DEFINING RELATIONS FOR MINIMAL UNITARY QUANTUM AFFINE W-ALGEBRAS

. 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.


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.
Quantum affine W -algebras W k (g, x, f ), are simple vertex algebras constructed in [KRW03,KW04], starting from a datum (g, x, f ) and k ∈ C.Here g = g 0⊕g 1 is a basic Lie superalgebra, i.e. g is simple, its even part g 0 is a reductive Lie algebra and g carries an even invariant non-degenerate supersymmetric bilinear form (.|.), x is an ad-diagonalizable element of g 0 with eigenvalues in 1 2 Z, f ∈ g 0 is such that [x, f ] = −f and the eigenvalues of ad x on the centralizer g f of f in g are non-positive, and k is non-critical, i.e. k = −h ∨ , where h ∨ is the dual Coxeter number of g.Recall that W k (g, x, f ) is the unique simple quotient of the universal W -algebra, denoted by W k (g, x, f ), which is freely strongly generated by elements labeled by a basis of the centralizer of f in g [KW04].
In the paper [KMP23] we focused on minimal data (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 s, isomorphic to sl(2).The Virasoro, Neveu-Schwarz and N = 2 algebras cover all minimal Walgebras (associated with g = sl(2), spo(2|1), sl(2|1) respectively), such that the 0-eigenspace 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 g 0 is not abelian, and we found all non-critical k ∈ C for which the simple minimal W -algebras W k (g, x, f ), denoted henceforth by W min k (g), are unitary, along with the (partly conjectural) classification of unitary highest weight modules over their universal covers W k min (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 min k (g) is the simple affine vertex algebra attached to the centralizer g ♮ in g 0 of s.If k is collapsing, then the unitarity of W min k (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 g, namely 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 min k (g) generated by g ♮ is integrable when viewed as a g ♮ -module.
The analysis of the unitarity of W min k (g) developed in [KMP23] led to the more general problem of classifying unitary highest weight modules over W min k (g); these results, regarding modules in the Neveu-Schwarz sector, are summarized in Section 2.
It was conjectured in [KMP23] that actually any unitary highest weight W k min (g)-module descends to W min k (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 (g) for these k (Theorem 3.1).We also find generators for the maximal ideal I k of W k min (g), so that W min k (g) = W k min (g)/I k (Theorem 4.1).Finally, for all non-critical k ∈ C in the unitarity range, we classify all irreducible highest weight modules over W min k (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 min k (g) all irreducible positive energy modules are highest weight modules (Theorem 6.9), whereas this statement does not hold for V k (g) (Remark 3.3).In the final section, we show examples of positive energy modules over V k (g) which are not highest weight and we outline an approach to more general representations of W min k (g), showing the existence of non-positive energy modules.
In the present paper we keep notation and terminology of [KMP23]; in particular, N and Z + denote the sets of positive and non-negative integer numbers, respectively.The base field is C.

Setup
2.1.Minimal W -algebras.Let g = g 0 ⊕ g 1 be a simple finite-dimensional Lie superalgebra over C with a reductive even part g 0 and a non-degenerate even supersymmetric invariant bilinear form (.|.).They are classified in [K77].Let s = {e, x, f } ⊂ g 0 be a minimal sl(2)triple, i.e. [x, e] = e, [x, f ] = −f, [e, f ] = x and the ad x gradation of g has the form Denote by g ♮ the centralizer of s in g 0 .Let W k min (g) be the associated to s and k ∈ C universal quantum affine W -algebra [KW04].This vertex algebra is strongly generated by fields J {a} , a ∈ g ♮ , G {v} , v ∈ g −1/2 , L of respective conformal weight 1, 3/2, 2 and λ-brackets explicitly given in [KW04].We normalize the bilinear form (.|.) by the condition (x|x) = 1 2 , and denote by h ∨ the corresponding dual Coxeter number.We shall assume throughout the paper that k = −h ∨ , i.e. k is non-critical.Then the vertex algebra W k min (g) has a unique maximal ideal I k , and we let W min k (g) = W k min (g)/I k be the corresponding simple vertex algebra.

Unitary conformal vertex algebras.
Let V be a conformal vertex algebra with conformal vector L = n∈Z L n z −n−2 .Let φ be a conjugate linear involution of V .A Hermitian form H( ., . ) on V is called φ-invariant if, for all a ∈ V , one has [KMP22] (2.2) Here the linear map A(z) : V → V ((z)) is defined by We say that V is unitary if there is a conjugate linear involution φ of V and a φ-invariant Hermitian form on V which is positive definite.

Unitary minimal W -algebras and unitary highest weight modules.
In classifying the unitary minimal W -algebras W min k (g), one first considers the levels k when W min k (g) is either C or an affine vertex algebra.Such levels are called collapsing levels.The triples (g, s, k) such that W min k (g) = C are described in Proposition 3.4 of [AKMPP18].Corollary 7.12 of [KMP22] provides the list of triples (g, s, k) such that W min k (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 (g) is unitary and k is not a collapsing level, then the parity of g is compatible with the ad x-gradation, i.e. the parity of the whole subspace g j is 2j mod 2. It follows from [KRW03], [KW04] that for each basic simple Lie superalgebra g there is at most one minimal datum (g, s) that is compatible with parity, and the complete list of the g which admit such a datum is as follows: sl(2|m) for m ≥ 3, psl(2|2), spo(2|m) for m ≥ 0, osp(4|m) for m > 2 even, D(2, 1; a), F (4), G(3).(2.5) Remark 2.1.Recall from [K77] that in the case of D(2, 1; a), the parameter a ranges over C \ {0, −1} (if a = 0, −1 the superalgebra is not simple); moreover the symmetric group S 3 generated by the tranformations a → −1−a, a → 1/a induces Lie superalgebra isomorphisms.
The even part g 0 of g in these cases is g ♮ ⊕s with g ♮ a reductive Lie algebra.In Propositions 7.1.and 7.2 of [KMP23] it is proven that the conjugate linear involutions of W min k (g) that fix the Virasoro vector L are in one-to-one correspondence with the conjugate linear involutions φ of g that fix pointwise the triple {e, x, f }.It is easy to see that, in order to have W min k (g) unitary, φ |g ♮ 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 φ ac exists in all the cases listed in (2.5).Since the cases 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 g from consideration.For the other cases, Theorem 1.4 of [KMP23] lists all the pairs (g, k) for which the φ ac -invariant hermitian form on W min k (g) is positive definite, making W min k (g) a unitary vertex algebra.We say that k belongs to the unitary range if k is not collapsing and W min k (g) is a unitary vertex algebra.
We have [KMP23] (2.6) where g ♮ i are simple summands of g ♮ , and either S = {1} or S = {1, 2}; the latter happens only for D(2, 1; m n ).Let h ♮ be a Cartan subalgebra of g ♮ , then h = h ♮ + Cx is a Cartan subalgebra of g 0 and of g.Let θ ∈ h * be the root of e, so that (θ|θ) = 2, and denote by θ i ∈ (h ♮ ) * the highest root of g ♮ i .Denote by M i (k) the levels of the affine Lie algebras g ♮ i in W k min (g).Then we have for i ∈ S [KMP23] where ξ ∈ (h ♮ ) * is the highest weight of the g ♮ -module g −1/2 (this module is irreducible, except for g = psl(2|2), when its two irreducible components turn out to be isomorphic).In [KMP23] it is shown that χ i = −2 when g = spo(2|3), and χ i = −1 otherwise.
In Table 1 we describe the unitary range in all cases, along with the subalgebras g ♮ , the numbers M i (k), and the dual Coxeter numbers h ∨ .
Table 1.Unitarity range, g ♮ , M i (k), and h ∨ We also make a specific choice for a set of simple roots Π of g in each case.In Table 2 we list our choice of Π = {α 1 , . ..} of g, ordered from left to right, as well as the invariant bilinear form (.|.) on h * , and the root θ.
Table 2. Simple roots Π, invariant form (.|.), and the highest root θ of g Let ∆ ♮ be the set of roots of (g ♮ , h ♮ ).We made our choice of Π so that Π ♮ = Π ∩ ∆ ♮ is a set of simple roots for g ♮ .Write g ♮ = n ♮ − ⊕ h ♮ ⊕ n ♮ + for the triangular decomposition of g ♮ corresponding to choosing Π ♮ as a set of simple roots.Note that α 1 is an isotropic root, and that we have (2.9) (α 1 ) |h ♮ = −ξ.
We parametrize the highest weight modules for W k min (g) following Section 7 of [KW04].For ν ∈ (h ♮ ) * and ℓ 0 ∈ C, let L W (ν, ℓ 0 ) denote the irreducible highest weight W k min (g)-module with highest weight (ν, ℓ 0 ) and highest weight vector v ν,ℓ 0 .This means that one has + .Let P + ⊂ (h ♮ ) * be the set of dominant integral weights for g ♮ and let (2.10) For ν ∈ P + , introduce the following number [KMP23] (2.12) where 2ρ ♮ is the sum of positive roots of g ♮ .The main result of [KMP23] is the following theorem.
(1) This module can be unitary only if the following conditions hold: and equality holds in Applying this theorem to L W (0, 0) = W min k (g) one recovers the unitarity range displayed in Table 1.

A technical result.
Let ∆ be the set of roots of g and let Π = {α 0 = δ − θ, α 1 , . ..} be a set of simple roots of the affine Lie superalgebra g.For an isotropic root β ∈ Π, we denote by r β ( Π) the set of simple roots in the set ∆ of roots of g obtained by the corresponding odd reflection.We denote by x α a root vector of g, attached to α ∈ ∆, and by w. the shifted action of W : w.λ = w(λ + ρ) − ρ.
Lemma 2.4.[KMP23,Lemma 11.3] Let Π ′ be a set of simple roots for ∆.Let M be a g-module and assume that m ∈ M is a singular vector with respect to Π ′ .If α ∈ Π ′ is an isotropic root and x −α m = 0, then x −α m is a singular vector with respect to r α ( Π ′ ).

Classification of irreducible highest weight representations of V k (g) in the unitary range
Denote by V k (g) and V k (g) the universal and simple affine vertex algebras of level k associated to g. Denote by L(λ) the irreducible highest weight Note that every highest weight module for V k (g) has highest weight ν h for some ν ∈ (h ♮ ) * and h ∈ C.
Theorem 3.1.Let k be in the unitary range.Then, up to isomorphism, the irreducible highest weight V k (g)-modules are as follows: (1) 1.By a case-wise verification one shows that the roots η i := δ − θ i are in Π ′ and, by (2.7) and (2.8), Hence the vector (x θ i ) Proof of Theorem 3.1.In [GK15, Sections 4 and 6] the characters of highest weight g-modules with highest weight kΛ 0 have been computed using only their integrability with respect to g ♮ , which implies that such modules are irreducible.It was deduced from this in [GS18, Theorem 5.3.1]that, if V k (g) is integrable as a g ♮ -module, then the V k (g)-modules, which are integrable over g ♮ , descend to V k (g).
By Lemma 3.2, we are left with proving that the modules listed in (1), (2) are g ♮ -integrable.Let v be a highest weight vector for L( ν h ).

Case (1):
v is a singular vector with respect to the set of simple roots r α 0 +α 1 r α 1 ( Π).Moreover, by (2.7),(2.8),and (2.9) we have for i ∈ S and we conclude as in the previous case.
We are left with proving that any irreducible highest weight module for V k (g) is of the form (1) or (2).Let L( ν h ) be an irreducible highest weight V k (g)-module.We prove that, necessarily, ν ∈ P + k .Indeed, the action of g ♮ on a highest weight vector v should be locally finite, so that ν v is a highest weight vector with respect to the set of simple roots Remark 3.3.All modules listed in Theorem 3.1 are of positive energy (the definition is recalled in Section 6), but there might exist positive energy V k (g)-modules outside of this list.In § 7.1 we present arguments for this claim in the case g = spo(2|3), k = −m/4, and m ≥ 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 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 (g)-module L( ν h ) with ν ∈ P + k non-extremal and h ∈ C arbitrary, from Theorem 3.1.Then L( ν h ) top contains an irreducible g 0-submodule E = U (g 0).v,where v is the highest weight vector of L( ν h ).Next we consider the Kac module E 1 , which is a suitable quotient of Ind g g0 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 (g)).Using Zhu's functor we conclude the following: • the module L( ν h ) is indecomposable and has an irreducible subquotient isomorphic to L( ν h ).
Hence non-highest weight positive energy modules for V k (g) exist.But still these modules are weight modules with finite-dimensional weight spaces.
Let us mention one important consequence.Take h ∈ 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 (g)), and therefore the category KL k [AMP22] is not semisimple.In the case g = spo(2|3), one can show that the category KL −m/4 (g) is not semisimple for m ≥ 4. 4. Explicit description of the maximal ideal of W k min (g) Let, as before, Π = {α 1 , . ..} be the set of simple roots for g given in Table 2, and I k be the maximal ideal of W k min (g).Also denote by J k the maximal ideal of V k (g).Set (4.1) If an irreducible highest weight W k min (g)-module L W (ν, ℓ 0 ) is unitary, then, restricted to the affine subalgebra V β k (g ♮ ) (see [KMP23,(7.4),(7.5)] for the definition of this subalgebra), it is unitary, hence a direct sum of irreducible integrable highest weight g ♮ -modules of levels M i (k), i ∈ S.But it is well-known [FZ92], [KWang92] that all these modules descend to the simple affine vertex algebra V β k (g ♮ ), and are annihilated by the elements v i .In particular, applying this argument to W min k (g) = L W (0, 0), we deduce that Theorem 4.1.Let k be in the unitary range.The maximal ideal I k is generated by the singular vector and by the singular vectors v i , i ∈ S (cf.(4.1)) in the other cases.
Introduce the following vectors in V k (g), where i ∈ S: Proposition 4.3.The vectors s i are singular in the universal affine vertex algebra V k (g) and generate J k .
Proof.Recall that α 0 = δ − θ.Since (x α 1 ) (0) 1 = 0, 1 is singular also for r α 1 ( Π).Since (x θ−α 1 ) (−1) 1 is nonzero, it follows from Lemma 2.4 that ( Since δ − θ i is in Π ′ and, by (2.7) and (2.8), we see that w i is singular for Π ′ .Since V k (g)/ w i is integrable with respect to g ♮ , 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 (g).The weight of u i is kΛ 0 − s ′ η i , where We now observe that the fact that u i = 0 implies that w i is a singular vector for Having shown that w i are singular for r α 1 ( Π), it follows that the s i are either zero or singular vectors for Π. Let Since w i are singular vectors for r α 1 ( Π), we see that It follows that (x −α 1 ) (0) s i are nonzero multiples of w i , hence the s i generate the maximal proper ideal of V k (g), since the w i do.
Proof of Theorem 4.
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, g is of type D(2, 1; m n ) and k = mn m+n q, q ∈ N. H 0 (J k ) is the sum of two highest weight modules of weights (mqθ 1 , mq), (nqθ 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 We can clearly assume that the v's appearing are root vectors; let β v be the corresponding root, so that to be the pair (η v , −s); likewise, if u is a root vector of g ♮ , and η u is the corresponding root, define the weight of J {u} r as (η u , −r).Finally, declare that the weight of L n is (0, −n).Let ∆ W be the set of weights.Note that any element of ∆ W is a positive integral linear combination of Π W = {(−α 2 , 0), (−α 3 , 0), ( Since a 1 is negative, we have the required contradiction.

g = spo(2|3).
Introduce the following vector (4.11) Proposition 4.4.The vector r 1 is singular in the universal affine vertex algebra V k (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 (g).Let µ be weight of w 1 .An explicit calculation shows that (µ|α 0 + α 1 ) = −k − 1 2 = 0, hence u 1 = 0, and it is singular for r α 1 Π.So r 1 is either 0 or singular for Π.The first possibility does not occur, since Proof of Theorem 4.1 for g = spo(2|3).Arguing as in the previous subsection, by Proposition 4.4 it follows that the maximal ideal in W k min (g) generated by a singular vector v of weight (2(m − 2)ω 1 , m − 3/2), where ω 1 is the fundamental weight for sl 2 and m = M 1 (k) + 2. We observe that also the weight of v 1 (cf.(4.3)) is (2(m − 2)ω 1 , m − 3/2).Moreover, using (4.2) and the relations we see that v 1 ∈ I k , hence it is a multiple of v, so it is singular and generates I k . 5. Descending from W k min (g) to W min k (g) Let H 0 be the quantum Hamiltonian reduction functor from the category O k of g-modules of level k to the category of W k min (g)-modules [KW04].By [Ar05], it is exact.As in (3.1), for ν and L W (ν, ℓ 0 ) is the irreducible highest weight W k min (g)-module with highest weight (ν, ℓ 0 ) [KMP23], [KW04].
Theorem 5.1.Let k be in the unitary range.Then all irreducible highest weight W k min (g)modules L W (ν, ℓ 0 ) with ℓ 0 ∈ C when ν ∈ P + k is not extremal, and A simple application of Theorem 5.1 is the proof of Conjecture 4 in [KMP23].
6. Classification of irreducible highest weight representations of W min k (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 min k (g).Remark 6.2.Combining (5.2) and Theorem 6.1, we have proven that the irreducible highest weight modules of W min k (g) are precisely the non-zero images of the irreducible V k (g) under Hamiltonian reduction.
We need to recall some well-known facts about Zhu algebras in the super case [KWang92].
V n be a conformal vertex algebra, graded by the eigenspaces of L 0 , with the parity p(V n ) ≡ 2n mod 2. For a ∈ V n we write deg a = n.Define bilinear maps Denote by O(V ) ⊂ V the C-span of elements of the form a • 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 ∈ V in Zhu(V ), then for all a, b ∈ V 0 the following relations hold: 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 + R ≥0 for some h ∈ C: 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) = j a M j z −j−deg a one associates a M 0 |M h .By the above construction of the Zhu algebra, it follows from [KW04, Theorem 7.1] that (6.2) Under the correspondence between irreducible positive energy W k min (g)-modules and irreducible modules over its Zhu algebra, the module L W (ν, h) goes to the irreducible highest weight g ♮ -module with highest weight ν on which L acts by the scalar h, which we denote by V (ν, h).It follows from (6.2) that (6.3) where J(g) is a 2-sided ideal of the associative algebra C[L] ⊗ U (g ♮ ).So any non-zero element in J(g) imposes a condition on the highest weight (ν, h) of the Zhu(W min k (g))-module V (ν, h).
Proof of Theorem 6.1: the case g = spo(2|3).First we present a proof in the case g = spo(2|3), which gives a motivation for the proof in the general case.
Denote by L k [j, q] the irreducible highest weight W k -module of level k generated by a highest weight vector v j,q , such that for n ∈ Z ≥0 : Here we use notation for the generators of W k as in [KW04, Section 8.5].Note that U (sl(2))v j,q = V (jω 1 ) is the irreducible highest weight sl(2)-module with highest weight jω 1 .
Proof.By Theorem 4.1, we have that ( 0 acts as a derivation, using (6.1) we get that Using λ-bracket formulas from [KW04, Section 8.5] we get that the following relations hold in W k : Substituting (6.6), (6.7), (6.8) into (6.5) and collecting [J − ] m−3 , we get (6.4),where Proposition 6.4.Let L k [j, q] be an irreducible highest weight W k -module.Then it is isomorphic to exactly one of the following modules: where V (jω 1 ) is the (j + 1)-dimensional irreducible sl(2)-module with highest weight jω 1 , and acts non-trivially on L k [j, q] top .Hence there exists w ∈ L k [j, q] top such that w ′ = (J − ) m−3 (0) w is a lowest weight vector for sl(2), i.e.
Then we have, by Lemma 6.3: hence, when j = m − 3 or m − 2, we have that q = j 4 .
Since the modules appearing in Theorem 6.1 in case of 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 g = spo(2|3).We first illustrate the strategy of the proof in the case g = psl(2|2).We use notation of [KW04,Section 8.4]. Set ) is a vertex subalgebra of W min k generated by J ± , J 0 .The odd generators of conformal weight 3/2 are G ± , G ± .By Theorem 4.1 the maximal ideal I k is generated by the singular vector (J + ) m+1 Using the definition of the Zhu algebra Zhu(W k min ), we have that for Using (6.10) and the λ-bracket formulas given in [KW04, Section 8.4], we get ], (6.12)Using (6.9), (6.11),(6.12)and (6.1), we find Then it is isomorphic to exactly one of the following modules: , where V (jω 1 ) is the (j + 1)-dimensional irreducible sl(2)-module with highest weight jω 1 , and ] is a highest weight vector for sl(2).We get by (6.10) This implies that for j = m we need to have ℓ 0 = m 2 .Proposition 6.5 proves, in particular, Theorem 6.1 for g = psl(2|2).We now deal with the general case g = spo(2|3).We shall see that a relation similar to (6.13) holds in Zhu(W k min (g)).We do not need a very precise expression for Ω: what is really relevant is the fact that the action of Ω gives a relation which is linear in ℓ 0 .
We start by observing that there exist two odd positive roots γ 1 , γ 2 such that (6.14) This fact can be verified directly by looking at Table 3.Using (6.14) and the explicit expression for [G {u} λ G {v} ] given in [AKMPP18, (1.1)] we find where c 1 , c 2 , c 3 are constants independent of k.Here a ♮ denotes the orthogonal projection of a ∈ g to g ♮ and {u s }, {u s } are basis of g 1/2 dual with respect to the bilinear form (6.17) Set x θ i −γ j = [x −γ j , x θ i ], j = 1, 2. Recall that , by [KW04, Theorem 2.1 (e)], if u ∈ g −1/2 and a ∈ g ♮ , then (6.18) [J Note that all the constants involved in (6.24) do not depend on k; moreover, by (6.14), (6.25) Substituting (6.23) in (6.22) we obtain Proposition 6.8.In Zhu(W k min (g)), we have Proof of Theorem 6.1.We have to prove that if L W (ν, ℓ 0 ) is a irreducible highest weight W min k (g)-module, then the pair (ν, ℓ 0 ) is among those listed in the statement.Note that there is a non-zero vertex algebra homomorphism Θ : If ν is not extremal, we are done.Assume therefore that ν is extremal.The action of the Zhu algebra on the top component V (ν, ℓ 0 ) of L W (ν, ℓ 0 ) is given by the action of the elements [J {a} ] on the irreducible finite-dimensional g ♮ -module V (ν, ℓ 0 ) of highest weight ν, while [L] acts as the multiplication by ℓ 0 .Consider the sl(2)- By sl(2)-theory applied to U (sl(2))v where v ∈ V (ν, ℓ 0 ) is a highest weight vector, we see that v is a non-zero multiple of x where, by (6.24), Γ i (ℓ 0 ) = −c 2 (k + h ∨ )ℓ 0 + b and b ∈ C is independent of ℓ 0 , and c 2 = 0 by (6.25).Since, by construction, [J {x θ i } ] M i (k) v ′ = 0, we find from (6.27) that Γ i (ℓ 0 ) = 0. Since L W (ν, A(k, ν)) is a representation of W min k (g), then Γ i (A(k, ν)) = 0.But Γ i is a linear relation, so ℓ 0 = A(k, ν) is the unique solution of the equation Γ i (ℓ 0 ) = 0.
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 min k (g)-modules are all the irreducible positive energy representations.
) is a semi-simple finite-dimensional associative algebra.Let M be an irreducible positive energy W min k (g)-module.Then the top component M top is an irreducible module for Zhu's algebra Zhu(W min k (g)).By using the embedding V k (g ♮ ) → W min k (g) we see that M top is also a Zhu(V k (g ♮ ))-module.Since [L] is a central element in Zhu(W min k (g)), it acts on M top as multiplication by a scalar, hence, by (6.3), M top remains irreducible when restricted to g ♮ and therefore it is an irreducible Zhu(V k (g ♮ ))-module.Since Zhu(V k (g ♮ )) 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 V k (g ♮ )-vector w, which is then a highest weight vector for the action of W min k (g).Therefore W min k (g)w is a highest weight submodule of M .Irreducibility of M implies that M = W min k (g)w is a highest weight W min k (g)-module.
7. On positive energy modules over V k (g) and W min k (g) By Theorem 6.9, the irreducible highest weight W min k (g)-modules are all the irreducible positive energy representations.In this section we show, by contrast, that V k (g) admits positive energy modules which are not highest weight, and we show that W min k (g) admits non-positive energy modules.
The V k (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 (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 (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 βγ ghost vertex algebra).These relaxed modules were previously studied in [RW, AW22,AdP19].
On the contrary, the vertex algebra W min k (g) does not have non-highest weight positive energy modules.The non-highest weight relaxed highest weight V k (g)-modules are either mapped to zero by quantum Hamiltonian reduction, so they don't contribute to W min k (g)modules; or they are mapped to highest weight W min k (g)-modules.This connection deserves a more detailed analysis in the future.
We shall see below that W min k (g) contains non-positive energy modules having all infinite dimensional weight spaces.7.1.Positive energy modules over V k (g) which are not highest weight.We follow the notation and results of [AMPP20, Section 6].Consider the superspace C m|2n equipped with the standard supersymmetric form •, • m|2n given in [K77].Let V = ΠC m|2n , where Π is the parity reversing functor.Let M (m|2n) be the universal vertex algebra generated by V with λ-bracket Let {e i } be the standard basis of V and let {e i } be its dual basis with respect to •, • (i.e. e i , e j = δ ij ).In this basis the λ-brackets are given by The vertex algebra M (m|n) is called the Weyl-Clifford vertex algebra and, in the physics terminology, the βγbc 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: 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 U n (a), whose top components are isomorphic to It was proved in [AMPP20, Section 6] that the simple affine vertex algebra V −1/2 (spo(2r|s)) is realized as a fixed point subalgebra M + (s|2r) of M (s|2r) .In particular, for g = spo(2|3), we have From Proposition 4.4 below it follows that This easily implies that In this way, we get a non-zero vertex algebra homomorphism Φ (m) : for each m ≥ 4 even.Now using positive energy modules U m (a) for M (m) , we construct M ( 3m 2 |m) -modules F ( 3m 2 ) ⊗ U m (a), which by restriction become V −m/4 (g)-modules.Since their top components are not highest weight g-modules, we have constructed a family of non highest weight, positive energy V −m/4 (g)-modules.Interestingly, these modules are still integrable for g ♮ .We shall present details and irreducibility analysis in our forthcoming publications.
7.2.Non-positive energy modules over W min k (g).For a conformal vertex algebra V let E(V ) be the category of all (weak) V -modules, on which L 0 is diagonalizable, and let E + (V ) be the category of positive energy modules in E where k is from the unitary range for g.
be the subcategory of E k consisting of weight modules with finite multiplicities.Theorem 6.9 shows that each irreducible module in E + k belongs to the category E f in k .One very interesting question is to see if there are weight representations of W min k (g) outside of the category E + k .We believe that there are no such modules in E f in k : Conjecture 7.1.The irreducible highest weight W min k (g)-modules are all the irreducible representations in the category E f in k .However, weight modules which are not in E + k do exist.In order to see this, we use Kac-Wakimoto free field realization of W k min (g) [KW04, Theorem 5.2], which gives a vertex algebra homomorphism Ψ : W k min (g) → H ⊗ V k (g ♮ ) ⊗ F (g 1/2 ), where • H is the Heisenberg vertex algebra of rank 1 generated by a field a such that [a λ a] = λ; ) is a fermionic vertex algebra.
The map Ψ ′ induces a map Ψ : ).We first prove that the image of Ψ is simple.Proposition 7.3.Assume that M i (k) + χ i ∈ Z ≥0 for all i.Then Ψ(W k min (g)) = W min k (g).Proof.As shown in the proof of Theorem 4.1, the vectors v i generate the maximal ideal in W k min (g) (although v 1 is not a singular vector when g = spo(2|3)).Hence it suffices to check that in V 1 (Ca) ⊗ V α k (g ♮ ) ⊗ F (g 1/2 ) we have: ).Since M i (k)+ χ i ∈ Z ≥0 , the vertex algebra V α k (g ♮ )⊗ F (g 1/2 ) is unitary, hence it is completely reducible, so Im Θ is simple and in turn (J {x θ i } (−1) ) M i (k)+1 1 = 0. We have therefore proved the following result.
Theorem 7.4.If k is non-collapsing and lies in the unitary range, then Ψ induces a nontrivial homomorphism of vertex algebras . Since V k (g ♮ ) and F (g 1/2 ) are rational vertex algebras, they don't have non-positive energy modules.But the Heisenberg vertex algebra H is not rational and it has very rich representation theory: (fin) The category of H-modules with finite-dimensional weight spaces E f in (H) is semisimple and every irreducible H-module in that category is of highest weight.The category E f in (H) concides with the category of modules in E + (H) of finite length.(inf) The category E(H) admits irreducible H-modules with infinite-dimensional weight spaces [FGM14].
Remark 7.5.Using Theorem 7.4 and the representation theory of the Heisenberg vertex algebra H one can prove the following facts: assume that E 1 is an irreducible module in E(H).Let E 2 be a V k (g ♮ )-module.Then we have: (1) E 1 ⊗ E 2 ⊗ F (g 1/2 ) lies in E k .It lies in the category E + k if and only if E 1 lies in the category E + (H).
(2) If E 1 is a weight H-module with infinite-dimensional weight spaces then E 1 ⊗ E 2 ⊗ F (g 1/2 ) is a non-positive energy weight W min k (g)-module with infinite-dimensional weight spaces.
(3) If E 1 is a non-weight H-module, then E 1 ⊗ E 2 ⊗ F (g 1/2 ) is a non-weight W min k (g)module.In particular, we can construct analogs of non-weight V k (g)-modules from Remark 3.3.
Further details on applications of Theorem 7.4 and Remark 7.5 will appear elsewhere.