Communications in Mathematical Physics Unitarity of Minimal W-Algebras and Their Representations I

Webegin a systematic studyof unitary representations ofminimalW -algebras. In particular, we classify unitary minimal W -algebras and make substantial progress in classification of their unitary irreducible highest weight modules. We also compute the characters of these modules.


Introduction
In the present paper we study unitarity of minimal W -algebras and of their representations. Minimal W -algebras are the simplest conformal vertex algebras among the simple vertex algebras W k (g, x, f ), constructed in [18,20], associated to a datum (g, x, f ) and k ∈ R. Here g = g0 ⊕ g1 is a basic Lie superalgebra, i.e. g is simple, its even part g0 is a reductive Lie algebra and g carries an even invariant non-degenerate supersymmetric bilinear form (.|.), x is an ad-diagonalizable element of g0 with eigenvalues in 1 2 Z, f ∈ g0 is such that 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 [20].
We also proved in [16,Proposition 7.4] that if φ is a conjugate linear involution of W k (g, x, f ), this vertex algebra carries a non-zero φ-invariant Hermitian form H (·, ·) for all k = −h ∨ if and only if (g, x, f ) is a Dynkin datum; moreover, such H is unique, up to a real constant factor, and we normalize it by the condition H (1, 1) = 1. A module M for a vertex algebra V is called unitary if there is a conjugate linear involution φ of V such that there is a positive definite φ-invariant Hermitian form on M. The vertex algebra V is called unitary if the adjoint module is.
For some levels k the vertex algebra W k (g, x, f ) is trivial, i.e. isomorphic to C; then it is trivially unitary. Another easy case is when W k (g, x, f ) "collapses" to the affine part. In both cases we will say that k is collapsing level.
In the case of a Dynkin datum let g be the centralizer of the sl 2 subalgebra s = span {e, x, f } in g0; it is a reductive subalgebra. If φ satisfies the first two conditions in (1.1), it fixes e, x, f , hence φ(g ) = g . It is easy to see that unitarity of W k (g, x, f ) implies, when k is not collapsing, that φ |[g ,g ] is a compact involution.
In the present paper we consider only minimal data (g, x, f ), defined by the property that for the ad x-gradation g = j∈ 1 2 Z g j one has g j = 0 if | j| > 1, and g −1 = C f. (1.2) In this case (g, x, f ) is automatically a Dynkin datum. The corresponding W -algebra is called minimal. The element f ∈ g is a root vector attached to a root −θ of g, and we shall normalize the invariant bilinear form on g by the usual condition (θ |θ) = 2, which is equivalent to (x|x) = 1 2 . Recall that the dual Coxeter number h ∨ of g is half of the eigenvalue of its Casimir element of g, attached to the bilinear form (.|.). We shall denote by W min k (g) the minimal W -algebra, corresponding to g and k = −h ∨ , and by W k min (g) the corresponding universal W -algebra. We proved in [16,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 2 j mod 2.
It follows from [18], [20] that for each basic simple Lie superalgebra g there is at most one minimal Dynkin datum, 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) for a ∈ C, F(4), G (3). (1. 3) The even part g0 of g in this case is isomorphic to the direct sum of the reductive Lie algebra g and s ∼ = sl 2 . One of our conjectures (see Conjecture 4 in Sect. 8) 1 states that any unitary W k min (g)module descends to W min k (g). In fact, it is tempting to conjecture that for any conformal vertex algebra V any unitary V -module descends to the simple quotient of V .
It turns out (cf. Proposition 7.2) that a conjugate linear involution of the universal minimal W -algebra W k min (g) at non-collapsing level k is necessarily induced by a conjugate linear involution φ of g. Moreover, by Proposition 8.9, if W k min (g) admits a unitary highest weight module and k is not collapsing, then g has to be semisimple. As explained above, the involution φ of g must be almost compact, according to the following definition. Indeed (i) is equivalent to the first two requirements in (1.1), and the third requirement in (1.1) follows from Lemma 3.1 in Sect. 3. So, in order to study unitarity of highest weight modules, it is not restrictive to assume that the conjugate linear involution of W k min (g) is induced by an almost compact conjugate linear involution of g.
We prove in Sects. 3 and 4 that an almost compact conjugate linear involution φ exists for all g from the list (1.3), except that a must lie in R in case of D(2, 1; a), and is essentially unique.
It was shown in [20] that the central charge of W min k (g) equals ( 1.4) Here is another useful way to write this formula: Recall that the most important superconformal algebras in conformal field theory are the simple minimal W -algebras or are obtained from them by a simple modification: (a) W min k (spo(2|N )) is the Virasoro vertex algebra for N = 0, the Neveu-Schwarz vertex algebra for N = 1, the N = 2 vertex algebra for N = 2, and becomes the N = 3 vertex algebra after tensoring with one fermion; it is the Bershadsky-Knizhnik algebra for N > 3; The following theorem is a result of several papers, published in the 80s in physics and mathematics literature, see e.g. [5] for references.
The above three cases cover all minimal W -algebras, associated with g, such that the eigenspace g 0 of ad x is abelian. Thus, we may assume that g 0 is not abelian.
In order to study unitarity of the simple minimal W -algebra W min k (g), one needs to consider the more general framework of representation theory of universal minimal Walgebras W k min (g). Of course, unitarity of W k min (g) is equivalent to that of W min k (g). It is therefore natural to study unitarity of irreducible W k min (g)-modules. For that purpose, we take, in Sect. 6, a long detour to develop a general theory of invariant Hermitian forms on modules over the vertex algebra of free bosons, which will be eventually applied to our main object of interest. As a byproduct we obtain a field theoretic version of the Fairlie construction, which yields explicit models of unitary representations of the Virasoro algebra for certain values of the highest weight (cf. [17, 3.4], Example 6.9).
We consider in Sect. 9 the free field realization : W k min (g) → V k = V k+h ∨ (Cx) ⊗ V α k (g ) ⊗ F(g 1/2 ) introduced in [20] (here V γ (a) denotes the universal affine vertex algebra associated to the Lie algebra a and to a 2-cocycle γ , α k is the 2-cocycle defined in (7.24), and F(g 1/2 ) is the fermionic vertex algebra "attached" to g 1/2 ). Let M(μ) be the Verma module of highest weight μ ∈ C for the bosonic vertex algebra V k+h ∨ (Cx) and consider the V k -module N (μ) = M(μ) ⊗ V α k (g ) ⊗ F(g 1/2 ). Applying to N (μ) results from Sect. 6, we obtain in Proposition 9.2 a generalization of the Fairlie construction to universal minimal W -algebras.
The conformal vertex algebras (W k min (g), L) and (V k , L(0)) (see (6.29)) both admit Hermitian invariant forms H (·, ·) W and H (·, ·) f ree , respectively. Unfortunately, the embedding is not conformal, i.e., (L) = L(0), in particular is not an isometry (which was erroneously claimed in [14]). So, though the vertex algebra V k is unitary, this does not imply the unitarity of W k min (g). A few explicit computations suggest the following conjecture, which we were unable to prove.
We start the study of unitary modules over minimal W -algebras in Sect. 8 by introducing the irreducible highest weight W k min (g)-modules L W (ν, 0 ) with highest weight (ν, 0 ), where ν is a real weight of g and 0 ∈ R is the minimal eigenvalue of L 0 . We prove that L W (ν, 0 ) admits a φ-invariant nondegenerate Hermitian form (unique up to normalization), see Lemma 8.1. In Sect. 8 we also determine necessary conditions for the unitarity of L W (ν, 0 ). Part of the necessary conditions is displayed in Proposition 8.5. They say that unitarity of L W (ν, 0 ) implies that the levels M i (k) of the affine Lie algebras g i in W k min (g) (given in Table 2, Sect. 7), where g i are the simple components of g , are non-negative integers, ν is dominant integral of levels M i (k), and the inequality (1.9) below holds. Proposition 8.8 provides a further necessary condition, which says that (1.9) must be an equality when ν is an "extremal" weight. See Theorem 1.3 (1) below for a precise statement.
In Sect. 10, using the generalization of the Fairlie construction, developed in Sect. 9, we prove a partial converse result: if M i (k) + χ i ∈ Z + , where χ i are negative integers, displayed in Table 2, and ν is dominant integral weight for g which is not extremal, then the W k min (g)-module L W (ν, 0 ) is unitary for l 0 sufficiently large, see Proposition 10.2. In Sect. 11 we prove our central Theorem 11.1, which claims that actually Proposition 10.2 holds for l 0 satisfying the inequality (1.9), provided that ν is not extremal. This is established by the following construction. Let g be the affinization of g. We introduce in (11.4) a highest weight module M( ν h ) over g, whose highest weight ν h depends on h ∈ C, with the following two properties (2) For the quantum Hamiltonian reduction functor H 0 , the W k min (g)-module H 0 (M( ν)) admits a Hermitian form, depending polynomially on h.
Let us state our main results. First of all, if g = sl(2|m) with m ≥ 3 or osp(4|m) with m ≥ 2 even, then none of the W k min (g)-modules L W (ν, 0 ) are unitary for a noncollapsing level k. For the remaining g from the list (1.3) the Lie algebra g is semisimple (actually simple, except for g = D(2, 1; a), when g = sl 2 ⊕sl 2 ). Let θ ∨ i be the coroots of the highest roots θ i of the simple components g i of g . Let 2ρ be the sum of positive roots of g , and let ξ be a highest weight of the g -module g −1/2 (this module is irreducible, except for g = psl(2|2) when it is C 2 ⊕ C 2 ). Let ν be a dominant integral weight for g and l 0 ∈ R. We prove the following theorem.
(1) This module can be unitary only if the following conditions hold: and equality holds in (1.9) (1.9). In other words, the necessary conditions of unitarity in Theorem 1.3 (1) are sufficient.
We were able to prove this conjecture only for g = psl(2|2) and spo(2|3), obtaining thereby a complete classification of unitary simple highest weight W k min (g)-modules in these two cases. Note that papers [3,4,21] respectively claim (without proof) these results.
Since ν = 0 is extremal iff k is collapsing, we obtain the following complete classification of minimal simple unitary W -algebras: . This result, along with all known results on unitarity of vertex algebras, leads to the following general conjecture.

Conjecture 3. A CFT type vertex operator algebra admitting a invariant Hermitian form and having a unitary module is unitary.
In the final Sect. 14 we provide character formulas for all unitary W k min (g)-modules L W (ν, 0 ), which are obtained by applying the quantum Hamiltonian reduction to the corresponding irreducible highest weight modules over the affinization g of g. There are two cases to consider. In the first case, called massive (or typical), when inequality (1.9) is strict, this character formula is easy to prove (see the proof of Proposition 11.5), which leads to the character formula (14.5). In the second case, called massless (or atypical), when the inequality (1.9) is equality, there is a general KW-formula for maximally atypical tame integrable g-modules, conjectured in [19] and proved in [7] for all g in question, except for g = D(2, 1; m n ), ν = 0, which leads to the character formula (14.6). Character formulas were also given in [4] (resp. [21]) for the N = 4 superconformal algebra (resp. for W k min (spo(2|3)), hence for the N = 3 superconformal algebra). The proofs given in these papers are incomplete since they assume that their list of singular vectors is complete and that in the usual argument of inclusion-exclusion of Verma modules subsingular vectors cancel out. Their formulas for both massive and massless representations coincide with (14.5) and (14.6), respectively.
In our next paper of this series we will study unitarity of twisted representations of minimal W -algebras.
Throughout the paper the base field is C, and Z + and N stand for the set of non negative and positive integers, respectively.

Basic Lie superalgebras.
Let g = g0 ⊕ g1 be a basic finite-dimensional Lie superalgebra over C as in (1.3). Choose a Cartan subalgebra h of g0. It is a maximal ad-diagonalizable subalgebra of g, for which the root space decomposition is of the form where ⊂ h * \ {0} is the set of roots. In all cases, except for g ∼ = psl(2|2), the root spaces have dimension 1. In the case g = psl(2|2) one can achieve this property by embedding in pgl(2|2) and replacing (2.1) by the root space decomposition with respect to a Cartan subalgebra of pgl(2|2), which we will do. Let + be a subset of positive roots and = {α 1 , . . . , α r } be the corresponding set of simple roots. We will denote by 0 , 1 , the sets of even and odd simple roots, respectively. For each α ∈ + choose X α ∈ g α and X −α ∈ g −α such that (X α |X −α ) = 1, . . , r } generates g, and satisfies the following relations (2. 2) The Lie superalgebrag on generators {e i , f i , h α i | i = 1, . . . , r } subject to relations (2.2) is a (infinite-dimensional) Z-graded Lie algebra, where the grading is defined by deg h α i = 0, deg e i = − deg f i = 1, with a unique Z-graded maximal ideal, and g is the quotient ofg by this ideal. We assume that (α i |α j ) ∈ R for all α i , α j ∈ .

Conjugate linear involutions and real forms.
In the above setting, given a collection of complex numbers = {λ 1 , . . . , λ r } such that λ i ∈ √ −1R if α i is an odd root and λ i ∈ R if α i is an even root, we can define an antilinear involution ω : g → g setting Since ω preserves relations (2.2), it induces an antilinear involution ofg, and, since ω preserves the Z-grading ofg, it preserves its unique maximal ideal, hence it induces an antilinear involution of g. Set σ α = −1 if α is an odd negative root and σ α = 1 otherwise, so that (X α |X −α ) = σ α . Let ξ α = sgn(α|α) if α is an even root, 1 i fα is an odd root.
Then in [8, (4.13), (4.15)] it is proven (using results from [9]), that one can choose root vectors X α in such a way that We shall call this a good choice of root vectors.

Invariant Hermitian forms on vertex algebras.
Let V be a conformal vertex algebra with conformal vector L = n∈Z L n z −n−2 (see [16] for the definition and undefined notation). 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 [16] H Here the linear map

8)
a stands for the L 0 -eigenvalue of a, and

The Almost Compact Conjugate Linear Involution of g
From now on we let g be a basic simple finite-dimensional Lie superalgebra such that where s ∼ = sl 2 and g is the centralizer of s in g. This corresponds to consider g as in Table 2 of [20]. We will also assume that g is not abelian; this condition rules out g = spo(2|m), m = 0, 1, 2. The explicit list is given in the leftmost column of Table 1. Note that sl(2|1) and osp(4|2) are missing there since sl(2|1) ∼ = spo(2|2) and osp(4|2) ∼ = D(2, 1; a) with a = 1, −2 or − 1 2 . First, we prove the simple lemma mentioned in the Introduction, which states that the first two conditions of (1.1) imply the third one.  Proof. Note that (φ(a)|φ(b)) is an invariant supersymmetric bilinear form as well, hence it is proportional to (a|b) since g is simple. Due to (3.2) these two bilinear forms coincide.
We now discuss the existence of an almost compact involution of g (see Definition 1.1). Proof. Choose a Cartan subalgebra t of g0. We observe that if we prove the existence of an almost compact involution φ for a special choice of {e, x, f }, then an almost compact involution exists for any choice of the sl 2 -triple. Indeed, if {e , x , f } is another sl 2triple, then there is an inner automorphism ψ of s mapping {e, x, f } to {e , x , f }, which extends to an inner automorphism of g. Therefore φ = ψφψ −1 is an almost compact involution for {e , x , f }. The construction of {e, x, f } and φ and the verification of properties (i)-(iii) in Definition 1.1 will be done in four steps: (1) make a suitable choice of positive roots for g with respect to t; (2) define φ by specializing (2.3); (3) construct {e, f, x} and verify that φ( f ) = f, φ(x) = x, φ(e) = e; (4) check that φ is a compact involution for g ; Step 1. We need some preparation. Let be the set of roots of g with respect to the Cartan subalgebra t ∩ g . Let {±θ } be the t ∩ s-roots of s. Then R0 = {±θ } ∪ is the set of roots of g0 with respect to t.
Let R be the set of roots of g with respect to t, let R + be the subset of positive roots whose corresponding set of simple roots S = {α 1 , . . . , α r } is displayed in Table 1.
Note that θ is the highest root of R. (3.5) If θ = r i=1 m i α i , then, by our special choice of + , we have either m i = 2 for exactly one odd simple root α i , or m i = m j = 1 for exactly two odd distinct simple roots α i , α j (this corresponds to the fact that R + is distinguished, in the terminology of [8]). By (2.4) we have φ fixes e, f, x. One checks directly that {e, f, x} is an sl 2 -triple.
Step 4. Endow g with the Z-grading which assigns degree 0 to h ∈ t and to e i and f i if α i is even, and degree 1 to e i and degree −1 to f i , if α i is odd. A direct check on Table 1 shows that q 0 = g . Recall from [8,Proposition 4.5] that the fixed points of φ in q 0 are a compact form of q 0 if and only if λ i (α i |α i ) < 0 for all α i ∈ S \ S 1 .

Explicit Expressions for Almost Compact Real Forms
In this section we exhibit explicitly an almost compact involution φ in each case and discuss its uniqueness. If φ is an almost compact involution of g, we denote by g ac the corresponding real form (the fixed point set of φ). We can define g ac by specifying a real form g ac 0 of g0 and a real form g ac 1 of g1. (1) g = spo(2|m). Then g0 = sl 2 ⊕ so m and g1 = C 2 ⊗ C m as g0-module. We set Explicitly, let B be a non-degenerate R-valued bilinear form of the superspace R 2|m with matrix Then for g = spo(2|m) we have: (2) g = psl(2|2). Let H be a C-valued non-degenerate sesquilinear form on the superspace C 2|2 whose matrix is diag( Then Explicitly, we have g0 = sl 2 ⊕ sl 2 and g1 = 0 B C 0 | B, C ∈ M 2,2 (C) as a g0-module. Theng (3) g = D(2, 1; a). Then g0 = sl 2 ⊕sl 2 ⊕sl 2 = so(4, C)⊕sl 2 and g1 = C 2 ⊗C 2 ⊗C 2 = C 4 ⊗ C 2 as g0-module. We set To get an explicit realization, consider the contact Lie superalgebra (see [11] for more details) where t is an even variable and ξ i , 1 ≤ i ≤ 4, are odd variables. Introduce on the associative superalgebra K (1, 4) a Z-grading by letting deg t = 2, deg ξ i = 1, and the bracket (4, C), that g 1 is isomorphic to the standard representation C 4 of so(4, C) and that g 1 is isomorphic to 3 C 4 , so that K (1, 4) 1 = C 4 ⊕C 4 as so(4, C)-module. Also notice that {g 1 , Fix now a copyg b of an so(4, C)-module C 4 in C 4 ⊕C 4 , depending on a constant b ∈ R, as follows. Set, for 1 ≤ i ≤ 4, Let b ∈ R. Note that, setting ξ = ξ 1 ξ 2 ξ 3 ξ 4 , we have Hence, if we set Then g ac is an almost compact form of D(2, 1; 1+b 1−b ). To prove this, it suffices to calculate the Cartan matrix for a choice of Chevalley generators of the complexification of g ac . Fix a Cartan subalgebra in g = so(4, C) as the span of v 2 = − √ −1ξ 1 ξ 2 , v 3 = − √ −1ξ 3 ξ 4 . Set v 1 = t; then {v 1 , v 2 , v 3 } is a basis of a Cartan subalgebra of g. Let { 1 , 2 , 3 } be the dual basis to {v 1 , v 2 , v 3 }. One can choose {α 1 = 2 − 1 , α 2 = 1 − 3 , α 3 = 1 + 3 } as a set of simple roots. The associated Chevalley generators are and the corresponding Cartan matrix, normalized as in [11], is Hence a = 1+b 1−b and therefore all a = −1 occur in this construction. Since this subalgebra is 17-dimensional, it is isomorphic to D(2, 1; a). Remark 4.1. Note that a = 0 for b = −1. In this case, D(2, 1; 0) contains a 11dimensional solvable ideal generated by f 1 , which is spanned by h 1 and the root vectors relative to roots having α 1 in their support. If we replace a i by a i /b and h 1 by h 1 /b, and let b tend to +∞, we recover also the Lie superalgebra of derivations of psl(2|2), and its almost compact real form.
(4) g = G(3). Then g0 = sl 2 ⊕ G 2 and g1 = C 2 ⊗ L min , where L min is the complex 7-dimensional irreducible representation of G 2 , and we let where G 2,0 is the real compact form of G 2 and L min,0 is the real 7-dimensional irreducible representation of G 2,0 whose complexification is L min . (5) g = F(4). Then g0 = sl 2 ⊕ so 7 and g1 = C 2 ⊗ spin 7 , where spin 7 is the complex spinor representation of so 7 , and we let where spin(R 7 ) is the spinor representation of the compact group so 7 (R).
It is proved in [11,Proposition 5.3.2] that in both cases (4) and (5) g ac = g ac 0 ⊕ g ac 1 is an almost compact form of g.

Uniqueness of the almost compact involution.
Proposition 4.2. An almost compact involution is uniquely determined up to a sign by its action on g 0 , provided that the g 0 -module g 1/2 is irreducible.
Proof. If there are two different extensions of the compact involution, then their ratio ψ, say, is identical on g 0 , hence, by Schur's lemma, ψ acts as a scalar on g −1/2 . Since φ( f ) = f , we conclude that this scalar is ±1.
is again an almost compact involution.

The Bilinear Form ·, · on g −1/2
Let s = {e, x, f } be an sl 2 -triple as in Proposition 3.2. Consider the following symmetric bilinear forms on g −1/2 and g 1/2 respectively: We want to prove the following Proposition 5.1. We can choose an almost compact involution such that the bilinear form ., . is positive definite on g ac ∩ g −1/2 . In particular, the Hermitian form φ(u), v (resp. φ(u), v ne ) is positive definite (resp, negative definite) on g ac ∩ g −1/2 (resp. g ac ∩ g 1/2 ).
The proof requires a detailed analysis of the action of an almost compact involution on g −1/2 . Define structure constants N α,β for a good choice of root vectors (see Sect. 2.2) by the relation Observe that {X −θ , X θ , 1 2 h θ } is a sl 2 -triple in s. Let g = CX θ ⊕g 1/2 ⊕g 0 ⊕g −1/2 ⊕ CX −θ be the decomposition into ad 1 2 h θ eigenspaces. By the sl 2 representation theory, ad X ±θ : g ∓1/2 →g ±1/2 is an isomorphism of g -modules. Moreover, by our choice of R + in Sect. 3, the roots ofg −1/2 (resp.g 1/2 ) are precisely the negative (resp. positive) odd roots. In particular, the map α → −θ + α defines a bijection between the positive and negative odd roots. We shall need the following properties.

Lemma 5.2. For a positive odd root α we have
In particular N θ,α is real.
Arguing as in Proposition 3.2, we can assume in the proof of Proposition 5.1 that {e, x, f } is the sl 2 -triple defined in (3.5); ad x defines on g a minimal grading Note that Note that, since φ(x) = x, φ(u α ) has to belong to g −1/2 . This forces and (5.9) becomes (5.8).
Proof of Proposition 5.1.
where α runs over the positive odd roots. It is clear that v α ∈ r. We want to prove that the vectors v α form an orthogonal basis of r. We need two auxiliary computations: To prove (5.11) use (5.4): To prove (5.12) use (5.11): Therefore by (5.4) and (5.10) v α , v β = 2δ α,β .

A General Theory of Invariant Hermitian Forms on Modules Over the Vertex Algebra of Free Boson and the Fairlie Construction
Consider the infinite dimensional Heisenberg Lie algebra It is well known that M(0) carries a simple vertex algebra structure, called the vertex algebra of free boson, which we denote by V 1 (Ca), and that M(μ) is a simple module over the vertex algebra V 1 (Ca). Moreover, V 1 (Ca) is the universal enveloping vertex algebra of the nonlinear Lie conformal algebra We introduce conformal weight on V 1 (Ca) by letting a = 1, and for v ∈ V 1 (Ca) we write the corresponding quantum field as Fix t ∈ C and set It is an energy-momentum element for all t. Set . By the −1-st product identity : aa : In particular On the other hand, by the commutator formula, Let M(μ) n be the eigenspace for the energy operator H (t) corresponding to the eigenvalue n + 1 2 μ 2 − tμ. Since Thus This shows that M(μ) is a positive energy V 1 (Ca)-module, i.e. real parts of the eigenvalues of H (t) are bounded below. Moreover its minimal energy subspace is Lemma 6.1.
e − 2t n z n a n . (6.5) Proof. Identify V 1 (Ca) with the polynomial algebra in infinitely many variables using (6.4) with μ = 0: Since L(t) 1 1 = 0 and a n 1 = 0 if n > 0, both L(t) 1 and a n for n > 0 act as derivations of the algebra P under our identification. It follows that both sides of (6.5) are automorphisms of P. It is therefore enough to check the equality only on the generators a −n . We need the following formulae: Applying these formulae we find It follows that To conclude we only need to check that, if n ≥ 1, then We prove this by induction on n. If n = 1 the formula reads Using (6.7) with t = 0 we see that the latter formula is equivalent to which is just (6.7). If n > 1 and m = 1, then If n > 1 and m > 1, then Let φ be the conjugate linear involution of the vector space Ca defined by φ(a) = −a. Assume from now on that t ∈ √ −1R. This assumption is necessary since, in order to apply the results of [16], we need to assume φ(L(t)) = L(t). Set (cf. (2.7)) ) be the canonical projection to the Zhu algebra (see e.g. [16,Section 2]). Let ω be the conjugate linear It is proven in [16, Proposition 6.1] that ω is indeed well-defined.
Proof. By Lemma 6.1, since g(a) = a and L(0) 1 a = 0, By abuse of terminology, we shall call H (·, ·) an L-invariant Hermitian form, where L is the conformal vector of V . If μ ∈ C we denote by (μ) and (μ) the real and imaginary part of μ, respectively.

Proposition 6.3. There is a non-zero L(t)-invariant Hermitian form on M(μ) if and
Proof. Let (·, ·) be the unique Hermitian form on Cv μ such that (v μ , v μ ) = 1. By Proposition 6.7 of [16], there is a non-zero L(t)-invariant Hermitian form on M(μ) if and only if (·, ·) is an ω-invariant Hermitian form on Cv μ . By Lemma 6.2, that is equivalent to hence the statement.
We denote by H μ the unique L(

Lemma 6.4. If m, m ∈ M(μ), then
Proof. By invariance of the Hermitian form, The last two steps follow by (6.10) and the fact that we get the result.
It is now easy to compute the invariant form in the basis (6.4): It follows that the basis is orthogonal and In particular the form is positive definite and its values on the chosen basis do not depend on μ.
We now want to describe explicitly the action of V 1 (Ca) under this identification. We need the following result: Lemma 6.5. If t ∈ √ −1R, then z 2H (0) e z n a n z −2H (0) = e z −n a n (6.14) e tz n a n g = ge t (−z) n a n (6.15) For the second formula note that so, since t is purely imaginary, e tz n a n g(b) = In particular the fields Proof. By definition, Using (6.5) we can write ) n z n a n so, by Lemma 6.5, We also set , we need to check that It is enough to check this for b = a I 1. Using (6.9), we can write It follows that hence we need to check that Indeed, setting t 0 = 2(−t + √ −1 (μ)) and letting q be the number of j such that i j = 0, as wished.

It follows that
In particular, if μ is real, Proof. We first prove that so (6.25) follows.
To prove (6.23) write e − 2 p n z n a n , we find that In other words In particular, if μ ∈ R, we have and, setting s = 2t, (6.27) becomes Moreover, (T a) μ n = −(n + 1)a μ n , hence, substituting in (6.28), we obtain by the Borcherds commutator formula, Finally, since L(s) is quasiprimary for L(s) and g(L(s)) = L(s), by (6.24) we have We now extend the previous analysis of invariant Hermitian forms on bosons to the case of the vertex algebra The arguments developed in this section for V 1 (Ca) can be carried out in the same way in the more general setting of the vertex algebra where V is any conformal vertex algebra. In particular, we have As before, we can The following is the generalization of Lemma 6.7. The proof is the same.

λ-brackets and conjugate linear involutions.
Let, as before, g be a basic classical Lie superalgebra, and x ∈ g be an element, for which ad x is diagonalizable with eigenvalues in 1 2 Z, the ad x-gradation of g satisfies (1. 2) with some f ∈ g −1 and is compatible with the parity of g. Then for some e ∈ g 1 , {e, x, f } is an sl 2 -triple as in Proposition 3.2, i.e. (3.1) holds with g the centralizer of f in g. Recall that the invariant bilinear form (.|.) on g is normalized by the condition (x|x) = 1 2 , and we have the orthogonal direct sum of ideals Choose a Cartan subalgebra h of g , so that, by (7.1), h = Cx ⊕h is a Cartan subalgebra of g 0 (and of g). Let be the decomposition of g into the direct sum of ideals, where g 0 is the center and the g i are simple for i > 0. Let h ∨ be the dual Coxeter number of g, and denote byh ∨ i half of the eigenvalue of the Casimir element of g i with respect to (.|.) |g i ×g i , when acting on g i . Note thath ∨ 0 = 0. In [18] the authors introduced (as a special case of a more general construction) the universal minimal W -algebra W k min (g), whose simple quotient is W min k (g), attached to the grading (5.6). This is a vertex algebra strongly and freely generated by elements L, J {v} where v runs over a basis of g , G {u} where u runs over a basis of g −1/2 , with the following λ-brackets ( [20, Theorem 5.1]): L is a Virasoro element (conformal vector) with central charge c(k) given by (1.4), J {u} are primary of conformal weight 1, G {v} are primary of conformal weight 3 2 , and Furthermore, the most explicit formula for the λ-bracket between the G {u} is given in [1, (1.1)] and in [20, Theorem 5.1 (e)]. We will need both formulas: where {u α } and {u α } (resp. {w γ }, {w γ }) are dual bases of g (resp. g 1/2 ) with respect to (.|.) (resp. with respect to ·, · ne ), a → a i (resp. a → a ) for a ∈ g 0 is the orthogonal projection to g i (resp g ), p(k) is the monic quadratic polynomial proportional to (7.28), introduced in [1, Table 4], and thoroughly investigated in [15], and Table 2 below for the values ofh ∨ i ). The following proposition is a special case of [16,Lemma 7.3], in view of Lemma 3.1.
extends to a conjugate linear involution of the vertex algebra W k min (g). The following result is a sort of converse to Proposition 7.1.

Proposition 7.2.
Assume that k ∈ R is non-collapsing. Let ψ be a conjugate linear involution of W k min (g). Then there exists a conjugate linear involution φ of g satisfying (1.1) such that ψ is the conjugate linear involution induced by φ.
Proof. If a, b ∈ g , define φ(a) by Since ψ is a vertex algebra conjugate linear automorphism, (7.9) equals (7.10), so that φ is a conjugate linear involution of g , and we have Since k is not collapsing, relations (7.22), (7.28) and (7.11) imply that We now prove that there is a unique extension of φ to a conjugate linear automorphism of g fixing e, x, and f . Note that φ(g −1/2 ) ⊂ g −1/2 and that g 1/2 = [e, g −1/2 ]. In particular, setting φ( , v ∈ g −1/2 , we extend φ to a conjugate linear bijection g → g. In particular, φ is unique. It remains to prove that it is a conjugate linear automorphism. Note first that, by (7.1), equation (7.12) holds for a, b ∈ g 0 . Consider elements where α, α , β, β , γ, γ ∈ C, u, u ∈ g 1/2 , v, v ∈ g −1/2 , a, a ∈ g . Then (7.14) Hence (7.13) equals (7.14), provided the following equalities hold (7.20) Relation (7.3) implies at once (7.19). To prove (7.18) Since p(k) = 0 (k is not collapsing) and k is real, we have the claim. Now we prove (7.20). Here and in the following Next we prove (7.17). We have to prove that By (7.6) On the other hand Since φ is an automorphism of g there is a permutation i → i such that φ( To conclude we have to check the x-component: Since (1.1) holds on g 0 , we have Next, we prove (7.16 It remains to check that for a, b ∈ g. We already observed that this relation holds for a, b ∈ g 0 and it is obvious that (φ(e)|φ( f )) = (e| f ). We now compute for u ∈ g 1/2 , v ∈ g −1/2 , By Proposition 3.2 there is a conjugate linear involution φ on g such that φ(x) = x, φ( f ) = f and (g ) φ is a compact real form of g , hence, by Proposition 7.1, φ induces a conjugate linear involution of the vertex algebra W k min (g), and descends to a conjugate linear involution of its unique simple quotient W min k (g), which we again denote by φ. By [16,Proposition 7.4 (b)], W k min (g) admits a unique φ-invariant Hermitian form H (·, ·) such that H (1, 1) = 1. Recall that if k + h ∨ = 0 then the kernel of H (·, ·) is the unique maximal ideal of W k min (g), hence H (·, ·) descends to a non-degenerate φ-invariant Hermitian form on W min k (g), which we again denote by H (·, ·). We need to fix notation for affine vertex algebras. Let a be a Lie superalgebra equipped with a nondegenerate invariant supersymmetric bilinear form B. The universal affine vertex algebra V B (a) is the universal enveloping vertex algebra of the Lie conformal superalgebra R = (C[T ] ⊗ a) ⊕ C with λ-bracket given by λB(a, b), a, b ∈ a.
In the following, we shall say that a vertex algebra V is an affine vertex algebra if it is a quotient of some V B (a). If a is simple Lie algebra, we denote by (.|.) a the normalized invariant bilinear form on a, defined by the condition (α|α) a = 2 for a long root α. Then B = k(.|.) a , and we simply write V k (a). If k = −h ∨ , then V k (a) has a unique simple quotient, which will be denoted by V k (a).
Let ψ be a conjugate linear involution of a such that (ψ(x)|ψ(y)) = (x|y). By [16, §5.3] there exists a unique ψ-invariant Hermitian form H a on V k (a). The kernel of H a is the maximal ideal of V k (a), hence H a descends to V k (a).

Some numerical information.
Recall the decomposition (7.2) of the Lie algebra g , and that we assume that g is not abelian, i.e. s ≥ 1 in (7.2). Let θ i be the highest root of the simple component g i for i > 0. Set where 2 (a|b), hence, formula (7.5) can be written as In other words, the vertex subalgebra of W k min generated by J {a} , a ∈ g , is i≥0 V M i (k) (g i ).
Closely related to the vertex algebra W k min (g) is the universal affine vertex algebra V α k (g 0 ) (see [20, (5.16) a, b)) , (7.24) and where κ g 0 denotes the Killing form of g 0 . Note that We have another formula for the cocycle α k , closely related to (7.23): where The relevant data for computing the M i (k) and χ i are collected in Table 2, where their explicit values are also displayed. Note that M 0 (k) = k + 1 2 h ∨ . As in the Introduction, denote by ξ ∈ (h ) * a highest weight of the g -module g −1/2 .

Lemma 7.3. For i ≥ 1 we have
with the exception of χ 1 for g = osp(4|m).
Proof. The weights ξ are restrictions to h of the maximal odd roots of g; they are listed in Table 3, together with the maximal roots θ i . Relation (7.27) is then checked directly using the data in Tables 1, 2, 3.
Recall from [1] that a level k is collapsing for W min k (g) if W min k (g) is a subalgebra of the simple affine vertex algebra V β k (g ).
We summarize in the following result the content of Theorem 3.3 and Proposition 3.4 of [1] relevant to our setting. We say that an ideal in g is a component of g if it is simple or 1-dimensional. Highest odd roots  1; a) and k is collapsing, then W min Remark 7.5. If M i (k) ∈ Z + for all i ≥ 1, g = osp(4|m) and M i (k) < −χ i for some i ≥ 1, then k is a collapsing level (or critical). This is clear by looking at Table 2.

Necessary Conditions for Unitarity of Modules Over W k min (g)
We assume that g is from the list (1.3); in particular, g is a reductive Lie algebra. We parametrize the highest weight modules for W k min (g) following Sect. 7 of [20]. Let h be a Cartan subalgebra of g , and choose a triangular decomposition g = n − ⊕h ⊕n + . For ν ∈ (h ) * and l 0 ∈ C, let L W (ν, 0 ) (resp. M W (ν, 0 ) ) denote the irreducible highest weight (resp. Verma) W k min (g)-module with highest weight (ν, 0 ) and highest weight vector v ν, 0 . This means that one has Let φ is an almost compact conjugate linear involution of g (see Definition 1.1); in particular, the fixed points set g R of φ |g is a compact Lie algebra (the adjoint group is compact). Set h R = g R ∩ h . Recall that ν ∈ (h R ) * is said to be purely imaginary if ν(h R ) ⊂ √ −1R. It is well-known that if α is a root of g and ν is purely imaginary then ν(α) ∈ R. Lemma 8.1. Assume that l 0 ∈ R and that ν is purely imaginary. Then L W (ν, 0 ) admits a unique φ-invariant nondegenerate Hermitian form H ( . , . ) such that H (v ν, 0 , v ν, 0 ) = 1.

Proof. It is enough to show that the Verma module M W (ν, 0 ) admits a φ-invariant
If v ∈ M W (ν, 0 ), m > 0, and u ∈ g , then Similarly we see that, if u ∈ g −1/2 , then On the other hand, if u ∈ n 0+ , then, since φ(u) ∈ n 0− , If h ∈ h R , then, since ν(h) is purely imaginary, Let us check that this form is φ-invariant: write Y ν, 0 for the field Y M W (ν, 0 ) andY ν, 0 for the field Y M W (ν, 0 ) ∨ . Then ·) is positive definite. The vertex algebra W min k (g) is called unitary if its adjoint module is unitary.

)). It follows that
In particular, since v is a highest weight vector for the g -module g −1/2 , we have Substituting (8.9) into (8.7) we obtain as claimed.

Remark 8.4.
Let v ∈ g −1/2 be as in Lemma 8.3 and u a root vector for the root θ i . Then Indeed, Let P + ⊂ (h ) * be the set of dominant integral weights for g and let Recall that ξ ∈ (h ) * is a highest weight of the g -module g −1/2 . Introduce the following number Proof. In order to prove that M i (k) ∈ Z + for all i ≥ 1 and ν ∈ P + k , it is enough to observe that, if L W (ν, 0 ) is a unitary module over W k min (g), then, in particular, V β k (g )v ν, 0 is a unitary module over V β k (g ), hence ν ∈ P + k [12], which is non-empty if and only if M i (k) ∈ Z + for all i ≥ 1.
To prove the second claim recall that, by Proposition 5.1, the Hermitian form φ(.), . is positive definite on g −1/2 . Since k + h ∨ < 0, we obtain from (8.1) that as claimed.
Consider the short exact sequence If a W k min (g)-module L W (ν, 0 ) is unitary, then, restricted to the subalgebra V β k (g ) it is unitary, hence a direct sum of irreducible integrable highest weight g -modules of levels M i (k), i ≥ 1. But it is well known that all these modules descend to V β k (g ). Also, all these modules are annihilated by the elements Proof. Let u be a root vector for ξ . Then G {u} −1/2 v ν, 0 is a singular vector for V β k (g ). Since L W (ν, 0 ) is unitary, all vectors that are singular for V β k (g ) should have weight in P + k . By the assumption, we have G {u} −1/2 v ν, 0 = 0, hence the norm of this vector is 0, and we can apply (8.1).
Consider the remaining case g = D(2, 1; a). By this we mean the contragredient Lie superalgebra with Cartan matrix  Table 1, corresponds to the choice θ = 2 1 , so that (2 1 |2 1 ) = 2. If we choose θ = 2 2 , then the bilinear form (.|.) is given by Then a = − m m+n , m, n ∈ N, m and n are coprime (i. e. a ∈ Q, −1 < a < 0) and in turn k ∈ − mn m+n N. If we choose θ = 2 3 , then the bilinear form (.|.) is given by Then a = − m+n m , m, n ∈ N, m and n are coprime (i. e. a ∈ Q, a < −1) and in turn k ∈ − mn m+n N. Recall that one obtains isomorphic superalgebras of the family D(2, 1; a), a = 0, −1, under the action of the group S 3 , generated by the transformations a → 1/a, a → −1 − a. These transformations permute transitively the domains Q >0 , Q >−1 ∩ Q <0 and Q <−1 , which correspond to the above three cases. Corollary 8.13. If k is from the unitarity range for W k min (g), then k + h ∨ is a negative rational number.

Free Field Realization of Minimal W -Algebras
) be the free field realization introduced in [20, Theorem 5.2]; it is explicitly given on the generators of W k min (g) by Recall that F(g 1/2 ) is the universal enveloping vertex algebra of the (non-linear) Lie conformal superalgebra C[T ] ⊗ g 1/2 with [a λ b] = a, b ne 1, a, b ∈ g 1/2 , and { α } α∈S 1/2 , { α } α∈S 1/2 are dual bases of g 1/2 with respect to ., . ne .
We now apply the results of Sect. 6 to V k+h ∨ (Cx). By Corollary 8.13 unitarity of W k min (g) implies k + h ∨ < 0. Hence, using the normalization Recall that in Proposition 5.1 we proved that one can choose an almost compact involution φ of g that fixes pointwise the sl 2 -triple {e, x, f } in such a way that the Hermitian form φ(u), v ne on g 1/2 is negative definite. This conjugate linear involution induces a conjugate linear involution of W k min (g) and of V α k (g 0 ) ⊗ F(g 1/2 ) as well, both denoted again by φ. It is readily checked, using (9.1), (9.2), and (9.3), that The conformal vector of the vertex algebra V k is Here {u α } α∈S and {u α } α∈S are dual bases of g with respect to the bilinear form (.|.) restricted to g . Recall that L g is the conformal vector of V α k (g ) and L F is the conformal vector of F(g 1/2 ). Let It follows from (9.3) and (9.6) that (L) = L(s k ) + L = L f ree + s k T (a), (9.8) where L = L g + L F , and L(s) = 1 2 : aa : +sT a, L(s) = L(s) + L, cf. (6.1) and (6.29), respectively. Note , and (L) = L(s) (cf. (6.28), (6.29)).
Given μ ∈ C, let M(μ) be the irreducible V 1 (Ca)-module with highest weight μ, and consider the V k -module Recall that V carries a φ-invariant Hermitian form H g ⊗ H F , which is positive definite. Recall also that, by Proposition 6.3, the V 1 (Ca)-module M(μ) carries a unique L(t)invariant Hermitian form, provided that t = √ −1 (μ), which is positive definite. This Hermitian form, normalized by the condition that the norm of the highest weight vector equals 1, was denoted by H μ . Hence we have a φ-invariant positive definite Hermitian form H μ ( . , . ) ⊗ H g (. , .) ⊗ H F (. , .) on N (μ), which we denote by (·, ·) μ .
It follows from Proposition 6.10 that, restricting the fields Y μ,t (−, z) from V k to (W k min (g)), one equips N (μ) with a structure of a W k min (g)-module. We now explicitly describe this action of the generators of W k min (g) on N (μ).
As an application of Proposition 9.1, we obtain a generalization of the Fairlie construction to minimal W -algebras. = s k (cf. (9.7)) and

Sufficient Conditions for Unitarity of Modules Over W k min (g)
Due to the Proposition 8.9 (a), we may assume in this section that g = sl(2|m) and osp(4|m), m > 2. Then, in particular, g = ⊕ i≥1 g i is the decomposition of g into simple ideals, and the χ i are given by (7.27).
Proposition 10.1. Assume that k +h ∨ = 0. Then there exists a unitary module L W (ν, 0 ) over W k min (g) if and only if M i (k) ∈ Z + for all i and ν ∈ P + k . Proof. One implication has been already proven in Proposition 8.5. To show that the converse implication also holds, assume M i (k) ∈ Z + for all i. Recall (see (7.25)) that the cocycle α k is given by Assume first that M i (k) + χ i ∈ Z + for all i. Then the simple quotient V α k (g ) of V α k (g ) is unitary, since it is an integrable g -module [11]. Next, the vertex algebra F(g 1/2 ) is unitary due to Proposition 5.1 and [16, §5.1]. Finally, the V 1 (Ca)-module M(s) , where s is given by (9.7), is unitary by the observation following Lemma 6.4. Consider the unitary W k min (g)-module M(s) ⊗ V α k (g ) ⊗ F(g 1/2 ), and its submodule Since the Hermitian form H s ( . , .) is L(s)-invariant and (L) = L(s), we see that U admits a φ-invariant Hermitian positive definite form, thus U is a unitary highest weight module for W k min (g). Now we look at the missing cases, where there is i such that 0 ≤ M i (k) < −χ i , described in Remark 7.5. Assume first that g is simple. If χ 1 = −1 then the only possible value is M 1 (k) = 0, so, W min k (g) = C, by Theorem 7.4 (1) (a). In the case of g = spo(2|3) one should consider the cases M 1 (k) = 1 and M 1 (k) = 0: in the former case k = − h ∨ 1 2 − 1, hence Theorem 7.4 (1) (b) applies and W min k (spo(2|3)) = V 1 (sl (2)), whereas in the latter case k + h ∨ = 0. If g is semisimple but not simple, then g = D(2, 1; a). In this case we have to consider only the case in which either M 1 (k) or M 2 (k) is zero. If M 1 (k) = 0 (resp. M 2 (k) = 0) then, by Theorem 7.4 (2), W min k (D(2, 1; a)) = V M 2 (k) (sl(2)) (resp. = V M 1 (k) (sl(2))).
We now generalize the construction given in the proof of Proposition 10.1 to provide families of unitary representations. For ν ∈ P + k introduce the following number Proof. Let L (ν) be the irreducible highest weight V α k (g )-module of highest weight ν and let v ν be a highest weight vector. Fix μ ∈ R and set where s = s k is given by formula (9.7). Note that the Hermitian form (·, ·) μ+s is L(s)- [12]. Thus N (μ, ν) is a unitary representation of W k min (g). We now compute the highest weight of N (μ, ν). Recall that By the −1-st product identity, It follows that N (μ, ν) = L W (ν, 0 ) for some 0 . We now compute 0 : so that, using (9.7), Hence 0 ≥ B(k, ν). , ν), we see that the module L W (ν, 0 ) = N (μ, ν) is unitary.

Unitarity of Minimal W -Algebras and Modules Over Them
The main result of this paper is the following. Theorem 11.1. Let k = −h ∨ , and recall the number A(k, ν) given by (8.11). If k lies in the unitary range (hence M i (k) ∈ Z + for i ≥ 1), then the W k min (g)-module L W (ν, 0 ) is unitary for all non extremal ν ∈ P + k and 0 ≥ A(k, ν). In the rest of this section we give a proof of these results. First, by Proposition 8.9 (a), we may exclude g = sl(2|m), m > 2, from consideration, so that g is semisimple and by Proposition 8.5, conditions M i (k) ∈ Z + are necessary for unitarity, hence we shall assume that these conditions hold. Let Let ⊂ h * be the set of roots of g. As a subset of simple roots where is the set of simple roots for g given in Table 1. We denote by + the corresponding set of positive roots and by ρ ∈ h * the corresponding ρ-vector.
For ν ∈ P + k and h ∈ C, set Let p be the parabolic subalgebra of g with Levi factor h + g and the nilradical u + = α∈ + \ g α . Set u − = α∈ + \ g −α . Let V (ν) denote the irreducible g -module with highest weight ν and extend the g action to p by letting u + act trivially; x, K , and d act by h, k, and 0 respectively. Let M ( ν h ) be the corresponding generalized Verma module for g, i.e.
We denote by v ν h a highest weight vector for M ( ν h ) . If μ ∈ h * and M is a g-module, we denote by M μ the corresponding weight space. Let If α ∈ is a non-isotropic root, denote by s α ∈ End( h * ) the corresponding reflection and the group generated by them by W . If β ∈ \ Zδ is an odd isotropic root, we let r β denote the corresponding odd reflection. We denote by x α a root vector attached to α ∈ . Denote by w. the shifted action of W : w.λ = w(λ + ρ) − ρ.
Lemma 11.3. Let 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 . If α j ∈ is an isotropic root and x −α j m = 0, then x −α j m is a singular vector with respect to r α j ( ).
Lemma 11.6. Let k be in the unitarity range and let A(k, ν) be as in (8.11). Assume that ν is not extremal. Then h n, m (k, ν) ≤ A(k, ν), (11.10) h m,γ (k, ν) ≤ A(k, ν). (11.11) Proof. First we prove (11.10). Plugging (11.8) into (11.10) we get which is equivalent to Since k + h ∨ < 0, it is enough to check (11.12) with m = 1, n = 1/ . In the case (k + 1) ≤ 2(ξ |ν), (11.12) reads Looking at the values of h ∨ in Table 2, we see that the L.H.S. of (11.13) is non-negative. Now we prove that (ξ |ν) ≤ 0. Indeed, from Table 1 we deduce that the restriction of (.|.) to the real span of is negative definite. From Tables 1 and 3 one checks that ξ is a linear combination with non-negative coefficients of simple roots of g ; since ν is dominant, if α ∈ is a simple root then ν(α ∨ ) ≥ 0, hence (ν|α) ≤ 0 since (α|α) < 0. In the case (k + 1) ≥ 2(ξ |ν) we have to prove that (11.14) The non-extremality condition means that 15) hence it is enough to prove that Note that θ i = ξ + β i , where, as above, β i is a linear combination with non-negative coefficients of simple roots of g . Therefore (11.16) can be written as , which is clearly verified, since the left hand side is negative and the right hand side is positive (use the data in Table 2). Now we prove (11.11). Substituting (11.8) in it we obtain which is equivalent to (11.17) Table 4. Data employed in the proof of Lemma 11.6 Recall that, even though g −1/2 can be reducible as a g -module, all irreducible components have the same highest weight ξ . It follows that A direct check on Table 4 shows that 2 max γ ∈ (ρ |γ ) + h ∨ = 1. (11.19) Note that, by (11.18) and (11.19) and (11.17) reads which is clearly true. Now consider the case The inequality (11.17) becomes which is implied by . (11.22) If γ = −ξ , then the left hand side of (11.22) is hence (11.21) implies that both members of (11.22) are zero.
To prove our claim, assume that there is m > 1/2 such that Taking (11.15) into account, we are done if we prove that The next to last inequality in (11.31) follows from Table 2; more precisely, the strict inequality holds in all cases except for spo(2|3). The last inequality in (11.31) uses that (ν + ξ |θ i ) ≤ 0. For g = spo(2|3) the last inequality in (11.31) is strict, hence (11.30) is proven in all cases.
Let H 0 denote the quantum Hamiltonian reduction functor, from the category O of gmodules of level k to the category of W k min (g)-modules. Recall that, for a g-module M, H 0 (M) is the zeroth homology of the complex (M ⊗ F(g, x, f ), d 0 ) defined in [18]. Recall that the functor H 0 maps Verma modules to Verma modules [20,Theorem 6.3] and it is exact [2,Corollary 6.7.3]. By [20,Lemma 7.3 (b)], if M is a highest weight module over g of highest weight ∈ h * , H 0 (M) is either zero or a highest weight module over W k min (g) of highest weight (ν, ) with Remark 11.7. Let L( ) denote the irreducible g-module of highest weight ∈ h * . By Arakawa's theorem [2, Main Theorem] H 0 (L( )) is either irreducible or zero, and it is zero if and only if ( |α 0 ) = n 2 (α 0 |α 0 ), n ∈ Z + . In particular, if (11.5) holds, then H 0 (M( ν h )) is a non-zero highest weight module of highest weight (ν, (h)), where where ν, are given by (11.44). From now on we assume • k is in the unitarity range;

Lemma 11.8. Let h, h be the solutions of the equation
Proof. Recalling that hence h = (k + 1 − n)/2 and h = (k + n + 1)/2 so that  F(g, x, f ))) = 0 if j = 0. Thus, using Euler-Poincaré character, the fact that H 0 maps Verma modules over g to Verma modules over W k min (g), and (ii) in Proposition 11.5, we find that (11.46) holds.

If ζ (y) ∈ Hom C[y] (C[y] ⊗ h, C[y]) is a weight of
End of proof of Theorem 11.1 and Corollary 11.2. We may assume that the level is not collapsing, so that M i (k) + χ i ∈ Z + by Remark 7.5. Then, by Proposition 10.2, the Hermitian form on L W (ν, 0 ) is positive definite for 0 0. By Lemma 11.10, N (μ, ν). It follows that the Hermitian form is positive definite for 0 > A(k, ν), hence positive semidefinite for 0 = A(k, ν).

Explicit Necessary Conditions and Sufficient Conditions of Unitarity
Looking for the pairs (ν, 0 ), ν ∈ P + k , 0 ∈ R, such that L W (ν, 0 ) is a unitary W k min (g)-module for k in the unitarity range, we rewrite for each case (excluding the trivial case (1)) the conditions in terms of the parameters M i = M i (k) from Table 2. Namely, we provide the necessary and sufficient conditions of unitarity of L W (ν, 0 ) for a non-extremal weight ν, given by Theorem 11.1, and the necessary condition of unitarity for an extremal weight ν, given by Proposition 8.8. We also provide explicit expressions for the cocycle α k and the central charge c of L. Recall the invariant bilinear form (.|.) i on g i , introduced in Sect. 7.
If ν is dominant integral, ν(θ ∨ 1 ) ≤ M 1 − 1, then the necessary and sufficient condition for unitarity is where r = (ω 1 |ν) , and ω 1 is the highest weight of the standard representation of so(m).

Unitarity for Extremal Modules Over the N = 3, N = 4 and big N = 4 Superconformal Algebras
A module L W (ν, 0 ) for W k min (g) is called extremal if the weight ν is extremal (see Definition 8.7). In this section we give a partial solution of Conjecture 2 for some g. Namely, g will be either spo(2|3), or psl(2|2), or D(2, 1; a), so that W k min (g) is related to the N = 3, N = 4 and big N = 4 superconformal algebra, respectively. Recall from [20,Section 8] that in these cases, up to adding a suitable number of bosons and fermions, it is always possible to make the λ-brackets between the generating fields linear, hence the span of their Fourier coefficients gets endowed with a Lie superalgebra structure, called the N = 3, N = 4 and big N = 4 superconformal algebra respectively.
Recall that, by Proposition 8.8, for each extremal weight ν there is at most one 0 for which the extremal module L W (ν, 0 ) is unitary, hence for each extremal ν it suffices to construct one such unitary module.
The unitarity of L W (θ 1 /2, 1/2) is proved by constructing this module as a submodule of a manifestly unitary module. This is achieved by using the free field realization of W −2 min ( psl(2|2)) given in [3], in terms of four bosonic fields and four fermionic fields, which we now describe. Let F be the vertex algebra generated by four even fields a i , 1 ≤ i ≤ 4 and four odd fields b i , 1 ≤ i ≤ 4 with λ-bracket There is an homomorphism F F R : W −2 min ( psl(2|2)) → F given by We define a conjugate linear involution ψ on F by  ( psl(2|2))-module. An easy calculation shows that v = b 1 + √ −1b 2 is a singular vector for W −2 min ( psl(2|2)), thus v generates a unitary highest weight representation L W (ν, 0 ) of W −2 min ( psl(2|2)). Clearly F F R(L) 0 v = 1 2 v, while J 0 v = v, hence ν = 1 2 θ 1 and 0 = 1 2 . This proves that the highest weight module corresponding the extremal weight ν = 1 2 θ 1 is indeed unitary.
13.3. g = D(2, 1; m n ). In this case we are able to prove unitarity only in the very special case when either m = 1 of n = 1.
If n = 1, then the unitarity range is {− m m+1 N | N ∈ N}. Take N = 1 and observe that W − m m+1 min (D(2, 1; m)) collapses to V m−1 (sl (2)). In this case there is only one extremal weight ν = m−1 2 α 2 , which gives rise to a unitary representation since it is integrable. The case m = 1 is dealt with in a similar way, switching the roles of α 2 , α 3 .

Characters of the Irreducible Unitary W k min (g)-Modules
Recall that, for ∈ h * , we denoted by M W ( ) the Verma module M W (ν, ), where (ν, ) is given by (11.44). It follows from [20, (6.11)], that ch M W ( ) = e ν q F N S (q), (14.1) where q = e (0, 1) and (1 − q n ) rankg +1 α∈ + ((1 − q n−1 e −α )(1 − q n e α ))) . (14.2) In particular, where (h) is given by (11.45). The characters of unitary W k min (g)-modules L W (ν, 0 ) are computed by applying the quantum Hamiltonian reduction to the irreducible highest weight g-modules L( ν h ), where ν ∈ P + k and 0 = (h), and using the argument in the proof of Theorem 11.9, which is based on Remark 11.7. There are two cases to consider in computation of their characters. First, if the weight ν h is typical, i.e. conditions (11.5) hold, then ch L( ν h ) is given by the R.H.S. of (11.6), by Proposition 11.5.

Remark 14.2.
It is still an open problem whether in the case g = D(2, 1; m n ) formula (14.4) holds for an arbitrary ν ∈ P + k . Remark 14.3. For the N = 4 superconformal algebra, formula (14.5) appears, in a different form, in [4, formula (14)], where it has been derived in a non-rigorous way. To establish a dictionary to match the two formulas first observe that a parameter y occurs in the formulas of [4] corresponding to an extra U (1)-symmetry that we do not consider, hence, to compare the formulas, we set y = 1. Next recall that in this case W is of type A (1) 1 , hence its elements are of the form u i = s 0 s 1 · · · i factors or u i = s 1 s 0 · · · i factors (set u 0 = u 0 = I d). In the notation of [4], the pairs (a n , b n ) corresponding to the α-series (resp. β-series) in formula (12) of [4] match exactly the pairs (ν, ) given in (11.44) for the weight = u i . ν h (resp. = u i . ν h ). The factor F N S (θ, 1) translates precisely to (14.3) according to the dictionary e δ 1 −δ 2 ↔ e √ −1θ .
The character formula (14.6) corresponds to the formula (26) in [4] for the character of "massless" representations. To show this, we first remark that, if γ ∈ Z + 1 , then where (ν, ) is given by (11.44). In particular hence, using formula (14.1), and we obtain that ) .
In the massless case, the character formula (14.6) corresponds to formula (4.6.1) in [21], hence Theorem 14.1 provides a proof of it, since formula (14.4) holds in this case, due to [7,Subsection 12.3].
Note added in proof: Conjecture 4 has been proved in a joint paper with Drazen Adamović.