]=[a_{(-1)}b]. \end{aligned}$$

(6.1)

Recall that a module M over a conformal vertex algebra is called a positive energy module if $L_0$ is diagonalizable on M and all its eigenvalues lie in $h+{\mathbb {R}}_{\ge 0}$ for some $h\in {\mathbb {C}}$:

$$\begin{aligned} M=\bigoplus _{j\in h+{\mathbb {R}}_{\ge 0}}M_j,\quad M_h\ne \{0\}. \end{aligned}$$

The subspace $M_h$ is called the top component of M.

Recall [FZ92, KWang92] that there is one-to-one correspondence between irreducible positive energy V–modules and irreducible modules over the Zhu algebra Zhu(V), which associates to a V-module M the Zhu(V)-module $M_h$. Namely, to $Y^M(a,z)=\sum _j a^M_jz^{-j-\deg a}$ one associates ${a_0^M}_{|M_h}$. By the above construction of the Zhu algebra, it follows from [KW04, Theorem 7.1] that

$$\begin{aligned} Zhu(W^k_{\min }({\mathfrak {g}}))\simeq {\mathbb {C}}[L]\otimes U({\mathfrak {g}}^\natural ). \end{aligned}$$

(6.2)

Under the correspondence between irreducible positive energy $W^k_{\min }({\mathfrak {g}})$-modules and irreducible modules over its Zhu algebra, the module $L^W(\nu ,h)$ goes to the irreducible highest weight ${\mathfrak {g}}^\natural$-module with highest weight $\nu$ on which L acts by the scalar h, which we denote by $V(\nu ,h)$. It follows from (6.2) that

$$\begin{aligned} Zhu(W_k^{\min }({\mathfrak {g}}))\simeq \left( {\mathbb {C}}[L]\otimes U({\mathfrak {g}}^\natural )\right) /J({\mathfrak {g}}), \end{aligned}$$

(6.3)

where $J({\mathfrak {g}})$ is a 2-sided ideal of the associative algebra ${\mathbb {C}}[L]\otimes U({\mathfrak {g}}^\natural )$. So any non-zero element in $J({\mathfrak {g}})$ imposes a condition on the highest weight $(\nu ,h)$ of the $Zhu(W_k^{\min }({\mathfrak {g}}))$-module $V(\nu ,h)$.

Proof of Theorem 6.1: the case ${\mathfrak {g}}= spo(2\vert 3)$.

First we present a proof in the case ${\mathfrak {g}}= spo(2 \vert 3)$, which gives a motivation for the proof in the general case.

Let $k=-\frac{m}{4},$ where $m\in {\mathbb {Z}}_{\ge 3}$, so that $M_1(k)=m-2$ and $P^+_k=\{j\omega _1\mid 0\le j\le m-2\}$ (here $\omega _1$ is the fundamental weight for ${\mathfrak {g}}^\natural \cong sl_2$). Set ${\mathcal {W}}^k = W_{\min }^{k}(spo(2|3)),$ ${\mathcal {W}}_k = W^{\min }_{k}(spo(2|3))$.

Denote by $L_k [j, q]$ the irreducible highest weight $\mathcal W_k$-module of level k generated by a highest weight vector $v_{j,q}$, such that for $n \in {{\mathbb {Z}}}_{\ge 0}$:

$$\begin{aligned}{} & {} L_n v_{j,q} = q \delta _{n,0}v_{j,q}, \ G^{+}_{(n+1/2)}v_{j,q} = G^{-}_{(n+1/2)}v_{j,q} = G^{0}_{(n+1/2)}v_{j,q} =0 \\{} & {} J^0_{(n)}v_{j,q} = j \delta _{n,0} v_{j,q}, \ J^+_{(n)}v_{j,q} = J^-_{(n+1)}v_{j,q}=0. \end{aligned}$$

Here we use notation for the generators of ${\mathcal {W}}_k$ as in [KW04, Section 8.5]. Note that $U(sl(2)) v_{j,q}=V (j \omega _1)$ is the irreducible highest weight sl(2)–module with highest weight $j \omega _1$.

Lemma 6.3

Set $\Omega = -\tfrac{m-2}{ 4} ( [L] +\frac{[J^0]}{4} ) + \tfrac{1}{8} [J^+]*[J^-]$. Then

$$\begin{aligned} \Omega * [J^-] ^{m-3} \in J({\mathfrak {g}}). \end{aligned}$$

(6.4)

Proof

By Theorem 4.1, we have that $(J^-)_{(-1)}^{m-3} G^- \in I^k$, hence $[G^+ _{(0)}(J^-)_{(-1)}^{m-3} G^-] =0$. Since $G^+ _0$ acts as a derivation, using (6.1) we get that

$$\begin{aligned}&0=[(G^+)_{(0)}(J^-)_{(-1)}^{m-3} G^-] = [(G^0)_{(-1)} (J^-)_{(-1)}^{m-4} G^-] + \cdots +\nonumber \\&[(J^-)_{(-1)}^{m-5} (G^{0}) _{(-1)}(J^-)_{(-1)} G^-] + [(J^-)_{(-1)}^{m-4} (G^+)_{(0)} G^-] \nonumber \\&\quad = (m-3) [ (J^-)_{(-1)}^{m-4} (G^0)_{(-1)} G^-] + \tfrac{ (m-3) (m-4)}{2} [ (J^-)_{(-1)}^{m-5} (G^-) _{(-2)} G^-] \nonumber \\ {}&\qquad + [ (J^-)_{(-1)}^{m-4}(G^+)_{(0)} G^-] \nonumber \\&\quad = (m-3) [(G^0)_{(-1)} G^+] * [J^{-}] ^{m-4} + \tfrac{ (m-3) (m-4)}{2} [ (G^-)_{(-2)} G^-] * [J^{-}] ^{m-5}\nonumber \\&\qquad + [(G^+)_ {(0)} G^-] *[J^{-}] ^{m-3}. \end{aligned}$$

(6.5)

Using $\lambda$–bracket formulas from [KW04, Section 8.5] we get that the following relations hold in ${\mathcal {W}}^k$:

$$\begin{aligned} (G^+)_{(0)} G^-&= \tfrac{2k+1}{2} L + \tfrac{1}{8} :J^- J^+: +\tfrac{k+1}{4} \partial J^0, \\ (G^0)_{(0)} G^-&= \tfrac{1}{8} :J^- J^0: -\tfrac{k+1}{2} \partial J^-, \\ (G^-) _{(0)}G^-&= - \tfrac{1}{4} :J^- J^-:. \end{aligned}$$

This implies, in $Zhu({\mathcal {W}}^k)$,

$$\begin{aligned}{}[G^+ _{(0)}G^-]&= \tfrac{2k+1}{2} ( [L] +\tfrac{1}{4 (2 k+1)} [J^+]* [J^-] -\tfrac{k+1}{2 (2k+1)} [J^0] ), \end{aligned}$$

(6.6)

$$\begin{aligned}&= - [(G^0)_{(0)} G^ -] = -\tfrac{1}{8} [J^0] *[J^-] - \tfrac{k+1}{2} [J^-]= - \tfrac{1}{8} ( [J^0] +4(k+1) ) *[J^- ], \end{aligned}$$

(6.7)

$$\begin{aligned}&= - [(G^-)_{(-1)} G^- ] = [(G^-)_{(0)} G^- ] = - \tfrac{1}{4} [J^-] * [J^-]. \end{aligned}$$

(6.8)

Substituting (6.6), (6.7), (6.8) into (6.5) and collecting $[J^-]^{m-3}$, we get (6.4), where

$$\begin{aligned} \Omega= & {} \tfrac{2k+1}{ 2} [L] + \tfrac{1}{8} [J^+]*[J^-] - \tfrac{k+1}{4} [J^0] - \tfrac{ (m-3) (m-4)}{8} - \tfrac{m-3}{8} ([J^0] + 4 (k+1) ). \\= & {} - \tfrac{m-2}{ 4} [L]+ \tfrac{1}{8} [J^+]*[J^-] + \tfrac{m-4}{16} [J^0] - \tfrac{ (m-3) (m-4)}{8} - \tfrac{m-3}{8} ([J^0] - m+4) \\= & {} -\tfrac{m-2}{ 4} [L] -\tfrac{m-2}{16} [J^0] + \tfrac{1}{8} [J^+]*[J^-]. \end{aligned}$$

$\square$

Proposition 6.4

Let $L_{k}[j,q]$ be an irreducible highest weight ${\mathcal {W}}_{k}$–module. Then it is isomorphic to exactly one of the following modules:

• $L_{k}[j, q]$ with $0 \le j \le m-4$ and $q \in {{\mathbb {C}}}$;

• $L_{k} [m-3, \frac{m-3}{4}]$;

• $L_{k} [m-2, \frac{m-2}{4}]$.

Proof

First, $j \in \{0, \dots , m-2\}$ and the top component is $L_k[j, q]_{top} =V(j \omega _1) \otimes {{\mathbb {C}}}_q,$ where $V(j \omega _1)$ is the $(j+1)$–dimensional irreducible sl(2)–module with highest weight $j \omega _1$, and ${\mathbb {C}}_q$ the 1-dimensional ${{\mathbb {C}}}[L]$–module on which $L_0$–acts as multiplication with $q \in {{\mathbb {C}}}$. If $0 \le j \le m-4$, then $[J^-] ^{m-3}$ acts trivially on $L_k[j, q]_{top}$ for each $q \in {{\mathbb {C}}}$, hence the same holds for the action of $[{G^+}_{(0)} G^-]* [J^-] ^{m-3}$. If $m-3\le j \le m-2$, then $(J^-)_{(0)}^{m-3}$ acts non-trivially on $L_k[j, q]_{top}$. Hence there exists $w \in L_k[j, q]_{top}$ such that $w' =(J^-)_{(0)}^{m-3} w$ is a lowest weight vector for sl(2), i.e.

$$\begin{aligned} (J^0)_{(0)}w' = -j w',\ (J^-)_{(0)} w' = 0. \end{aligned}$$

Then we have, by Lemma 6.3:

$$\begin{aligned} 0= & {} ( \Omega * [J^-] ^ {m-3} )w \\= & {} \left( -\tfrac{m-2}{4} ( L_0 + \tfrac{1}{4}(J^0)_{(0)} ) +\tfrac{1}{8} (J^+)_{(0)} (J^-)_{(0)} \right) (J^-)_{(0)}^{m-3} w \\ {}= & {} \left( -\tfrac{m-2}{4} ( L_0 + \tfrac{ 1}{4} (J^0)_{(0)}) +\tfrac{1}{8} (J^+)_{(0)} (J^-)_{(0)} \right) w' \\= & {} -\tfrac{m-2}{4} (q -\tfrac{j}{4})w', \end{aligned}$$

hence, when $j=m-3$ or $m-2$, we have that $q= \tfrac{j}{4}$. $\square$

Since the modules appearing in Theorem 6.1 in case of ${\mathfrak {g}}=spo(2|3)$ are exactly those listed in Proposition 6.4, Theorem 6.1 is proved in this case.

Proof of Theorem 6.1: the case ${\mathfrak {g}}\ne spo(2 \vert 3)$.

We first illustrate the strategy of the proof in the case ${\mathfrak {g}}=psl(2|2)$. We use notation of [KW04, Section 8.4].

Set ${\mathcal {W}}_{\min }^{k} = W^{k}_{\min }(psl(2\vert 2))$ and ${\mathcal {W}}^{\min }_{k} = W^{\min }_{k}(psl(2\vert 2))$.

Let $m = -k-1=M_1(k) \in {{\mathbb {Z}}}_{>0}$. Then $V_k(sl(2))$ is a vertex subalgebra of ${\mathcal {W}}^{\min }_{k}$ generated by $J^{\pm }, J^0$. The odd generators of conformal weight 3/2 are $G^{\pm }, {\overline{G}}^{\pm }$. By Theorem 4.1 the maximal ideal $I^k$ is generated by the singular vector $(J^+)_{(-1)}^{m+1} \textbf{1}$. Then $(G^-)_{(0)} ({\overline{G}}^{-})_{(0)}(J^+_{(-1)})^{m+1} \textbf{1} \in I^k$. By the $\lambda$-bracket formulas given in [KW04, Section 8.4] we have

$$\begin{aligned} (G^-) _{(0)} {\overline{G}}^{-}_{(0)} (J^+_{(-1)})^{m+1} \textbf{1}&= (m+1) (G^-) _{(0)} (J^+_{(-1)}) ^{m} {\overline{G}}^+ \nonumber \\ {}&=- (m+1) m (J^+_{(-1)}) ^{m-1} (G^+)_{(-1)} {\overline{G}}^+ + (m+1) (J^+_{(-1)}) ^{m} (G^-) _{(0)} {\overline{G}}^+. \end{aligned}$$

(6.9)

Using the definition of the Zhu algebra $Zhu({\mathcal {W}}^k_{\min })$, we have that for $G =G^{\pm }$ or $G ={\overline{G}}^{\pm }$ and each $v \in {\mathcal {W}}^k_{\min }$:

$$\begin{aligned} \text{ Res}_z \frac{(z+1) ^{\deg G-1/2}}{z} G(z) v = (G_{(-1)} + G_{(0)} ) v = 0 \ \text {in } Zhu({\mathcal {W}}^k_{\min }). \end{aligned}$$

(6.10)

Using (6.10) and the $\lambda$-bracket formulas given in [KW04, Section 8.4], we get

$$\begin{aligned}&[ (G^+)_{(-1)} {\overline{G}}^+] = - [(G^+)_{(0)} {\overline{G}}^+] =- [(J^+)_{(-2)}{} \textbf{1}] = [J^+], \end{aligned}$$

(6.11)

$$\begin{aligned}&[ (G^-) _{(0)} {\overline{G}}^+] = [L] - \tfrac{1}{2} [(J^{0})_{(-2)}{} \textbf{1}] = [L] + \tfrac{1}{2} [J^0], \end{aligned}$$

(6.12)

Using (6.9), (6.11),(6.12) and (6.1), we find

$$\begin{aligned}{}[(G^-)_{(0)} ({\overline{G}}^{-})_{(0)} (J^+_{(-1)})^{m+1} \textbf{1}]&= - m (m+1) [J^+ ]^m + (m+1) ([L] + \tfrac{1}{2} [J^0] ) * [J^+]^{m} \\&= (m+1) \left( [L] + \tfrac{1}{2} [J^0] -m \right) * [J^+]^{m} \end{aligned}$$

so that, setting $\Omega = [L] + \tfrac{1}{2} [J^0] -m$, we have from (6.10)

$$\begin{aligned} \Omega *[J^+]^m\in J({\mathfrak {g}}). \end{aligned}$$

(6.13)

Proposition 6.5

Let $L^{N4} _{k}[j,\ell _0]$ be an irreducible highest weight ${\mathcal {W}}^{\min }_{k}$–module. Then it is isomorphic to exactly one of the following modules:

• $L^{N4}_{k}[j, \ell _0]$ with $0 \le j \le m-1$ and $\ell _0 \in {{\mathbb {C}}}$;

• $L^{N4}_{k} [m, \frac{m}{2}]$.

Proof

Let $v[j,\ell _0]$ be a highest weight vector for $L^{N4} _{k}[j,\ell _0]$. Then $j \in \{0, \dots , m\}$ and the top component is $L^{N4}_k[j, \ell _0]_{top} =V(j \omega _1) \otimes {{\mathbb {C}}}_{\ell _0},$ where $V(j \omega _1)$ is the $(j+1)$–dimensional irreducible sl(2)–module with highest weight $j \omega _1$, and ${{\mathbb {C}}}_{\ell _0}$ the 1-dimensional ${{\mathbb {C}}}[L]$–module on which L(0)–acts as multiplication by $\ell _0 \in {{\mathbb {C}}}$. If $0 \le j \le m-1$, then $\Omega * [J^+] ^{m}$ acts trivially on $L^{N4}_k[j, \ell _0]_{top}$ for each $\ell _0 \in {{\mathbb {C}}}$.

If $j=m$, then $(J^+_{(0)})^{m}$ acts non-trivially on $L^{N4}_k[j, \ell _0]_{top}$. Then there exists $w \in L^{N4} _k[j, \ell _0]_{top}$ such that $(J^+_{(0)})^m w = v[j,\ell _0]$ is a highest weight vector for sl(2). We get by (6.10)

$$\begin{aligned} 0= & {} ( \Omega * [J^+] ^ {m} ) w \\= & {} (L_{(0)}+ \tfrac{1}{2} (J^0_{(0)})= -m (J^+_{(0)})^{m} w \\= & {} \left( \ell _0-m/2 \right) v[j,\ell _0]. \end{aligned}$$

This implies that for $j=m$ we need to have $\ell _0= \tfrac{m}{2}$. $\square$

Proposition 6.5 proves, in particular, Theorem 6.1 for ${\mathfrak {g}}=psl(2|2)$. We now deal with the general case ${\mathfrak {g}}\ne spo(2|3)$. We shall see that a relation similar to (6.13) holds in $Zhu(W^k_{\min }({\mathfrak {g}}))$. We do not need a very precise expression for $\Omega$: what is really relevant is the fact that the action of $\Omega$ gives a relation which is linear in $\ell _0$.

We start by observing that there exist two odd positive roots $\gamma _1,\gamma _2$ such that

$$\begin{aligned} \theta -\gamma _1-\gamma _2=-\theta _i,\ i\in S. \end{aligned}$$

(6.14)

This fact can be verified directly by looking at Table 3.

Table 3 Root data

Using (6.14) and the explicit expression for $[{G^{\{u\}}}_{\lambda }G^{\{v\}}]$ given in [AKMPP18, (1.1)] we find

$$\begin{aligned}&{G^{\{x_{\theta _i-\gamma _1}\}}}_{(0)}G^{\{x_{\theta _i-\gamma _2}\}}= \sum _{s=1}^{\dim {\mathfrak {g}}_{1/2}}:J^{\{[x_{\theta _i-\gamma _1},u^s]^\natural \}}J^{\{[u_s,x_{\theta _i-\gamma _2}]^\natural \}}: +2c_1(k+1)\partial J^{\{x_{\theta _i}\}}, \end{aligned}$$

(6.15)

$$\begin{aligned}&{G^{\{x_{-\gamma _1}\}}}_{(0)}G^{\{x_{\theta _i-\gamma _2}\}}=-2(k+h^\vee ) c_2L+c_2\sum _{\alpha =1}^{\dim {\mathfrak {g}}^\natural } :J^{\{a^\alpha \}}J^{\{a_\alpha \}}:+\nonumber \\&\sum _{s=1}^{\dim {\mathfrak {g}}_{1/2}}:J^{\{[x_{-\gamma _1},u^s]^\natural \}}J^{\{[u_s,x_{\theta _i-\gamma _2}]^\natural \}}: +2(k+1)c_3\partial J^{\{h^\natural _{-\theta _i+\gamma _2}\}}, \end{aligned}$$

(6.16)

where $c_1,c_2,c_3$ are constants independent of k. Here $a^\natural$ denotes the orthogonal projection of $a\in {\mathfrak {g}}$ to ${\mathfrak {g}}^\natural$ and $\{u_s\},\, \{u^s\}$ are basis of ${\mathfrak {g}}_{1/2}$ dual with respect to the bilinear form

$$\begin{aligned} \langle u,v\rangle =(x_{-\theta }|[u,v]). \end{aligned}$$

(6.17)

Set $x_{\theta _i-\gamma _j}=[x_{-\gamma _j},x_{\theta _i}],\,j=1,2$. Recall that, by [KW04, Theorem 2.1 (e)], if $u\in {\mathfrak {g}}_{-1/2}$ and $a\in {\mathfrak {g}}^\natural$, then

$$\begin{aligned}{}[J^{\{a\}}_{(-1)}, G^{\{u\}}_{(0)}]=G^{\{[a,u]\}}_{(-1)}. \end{aligned}$$

(6.18)

Lemma 6.6

If ${\mathfrak {g}}\ne spo(2|3)$, $N\in {\mathbb {Z}}_{\ge 1}$ and $M\in {\mathbb {Z}}_{\ge 2}$, then, for $i\in S$ we have

$$\begin{aligned}&[G_{(0)}^{\{x_{-\gamma _j}\}},(J_{(-1)}^{\{x_{\theta _i}\}})^N] =-N(J_{(-1)}^{\{x_{\theta _i}\}})^{N-1}G_{(-1)}^{\{x_{\theta _i-\gamma _j}\}},\quad j=1,2. \end{aligned}$$

(6.19)

$$\begin{aligned}&(G^{\{x_{-\gamma _1}\}})_{(0)} (G^{\{x_{-\gamma _2}\}})_{(0)}(J^{\{x_{\theta _i}\}}_{(-1)})^{M}\textbf{1}\nonumber \\&\quad =M\left( (M-1)(J^{\{x_{\theta _i}\}}_{(-1)})^{M-2}(G^{\{x_{\theta _i-\gamma _1}\}})_{(-1)} (G^{\{x_{\theta _i-\gamma _2}\}})_{(-1)}\textbf{1}\right. \nonumber \\&\qquad -\left. (J^{\{x_{\theta _i}\}}_{(-1)})^{M-1}(G^{\{x_{-\gamma _1}\}})_{(0)}(G^{\{x_{\theta _i-\gamma _2}\}})\right) . \end{aligned}$$

(6.20)

Proof

We prove (6.19) by induction on N, with base $N=1$ given by (6.18). If $N>1$ we have

$$\begin{aligned}&[G_{(0)}^{\{x_{-\gamma _j}\}},(J_{(-1)}^{\{x_{\theta _i}\}})^N]=[G_{(0)}^{\{x_{-\gamma _j}\}}, (J_{(-1)}^{\{x_{\theta _i}\}})^{N-1}](J_{(-1)}^{\{x_{\theta _i}\}})+ (J_{(-1)}^{\{x_{\theta _i}\}})^{N-1}[G_{(0)}^{\{x_{-\gamma _j}\}},(J_{(-1)}^{\{x_{\theta _i}\}})]\\&\quad =(1-N)(J_{(-1)}^{\{x_{\theta _i}\}})^{N-2}G_{(-1)}^{\{x_{\theta _i-\gamma _j}\}} (J_{(-1)}^{\{x_{\theta _i}\}})-(J_{(-1)}^{\{x_{\theta _i}\}})^{N-1}G_{(-1)}^{\{x_{\theta _i-\gamma _j}\}}\\&\quad =-N(J_{(-1)}^{\{x_{\theta _i}\}})^{N-1}G_{(-1)}^{\{x_{\theta _i-\gamma _j}\}}. \end{aligned}$$

In the final equality we have used that $2\theta _i-\gamma _j\notin \Delta$ for ${\mathfrak {g}}\ne spo(2|3)$.

We now prove (6.20):

$$\begin{aligned} (G^{\{x_{-\gamma _1}\}})_{(0)} (G^{\{x_{-\gamma _2}\}})_{(0)}(J^{\{x_{\theta _i}\}}_{(-1)})^{M}\textbf{1}&= -M(G^{\{x_{-\gamma _1}\}})_{(0)} (J^{\{x_{\theta _i}\}}_{(-1)})^{M-1} (G^{\{x_{\theta _i-\gamma _2}\}})_{(-1)}\textbf{1}\\&=M(M-1) (J^{\{x_{\theta _i}\}}_{(-1)})^{M-2}(G^{\{x_{\theta _i -\gamma _1}\}})_{(-1)}(G^{\{x_{\theta _i-\gamma _2}\}})_{(-1)}\textbf{1}\\&\quad -M(J^{\{x_{\theta _i}\}}_{(-1)})^{M-1}(G^{\{x_{-\gamma _1}\}})_{(0)} (G^{\{x_{\theta _i-\gamma _2}\}})\textbf{1}. \end{aligned}$$

$\square$

By Theorem 4.1, if $M=M_i(k)+1$, then the element displayed in (6.20) lies in $I^k$, hence its projection to $Zhu(W_k^{\min }({\mathfrak {g}}))$ lies in $J({\mathfrak {g}})$. Next we calculate explicitly this projection. First remark that, by (6.10),

$$\begin{aligned}{}[(G^{\{x_{\theta _i-\gamma _1}\}})_{(-1)} (G^{\{x_{\theta _i-\gamma _2}\}})_{(-1)}\textbf{1}]=-[(G^{\{x_{\theta _i-\gamma _1}\}})_{(0)} (G^{\{x_{\theta _i-\gamma _2}\}})]. \end{aligned}$$

(6.21)

Substituting (6.21) into (6.20) and using (6.15), (6.16), we get

$$\begin{aligned}&\frac{1}{M_i(k)+1}[(G^{\{-\gamma _1\}})_{(0)} (G^{\{-\gamma _2\}})_{(0)}(J^{\{x_{\theta _i}\}}_{(-1)})^{M_i(k)+1}\textbf{1}]=\nonumber \\&\quad -M_i(k)\left( \sum _{\gamma =1}^{\dim {\mathfrak {g}}_{1/2}}[J^{\{[u_s,x_{\theta _i -\gamma _2}]^\natural \}}]*[J^{\{[x_{\theta _i-\gamma _1},u^s]^\natural \}}]*[J^{\{x_{\theta _i}\}}]^{M_i(k)-1} -2(k+1)c_1[J^{\{x_{\theta _i}\}}]^{M_i(k)}\right) \nonumber \\&\quad +c_2\left( -(k+h^\vee )[L]+\tfrac{1}{2}\sum _{\alpha =1}^{\dim {\mathfrak {g}}^\natural } [J^{\{a_\alpha \}}][J^{\{a^\alpha \}}]\right) *[J^{\{x_{\theta _i}\}}]^{M_i(k)}+\nonumber \\&-\left( \sum _{s=1}^{\dim {\mathfrak {g}}_{1/2}}[J^{\{[u_s,x_{\theta _i-\gamma _2}]^\natural \}}] *[J^{\{[x_{-\gamma _1},u^s]^\natural \}}] +2(k+1)c_3[\partial J^{\{h^\natural _{-\theta _i+\gamma _2}\}}]\right) *[J^{\{x_{\theta _i}\}}]^{M_i(k)}. \end{aligned}$$

(6.22)

To finish the calculation we need the following fact.

Lemma 6.7

For suitable constants $d_1,d_2,d_3$, independent of k, we have

$$\begin{aligned} \sum _{s=1}^{\dim {\mathfrak {g}}_{1/2}}[J^{\{[u_s,x_{\theta _i -\gamma _2}]^\natural \}}]*[J^{\{[x_{\theta _i-\gamma _1},u^s]^\natural \}}]&=d_1\, [J^{\{h^\natural _{-\theta _i+\gamma _2}\}}]*[J^{\{x_{\theta _i}\}}]+ d_2\, [J^{\{x_{\theta _i}\}}]*[J^{\{h^\natural _{-\theta _i+\gamma _1}\}}]\nonumber \\&= (d_1\, [J^{\{h^\natural _{-\theta _i+\gamma _2}\}}]+ d_2\,[J^{\{h^\natural _{-\theta _i+\gamma _1}\}}]+d_3)*[J^{\{x_{\theta _i}\}}]. \end{aligned}$$

(6.23)

Proof

We can assume that $u_s, u^s$ are root vectors. A direct inspection shows that $[u_s,x_{\theta _i-\gamma _2}]^\natural ,$ $[x_{\theta _i-\gamma _1},u^s]^\natural$ are both non-zero only if $u_s\in {\mathfrak {g}}_\alpha ,\,u^s\in {\mathfrak {g}}_\beta$ with either $-\alpha =\theta _i-\gamma _{2}$ and $\beta =\gamma _1$ or $\alpha =\gamma _{2}$ and $-\beta =\theta _i-\gamma _{1}$. $\square$

Set

$$\begin{aligned} \Omega _i&=(-M_i(k)(d_1\, [J^{\{h^\natural _{-\theta _i+\gamma _2}\}}]+ d_2\,[J^{\{h^\natural _{-\theta _i+\gamma _1}\}}]+d_3) -c_2(k+h^\vee )[L] \nonumber \\&+\tfrac{1}{2}c_2\sum _{\alpha =1}^{\dim {\mathfrak {g}}^\natural } [J^{\{a_\alpha \}}][J^{\{a^\alpha \}}]+\sum _{s=1}^{\dim {\mathfrak {g}}_{1/2}}[J^{\{[u_s, x_{\theta _i-\gamma _2}]^\natural \}}]*[J^{\{[x_{-\gamma _1},u^s]^\natural \}}] -2(k+1)c_3[J^{\{h^\natural _{-\theta _i+\gamma _2}\}}]). \end{aligned}$$

(6.24)

Note that all the constants involved in (6.24) do not depend on k; moreover, by (6.14),

$$\begin{aligned} c_2=(x_\theta |[x_{\theta _i-\gamma _1},x_{\theta _i-\gamma _2}])\ne 0. \end{aligned}$$

(6.25)

Substituting (6.23) in (6.22) we obtain

Proposition 6.8

In $Zhu(W^k_{\min }({\mathfrak {g}}))$, we have

$$\begin{aligned} \Omega _i*[J^{\{x_{\theta _i}\}}]^{M_i(k)}\in J({\mathfrak {g}}),\ i\in S. \end{aligned}$$

(6.26)

Proof of Theorem 6.1

We have to prove that if $L^W(\nu ,\ell _0)$ is a irreducible highest weight $W_k^{\min }({\mathfrak {g}})$-module, then the pair $(\nu ,\ell _0)$ is among those listed in the statement. Note that there is a non-zero vertex algebra homomorphism $\Theta :V^{M_i(k)}({\mathfrak {g}}^\natural _i)\rightarrow W_k^{\min }({\mathfrak {g}})$. Since $W_k^{\min }({\mathfrak {g}})$ is unitary, we have that $\Theta (V^{M_i(k)}({\mathfrak {g}}^\natural _i))=V_{M_i(k)}({\mathfrak {g}}^\natural )$. In particular, $L^W(\nu ,\ell _0)$ is integrable as a $V_{M_i(k)}({\mathfrak {g}}^\natural _i)$-module, hence $\nu \in P^+_k$. If $\nu$ is not extremal, we are done. Assume therefore that $\nu$ is extremal. The action of the Zhu algebra on the top component $V(\nu ,\ell _0)$ of $L^W(\nu ,\ell _0)$ is given by the action of the elements $[J^{\{a\}}]$ on the irreducible finite-dimensional ${\mathfrak {g}}^\natural$-module $V(\nu , \ell _0)$ of highest weight $\nu$, while [L] acts as the multiplication by $\ell _0$. Consider the sl(2)-triple $\{x_{\theta _i},h_{\theta _i},x_{-\theta _i}\}$. Since $\nu$ is extremal, $\nu (\theta _i^\vee )=M_i(k)$ for some $i\in S$. By sl(2)-theory applied to U(sl(2))v where $v\in V(\nu ,\ell _0)$ is a highest weight vector, we see that v is a non-zero multiple of $x_{\theta _i}^{M_i(k)}v'$, with $v'=x_{-\theta _i}^{M_i(k)}v$. It follows that

$$\begin{aligned} 0=\Omega _i [J^{\{x_{\theta _i}\}}]^{M_i(k)}v'=\Gamma _i(\ell _0) [J^{\{x_{\theta _i}\}}]^{M_i(k)}v', \end{aligned}$$

(6.27)

where, by (6.24), $\Gamma _i(\ell _0)=-c_2(k+h^\vee )\ell _0+b$ and $b\in {\mathbb {C}}$ is independent of $\ell _0$, and $c_2\ne 0$ by (6.25). Since, by construction, $[J^{\{x_{\theta _i}\}}]^{M_i(k)}v'\ne 0$, we find from (6.27) that $\Gamma _i(\ell _0)=0$. Since $L^W(\nu ,A(k,\nu ))$ is a representation of $W_k^{\min }({\mathfrak {g}})$, then $\Gamma _i(A(k,\nu ))=0$. But $\Gamma _i$ is a linear relation, so $\ell _0=A(k,\nu )$ is the unique solution of the equation $\Gamma _i(\ell _0)=0$. $\square$

A further refinement of Theorem 6.1 is the following result.

Theorem 6.9

Let k be in the unitary range. The irreducible highest weight $W_k^{\min }({\mathfrak {g}})$-modules are all the irreducible positive energy representations.

Proof

Let ${\mathcal {V}}_k({\mathfrak {g}}^{\natural }) = \bigotimes _{i\in S} V_{M_i(k)}({\mathfrak {g}}^{\natural }_i)$. Since each $V_{M_i(k)}({\mathfrak {g}}^{\natural }_i)$ is a rational vertex algebra, then $Zhu({\mathcal {V}}_k({\mathfrak {g}}^{\natural })) = \bigotimes _i Zhu(V_{M_i(k)}({\mathfrak {g}}^{\natural }_i) )$ is a semi-simple finite-dimensional associative algebra.

Let M be an irreducible positive energy $W_k^{\min }({\mathfrak {g}})$–module. Then the top component $M_{top}$ is an irreducible module for Zhu’s algebra $Zhu(W_k^{\min }({\mathfrak {g}}))$. By using the embedding ${\mathcal {V}}_k({\mathfrak {g}}^{\natural }) \rightarrow W_k^{\min }({\mathfrak {g}})$ we see that $M_{top}$ is also a $Zhu({\mathcal {V}}_k({\mathfrak {g}}^{\natural }) )$–module. Since [L] is a central element in $Zhu(W_k^{\min }({\mathfrak {g}}))$, it acts on $M_{top}$ as multiplication by a scalar, hence, by (6.3), $M_{top}$ remains irreducible when restricted to ${\mathfrak {g}}^\natural$ and therefore it is an irreducible $Zhu({\mathcal {V}}_k({\mathfrak {g}}^{\natural }) )$–module. Since $Zhu(\mathcal V_k({\mathfrak {g}}^{\natural }) )$ is a semi-simple finite-dimensional associative algebra, we conclude that $M_{top}$ is finite-dimensional. This implies that $M_{top}$ contains a highest weight ${\mathcal {V}}_k({\mathfrak {g}}^{\natural })$–vector w, which is then a highest weight vector for the action of $W_k^{\min }({\mathfrak {g}})$. Therefore $W_k^{\min }({\mathfrak {g}})w$ is a highest weight submodule of M. Irreducibility of M implies that $M =W_k^{\min }({\mathfrak {g}})w$ is a highest weight $W_k^{\min }({\mathfrak {g}})$–module. $\square$

7 On Positive Energy Modules Over $V_k({\mathfrak {g}})$ and $W_k^{\min }({\mathfrak {g}})$

By Theorem 6.9, the irreducible highest weight $W_k^{\min }({\mathfrak {g}})$-modules are all the irreducible positive energy representations. In this section we show, by contrast, that $V_k({\mathfrak {g}})$ admits positive energy modules which are not highest weight, and we show that $W_k^{\min }({\mathfrak {g}})$ admits non-positive energy modules.

The $V_k({\mathfrak {g}})$–modules which we construct belong to class of modules called relaxed highest weight modules. These modules appear in [AdM95, FST, LMRS, R], in the context of the representation theory of $V_k(\mathfrak {sl}_2)$; later they have been systematically studied in [KR1, KR2, KaRW] for higher rank cases. We will show the existence of such modules for $V_k({\mathfrak {g}})$ for some values of k belonging to the unitary range. We explore a free field realization which enables us to construct relaxed highest weight modules from relaxed highest weight modules over the Weyl vertex algebra (also called $\beta \gamma$ ghost vertex algebra). These relaxed modules were previously studied in [RW, AW22, AdP19].

On the contrary, the vertex algebra $W_k^{\min }({\mathfrak {g}})$ does not have non-highest weight positive energy modules. The non-highest weight relaxed highest weight $V_k({\mathfrak {g}})$–modules are either mapped to zero by quantum Hamiltonian reduction, so they don’t contribute to $W_k^{\min }({\mathfrak {g}})$–modules; or they are mapped to highest weight $W_k^{\min }({\mathfrak {g}})$–modules. This connection deserves a more detailed analysis in the future.

We shall see below that $W_k^{\min }({\mathfrak {g}})$ contains non-positive energy modules having all infinite dimensional weight spaces.

7.1 Positive energy modules over $V_k({\mathfrak {g}})$ which are not highest weight

We follow the notation and results of [AMPP20, Section 6].

Consider the superspace ${\mathbb {C}}^{m|2n}$ equipped with the standard supersymmetric form $\langle \cdot ,\cdot \rangle _{m|2n}$ given in [K77]. Let $V=\Pi {\mathbb {C}}^{m|2n}$, where $\Pi$ is the parity reversing functor. Let $M_{(m|2n)}$ be the universal vertex algebra generated by V with $\lambda$–bracket

$$\begin{aligned}{}[v_\lambda w]=\langle w,v\rangle . \end{aligned}$$

(7.1)

Let $\{e_i\}$ be the standard basis of V and let $\{e^i\}$ be its dual basis with respect to $\langle \cdot ,\cdot \rangle$ (i. e. $\langle e_i,e^j\rangle =\delta _{ij}$). In this basis the $\lambda$-brackets are given by

$$\begin{aligned}{}[{e_h}_\lambda e_{m-k+1}]=\delta _{hk},\quad [{e_{m+i}}_\lambda e_{m+2n-j+1}]=-\delta _{ij},\quad [{e_{m+n+i}}_\lambda e_{m+n-j+1}]=\delta _{ij}, \end{aligned}$$

for $h,k=1,\ldots m,\,\,i,j= 1\ldots ,n$.

The vertex algebra $M_{(m \vert n)}$ is called the Weyl–Clifford vertex algebra and, in the physics terminology, the $\beta \gamma b c$ system (cf. [FMS86]). In the case $m= 0$ (resp. $n=0$), we have the Weyl vertex algebra $M_{(n)}:= M_{(0, 2n)}$, (resp. the Clifford vertex algebra $F_{(m)}:= M_{(m, 0)}$). Clearly, we have the isomorphism:

$$\begin{aligned} M_{(m|2n)} \cong F_{(m)} \otimes M_{(n)}. \end{aligned}$$

It was proved in [AdP19, AW22, RW] that the Weyl vertex algebra $M_{(n)}$ has a remarkable family of irreducible positive energy modules, which we denote by $\widetilde{U_n}(\textbf{a})$, whose top components are isomorphic to

$$\begin{aligned} U_n(\textbf{a}) = x_1 ^{a_1} \cdots x_n ^{a_n} {{\mathbb {C}}}[ x_1, \ldots , x_n, x_1 ^{-1}, \ldots , x_{n} ^{-1}], \quad \text{ for } \ \textbf{a} =(a_1, \ldots , a_n) \in ({{\mathbb {C}}} \setminus {{\mathbb {Z}}})^n. \end{aligned}$$

It was proved in [AMPP20, Section 6] that the simple affine vertex algebra $V_{-1/2} (spo(2r \vert s))$ is realized as a fixed point subalgebra $M_{(s \vert 2 r)}^+$ of $M_{(s \vert 2 r)}$. In particular, for ${\mathfrak {g}}=spo(2|3)$, we have

$$\begin{aligned} V_{-1/2} ({\mathfrak {g}}) = M_{(3 \vert 2 )}^+. \end{aligned}$$

From Proposition 4.4 it follows that

$$\begin{aligned} V_{-m/4} ({\mathfrak {g}}) = \frac{ V^{-m/4} ({\mathfrak {g}}) }{ V^{-m/4} ({\mathfrak {g}})((x_{\theta _1})_{(-1)})^{m+1}\textbf{1}}. \end{aligned}$$

This easily implies that

$$\begin{aligned} V_{-m/4 }({\mathfrak {g}}) \hookrightarrow V_{-1/2} ({\mathfrak {g}}) ^{\otimes \tfrac{m}{2} } \hookrightarrow M_{(3 \vert 2 )}^{\otimes \tfrac{m}{2}} \cong M_{( \tfrac{3m}{2} \vert m)}. \end{aligned}$$

In this way, we get a non-zero vertex algebra homomorphism $\Phi ^{(m)}: V_{-m/4 }({\mathfrak {g}}) \rightarrow M_{( \frac{3\,m}{2} \vert m)}$ for each $m\ge 4$ even. Now using positive energy modules $\widetilde{U_m}(\textbf{a})$ for $M_{(m)}$, we construct $M_{( \frac{3\,m}{2} \vert m)}$–modules $F_{( \frac{3\,m}{2}) } \otimes \widetilde{U_m}(\textbf{a})$, which by restriction become $V_{-m/4 }({\mathfrak {g}})$–modules. Since their top components are not highest weight ${\mathfrak {g}}$–modules, we have constructed a family of non highest weight, positive energy $V_{-m/4 }({\mathfrak {g}})$–modules. Interestingly, these modules are still integrable for ${\widehat{{\mathfrak {g}}}}^\natural$. We shall present details and irreducibility analysis in our forthcoming publications.

7.2 Non-positive energy modules over $W_k^{min}({\mathfrak {g}})$

For a conformal vertex algebra V let ${\mathcal {E}}(V)$ be the category of all (weak) V–modules, on which $L_0$ is diagonalizable, and let ${\mathcal {E}}^+(V)$ be the category of positive energy modules in ${\mathcal {E}}(V)$. Set ${\mathcal {E}}_k = {\mathcal {E}} (W_k^{min}({\mathfrak {g}}))$, ${\mathcal {E}}_k ^+ = {\mathcal {E}} ^+(W_k^{min}({\mathfrak {g}}))$, where k is from the unitary range for ${\mathfrak {g}}$.

A $W_k^{min}({\mathfrak {g}})$–module M is called a weight $W_k^{min}({\mathfrak {g}})$–module if it is a weight module for $\widehat{{\mathfrak {g}}}^{\natural }$.

Let ${\mathcal {E}}_k^{fin}$ be the subcategory of ${\mathcal {E}}_k$ consisting of weight modules with finite multiplicities.

Theorem 6.9 shows that each irreducible module in $\mathcal E_k ^+$ belongs to the category $\mathcal {E}_k ^{fin}$. One very interesting question is to see if there are weight representations of $W^{min}_k({\mathfrak {g}})$ outside of the category ${\mathcal {E}}_k ^+$. We believe that there are no such modules in $\mathcal {E}_k ^{fin}$:

Conjecture 7.1

The irreducible highest weight $W_k^{\min }({\mathfrak {g}})$-modules are all the irreducible representations in the category $\mathcal {E}_k ^{fin}$.

However, weight modules which are not in ${\mathcal {E}}_k ^+$ do exist. In order to see this, we use Kac–Wakimoto free field realization of $W_{min}^k({\mathfrak {g}})$ [KW04, Theorem 5.2], which gives a vertex algebra homomorphism

$$\begin{aligned} \Psi :W^k_{\min }({\mathfrak {g}})\rightarrow {\mathcal {H}} \otimes \mathbb V^k({\mathfrak {g}}^\natural )\otimes F({\mathfrak {g}}_{1/2}), \end{aligned}$$

where

• ${\mathcal {H}}$ is the Heisenberg vertex algebra of rank 1 generated by a field a such that $[a_{\lambda } a] = \lambda$;

• ${\mathbb {V}}^k({\mathfrak {g}}^{\natural }) = \bigotimes _{i\in S} V^{M_i(k)+\chi _i}({\mathfrak {g}}^{\natural }_i)$;

• $F({\mathfrak {g}}_{1/2})$ is a fermionic vertex algebra.

Lemma 7.2

There is a embedding of vertex algebras $\Phi : \bigotimes _{i\in S}V_{-\chi _i} ({\mathfrak {g}}_i^{\natural }) \rightarrow F({\mathfrak {g}}_{1/2})$ uniquely determined by

$$\begin{aligned} b \mapsto \tfrac{1}{2} \sum _{ \alpha \in A} : \Phi ^{\alpha } \Phi _{[u_{\alpha }, b]}:,\quad b\in {\mathfrak {g}}^\natural . \end{aligned}$$

(7.2)

Proof

We start with Kac–Wakimoto free field realization [KW04, Theorem 5.1]

$$\begin{aligned} \Psi ':W^k_{\min }({\mathfrak {g}})\rightarrow \mathcal V=V^{1}({\mathbb {C}}a)\otimes V^{\alpha _k}({\mathfrak {g}}^\natural )\otimes F({\mathfrak {g}}_{1/2}), \end{aligned}$$

where

$$\begin{aligned} \alpha _k(a, b)=\delta _{i,j} (M_i(k)+\chi _i)(a|b)_i^\natural \text { for }a\in {\mathfrak {g}}^\natural _i, b\in {\mathfrak {g}}^\natural _j,\ i,j\ge 0. \end{aligned}$$

Choose $\mu \in {\mathbb {C}}$ such that the ${\mathcal {V}}$-invariant Hermitian form on $M(\mu )\otimes V^{\alpha _k}({\mathfrak {g}}^\natural )\otimes F({\mathfrak {g}}_{1/2})$ is invariant for the action of $W^k_{\min }({\mathfrak {g}})$. Since $\Psi '(J^{\{b\}})\in V^{\alpha _k}({\mathfrak {g}}^\natural )\otimes F({\mathfrak {g}}_{1/2})$, it follows that the map $J^{\{b\}}\mapsto \Psi '(J^{\{b\}})_{(-1)}(v_\mu \otimes \textbf{1}\otimes \textbf{1}),\,b\in {\mathfrak {g}}_i^\natural$, extends to a map

$$\begin{aligned} \bigotimes _{i\in S}V^{M_i(k)}({\mathfrak {g}}^\natural _i)\rightarrow {\mathbb {C}}v_\mu \otimes V_{\alpha _k}({\mathfrak {g}}^\natural )\otimes F({\mathfrak {g}}_{1/2}) \end{aligned}$$

(7.3)

that, identifying ${\mathbb {C}}v_\mu \otimes V_{\alpha _k}({\mathfrak {g}}^\natural )\otimes F({\mathfrak {g}}_{1/2})$ with $V_{\alpha _k}({\mathfrak {g}}^\natural )\otimes F({\mathfrak {g}}_{1/2})$ in the obvious way, is a vertex algebra homomorphism. Now plug $\alpha _i(k)=0$, i.e. $M_i(k)=-\chi _i$, into (7.3). We get the homomorphism $\Phi ':\bigotimes _iV^{-\chi _i} ({\mathfrak {g}}_i^{\natural }) \rightarrow F({\mathfrak {g}}_{1/2})$ explicitly given by (7.2). Since $F({\mathfrak {g}}_{1/2})$ is unitary, it is completely reducible, hence the image of $\Phi '$ is simple. Therefore $\Phi '$ descends to $\Phi$. $\square$

The map $\Psi '$ induces a map $\Psi : W^k_{\min }({\mathfrak {g}})\rightarrow V^{1}({\mathbb {C}}a)\otimes V_{\alpha _k}({\mathfrak {g}}^\natural )\otimes F({\mathfrak {g}}_{1/2})$. We first prove that the image of $\Psi$ is simple.

Proposition 7.3

Assume that $M_i(k) + \chi _i \in {{\mathbb {Z}}}_{\ge 0}$ for all i. Then $\Psi (W^k_{\min }({\mathfrak {g}}))=W_k^{\min }({\mathfrak {g}})$.

Proof

As shown in the proof of Theorem 4.1, the vectors $v_i$ generate the maximal ideal in $W^k_{\min }({\mathfrak {g}})$ (although $v_1$ is not a singular vector when ${\mathfrak {g}}=spo(2|3)$). Hence it suffices to check that in $V^{1}({\mathbb {C}}a)\otimes V_{\alpha _k}({\mathfrak {g}}^\natural )\otimes F({\mathfrak {g}}_{1/2})$ we have:

$$\begin{aligned} (J^{\{x_{\theta _i}\}}_{(-1)})^{M_i(k)+1}\textbf{1}=0 \quad i \ge 1. \end{aligned}$$

From Kac–Wakimoto free field realization and Lemma 7.2 we get that the map $b \mapsto b + \tfrac{1}{2} \sum _{ \alpha \in A}: \Phi ^{\alpha } \Phi _{[u_{\alpha }, b]}:$ induces a homomorphism

$$\begin{aligned} \Theta :=\Phi \circ \Psi _{|V^{M_i(k) } }: V^{M_i(k) } ({\mathfrak {g}}^{\natural } _i) \rightarrow V_{M_i(k) + \chi _i }({\mathfrak {g}}^{\natural } _i) \otimes V_{-\chi _i} ({\mathfrak {g}}^{\natural } _i) \subset V_{\alpha _k}({\mathfrak {g}}^\natural )\otimes F({\mathfrak {g}}_{1/2}). \end{aligned}$$

Since $M_i(k) + \chi _i \in {{\mathbb {Z}}}_{\ge 0}$, the vertex algebra $V_{\alpha _k}({\mathfrak {g}}^\natural )\otimes F({\mathfrak {g}}_{1/2})$ is unitary, hence it is completely reducible, so $Im\,\Theta$ is simple and in turn $(J^{\{x_{\theta _i}\}}_{(-1)})^{M_i(k)+1}\textbf{1}=0$. $\square$

We have therefore proved the following result.

Theorem 7.4

If k is non-collapsing and lies in the unitary range, then $\Psi$ induces a non-trivial homomorphism of vertex algebras

$$\begin{aligned} {\widetilde{\Psi }}:W_k^{\min }({\mathfrak {g}})\rightarrow {\mathcal {H}} \otimes \mathbb V_k({\mathfrak {g}}^\natural )\otimes F({\mathfrak {g}}_{1/2}), \end{aligned}$$

where ${\mathbb {V}}_k({\mathfrak {g}}^{\natural }) = \bigotimes _{i\in S} V_{M_i(k)+\chi _i}({\mathfrak {g}}^{\natural }_i)$.

Since ${\mathbb {V}}_k({\mathfrak {g}}^{\natural })$ and $F({\mathfrak {g}}_{1/2})$ are rational vertex algebras, they don’t have non-positive energy modules. But the Heisenberg vertex algebra ${\mathcal {H}}$ is not rational and it has very rich representation theory:

• (fin) The category of ${\mathcal {H}}$–modules with finite-dimensional weight spaces ${\mathcal {E}} ^{fin} ({\mathcal {H}})$ is semi-simple and every irreducible ${\mathcal {H}}$–module in that category is of highest weight. The category ${\mathcal {E}} ^{fin} ({\mathcal {H}})$ concides with the category of modules in $\mathcal E^+ ({\mathcal {H}})$ of finite length.

• (inf) The category ${\mathcal {E}}({\mathcal {H}})$ admits irreducible ${\mathcal {H}}$–modules with infinite-dimensional weight spaces [FGM14].

Remark 7.5

Using Theorem 7.4 and the representation theory of the Heisenberg vertex algebra ${\mathcal {H}}$ one can prove the following facts: assume that $E_1$ is an irreducible module in $\mathcal E({\mathcal {H}})$. Let $E_2$ be a ${\mathbb {V}}_k({\mathfrak {g}}^{\natural })$-module. Then we have:

• (1) $E_1 \otimes E_2 \otimes F({\mathfrak {g}}_{1/2})$ lies in $\mathcal E_k$. It lies in the category ${\mathcal {E}}^+ _k$ if and only if $E_1$ lies in the category ${\mathcal {E}}^+({\mathcal {H}})$.

• (2) If $E_1$ is a weight ${\mathcal {H}}$-module with infinite-dimensional weight spaces then $E_1 \otimes E_2 \otimes F({\mathfrak {g}}_{1/2})$ is a non-positive energy weight $W_k^{min}({\mathfrak {g}})$–module with infinite-dimensional weight spaces.

• (3) If $E_1$ is a non-weight ${\mathcal {H}}$-module, then $E_1 \otimes E_2 \otimes F({\mathfrak {g}}_{1/2})$ is a non-weight $W_k^{min}({\mathfrak {g}})$-module. In particular, we can construct analogs of non-weight $V_k({\mathfrak {g}})$–modules from Remark 3.3.

Further details on applications of Theorem 7.4 and Remark 7.5 will appear elsewhere.