Conformal embeddings of affine vertex algebras in minimal W -algebras II: decompositions
Drazen Adamovic, Victor G. Kac, Pierluigi Moseneder Frajria, Paolo Papi, Ozren Perse
aa r X i v : . [ m a t h . R T ] A p r CONFORMAL EMBEDDINGS OF AFFINE VERTEXALGEBRAS IN MINIMAL W -ALGEBRAS II:DECOMPOSITIONS DRAˇZEN ADAMOVI ´C, VICTOR G. KAC, PIERLUIGI M ¨OSENEDER FRAJRIA,PAOLO PAPI, AND OZREN PERˇSE
Abstract.
We present methods for computing the explicit decompo-sition of the minimal simple affine W -algebra W k ( g , θ ) as a module forits maximal affine subalgebra V k ( g ♮ ) at a conformal level k , that is,whenever the Virasoro vectors of W k ( g , θ ) and V k ( g ♮ ) coincide. A par-ticular emphasis is given on the application of affine fusion rules tothe determination of branching rules. In almost all cases when g ♮ is asemisimple Lie algebra, we show that, for a suitable conformal level k , W k ( g , θ ) is isomorphic to an extension of V k ( g ♮ ) by its simple module.We are able to prove that in certain cases W k ( g , θ ) is a simple currentextension of V k ( g ♮ ). In order to analyze more complicated non simplecurrent extensions at conformal levels, we present an explicit realiza-tion of the simple W -algebra W k ( sl (4) , θ ) at k = − /
3. We prove, asconjectured in [3], that W k ( sl (4) , θ ) is isomorphic to the vertex alge-bra R (3) , and construct infinitely many singular vectors using screen-ing operators. We also construct a new family of simple current mod-ules for the vertex algebra V k ( sl ( n )) at certain admissible levels and for V k ( sl ( m | n )) , m = n, m, n ≥ Contents
1. Introduction 22. Preliminaries 52.1. Vertex algebras 52.2. Intertwining operators and fusion rules 52.3. Commutants 62.4. Zhu algebras 72.5. Affine vertex algebras 72.6. Heisenberg vertex algebras 82.7. Rank one lattice vertex algebra V Z α
93. Minimal quantum affine W -algebras 104. A classification of conformal levels from [10] 135. Some results on admissible affine vertex algebras 155.1. Fusion rules for certain affine vertex algebras 16 Date : March 7, 2018.2010
Mathematics Subject Classification.
Primary 17B69; Secondary 17B20, 17B65.
Key words and phrases. conformal embedding, vertex operator algebra, central charge. V k ( sl (2)) 176. Semisimplicity of conformal embeddings 196.1. Semisimplicity with nontrivial center of g ♮ g ♮ has trivial center 236.3. Proof of Theorem 6.9 317. The vertex algebra R (3) and proof of Theorem 6.5 337.1. Definition of R (3) λ -brackets for R (3) W k ( sl (4) , θ ) → R (3) R (3) and proof of Theorem 7.1 377.5. d gl (2)-singular vectors in R (3) g ♮ is a Lie algebra 399. Explicit decompositions from Theorem 6.4: g ♮ is not a Liealgebra 439.1. Simple current V k ( sl ( m | n ))–modules 439.2. The decomposition for W k ( sl (6 | , θ ), k = −
2. 46References 471.
Introduction
Let g = g ⊕ g be a basic simple Lie superalgebra. Choose a Cartansubalgebra h for g ¯0 and let ∆ be the set of roots. Choose a subset ofpositive roots such that the minimal root − θ is even. Let W k ( g , θ ) be thesimple minimal affine W -algebra at level k associated to ( g , θ ) [30], [31], [10].Let V k ( g ♮ ) be its maximal affine subalgebra (see (3.2) and Definition 4.1). In[10] we classified the levels k such that V k ( g ♮ ) is conformally embedded into W k ( g , θ ), i.e. such that the Virasoro vectors of W k ( g , θ ) and V k ( g ♮ ) coincide.We call these levels the conformal levels . We proved that, if k is a conformallevel, then W k ( g , θ ) either collapses to V k ( g ♮ ) or k ∈ {− h ∨ , − h ∨ − } . (see Section 4 with review of main results from [10]).In the present paper, we study the decomposition of W k ( g , θ ) as a V k ( g ♮ )-module when k is a conformal level. It turns out that we can use methodssimilar to those we developed for studying conformal embeddings of affinevertex algebras in [9] (see also [7], [33]). Our main technical tool is therepresentation theory of affine vertex algebras at admissible and negativeinteger levels, in particular fusion rules for V k ( g ♮ )-modules, as we shall ex-plain below.As in [9], it is natural to discuss the case in which g ♮ is semisimple sepa-rately from the case in which g ♮ has a nontrivial center. In the latter case, ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 3 one has the eigenspace decomposition for the action of the center: W k ( g , θ ) ∼ = M i ∈ Z W k ( g , θ ) ( i ) , and we prove the following result (see Theorems 6.4 and 6.5). Theorem 1.1.
Consider the conformal embeddings of V k ( g ♮ ) into W k ( g , θ ) : (1) g = sl ( n ) or g = sl (2 | n ) ( n ≥ ), g = sl ( m | n ) , m > , m = n + 3 , n + 2 , n, n − , conformal level k = − h ∨ − ; (2) g = sl ( n ) ( n = 5 or n ≥ ), g = sl (2 | n ) ( n ≥ , g = sl ( m | n ) , m > , m = n + 6 , n + 4 , n + 2 , n , conformal level k = − h ∨ . (3) g = sl (4) , conformal level k = − h ∨ = − .Then V k ( g ♮ ) is simple and, in cases (1), (2), each W k ( g , θ ) ( i ) is an irre-ducible V k ( g ♮ ) -module, while in case (3) each W k ( g , θ ) ( i ) is an infinite sumof irreducible V k ( g ♮ ) -modules. It is quite surprising that there is only one case when each W k ( g , θ ) ( i ) is an infinite sum of irreducible V k ( g ♮ )-modules, namely the conformal em-bedding of V k ( gl (2)) into W k ( sl (4) , θ ) and k = − h ∨ = − /
3. This isrelated to the new explicit realization of W k ( sl (4) , θ ), conjectured in [3], asthe vertex operator algebra R (3) . This conjecture is proven in Section 7 ofthe present paper. Our realization is based on the logarithmic extensionof the Wakimoto modules for V − / ( gl (2)) by singular vectors constructedusing screening operators. The simplicity of R (3) is proved by construct-ing certain relations in the corresponding Zhu algebra. A generalization ofthis construction, together with more applications related to vertex algebrasappearing in LCFT will be considered in [4].Theorem 1.1 is proved by studying fusion rules for V k ( g ♮ )-modules. Recallthat fusion rules (or fusion coefficients) are the dimensions of certain spacesof intertwining operators. The fact that a certain fusion coefficient is zeroimplies that a given V k ( g ♮ )-module cannot appear in the decomposition of W k ( g , θ ) ( i ) . In many cases this is the only information we need to establishboth the simplicity of V k ( g ♮ ) and the semisimplicity of W k ( g , θ ) as a V k ( g ♮ )-module: see Theorem 6.2. This information, very often, is obtained just fromthe decomposition of tensor products of certain simple finite dimensional g ♮ -modules.In the cases when we are able to compute precisely the fusion rules, weare also able to describe explicitly the decomposition of W k ( g , θ ) as a V k ( g ♮ )-module. Indeed, we prove that the modules W k ( g , θ ) ( i ) are simple currentsin a suitable category of V k ( g ♮ )-modules in the following instances: eitherwhen V k ( g ♮ ) contains a subalgebra which is an admissible affine vertex al-gebra of type A (cf. Theorem 8.1), or in the case of the affine W -algebras W − ( sl (5) , θ ) (cf. Corollary 8.3), W − ( sl ( n + 5 | n ) , θ ) (cf. Corollary 9.3, Re-mark 9.2). We believe that this property of the modules W k ( g , θ ) ( i ) holds inall cases (1) and (2) from Theorem 1.1. ADAMOVI´C, KAC, M ¨OSENEDER, PAPI, PERˇSE
As a byproduct, we construct a new family of simple current modules forthe vertex superalgebra V k ( sl ( m | n )) which belong to the category KL k (cf.Theorem 9.1).Next we consider the cases when g ♮ is a semisimple Lie algebra. Then wehave a natural decomposition W k ( g , θ ) = W k ( g , θ ) ¯0 ⊕ W k ( g , θ ) ¯1 , according to the parity of twice the conformal weight.The subspaces W k ( g , θ ) ¯ i are naturally V k ( g ♮ )-modules, so we are reducedto compute the decompositions of the W k ( g , θ ) ¯ i . We solve our problem inmany cases by showing that the subspaces W k ( g , θ ) ¯ i are actually irreducibleas V k ( g ♮ )-modules: Theorem 1.2.
Consider the following conformal embeddings of V k ( g ♮ ) into W k ( g , θ ) [10] : (1) g = so ( n ) ( n ≥ , n = 11 ), g = osp (4 | n ) ( n ≥ ), g = osp ( m | n ) ( m ≥ , m = n + r , r ∈ {− , , , , , , , } ) or g is of type G , F E , E , E or g is a Lie superalgebra D (2 , , a ) ( a = 1 , / , F (4) and k = − h ∨ − . (2) g = sp ( n ) ( n ≥ ), g = spo (2 | m ) ( m ≥ , m = 4 ), g = spo ( n | m ) ( n ≥ , m = n + 2 , n, n − , n − ) and k = − h ∨ .Then V k ( g ♮ ) is simple and each W k ( g , θ ) ¯ i , i = 0 , , is an irreducible V k ( g ♮ ) -module. In most cases, the proof of this theorem follows quite easily combininginformation on fusion rules coming from tensor product decompositions of g ♮ -modules with basic principles of vertex algebra theory, such as Galoistheory: see Theorem 6.8. In some cases our proof is technically involvedand it uses the representation theory of the admissible affine vertex algebra V − / ( sl (2)): see Theorem 6.9.We also believe that the conformal embeddings listed in Theorem 1.2provide all cases where W k ( g , θ ) is isomorphic to an extension of V k ( g ♮ ) byits simple module. This result is also interesting from the perspective ofextensions of affine vertex algebras, since it gives a realization of a broadlist of extensions which (so far) cannot be constructed by other methods.In our forthcoming papers we shall consider embeddings of V k ( g ♮ ) into W k ( g , θ ) when critical levels appear. Acknowledgments.
Draˇzen Adamovi´c and Ozren Perˇse are partiallysupported by the Croatian Science Foundation under the project 2634 and bythe Croatian Scientific Centre of Excellence QuantixLie. Pierluigi M¨osenederFrajria and Paolo Papi are partially supported by PRIN project “Spazi diModuli e Teoria di Lie”.
Notation.
The base field is C . As usual, tensor product of a fam-ily of vector spaces ranging over the empty set is meant to be C . For ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 5 a vector superspace V = V ¯0 ⊕ V ¯1 we set Dim V = dim V ¯0 | dim V ¯1 andsdim V = dim V ¯0 − dim V ¯1 . 2. Preliminaries
Vertex algebras.
Here we fix notation about vertex algebras (cf.[28]). Let V be a vertex algebra, with vacuum vector . The vertex op-erator corresponding to the state a ∈ V is denoted by Y ( a, z ) = X n ∈ Z a ( n ) z − n − , where a ( n ) ∈ End V .
We frequently use the notation of λ -bracket and normally ordered product:[ a λ b ] = X i ≥ λ i i ! ( a ( i ) b ) , : ab := a ( − b. If V admits a Virasoro(=conformal) vector and ∆ a is the conformal weightof a , then we also write the corresponding vertex operator as Y ( a, z ) = X m ∈ Z − ∆ a a m z − m − ∆ a , so that a ( n ) = a n − ∆ a +1 , n ∈ Z , a m = a ( m +∆ a − , m ∈ Z − ∆ a . The product of two subsets
A, B of V is A · B = span ( a ( n ) b | n ∈ Z , a ∈ A, b ∈ B ) . This product is associative (cf. [13]).2.2.
Intertwining operators and fusion rules.
Let V be a vertex alge-bra. A V -module is a vector superspace M endowed with a parity preservingmap Y M from V to the superspace of End ( M )-valued fields a Y M ( a, z ) = X n ∈ Z a M ( n ) z − n − such that(1) Y M ( | i , z ) = I M ,(2) for a, b ∈ V , m, n, k ∈ Z , X j ∈ N (cid:18) mj (cid:19) ( a ( n + j ) b ) M ( m + k − j ) = X j ∈ N ( − j (cid:18) nj (cid:19) ( a M ( m + n − j ) b M ( k + j ) − p ( a, b )( − n b M ( k + n − j ) a M ( m + j ) ) , ADAMOVI´C, KAC, M ¨OSENEDER, PAPI, PERˇSE
Given three V -modules M , M , M , an intertwining operator of type (cid:20) M M M (cid:21) (cf. [22], [23]) is a map I : a I ( a, z ) = P n ∈ Z a I ( n ) z − n − from M to the space of End ( M , M )-valued fields such that for a ∈ V , b ∈ M , m, n ∈ Z , X j ∈ N (cid:18) mj (cid:19) ( a M ( n + j ) b ) I ( m + k − j ) = X j ∈ N ( − j (cid:18) nj (cid:19) ( a M ( m + n − j ) b I ( k + j ) − p ( a, b )( − n b I ( k + n − j ) a M ( m + j ) ) . We let I (cid:0) M M M (cid:1) denote the space of intertwining operators of type (cid:20) M M M (cid:21) ,and set N M M ,M = dim I (cid:18) M M M (cid:19) . When N M M ,M is finite, it is usually called a fusion coefficient .Assume that in a category K of Z ≥ –graded V -modules, the irreduciblemodules { M i | i ∈ I } , where I is an index set, have the following properties(1) for every i, j ∈ I N M k M i ,M j is finite for any k ∈ I ;(2) N M k M i ,M j = 0 for all but finitely many k ∈ I .Then the algebra with basis { e i ∈ I } and product e i · e j = X k ∈ I N M k M i ,M j e k is called the fusion algebra of V, K . (Note that we consider only fusion rulesbetween irreducible modules).Let K be a category of V -modules. Let M , M be irreducible V -modulesin K . Given an irreducible V -module M in K , we will say that the fusionrule(2.1) M × M = M holds in K if N M M ,M = 1 and N RM ,M = 0 for any other irreducible V -module R in K which is not isomorphic to M .We say that an irreducible V -module M is a simple current in K if M is in K and, for every irreducible V -module M in K , there is an irreducible V -module M in K such that the fusion rule (2.1) holds in K (see [19]).2.3. Commutants.
Let U be a vertex subalgebra of V . Then the commu-tant of U in V (cf. [22]) is the following vertex subalgebra of V :Com( U, V ) = { v ∈ V | [ Y ( a, z ) , Y ( v, w )] = 0 , ∀ a ∈ U } . ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 7 In the case when U is an affine vertex algebra, say U = V k ( g ) (see below),it is easy to see thatCom( U, V ) = { v ∈ V | x ( n ) v = 0 ∀ x ∈ g , n ≥ } . Zhu algebras.
Assume that V is a vertex algebra, endowed with aconformal vector ω such that the conformal weights ∆ v of v ∈ V are in Z .Then V = M r ∈ Z V ( r ) , V ( r ) = { v ∈ V | ∆ v = r } . Set V ¯0 = M r ∈ Z V ( r ) , V ¯1 = M r ∈ + Z V ( r ) . We define two bilinear maps ∗ : V × V → V , ◦ : V × V → V as follows:for homogeneous a, b ∈ V , let a ∗ b = ( Res x Y ( a, x ) (1+ x ) ∆ a x b if a, b ∈ V ¯0 ,0 if a or b ∈ V ¯1 (2.2) a ◦ b = Res x Y ( a, x ) (1+ x ) ∆ a x b if a ∈ V ¯0 ,Res x Y ( a, x ) (1+ x ) ∆ a − x b if a ∈ V ¯1 . (2.3)Next, we extend ∗ and ◦ to V ⊗ V linearly, and denote by O ( V ) ⊂ V thelinear span of elements of the form a ◦ b , and by A ( V ) the quotient space V /O ( V ). The space A ( V ) has a unital associative algebra structure, withmultiplication induced by ∗ . The algebra A ( V ) is called the Zhu algebra of V . The image of v ∈ V , under the natural map V A ( V ) will be denotedby [ v ]. In the case when V ¯0 = V we get the usual definition of Zhu algebrafor vertex operator algebras.An important result, proven by Zhu [38] for Z -graded V and later byKac-Wang [35] for Z -graded V , is the following theorem. Theorem 2.1.
There is a one-to-one correspondence between irreducible Z ≥ -graded V -modules and irreducible A ( V ) -modules. Affine vertex algebras.
Let a be a Lie superalgebra equipped witha nondegenerate invariant supersymmetric bilinear form B . The universalaffine vertex algebra V B ( a ) is the universal enveloping vertex algebra of theLie conformal superalgebra R = ( C [ T ] ⊗ a ) ⊕ C with λ -bracket given by[ a λ b ] = [ a, b ] + λB ( a, b ) , a, b ∈ a . In the following, we shall say that a vertex algebra V is an affine vertexalgebra if it is a quotient of some V B ( a ).If a is simple or is a one dimensional abelian Lie algebra, then one usuallyfixes a nondegenerate invariant supersymmetric bilinear form ( ·|· ) on a . Anyinvariant supersymmetric bilinear form is therefore a constant multiple of( ·|· ). In particular, if B = k ( ·|· ) ( k ∈ C ), then we denote V B ( a ) by V k ( a ). ADAMOVI´C, KAC, M ¨OSENEDER, PAPI, PERˇSE
Let h ∨ a be half of the eigenvalue of the Casimir element corresponding to( ·|· ); if h ∨ a = − k , then V k ( a ) admits a unique irreducible quotient which wedenote by V k ( a ).In the same hypothesis, V k ( a ) is equipped with a Virasoro vector(2.4) ω a sug = 12( k + h ∨ a ) X i =1 : b i a i : , where { a i } is a basis of a and { b i } is its dual basis with respect to ( ·|· ). Ifa vertex algebra V is some quotient of V k ( a ), we will say that k is the levelof V .A module M for b a is said to be of level k if K acts on M by kI M . Finallyrecall that an irreducible highest weight module L ( λ ) , over an affine algebra b a is said to be admissible [26] if(1) ( λ + ρ )( α ) / ∈ { , − , − , . . . } , for each positive coroot α ;(2) the rational linear span of positive simple coroots equals the rationallinear span of the coroots which are integral valued on λ + ρ .2.6. Heisenberg vertex algebras.
In the special case when a is an abelianLie algebra, V B ( a ) is of course a Heisenberg vertex algebra. If this is thecase, we will denote V B ( a ) by M a ( B ). In the special case when a is onedimensional, then we can choose a basis { α } of a and the form ( ·|· ) so that( α | α ) = 1. With these choices we denote V k ( a ) by M α ( k ) or simply by M ( k ) when the reference to the generator α need not to be explicit. Thevertex algebra M ( k ) is called the universal Heisenberg vertex algebra oflevel k generated by α . Recall that, if k = 0, M α ( k ) is simple and that M ( k ) ∼ = M (1). The irreducible M α ( k )-modules are the modules M α ( k, s )(or simply M ( k, s )) generated by a vector v s with action, for n ∈ Z + , givenby α ( n ) v s = δ n, sv s .The irreducible modules of the Heisenberg vertex algebra M ( k ) are allsimple currents in the category of M ( k )-modules. Indeed we have the fol-lowing fusion rules (cf.[22]): M ( k, s ) × M ( k, s ) = M ( k, s + s ) ( s , s ∈ C ) . (2.5)If a is simple or one-dimensional even abelian with fixed nondegenerateinvariant supersymmetric bilinear form ( ·|· ), the affinization of a is the Liesuperalgebra b a = ( C [ t, t − ] ⊗ a ) ⊕ C d ⊕ C K where K is a central element,and d acts as td/dt . We choose the central element K so that[ t s ⊗ x, t r ⊗ y ] = t s + r ⊗ [ x, y ] + δ r, − s rK ( x | y ) . Let h be a Cartan subalgebra of a and b h = h ⊕ C K ⊕ C d a Cartan subalgebraof b a . Let Λ ∈ b h ∗ be defined by Λ ( K ) = 1, Λ ( h ) = Λ ( d ) = 0. We fix a setof simple roots for b a and denote by ρ ∈ b h ∗ a corresponding Weyl vector. Weshall denote by L a ( λ ) the irreducible highest weight V k ( a )-module of highestweight λ ∈ b h ∗ . Sometimes, if no confusion may arise, we simply write L ( λ ).Similarly, we shall denote by V a ( λ ) or simply by V ( λ ) the irreducible highest ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 9 weight a -module of highest weight λ ∈ h ∗ . Note that in the case when a isa one dimensional abelian Lie algebra C α , then M α ( k, s ) = L ( k Λ + sλ )where λ ∈ ( C α ) ∗ is defined by setting λ ( α ) = 1.2.7. Rank one lattice vertex algebra V Z α . Assume that L is an integral,positive definite lattice; let L ◦ be its dual lattice. Set V L to be the latticevertex algebra (cf. [28]) associated to L .The set of isomorphism classes of irreducible V L -modules is parametrizedby L ◦ /L (cf. [16]). Let V ¯ λ denote the irreducible V L -module correspondingto ¯ λ = λ + L ∈ L ◦ /L . Every irreducible V L -module is a simple current.We shall now consider rank one lattice vertex algebras. For n ∈ Z > ,let M ( n ) be the universal Heisenberg vertex algebra generated by α andlet F n denote the lattice vertex algebra V Z α = M ( n ) ⊗ C [ Z α ] associated tothe lattice L = Z α , h α, α i = n . The dual lattice of L is L o = n L . For i ∈ { , . . . , n − } , set F in = V in α + Z α . Then the set { F in | i = 0 , . . . , n − } provides a complete list of non-isomorphic irreducible F n -module. We choosethe following Virasoro vector in F n : ω F n = 12 n : αα : . As a M ( n )-module, F n decomposes as F n = M j ∈ Z M ( n ) e jα = M j ∈ Z M ( n, jn )(2.6)The following result is a consequence of the result of H. Li and X. Xu [37]on characterization of lattice vertex algebras. Proposition 2.2.
Assume that V = L i ∈ Z V i is a Z -graded vertex algebrasatisfying the following properties (1) V is a subalgebra of a simple vertex algebra W ; (2) there exists a Heisenberg vector α ∈ V such that V = M α ( n ) , and V i ∼ = M α ( n, in ) as a V -module.Then V is a simple vertex algebra and V ∼ = F n .Proof. The Main Theorem of [37] implies that a simple vertex algebra satis-fying condition (2) is isomorphic to F n . To prove simplicity, we first observethat Y ( v, z ) w = 0 ∀ v, w ∈ V , (2.7)which in our setting holds since W is simple. Now (2.7) and the fusion rules(2.5) imply that V i · V j = V i + j ( i, j ∈ Z ) . This implies that V is simple, and the claim follows. (cid:3) Minimal quantum affine W -algebras In this section we briefly recall some results of [30] and [10]. We include anexample which contains explicit λ –bracket formulas for W k ( sl (4) , θ ) whichwe shall need in Section 7.We first recall the definition of minimal affine W -algebras.Let g be a basic simple Lie superalgebra. Choose a Cartan subalgebra h for g ¯0 and let ∆ be the set of roots. Fix a minimal root − θ of g . (Aroot − θ is called minimal if it is even and there exists an additive function ϕ : ∆ → R such that ϕ | ∆ = 0 and ϕ ( θ ) > ϕ ( η ) , ∀ η ∈ ∆ \ { θ } ). We chooseroot vectors e θ and e − θ such that[ e θ , e − θ ] = x ∈ h , [ x, e ± θ ] = ± e ± θ . Due to the minimality of − θ , the eigenspace decomposition of ad x definesa minimal Z -gradation ([30, (5.1)]):(3.1) g = g − ⊕ g − / ⊕ g ⊕ g / ⊕ g , where g ± = C e ± θ . One has(3.2) g = g ♮ ⊕ C x, g ♮ = { a ∈ g | ( a | x ) = 0 } . For a given choice of a minimal root − θ , we normalize the invariant bilinearform ( ·|· ) on g by the condition(3.3) ( θ | θ ) = 2 . The dual Coxeter number h ∨ of the pair ( g , θ ) is defined to be half theeigenvalue of the Casimir operator of g corresponding to ( ·|· ).The complete list of the Lie superalgebras g ♮ , the g ♮ -modules g ± / (theyare isomorphic and self-dual), and h ∨ for all possible choices of g and of θ (up to isomorphism) is given in Tables 1,2,3 of [30], and it is as follows Table 1 g is a simple Lie algebra. g g ♮ g / h ∨ g g ♮ g / h ∨ sl ( n ) , n ≥ gl ( n − C n − ⊕ ( C n − ) ∗ n F sp (6) V C so ( n ) , n ≥ sl (2) ⊕ so ( n − C ⊗ C n − n − E sl (6) V C sp ( n ) , n ≥ sp ( n − C n − n/ E so (12) spin G sl (2) S C E E dim = 56 30Table 2 g is not a Lie algebra but g ♮ is and g ± / is purely odd ( m ≥ ). g g ♮ g / h ∨ g g ♮ g / h ∨ sl (2 | m ) , gl ( m ) C m ⊕ ( C m ) ∗ − m D (2 , a ) sl (2) ⊕ sl (2) C ⊗ C m = 2 psl (2 | sl (2) C ⊕ C F (4) so (7) spin − spo (2 | m ) so ( m ) C m − m/ G (3) G Dim = 0 | − / osp (4 | m ) sl (2) ⊕ sp ( m ) C ⊗ C m − m Table 3
ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 11 Both g and g ♮ are not Lie algebras ( m, n ≥ ). g g ♮ g / h ∨ sl ( m | n ), m = n, m > gl ( m − | n ) C m − | n ⊕ ( C m − | n ) ∗ m − npsl ( m | m ) , m > sl ( m − | m ) C m − | m ⊕ ( C m − | m ) ∗ spo ( n | m ) , n ≥ spo ( n − | m ) C n − | m / n − m ) + 1 osp ( m | n ) , m ≥ osp ( m − | n ) ⊕ sl (2) C m − | n ⊗ C m − n − F (4) D (2 ,
1; 2) Dim = 6 | G (3) osp (3 |
2) Dim = 4 | In this paper we shall exclude the case of g = sl ( n + 2 | n ), n >
0. In allother cases the Lie superalgebra g ♮ decomposes in a direct sum of ideals,called components of g ♮ :(3.4) g ♮ = M i ∈ I g ♮i , where each summand is either the (at most 1-dimensional) center of g ♮ oris a basic classical simple Lie superalgebra different from psl ( n | n ). We willalso exclude g = sl (2).It follows from the tables that the index set I has cardinality r = 0, 1, 2,or 3. The case r = 0, i.e. g ♮ = { } , happens if and only if g = spo (2 | { } ) we use I = { , , . . . , r − } (resp. I = { , . . . , r } ) as the index set, and denote the center of g ♮ by g ♮ .Let C g ♮i be the Casimir operator of g ♮i corresponding to ( ·|· ) | g ♮i × g ♮i . Wedefine the dual Coxeter number h ∨ ,i of g ♮i as half of the eigenvalue of C g ♮i acting on g ♮i (which is 0 if g ♮i is abelian). Their values are given in Table 4of [30].Let W k ( g , e − θ ) be the minimal W-algebras of level k studied in [30]. It isknown that, for k non-critical, i.e., k = − h ∨ , the vertex algebra W k ( g , e − θ )has a unique simple quotient, denoted by W k ( g , e − θ ).To simplify notation, we set W k ( g , θ ) = W k ( g , e − θ ) , W k ( g , θ ) = W k ( g , e − θ ) . Throughout the paper we shall assume that k = − h ∨ . In such a case, itis known that W k ( g , f ) has a Virasoro vector ω , [30, (2.2)] that has centralcharge [30, (5.7)](3.5) c ( g , k ) = k sdim g k + h ∨ − k + h ∨ − . It is proven in [30] that the universal minimal W-algebra W k ( g , θ ) is freelyand strongly generated by the elements J { a } ( a runs over a basis of g ♮ ), G { u } ( u runs over a basis of g − / ), and the Virasoro vector ω . Furthermore theelements J { a } (resp. G { u } ) are primary of conformal weight 1 (resp. 3 / ω . The λ -brackets satisfied by these generators have been given in [30] and, in a simplified form, in [10]. This simplified form reads:[ J { a } λ J { b } ] = J { [ a,b ] } + λ (cid:0) ( k + h ∨ / a | b ) − κ ( a, b ) (cid:1) , a, b ∈ g ♮ , (3.6) [ J { a } λ G { u } ] = G { [ a,u ] } , a ∈ g ♮ , u ∈ g − / . (3.7)[ G { u } λ G { v } ] = − k + h ∨ )( e θ | [ u, v ]) ω + ( e θ | [ u, v ]) dim g ♮ X α =1 : J { u α } J { u α } :(3.8)+ dim g / X γ =1 : J { [ u,u γ ] ♮ } J { [ u γ ,v ] ♮ } : +2( k + 1) ∂J { [[ e θ ,u ] ,v ] ♮ } + 4 λ X i ∈ I p ( k ) k i J { [[ e θ ,u ] ,v ] ♮i + 2 λ ( e θ | [ u, v ]) p ( k ) . Here κ is the Killing form of g ; { u α } (resp. { v γ } ) is a basis of g ♮ (resp. g / ) and { u α } (resp. { u γ } ) is the corresponding dual basis w.r.t. ( ·|· ) (respw.r.t. h· , ·i ne = ( e − θ | [ · , · ])), a ♮ is the image of a ∈ g under the orthogonalprojection of g on g ♮ , a ♮i is the projection of a ♮ on the i th minimal ideal g ♮i of g ♮ , k i = k + ( h ∨ − h ∨ ,i ), and p ( k ) is the monic polynomial given in thefollowing table [10]: Table 4 g p ( k ) g p ( k ) sl ( m | n ), n = m ( k + 1)( k + ( m − n ) / E ( k + 3)( k + 4) psl ( m | m ) k ( k + 1) E ( k + 4)( k + 6) osp ( m | n ) ( k + 2)( k + ( m − n − / E ( k + 6)( k + 10) spo ( n | m ) ( k + 1 / k + ( n − m + 4) / F ( k + 5 / k + 3) D (2 , a ) ( k − a )( k + 1 + a ) G ( k + 4 / k + 5 / F (4), g ♮ = so (7) ( k + 2 / k − / G (3), g ♮ = G ( k − / k + 3 / F (4), g ♮ = D (2 ,
1; 2) ( k + 3 / k + 1) G (3), g ♮ = osp (3 |
2) ( k + 2 / k + 4 / Note that the linear polynomials k i always divide p ( k ) so the coefficientsin (3.8) depend polynomially on k . Example 3.1.
Consider g = sl (4). Set c = 12 − − . In this case g ♮ = g ♮ ⊕ g ♮ with g ♮ = C c, g ♮ = A
00 0 0 | A ∈ sl (2) ≃ sl (2) , so g ♮ ≃ gl (2), while g − / = span ( e , , e , , e , , e , ). ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 13 The λ -brackets [ G { u } λ G { v } ] are as follows:[ G { e , } λ G { e , } ] = [ G { e , } λ G { e , } ] = 0[ G { e , } λ G { e , } ] = [ G { e , } λ G { e , } ] = 0[ G { e , } λ G { e , } ] = [ G { e , } λ G { e , } ] = 0[ G { e , } λ G { e , } ] = 2 : J { c } J { e , } : − ( k + 2) ∂J { e , } − λ k + 2) J { e , } [ G { e , } λ G { e , } ] = 2 : J { c } J { e , } : − ( k + 2) ∂J { e , } − λ k + 2) J { e , } [ G { e , } λ G { e , } ] =( k + 4) ω − J { e , } J { e , } : −
12 : J { e , − e , } J { e , − e , } : −
32 : J { c } J { c } : + : J { c } J { e , − e , } : +( k + 1) ∂J { c } − k ∂J { e , − e , } + λ k + 1) J { c } − λ ( k + 2) J { e , − e , } − λ ( k + 1)( k + 2) [ G { e , } λ G { e , } ] = − ( k + 4) ω + 2 : J { e , } J { e , } : + 12 : J { e , − e , } J { e , − e , } :+ 32 : J { c } J { c } : + : J { c } J { e , − e , } : +( k + 1) ∂J { c } + k ∂J { e , − e , } + λ k + 1) J { c } + λ ( k + 2) J { e , − e , } + λ ( k + 1)( k + 2) . A classification of conformal levels from [10]In this section we recall the definition of conformal embeddings of affinevertex subalgebras into minimal affine W –algebras and review results from[10] on the classification of conformal levels.Let V k ( g ♮ ) be the subalgebra of the vertex algebra W k ( g , θ ), generated by { J { a } | a ∈ g ♮ } . By (3.6), V k ( g ♮ ) is isomorphic to a universal affine vertexalgebra. More precisely, the map a J { a } extends to an isomorphism(4.1) V k ( g ♮ ) ≃ O i ∈ I V k i ( g ♮i ) . Definition 4.1.
We set V k ( g ♮ ) to be the image of V k ( g ♮ ) in W k ( g , θ ).Clearly we can write V k ( g ♮ ) ≃ O i ∈ I V k i ( g ♮i ) , where V k i ( g ♮i ) is some quotient (not necessarily simple) of V k i ( g ♮i ). If k i + h ∨ ,i = 0, then V k i ( g ♮i ) is equipped with the Virasoro vector ω g ♮i sug (cf. (2.4)). If k i + h ∨ ,i = 0 for all i , we set ω sug = X i ∈ I ω g ♮i sug . Define K = { k ∈ C | k + h ∨ = 0 , k i + h ∨ ,i = 0 whenever k i = 0 } . If k ∈ K we also set ω ′ sug = X i ∈ I : k i =0 ω g ♮i sug . We define c sug = central charge of ω ′ sug . Definition 4.2.
Assume k ∈ K . We say that V k ( g ♮ ) is conformally embed-ded in W k ( g , θ ) if ω ′ sug = ω . The level k is called a conformal level .If W k ( g , θ ) = V k ( g ♮ ), we say that k is a collapsing level . Remark 4.3.
The above definition of conformal level is slightly more gen-eral than the one given in the Introduction. Indeed it makes sense also when k i = h ∨ ,i = 0.Next we recall the classification of collapsing levels from [10]. Proposition 4.1. [10, Theorem 3.3]
The level k is collapsing if and only if p ( k ) = 0 where p is the polynomiallisted in the Table 4. The classification of non-collapsing conformal levels is given in Section 4of [10]. It may be summarized as follows.
Proposition 4.2. (I).
Assume that g ♮ is either zero or simple or 1-dimensional.If g = sl (3) , or g = spo ( n | n + 2) with n ≥ , g = spo ( n | n − with n ≥ , g = spo ( n | n − with n ≥ , then there are no non–collapsing conformallevels. In all other cases the non-collapsing conformal levels are (1) k = − h ∨ − if g is of type G , F , E , E , E , F (4)( g ♮ = so (7)) , G (3) ( g ♮ = G , osp (3 | ), or g = psl ( m | m ) ( m ≥ ); (2) k = − h ∨ if g = sp ( n ) ( n ≥ , or g = spo (2 | m ) ( m ≥ , or g = spo ( n | m ) ( n ≥ . (II). Assume that g ♮ = g ♮ ⊕ g ♮ with g ♮ ≃ C and g ♮ simple.If g = sl ( m | m − with m ≥ , then there are no non–collapsing conformallevels. In other cases the non–collapsing conformal levels are (1) k = − h ∨ if g = sl ( m | m +1) ( m ≥ ), and g = sl ( m | m − ( m ≥ ); ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 15 (2) k = − h ∨ and k = − h ∨ − in all other cases. (III). Assume that g ♮ = P ri =1 g ♮i with g ♮ ≃ sl (2) and r ≥ . If g = osp ( n + 5 | n ) with n ≥ or g = D (2 , a ) with a = , − , − , then there areno non–collapsing conformal levels. In the other cases the non–collapsingconformal levels are (1) k = − h ∨ − if g = D (2 , a ) ( a
6∈ { , − , − } ), g = osp ( n + 8 | n ) ( n ≥ ), g = osp ( n + 2 | n ) ( n ≥ ), g = osp ( n − | n ) ( n ≥ ); (2) k = − h ∨ if g = osp ( n + 7 | n ) ( n ≥ ), g = osp ( n + 1 | n ) ( n ≥ ); (3) k = − h ∨ and k = − h ∨ − in all other cases. It is important to observe that, if k is a conformal level, we have thefollowing identification of the Zhu algebra of W k ( g , θ ). Proposition 4.3.
Assume that k is a conformal non-collapsing level. Let J be any proper ideal in W k ( g , θ ) which contains ω − ω sug . Then there is asurjective homomorphism of associative algebras A ( V k ( g ♮ )) → A ( W k ( g , θ ) / J ) . In particular, A ( W k ( g , θ )) is isomorphic to a certain quotient of U ( g ♮ ) .Proof. Recall first that if a vertex algebra V is strongly generated by the set S ⊂ V , then Zhu’s algebra A ( V ) is generated by the set { [ a ] , a ∈ S } (cf. [1,Proposition 2.5], [17]). Since W k ( g , θ ) / J is strongly generated by the set { G { u } , u ∈ g − / } ∪ { J { x } , x ∈ g ♮ } , we have that A ( W k ( g , θ ) / J ) is generated by the set { [ G { u } ] , u ∈ g − / } ∪ { [ J { x } ] , x ∈ g ♮ } . On the other hand, since G { u } = G { u } ◦ ∈ O ( W k ( g , θ ) / J ), we have[ G { u } ] = 0 in A ( W k ( g , θ ) / J ) for every u ∈ g − / . Therefore, A ( W k ( g , θ ) / J )is only generated by the set { [ J { x } ] , x ∈ g ♮ } . This gives a surjective homo-morphism A ( V k ( g ♮ )) = U ( g ♮ ) → A ( W k ( g , θ ) / J ). (cid:3) We should also mention that a conjectural generalization of our resultsto conformal embeddings of affine vertex algebras into more general W –algebras have been recently proposed by T. Creutzig in [15].5. Some results on admissible affine vertex algebras
Assume g is a simple Lie superalgebra. Let O k be the category of b g -modules from the category O of level k . Let KL k be the subcategory of O k consisting of modules on which g acts locally finitely. Note that modulesfrom KL k are V k ( g )-modules. Moreover, every irreducible module M from KL k has finite-dimensional weight spaces with respect to ( ω g sug ) and admitsthe following Z ≥ –gradation: M = M n ∈ Z ≥ M ( n ) , ( ω g sug ) | M ( n ) ≡ ( n + h )Id ( h ∈ C ) , (cf. [34]; such modules are usually called ordinary modules in the the ter-minology of vertex operator algebra theory [20]). The graded component M (0) is usually called the lowest graded component.5.1. Fusion rules for certain affine vertex algebras.
The classificationof irreducible modules in the category O k for affine vertex algebras V k ( g ) atadmissible levels was conjectured in [5] and proved by Arakawa in [12]. Weneed the classification result in the subcategory KL k of the category O k . Definition 5.1.
We define KL k to be the category of all modules M in KL k which are V k ( g )-modules.The classification of irreducible modules in the category KL k coincideswith the classification of irreducible V k ( g )-modules having finite-dimensionalweight spaces with respect to ( ω g sug ) [5], [12].We restrict our attention to g = sl ( n ) with ( ·|· ) the trace form. We choosea set of positive roots for g and let ω i ∈ h ∗ ( i = 1 , . . . , n −
1) denote thecorresponding fundamental weights. Set Λ i = Λ + ω i . Recall from 2.2 thedefinition of fusion rules. Proposition 5.1.
Let k = − n , n ≥ .(1) The set { L sl (2 n − ( k Λ + Λ i ) | i = 0 , . . . , n − } (5.1) provides a complete list of irreducible V k +1 ( sl (2 n − -modules in the cate-gory KL k +1 .(2) The following fusion rule holds in KL k +1 : L sl (2 n − ( k Λ + Λ i ) × L sl (2 n − ( k Λ + Λ i ) = L sl (2 n − ( k Λ + Λ i ) where ≤ i , i , i ≤ n − are such that i + i ≡ i mod (2 n − . In particular, the modules in (5.1) are simple currents in the category KL k +1 .Proof. First we notice that the set of admissible weights of level k + 1 whichare dominant with respect to sl (2 n −
2) is { k Λ + Λ i | i = 0 , . . . , n − } . Now assertion (1) follows from the main result from [12].Assertion (2) follows from (1) and the fact that the tensor product V sl (2 n − ( ω i ) ⊗ V sl (2 n − ( ω i )contains a component V sl (2 n − ( ω i ) if and only if i + i ≡ i mod (2 n − . (cid:3) The proof of the following result is completely analogous to the proof ofProposition 5.1.
ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 17 Proposition 5.2.
Let k = ( n − / ∈ Z . Then(1) The set { L sl ( n ) ( − ( k + 2)Λ + Λ i ) | i = 0 , . . . , n − } (5.2) provides a complete list of irreducible V − k − ( sl ( n )) -modules in the category KL − k − .(2) The following fusion rules hold in KL − k − : L sl ( n ) ( − ( k + 2)Λ + Λ i ) × L sl ( n ) ( − ( k + 2)Λ + Λ i ) = L sl ( n ) ( − ( k + 2)Λ + Λ i ) where ≤ i , i , i ≤ n − are such that i + i ≡ i mod ( n ) . In particular, the modules in (5.2) are simple currents in the category KL − k − . Remark 5.2.
It is also interesting to notice that the fusion algebra gener-ated by irreducible modules for V / − n ( sl (2 n − KL / − n (resp. for V − n − ( sl ( n )) in the category KL − n − ) is isomorphic to thefusion algebra for the rational affine vertex algebra V ( sl (2 n − V ( sl ( n ))). Moreover, all irreducible modules in the KL k category for thesevertex algebras are simple currents.5.2. The vertex algebra V k ( sl (2)) . Recall that a level k is called admis-sible if k Λ is abmissible. If g = sl (2) then k is admissible if and only if k + 2 = pq , p, q ∈ N , ( p, q ) = 1, p ≥ e, h, f be the usual Chevalleygenerators for sl (2). Theorem 5.3.
Assume that k = pq − is an admissible level for c sl . Thenwe have:(1) [32, Corollary 1] . The maximal ideal in J k in V k ( sl (2)) is generated bya singular vector v λ of weight λ = ( k − p − + 2( p − .(2). The ideal J k is simple.Proof. We provide here a proof of (2) which uses Virasoro vertex algebrasand Hamiltonian reduction. This result can be also proved by using embed-ding diagrams for submodules of the Verma modules for c sl .Assume first that k / ∈ Z ≥ . Let V V ir ( c p,q ) be the universal Virasorovertex algebra of central charge c p,q = 1 − ( p − q ) pq . Then the maximal ideal in V V ir ( c p,q ) is irreducible and it is generated by a singular vector of conformalweight ( p − q −
1) (cf. [24], Theorem 4.2.1). So V V ir ( c p,q ) contains a uniqueideal which we shall denote by I p,q . Then L V ir ( c p,q ) = V V ir ( c p,q ) /I p,q is asimple vertex algebra.Recall that by quantum Hamiltonian reduction W k ( sl (2) , θ ) = V V ir ( c p,q ) . Let H V ir be the corresponding functor (denoted in [11] by H ∞ +0 f ), whichmaps V k ( sl (2))-modules to V V ir ( c p,q )-modules. Assume that I is a non-trivial, proper ideal in V k ( sl (2)). By using the main result of [11], we get that H V ir ( I ) = 0 , H V ir ( I ) = V V ir ( c p,q ). So H V ir ( I ) = I p,q . Since thefunctor H V ir is exact, we get that H V ir ( V k ( sl (2)) /I ) = V V ir ( c p,q , /I p,q = L V ir ( c p,q ) . By using again the exactness and non-triviality result of the functor H V ir we conclude that V k ( sl (2)) /I is simple. So I is the maximal ideal.If k ∈ Z ≥ , then the maximal ideal is J k = V k ( sl (2)) · ( e ( − ) k +1 and wehave H V ir ( J k ) = W k ( sl (2) , θ ) = V V ir ( c k +2 , ) = L V ir ( c k +2 , ) . Since H V ir ( J k ) is irreducible, the properties of the functor H V ir imply that J k is a simple ideal. (cid:3) It follows from [24], Theorem 9.1.2, that, if g is a simple Lie algebradifferent from sl (2), then the maximal ideal in V k ( g ) is either zero or it isnot simple.5.2.1. Representation theory of V − / ( sl (2)) . We now recall some knownfacts on the representation theory of the vertex algebra V − / ( sl (2)) (cf.[5] and Theorem 5.3).We first fix notation. Let L sl (2) ( λ ) be a highest weight V − / ( sl (2))-module with highest weight λ , and let v λ be the corresponding highest weightvector. Writing λ = − / + µ with µ ∈ h ∗ , we let N sl (2) ( λ ) denote thegeneralized Verma module induced from the simple sl (2)-module V sl (2) ( µ ).Let ω sl (2) sug be the Sugawara Virasoro vector for V − / ( sl (2)). For i ∈ Z ≥ we define the following weights: λ i = − ( i + 1 / + i Λ = − / + iω . Then one has:(1). The maximal ideal of V − / ( sl (2)) is generated by the singular vector v λ ∈ V − / ( sl (2)) of weight λ . In particular, V − / ( sl (2)) = V − / ( sl (2)) (cid:14) V − / ( sl (2)) · v λ . Moreover V − / ( sl (2)) · v λ is simple.(2). There is a singular vector v λ ∈ N sl (2) ( λ ) of weight λ such that L ( λ ) = N sl (2) ( λ ) (cid:14) V − / ( sl (2)) · v λ . Moreover V − / ( sl (2)) · v λ is simple.(3). L sl (2) ( λ i ), i = 0 ,
1, are irreducible V − / ( sl (2))-modules.Every V − / ( sl (2))-module from the category KL − is completely reducibleand isomorphic to a direct sum of certain copies of L sl (2) ( λ i ), i = 0 , KL − : L sl (2) ( λ ) × L sl (2) ( λ ) = V − / ( sl (2)) . (5.3) ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 19 This fusion rule follows from the tensor product decomposition V sl (2) ( ω ) ⊗ V sl (2) ( ω ) = V sl (2) (2 ω ) + V sl (2) (0)and the classification of irreducible modules for V − / ( sl (2))-modules from[5]). In particular, we only need to note that L sl (2) ( λ ) is not a V − / ( sl (2))-module. 6. Semisimplicity of conformal embeddings
The main goal of this section is to give criteria for establishing the simplic-ity of V k ( g ♮ ) together with the semisimplicity of W k ( g , θ ) as a V k ( g ♮ )-modulewhen k is a non-collapsing conformal level. We will give two separate cri-teria: one for the cases when g ♮ has a nontrivial center and another for thecases when g ♮ is centerless.6.1. Semisimplicity with nontrivial center of g ♮ . The next result col-lects some structural facts proven in [10, Proposition 4.6] describing thestructure of g − / as a g ♮ -module. Lemma 6.1.
Assume that g ♮ is a Lie algebra and g ♮ = { } (which happensonly for g = sl ( n ) or g = sl (2 | n ) , n = 2 ). Then (1) Dim g ♮ = 1 | . (2) A basis { c } of g ♮ can be chosen so that the eigenvalues of ad ( c ) actingon g − / are ± . (3) Let U + (resp. U − ) be the eigenspace for ad ( c ) | g − / correspondingto the eigenvalue (resp. − ). Then g − / = U + ⊕ U − with U ± irreducible finite dimensional mutually contragredient g ♮ -modules. By (3.7) and the above Lemma, J { c } (0) defines a Z -gradation on W k ( g , θ ): W k ( g , θ ) = M W k ( g , θ ) ( i ) , W k ( g , θ ) ( i ) = { v ∈ W k ( g , θ ) | J { c } (0) v = iv } . Recall that a primitive vector in a module M for an affine vertex algebrais a vector that is singular in some subquotient of M .In light of Lemma 6.1, we have that, in the Grothendieck group of finitedimensional representations of g ♮ , we can write U + ⊗ U − = V (0) + X ν i =0 V ( ν i ) . Theorem 6.2.
Assume that the embedding of V k ( g ♮ ) in W k ( g , θ ) is confor-mal and that W k ( g ) (0) does not contain V k ( g ♮ ) -primitive vectors of weight ν r .Then W k ( g ) (0) = V k ( g ♮ ) , V k ( g ♮ ) is a simple affine vertex algebra and W k ( g , θ ) ( i ) are simple V k ( g ♮ ) -modules. Proof.
Let U ± = span ( G { u } | u ∈ U ± ). Let A ± = V k ( g ♮ ) · U ± . We claimthat(6.1) A − · A + ⊂ V k ( g ♮ ) . To check this, it is enough to check for all n ∈ Z , u ∈ U + , and v ∈ U − ,that G { u } ( n ) G { v } ∈ V k ( g ♮ ). Assume that this is not the case. Then wecan choose n maximal such that there are u ∈ U + , v ∈ U − such that G { u } ( n ) G { v } / ∈ V k ( g ♮ ). Since the map φ : U + ⊗ U − → W k ( g ) / V k ( g ♮ ) , φ : u ⊗ v G { u } ( n ) G { v } + V k ( g ♮ )is g ♮ -equivariant, we can choose a weight vector w = P u i ⊗ v i ∈ U + ⊗ U − of weight ν such that φ ( w ) is a highest weight vector in φ ( U + ⊗ U − ).Since, by maximality of n , φ ( w ) is singular for V k ( g ♮ ), we have that y = P G { u i } ( n ) G { v i } is primitive in W k ( g , θ ) so, by our hypothesis, ν = 0. Sincethe embedding is conformal, y is an eigenvector for ω ′ sug and since φ ( w )is singular for V k ( g ♮ ) of weight 0, we see that its eigenvalue is zero. Sincethe embedding is conformal we have that y has conformal weight zero in W k ( g , θ ) so y ∈ C ⊂ V k ( g ♮ ), a contradiction.Since the embedding is conformal, W k ( g , θ ) is strongly generated by span n J { a } | a ∈ g ♮ o + U + + U − . It follows that W k ( g , θ ) (0) is contained in the sum of all fusion products oftype A · A · . . . · A r with A i ∈ { A + , A − , V k ( g ♮ ) } such that ♯ { i | A i = A + } = ♯ { i | A i = A − } . By the associativity of fusion products, we see that (6.1) implies that A · A · . . . · A r ⊂ V k ( g ♮ ), so W k ( g , θ ) (0) = V k ( g ♮ ). It follows that V k ( g ♮ ) is asimple affine vertex algebra and W k ( g , θ ) ( i ) is a simple V k ( g ♮ )-module for all i . (cid:3) If V ( µ ), µ ∈ ( h ♮ ) ∗ , is an irreducible g ♮ -module, then we can write(6.2) V ( µ ) = O j ∈ I V g ♮j ( µ j ) , where V g ♮j ( µ j ) is an irreducible g ♮j -module. Let ρ j be the Weyl vector in g ♮j (with respect to the positive system induced by the choice of positive rootsfor g ). Corollary 6.3.
If the embedding of V k ( g ♮ ) in W k ( g , θ ) is conformal and,for each irreducible subquotient V ( µ ) with µ = 0 of the g ♮ -module U + ⊗ U − ,we have X i ∈ I,k i =0 ( µ i , µ i + 2 ρ i )2( k i + h ∨ ,i ) Z + , (6.3) ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 21 then W k ( g ) (0) = V k ( g ♮ ) , V k ( g ♮ ) is a simple affine vertex algebra and the W k ( g , θ ) ( i ) are simple V k ( g ♮ ) -modules.Proof. In order to apply Theorem 6.2, we need to check that if µ = 0 and V ( µ ) is an irreducible subquotient of U + ⊗ U − , then there is no primitivevector v in W k ( g , θ ) (0) with weight µ . Since the embedding is conformal, ω ′ sug acts diagonally on W k ( g , θ ). In particular, we can assume that v is aneigenvector for ω ′ sug . Let N ⊂ M ⊂ W k ( g , θ ) be submodules such that v + N is a singular vector in M/N . Then v + N is an eigenvector for the actionof ω ′ sug on M/N and the corresponding eigenvalue is P ri =0 ,k i =0 ( µ i ,µ i +2 ρ i )2( k i + h ∨ ,i ) .It follows that the eigenvalue for ω ′ sug acting on v is P ri =0 ,k i =0 ( µ i ,µ i +2 ρ i )2( k i + h ∨ ,i ) .It is easy to check that the conformal weights of elements in W k ( g , θ ) (0) arepositive integers hence, since ω ′ sug = ω , P ri =0 ,k i =0 ( µ i ,µ i +2 ρ i )2( k i + h ∨ ,i ) must be in Z + ,a contradiction. (cid:3) We now apply Corollary 6.3 to the cases where g ♮ is a basic Lie super-algebra with nontrivial center. These can be read off from Tables 1–3 andcorrespond to taking g = sl ( n ) ( n ≥ g = sl (2 | n ) ( n ≥ n = 2) or g = sl ( m | n ) ( n = m > Theorem 6.4.
Assume that we are in the following cases of conformalembedding of V k ( g ♮ ) into W k ( g , θ ) . (1) g = sl ( n ) , n ≥ , conformal level k = − n − = − h ∨ − ; (2) g = sl ( n ) , n ≥ , n = 6 , conformal level k = − n = − h ∨ ; (3) g = sl (2 | n ) , n ≥ , conformal level k = n − = − h ∨ − ; (4) g = sl (2 | n ) , n ≥ , conformal level k = n − = − h ∨ . (5) g = sl ( m | n ) , m > , m = n + 3 , n + 2 , n, n − , conformal level k = n − m +12 = − h ∨ − ; (6) g = sl ( m | n ) , m > , m = n + 6 , n + 4 , n + 2 , n , conformal level k = n − m )3 = − h ∨ .Then V k ( g ♮ ) is a simple affine vertex algebra and W k ( g , θ ) ( i ) is an irre-ducible V k ( g ♮ ) -module for every i ∈ Z . In particular, W k ( g , θ ) is a semisim-ple V k ( g ♮ ) -module.Proof. We verify that the assumptions of Corollary 6.3 hold. In cases (1)and (2), g ♮ ≃ gl ( n −
2) = C Id ⊕ sl ( n − g ♮ ≃ C and g ♮ ≃ sl ( n − U + = C n and U − = ( C n ) ∗ . The tensor product U + ⊗ U − decomposes as V (0) ⊕ V ( µ ) with µ = 0 , µ = ω + ω n − . Moreover k = k + h ∨ / = 0 and k = k + 1 = 0. Since r X i =0 ,k i =0 ( µ i , µ i + 2 ρ i )2( k i + h ∨ ,i ) = X i =0 ( µ i , µ i + 2 ρ i )2( k i + h ∨ ,i ) = n − n + k − , we see that (6.3) holds in cases (1) and (2).In cases (3) and (4), g ♮ ≃ gl ( n ) = C Id ⊕ sl ( n ), hence g ♮ ≃ C and g ♮ = sl ( n ). Moreover U + = C n and U − = ( C n ) ∗ . The tensor product U + ⊗ U − decomposes as V (0) ⊕ V ( µ ) with µ = 0 , µ = ω + ω n − . Moreover k = k + h ∨ / = 0 and k = k + 1 = 0. Since r X i =0 ,k i =0 ( µ i , µ i + 2 ρ i )2( k i + h ∨ ,i ) = X i =0 ( µ i , µ i + 2 ρ i )2( k i + h ∨ ,i ) = nn + k , we see that (6.3) holds in cases (3) and (4).In cases (5) and (6) we have g ♮ ≃ gl ( m − | n ) = C Id ⊕ sl ( m − | n ) (recallthat we are assuming m = n + 2), hence g ♮ ≃ C and g ♮ = sl ( m − | n ).Moreover U + = C m − | n and U − = ( C m − n | n ) ∗ . Then, as g ♮ -modules U + ⊗ U − ≃ sl ( m − | n ) ⊕ C . With notation as in [25], choose { ǫ − δ , δ − δ , . . . , δ n − − δ n , δ n − ǫ , . . . ,ǫ m − − ǫ m } as simple roots for g , so that the highest root is even. Theset of positive roots induced on g ♮ has as simple roots { δ − δ , . . . , δ n − − δ n , δ n − ǫ , . . . , ǫ m − − ǫ m − } . Then we have µ = 0 , µ = δ − ǫ m − . Since2( ρ ) = − n ( ǫ + . . . + ǫ m − ) + ( m − δ + . . . + δ n ) , ρ ) = ( m − ǫ +( m − ǫ + · · · + (3 − m ) ǫ m − + ( n − δ + ( n − δ + · · · + (1 − n ) δ n , wehave that( δ − ǫ m − , ρ ) = − n + 1 + m + 2 − (3 − m ) − n = 2( m − n − . In case (5), we have k = − h ∨ − . Then k + h ∨ , = m − n − and ( µ, µ +2 ρ ) = ( δ − ǫ m − , ρ ) = 2( m − n − µ , µ + 2 ρ )2( k + h ∨ , ) = 2 m − n − m − n − − m − n − m = n + 3 , n + 2 , n, n − k = − h ∨ . Then k + h ∨ , = m − n − and( µ , µ + 2 ρ )2( k + h ∨ , ) = 3 m − n − m − n − m − n − m = n + 6 , n + 4 , n + 2 , n . (cid:3) In Section 8 we will discuss explicit decompositions for some occurrencesof cases (1) and (4) of Theorem 6.4, exploiting the fact that some of thelevels k i may be admissible for V k i ( g ♮i ). We shall determine explicitly thedecomposition of W k ( g , θ ) as a module for this admissible vertex algebra.We now list the cases which are not covered by Theorem 6.4. Recall that,if g = sl ( m | n ), then we excluded the case m = n + 2 from the beginning ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 23 while the case m = n had to be excluded because g = sl ( n | n ) is not simple.The remaining cases are(1) sl ( n − | n ), k = 1;(2) sl ( n + 3 | n ) n ≥ k = − sl ( n + 4 | n ), k = − ;(4) sl ( n + 6 | n ), k = − sl (2 |
1) = spo (2 | k = − .If g = sl ( n − | n ) and k = 1 then k + h ∨ = 0, so we have to exclude thiscase.By Theorem 3.3 of [10] (stated in this paper as Proposition 4.1), k = − g = sl ( n + 3 | n ). It follows that W − ( sl ( n + 3 | n )) = V − ( gl ( n + 1 | n )). If n = 0, we obtain W − ( sl (3)) = V − ( gl (1)), which is theHeisenberg vertex algebra V ( C c ).In the case g = sl (2 | k = − , W k ( g , θ ) is the simple N = 2 supercon-formal vertex algebra V N =2 c (cf. [28], [30], [2]) with central charge c = 1.In this case V k ( gl (1)) is the Heisenberg vertex algebra M ( − ), so we haveconformal embedding of M ( − ) into W k ( g , θ ). It is well-known that V N =2 c admits the free-field realization as the lattice vertex algebra F . Using thisrealization we see that each W k ( g , θ ) ( i ) is an irreducible M ( − )-module.Case (3) with n = 0 is of special interest: it turns out that W k ( g , θ ) isisomorphic to the vertex algebra R (3) introduced in [3]. This will require athoughtful discussion which will be performed in Section 7. There we willprove the following Theorem 6.5.
Let g be of type A with k = − . Then, for all i ∈ Z , W k ( g , θ ) ( i ) is an infinite sum of irreducible V k ( gl (2)) -modules. Remark 6.1.
The only remaining open case is g of type A with k = − W k ( g , θ ) where W k ( g , θ ) ( i ) is not a finite sum of irreducible V k ( g ♮ )-modules.6.2. Finite decomposition when g ♮ has trivial center. Recall that W k ( g , θ ) is a Z ≥ -graded vertex algebra by conformal weight. It admitsthe following natural Z -gradation W k ( g , θ ) = W k ( g , θ ) ¯0 ⊕ W k ( g , θ ) ¯1 , where W k ( g , θ ) ¯ i = { v ∈ W k ( g , θ ) | ∆ v ∈ i/ Z } . Similarly to what we have done in Section 6.1, we start by developing acriterion for checking when V k ( g ♮ ) = W k ( g , θ ) ¯0 , so that V k ( g ♮ ) is a simpleaffine vertex algebra, and W k ( g , θ ) ¯1 is an irreducible V k ( g ♮ )-module. Inparticular, in these cases, we have a finite decomposition of W k ( g , θ ) as a V k ( g ♮ )-module. One checks, browsing Tables 1–3, that when g ♮ has trivial center, then g − / is an irreducible g ♮ -module. Then, in the Grothendieck group of finitedimensional representations of g ♮ , we can write g − / ⊗ g − / = V (0) + X ν i =0 V ( ν i ) . Theorem 6.6.
Assume that the embedding of V k ( g ♮ ) in W k ( g , θ ) is con-formal and that W k ( g ) ¯0 does not contain V k ( g ♮ ) -primitive vectors of weight ν i .Then W k ( g ) ¯0 = V k ( g ♮ ) , V k ( g ♮ ) is a simple affine vertex algebra, and W k ( g , θ ) ¯1 is a simple V k ( g ♮ ) -module.Proof. Let U = span { G { u } | u ∈ g − / } . Let A = V k ( g ♮ ) · U . Arguing as inthe proof of Theorem 6.2, we have(6.4) A · A ⊂ V k ( g ♮ ) . From (6.4) we obtain that V k ( g ♮ ) + A is a vertex subalgebra of W k ( g , θ ).Since the embedding is conformal, W k ( g , θ ) is strongly generated by span ( J { a } | a ∈ g ♮ ) + U , hence V k ( g ♮ ) + A contains a set of generators for W k ( g , θ ). It follows that W k ( g , θ ) = V k ( g ♮ ) + A, W k ( g , θ ) ¯0 = V k ( g ♮ ) , W k ( g , θ ) ¯1 = V k ( g ♮ ) · U . The statement now follows by a simple application of quantum Galoistheory, for the cyclic group of order 2. (cid:3)
The same argument of Corollary 6.3 provides a numerical criterion for a fi-nite decomposition, actually, for a decomposition in a sum of two irreduciblesubmodules:
Corollary 6.7.
If the embedding of V k ( g ♮ ) in W k ( g , θ ) is conformal and,for any irreducible subquotient V ( µ ) of g − / ⊗ g − / with µ = 0 , we have(see (6.2) ) r X i =0 ,k i =0 ( µ i , µ i + 2 ρ i )2( k i + h ∨ ,i ) Z + , (6.5) then W k ( g ) ¯0 = V k ( g ♮ ) , V k ( g ♮ ) is a simple affine vertex algebra and W k ( g , θ ) ¯1 is an irreducible V k ( g ♮ ) -module. We now apply Corollary 6.7 to the cases where g ♮ is a basic Lie super-algebra. These can be read off from Tables 1–2 and correspond to taking g = so ( n ) ( n ≥ g = sp ( n ) ( n ≥ g = psl (2 | g = spo (2 | m ) ( m ≥ g = osp (4 | m ) ( m ≥ g = psl ( m | m ) ( m > g = spo ( m | n ) ( n ≥ g = osp ( m | n ) ( m ≥
5) or g of the following exceptional types: G , F , E , E , E , F (4), G (3), D (2 , a ). ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 25 Theorem 6.8.
Assume that we are in the following cases of conformalembedding of V k ( g ♮ ) into W k ( g , θ ) : (1) g = so ( n ) ( n ≥ , n = 11 ), g = osp (4 | n ) ( n ≥ ), g = osp ( m | n ) ( m ≥ , m = n + r , r ∈ {− , , , , , , , } ) or g is of type G , F E , E , E , F (4) and k = − h ∨ − . (2) g = sp ( n ) ( n ≥ ), g = spo (2 | m ) ( m ≥ , m = 4 ), g = spo ( n | m ) ( n ≥ , m = n + 2 , n, n − , n − ) and k = − h ∨ .Then W k ( g , θ ) ¯ i , i = 0 , , are irreducible V k ( g ♮ ) -modulesProof. We shall show case-by-case that the numerical criterion of Corollary6.7 holds. We start by listing all cases explicitly.(1) g is of type D n , n ≥ k = − h ∨ − = − n ;(2) g is of type B n , n ≥ n = 5, k = − h ∨ − = 1 − n ;(3) g is of type G , k = − h ∨ − = − / g is of type F , k = − h ∨ − = − g is of type E , k = − h ∨ − = − / g is of type E , k = − h ∨ − = − / g is of type E , k = − h ∨ − = − / g = F (4), k = − h ∨ − = 3 / g = osp (4 | n ), n ≥ k = − h ∨ − = n − / g is of type C n +1 , n ≥ k = − h ∨ = − ( n + 2);(11) g = spo (2 | n ), n ≥ k = − h ∨ = ( n − g = spo (2 | n + 1), n ≥ k = − h ∨ = ( n − / g = spo ( n | m ) n ≥ k = − h ∨ = m − n − ;(14) g = osp ( m | n ) m ≥ k = − h ∨ − = n − m +32 .If V ( µ ) is an irreducible representation of g ♮ , we set h µ = r X i =0 ,k i =0 ( µ i , µ i + 2 ρ i )2( k i + h ∨ ,i ) . For each case listed above we give g ♮ , g − / , and the decomposition of g − / ⊗ g − / in irreducible modules for g ♮ . Then we list all values h µ for allirreducible components V ( µ ) of g − / ⊗ g − / with µ = 0, showing that theyare not positive integers. We also exhibit the decomposition of W k ( g , θ ) asa V k ( g ♮ )-module. In cases (13)–(14) we will use the usual ǫ − δ notation forroots in Lie superalgebras explained e.g. in [25].Case (1): g ♮ of Type A × D n − , g − / = V A ( ω ) ⊗ V D n − ( ω ) g − / ⊗ g − / = ( V A (2 ω ) + V A (0)) ⊗ ( V D n − (0) + V D n − ( ω ) + V D n − (2 ω )) . Values of h µ ’s: h ω , = 43 , h ,ω = 4 n − n − , h , ω = 4 n − n − ,h ω ,ω = 43 + 4 n − n − , h ω , ω = 43 + 4 n − n − . These values are non-integral for n ≥ W k ( D n ) = L A ( − Λ ) ⊗ L D n − (( − n )Λ ) ⊕ L A ( − Λ + Λ ) ⊗ L D n − (( − n )Λ + Λ ) . Case (2): g ♮ of Type A × B n − , g − / = V A ( ω ) ⊗ V B n − ( ω ) g − / ⊗ g − / = ( V A (2 ω ) + V A (0)) ⊗ ( V B n − (0) + V B n − ( ω ) + V B n − (2 ω )) . Values of h µ ’s: h ω , = 43 , h ,ω = 2 n − n − , h , ω = 2 n − n − ,h ω ,ω = 43 + 2 n − n − , h ω , ω = 43 + 2 n − n − . These values are non-integral for n ≥ n = 5.Decomposition: W k ( B n ) = L A ( − Λ ) ⊗ L B n − ((3 − n )Λ ) ⊕ L A ( − Λ + Λ ) ⊗ L B n − ((2 − n )Λ + Λ ) . Case (3): g ♮ of Type A , g − / = V A (3 ω ), g − / ⊗ g − / = V A (6 ω ) + V A (4 ω ) + V A (2 ω ) + V A (0) . Values of h µ ’s: h iω = 25 i ( i + 1) / ∈ Z ( i = 1 , , . Decomposition: W k ( G ) = L A ( 12 Λ ) ⊕ L A ( −
52 Λ + 3Λ ) . Case (4): g ♮ of Type C , g − / = V C ( ω ) g − / ⊗ g − / = V C (0) + V C (2 ω ) + V C (2 ω ) . Values of h µ ’s: h ω = 85 , h ω = 185 . Decomposition: W k ( F ) = L C ( −
32 Λ ) ⊕ L C ( −
52 Λ + Λ ) . ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 27 Case (5): g ♮ of Type A , g − / = V A ( ω ) g − / ⊗ g − / = V A (0) + V A ( ω + ω ) + V A ( ω + ω ) + V A (2 ω ) . Values of h µ ’s: h ω + ω = 127 , h ω + ω = 207 , h ω = 247 . Decomposition: W k ( E ) = L A ( −
52 Λ ) ⊕ L A ( −
72 Λ + Λ ) . Case (6): g ♮ of Type D , g − / = V D ( ω ) g − / ⊗ g − / = V D (0) + V D ( ω ) + V D ( ω ) + V D (2 ω ) . Values of h µ ’s: h ω = 3611 , h ω = 3211 , h ω = 2011 . Decomposition: W k ( E ) = L D ( −
92 Λ ) ⊕ L D ( −
112 Λ + Λ ) . Case (7): g ♮ of Type E , g − / = V E ( ω ) g − / ⊗ g − / = V E (0) + V E ( ω ) + V E ( ω ) + V E (2 ω ) . Values of h µ ’s: h ω = 6019 , h ω = 5619 , h ω = 3619 . Decomposition: W k ( E ) = L E ( −
172 Λ ) ⊕ L E ( −
192 Λ + Λ ) . Case (8): g ♮ of Type B , g − / = V B ( ω ) g − / ⊗ g − / = V B ( ω ) ⊗ V B ( ω ) = V B (2 ω ) + V B ( ω ) + V B ( ω ) + V B (0) . Values of h µ ’s: h ω = 247 , h ω = 207 , h ω = 127 , which are not integers. Decomposition W k ( F (4)) = L B ( −
134 Λ ) ⊕ L B ( −
174 Λ + Λ ) . Case (9): g ♮ of Type A × C n , g − / = V A ( ω ) ⊗ V C n ( ω ) g − / ⊗ g − / = ( V A (2 ω ) + V A (0)) ⊗ ( V C n (0) + V C n ( ω ) + V C n (2 ω )) . Values of h µ ’s: h ω , = 43 , h ,ω = 4 n n + 1 , h , ω = 4 n + 42 n + 1 ,h ω ,ω = 43 + 4 n n + 1 , h ω , ω = 43 + 4 n + 42 n + 1 . which are not integers if n ≥ W k ( osp (4 | n )) = L A ( −
12 Λ ) ⊗ L C n ( − n + 34 Λ ) ⊕ L A ( −
32 Λ + Λ ) ⊗ L C n ( − n + 74 Λ + Λ ) . Case (10): g ♮ of Type C n , g − / = V C n ( ω ), g − / ⊗ g − / = V C n (2 ω ) + V C n ( ω ) + V C n (0) . Values of h µ ’s: h ω = 6( n + 1)2 n + 1 , h ω = 6 n n + 1 . For n ≥ h ω and h ω are non-integral.Decomposition: W k ( C n +1 ) = L C n ( − n + 56 Λ ) ⊕ L C n ( − n + 116 Λ + Λ ) . Case (11): g ♮ of Type D n , g − / = V D n ( ω ), g − / ⊗ g − / = V D n (0) ⊕ V D n (2 ω ) ⊕ V D n ( ω ) . Values of h µ ’s: h ω = 3 + 32 n − , h ω = 3 − n − . These values are non-integral for n ≥ W k ( spo (2 | n )) = L D n ( − n −
53 Λ ) ⊕ L D n ( − n −
23 Λ + Λ ) . Case (12): g ♮ of Type B n , g − / = ( V B n ( ω ) if n ≥ V A (2 ω ) if n = 1 , g − / ⊗ g − / = ( V B n (0) ⊕ V B n (2 ω ) ⊕ V B n ( ω ) if n ≥ V A (0) ⊕ V A (2 ω ) ⊕ V A (4 ω ) if n = 1 . Values of h µ ’s: h ω = 3 + 32 n , h ω = 3 − n . These values are non-integral for n ≥ ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 29 Decomposition: W k ( spo (2 | n + 1)) = L B n ( − n −
33 Λ ) ⊕ L B n ( − n + Λ ) , and for n = 1 W k ( spo (2 | L A ( −
23 Λ ) ⊕ L A ( −
83 Λ + 2Λ )Case (13) g ♮ = spo ( n − | m ), g − / = C n − | m . We have(6.6) g − / ⊗ g − / = S C n − | m ⊕ ∧ C n − | m . As g ♮ -module, the first summand in the r.h.s. of (6.6) is the adjoint rep-resentation of g ♮ (which is irreducible, since g ♮ is simple), and the secondsummand in the sum of a trivial representation and an irreducible summand.Fix in g the distinguished set of positive roots Π B if m is odd and Π D if m is even (notation as in [25, 4.4]). This choice induces on g ♮ a distinguishedset of positive roots and, with respect to it, the nonzero highest weights ofthe irreducible g ♮ -modules appearing in (6.6) are 2 δ , δ + δ . Values of h µ ’s: h δ = 3(1 + 1 n − m − , h δ + δ = 3(1 − n − m − . This values are integers if and only if m = n + 2 , n, n − , n − g ♮ = osp ( m − | n ) ⊕ sl (2), g − / = C m − | n ⊗ C . We have(6.7) g − / ⊗ g − / = ( ∧ C m − | n ⊕ S C m − | n ) ⊗ ( sl (2) ⊕ C ) . As osp ( m − | n )-modules, ∧ C m − | n is the adjoint representation (which isirreducible, since osp ( m − | n ) is simple), and S C m − | n is the sum of atrivial representation and an irreducible summand. If m = 2 t + 1 is odd, wefix in g the set of positive roots corresponding to the diagram [25, (4.20)]with α t odd isotropic, the short root odd non-isotropic, and the other rootseven. If m is even we choose the set of positive roots corpsonding to thediagram Π D of [25]. With respect to the induced set of positive roots for osp ( m − | n ) the highest weight of ∧ C m − | n is ǫ + ǫ and the highestweight of the nontrivial irreducible component of S C m − | n is 2 ǫ . Thehighest weight of sl (2) is ǫ − ǫ . Values of h µ ’s: h ǫ ,ǫ − ǫ = 103 + 2 m − n − , h ǫ + ǫ ,ǫ − ǫ = 103 − m − n − ,h ǫ , = 2(1 + 1 m − n − , h ǫ + ǫ , = 2(1 − m − n − , h ,ǫ − ǫ = 43 . This values are not in Z + for m, n in the range showed in the statement. (cid:3) We now list the cases that are not covered by Corollary 6.7. We list hereonly the cases where there is a non-collapsing conformal level as describedin Proposition 4.2.(1) g of type G (3), k = ;(2) g = D (2 , a ) ( a
6∈ { , − , − } ), k = ; (3) g = psl (2 | , k = ;(4) g = spo ( m + r | m ), ( m ≥ r ∈ { , } ), k = − r +23 ;(5) g = osp ( n + r | n ), ( r ∈ {− , , , , , } ), k = − r ;(6) g = osp ( m | n ), k = ( n − m + 2);(7) g = F (4), k = − g = G (3), k = − .Sometimes W k ( g , θ ) still decomposes finitely as a V k ( g ♮ )-module. More ex-plicitly, we have the following result: Theorem 6.9. V k ( g ♮ ) is a simple affine vertex algebra and W k ( g , θ ) ¯ i , i =0 , , are irreducible V k ( g ♮ ) -modules in the following cases: (1) g = so (8) , k = − ; (2) g = D (2 ,
1; 1) = osp (4 | , k = ; (3) g = D (2 ,
1; 1 / , k = . The proof of Theorem 6.9 requires some representation theory of thevertex algebra V − / ( sl (2)). Remark 6.2.
Let k = . In the case when k − aa , where a ∈ Q , is anadmissible level for c sl we also expect that W k ( g , θ ) is a finite sum of V k ( g ♮ )-modules, but the decomposition is more complicated. We think that themethods developed in [21] can be applied for this conformal embedding.Here we shall only consider the cases a = 1 and a = 1 /
4, where we canapply fusion rules for affine vertex algebras.We will prove cases (1), (2), (3) of Theorem 6.9 in Sections 6.3.1, 6.3.2,and 6.3.3, respectively.The next result shows that in case (3) above we have an infinite decom-position:
Theorem 6.10. If g = psl (2 | and k = then V k ( g ♮ ) is simple and W k ( g ) decomposes into an infinite direct sum of irreducible V k ( g ♮ ) -modules.Proof. In this case W k ( g , θ ) is isomorphic to the N = 4 superconformalvertex algebra V N =4 c with c = − (cid:3) Remark 6.3.
The remaining open cases are • g of type G (3), k = ; • g = D (2 , a ), ( a
6∈ { , − , − , , } ), k = ; • g = so (11), k = − • g of type B n ( n ≥ k = − n − ; • g of type D n ( n ≥ k = − n − ; • g = osp (4 | n ) ( n > n = 8), k = − − n . ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 31 Proof of Theorem 6.9.
As in Theorem 6.6, we set U = span { G { u } | u ∈ g − / } , A = V k ( g ♮ ) · U . In order to apply Theorem 6.6, we have to prove that A · A ⊂ V k ( g ♮ ) . We first prove that V k ( g ♮ ) is simple and that A is a simple V k ( g ♮ )-module.Since V k ( g ♮ ) is admissible, we have that A · A is completely reducible [32].Let A · A = P i M i be its decomposition into simple modules. We haveto show that the only summands appearing are vacuum modules. This isguaranteed by the fusion rules (5.3) presented in Subsection 5.2.1. Next weprovide details in each of the three cases.6.3.1. Proof of Theorem 6.9, case (1) . We claim that, as [ sl (2)-modules, W − / ( so (8) , θ ) = L sl (2) ( − Λ ) ⊗ ⊕ L sl (2) ( − Λ + Λ ) ⊗ . Recall that in this case g ♮ ≃ sl (2) ⊕ sl (2) ⊕ sl (2). Then A is a highest weight V − / ( sl (2)) ⊗ -module with highest weight vector v λ ⊗ v λ ⊗ v λ .Assume now that A is not irreducible. Then, as observed in Subsection5.2.1 (2), a quotient of N sl (2) ( λ ) is either irreducible or it is N sl (2) ( λ ) itself.It follows that A ∼ = N sl (2) ( λ ) ⊗ e L sl (2) ( λ ) ⊗ ee L sl (2) ( λ )where e L sl (2) ( λ ) and ee L sl (2) ( λ ) are certain quotients of N sl (2) ( λ ) .Let w = v λ ⊗ v λ ⊗ v λ , W = U ( g ♮ ) w ⊂ A. By using tensor product decompositions V sl (2) (3 ω ) ⊗ V sl (2) ( ω ) = V sl (2) (4 ω ) ⊕ V sl (2) (2 ω ) ,V sl (2) ( ω ) ⊗ V sl (2) ( ω ) = V sl (2) (2 ω ) ⊕ V sl (2) (0) , and arguing as in the proof of Theorem 6.6 we see that U · W cannot containprimitive vectors of conformal weight ≤
3. Since the conformal weight ofall elements of W equals the conformal weight of w which is and theconformal weight of all elements of U is , we conclude that U ( n ) W = 0 ( n ≥ . This implies that w is a non-trivial singular vector in W − / ( so (8) , θ ). Acontradiction. Therefore W = 0 and A ∼ = L ( λ ) ⊗ . We will now show that V − / ( g ♮ ) is simple. If not, since a quotient of V − / ( sl (2)) is either simple or V − / ( sl (2)) itself, we have V − / ( g ♮ ) = V − / ( sl (2)) ⊗ e V − / ( sl (2)) ⊗ ee V − / ( sl (2)) , where e V − / ( sl (2)) and ee V − / ( sl (2)) are certain quotients of V − / ( sl (2)). Set w = v λ ⊗ ⊗ , W = U ( g ♮ ) w ⊂ V − / ( g ♮ ) . By using fusion rules again we see that U (1) W = W = 0. So w is a singularvector in W − / ( so (8) , θ ), a contradiction.Therefore V − / ( g ♮ ) = V − / ( sl (2)) ⊗ . By using fusion rules (5.3) weeasily get that V − / ( g ♮ ) ⊕ A = V − / ( sl (2)) ⊗ ⊕ L ( λ ) ⊗ is a vertex subalgebra of W − / ( so (8) , θ ). Since this subalgebra contains allgenerators of W − / ( so (8) , θ ), the claim follows. (cid:3) Proof of Theorem 6.9, case (2) . The proof is similar to case (1).We claim that, as [ sl (2)-modules, W / ( osp (4 | , θ )= L sl (2) ( − Λ ) ⊗ L sl (2) ( − Λ ) ⊕ L sl (2) ( − Λ + Λ ) ⊗ L sl (2) ( − Λ + Λ ) . Let U , A be as in Subsection 6.3.1. Then A is a highest weight V / ( g ♮ )-module with highest weight vector v λ ⊗ v ˜ λ where ˜ λ = − Λ + Λ . Assumenow that V − / ( sl (2)) v λ is not simple. Then A ∼ = N sl (2) ( λ ) ⊗ e L sl (2) (˜ λ ) , where e L sl (2) (˜ λ ) is certain a quotient of N sl (2) (˜ λ ).Set w = v λ ⊗ v ˜ λ , W = U ( g ♮ ) w ⊂ A. The same argument of case (1), using fusion rules and evaluation of confor-mal weights, shows that w is a non-trivial singular vector in W / ( osp (4 | ,θ ), a contradiction. Therefore W = 0 and A ∼ = L ( λ ) ⊗ ˜ L (˜ λ ) . Assume next that V − / ( sl (2)) · is not simple. Then V / ( g ♮ ) = V − / ( sl (2)) ⊗ e V − / ( sl (2))where e V − / ( sl (2)) is a certain quotient of V − / ( sl (2)).Set w = v λ ⊗ , W = U ( g ♮ ) w ⊂ V / ( g ♮ ) . By using fusion rules again we see that U (1) W = W = 0. So w is a singularvector in W / ( osp (4 | , θ ). A contradiction.Therefore V / ( g ♮ ) = V − / ( sl (2)) ⊗ e V − / ( sl (2)). By using fusion rules(5.3) we easily get that V / ( g ♮ ) ⊕ A = W / ( osp (4 | , θ ) . In particular V / ( g ♮ ) is a simple vertex algebra and A is its simple module.The claim follows. (cid:3) ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 33 Proof of Theorem 6.9, case (3) . In case (3) we have W / ( D (2 ,
1; 1 / , θ ) = L sl (2) (Λ ) ⊗ L sl (2) ( − Λ ) ⊕ L sl (2) (Λ ) ⊗ L sl (2) ( − Λ + Λ ) , and the proof is completely analogous to that of case (2).7. The vertex algebra R (3) and proof of Theorem 6.5 In this section we will present an explicit realization of the vertex algebra W k ( sl (4) , θ ) and prove that it is isomorphic to the vertex algebra R (3) from[3]. In this way we prove Conjecture 2 from [3]. Then we apply this newrealization to construct explicitly infinitely many singular vectors in eachcharge component W ( i ) k , proving Theorem 6.5.7.1. Definition of R (3) . Let us first recall the definition of the vertex alge-bra R (3) introduced in Section 12 of [3]. Let V L = M (1) ⊗ C [ L ] be the gen-eralized lattice vertex algebra (cf. [14], [18]) associated to the (non-integral)lattice L = Z α + Z β + Z δ + Z ϕ, with non-zero inner products h α, α i = −h β, β i = 1 , h δ, δ i = −h ϕ, ϕ i = 23 . Set α = α + β , α = ( δ + ϕ ), α = ( δ − ϕ ), and D = Z α + Z α + Z α . Then D is an even integral lattice. We choose a bi-multiplicative 2-cocycle ε such that for every γ , γ ∈ D we have ε ( γ , γ ) ε ( γ , γ ) = ( − h γ ,γ i . We fix the following choice of the cocycle: ε ( α , α i ) = ε ( α i , α ) = ε ( α i , α i ) = 1 ( i = 1 , , ε ( α , α ) = − ε ( α , α ) = 1 . This cocycle can be extended to a 2-cocyle on L by bimultiplicativity. Thenwe have ε ( α + β − δ, α ) = ε ( α − α − α , α ) = 1 ,ε ( α + β − δ, α ) = ε ( α − α − α , α ) = − . Let C ε [ D ] be the twisted group algebra associated to the lattice D andcocycle ε . We consider the lattice type vertex algebra V extD = M (1) ⊗ C ε [ D ] , which is realized as a vertex subalgebra of V L . (Note that V extD containsthe complete Heisenberg vertex subalgebra M (1) of V L , and that the lattice D has three generators.) All calculations below will be done in this vertexalgebra. For γ ∈ D we define the following elements of the Heisenberg vertexalgebra M (1): S ( γ ) = 12 (( γ ( − ) + γ ( − ) , S ( γ ) = 16 ( γ − + 3 γ ( − γ ( − + 2 γ ( − ) . First we recall that the vertex subalgebra V of V extD , generated by e = e α + β , (7.1) h = − β + δ,f = ( −
23 ( α − + α ( − ) − α ( − δ ( − + 13 α ( − β ( − ) e − α − β ,j = ϕ, is an affine vertex algebra. More precisely, it is isomorphic to M ϕ ( − / ⊗ V − / ( sl (2)) (Note that k = − / V − / ( sl (2)) = V − / ( sl (2)), cf. [24]).Let Q = e α + β − δ (0) be the screening operator (cf. [3]). Note that Q isa derivation of the vertex algebra V extD . We also have that the SugawaraVirasoro vector ω V sug of V maps to (cid:18)
12 ( α − − α ( − − β − + β ( − ) + 34 ( δ − − δ ( − − ϕ − ) (cid:19) . We define R (3) to be the vertex subalgebra of V extD generated by the gener-ators of V and the following four even vectors of conformal weight 3 / E = e
32 ( δ + ϕ ) ,E = Qe
32 ( δ − ϕ ) = S ( α + β − δ ) e −
32 ( δ + ϕ )+ α + β ,F = f (0) E = − α ( − e − α − β + 32 ( δ + ϕ ) ,F = f (0) E = ( − α ( − S ( α + β − δ ) + S ( α + β − δ )) e −
32 ( δ + ϕ ) . The vertex algebra R (3) satisfies the following properties: • R (3) is integrable, as a module over sl (2). • R (3) has finite-dimensional weight spaces with respect to ( ω V sug ) .The conformal weights lie in Z ≥ . • R (3) is contained in the following subalgebra of V extD : M ⊗ Π(0) , where M is the Weyl vertex algebra (i.e., the algebra of symplecticbosons [28]) generated by a = e α + β , a ∗ = − α ( − e − α − β , and Π(0) is the ”half-lattice” vertex algebraΠ(0) = M δ,ϕ (1) ⊗ C [ Z δ + ϕ )2 ] ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 35 containing the Heisenberg vertex algebra M δ,ϕ (1) generated by δ and ϕ (cf. [3]).Let ( M ⊗ Π(0)) int denote the maximal sl (2)-integrable submodule of M ⊗ Π(0). It is clear that it is a vertex subalgebra of M ⊗ Π(0).We shall prove the following result.
Theorem 7.1. (1)
There is a vertex algebra homomorphism W k ( sl (4) , θ ) → R (3) . (2) R (3) is a simple vertex algebra, i.e, W k ( sl (4) , θ ) = R (3) . (3) R (3) ∼ = ( M ⊗ Π(0)) int . Remark 7.1.
Theorem 7.1 gives a positive answer to Conjecture 2 from [3].The representation theory of R ( p ) for p > C -cofinitevertex algebras appearing in LCFT (such as triplet vertex algebras) will bestudied in [4].7.2. λ -brackets for R (3) .Proposition 7.2. We have the following λ -brackets: [ E iλ E i ] = [ F iλ F i ] = 0 ( i = 1 , , [ E λ E ] = 3( ∂e + 3 : je :) + 6 λe, [ F λ F ] = − ∂f + 3 : jf :) − λf, [ E λ F ] = 0 , [ E λ F ] = − ω V sug + 12 ( ∂h + 3 : jh : − jj : − ∂j )))+3 λ ( − h + 5 j ) + 5 λ , [ E λ F ] = − ω V sug + 12 ( ∂h − jh : − jj : +5 ∂j ))) − λ ( h + 5 j ) + 5 λ , [ E λ F ] = 0 . Proof.
The proof uses the standard computations in lattice vertex algebras[28]. Let us discuss the calculation of [ E λ F ] and of [ E λ F ].For [ E λ F ], the only difficult part is to compute E F . We have E F = 10 ,E F = − h + 5 ϕ = − h + 5 j,E F = − α ( − ( δ + ϕ ) − α ( − ( α + β − δ ) , +10 S ( ( δ + ϕ )) + 9( α ( − + β ( − − δ ( − )( δ + ϕ )+3 S ( α + β − δ ) = − ω sug + 1 / h ( − + 3 ϕ ( − h ( − − ϕ − − ϕ ( − ) ) . For the calculation of [ E λ F ], we shall use the fact that Q is a derivationin the lattice vertex algebra V D . Set E = e
32 ( δ − ϕ ) ,F = QF = − ( − α ( − S ( α + β − δ ) + S ( α + β − δ )) e −
32 ( δ − ϕ ) . Note that the minus sign in front of the r.h.s. of the formula above comesfrom the cocycle computation ε ( α + β − δ, − α − β + 32 ( δ + ϕ )) = ε ( α − α − α , − α + α ) = − . Next, we have[ E λ F ] = Q [ e
32 ( δ − ϕ ) λ F ] − [ e
32 ( δ − ϕ ) λ QF ] = − [ E λ F ] . The calculation of [ E λ F ] is essentially the same as for [ E λ F ] (we justreplace j by − j ). Now we have[ E λ F ] = − [ E λ F ]= ( − ω sug + 12 ( ∂h − jh : − jj : +5 ∂j ))) + 3 λ ( − h − j ) + 5 λ )= − ω sug + 12 ( ∂h − jh : − jj : +5 ∂j ))) + 3 λ ( − h − j ) + 5 λ The claim follows. (cid:3)
The homomorphism
Φ : W k ( sl (4) , θ ) → R (3) . Recall from Exam-ple 3.1 that the vertex algebra W k ( sl (4) , θ ) is generated by the Virasorovector ω of central charge c ( k ) = 15 k/ ( k + 4) − k, four even generators J { e , } , J { e , } , J { e , − e , } , J { c } of conformal weight 1, and four even vectors G { e , } , G { e , } , G { e , } , G { e , } of conformal weight 3 / λ -brackets from Proposition 7.2 and λ -brackets for thevertex algebra W − / ( sl (4) , θ ) we get the following result: Proposition 7.3.
Let k = − / . There is a vertex algebra homomorphism Φ : W k ( sl (4) , θ ) → R (3) such that J { e , } e, J { e , } f, J { e , − e , } h, J { c } j,G { e , }
7→ √ E , G { e , }
7→ √ F , G { e , }
7→ √ E , G { e , }
7→ − √ F ,ω ω V sug . ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 37 Proof.
It is enough to check λ -brackets from Example 3.1 in the case k = − /
3. In particular, taking into account that ω = ω sug = 32 (2 : J { e , } J { e , } : − ∂J { e , − e , } + 12 : J { e , − e , } J { e , − e , } :) −
34 : J { c } J { c } : , we get[ G { e , } λ G { e , } ] = 23 ω − J { c } J { c } : + 13 ∂J { e , − e , } − ∂J { c } + : J { c } J { e , − e , } : + 23 λ ( − J { c } + J { e , − e , } ) − λ = −
29 ( − ω V sug + 12 ( ∂h + 3 : jh : − j − ∂j ))) + 3 λ ( − h + 5 j ) + 5 λ ) . = −
29 [ E λ F ] . All other λ -brackets are checked similarly. (cid:3) Proposition 7.3 implies that R (3) is conformally embedded into a certainquotient of W k ( sl (4) , θ ). In the following subsection we will prove that R (3) is isomorphic to W k ( sl (4) , θ ).7.4. Simplicity of R (3) and proof of Theorem 7.1. Our proof of sim-plicity is similar to the proof of simplicity of the N = 4 superconformalvertex algebra realized in [3]. As a tool we shall use the theory of Zhu alge-bras associated to the Neveu-Schwarz sector of Z ≥ -graded vertex algebras.Let V is a Z ≥ -graded vertex algebra and A ( V ) = V /O ( V ) the associatedZhu algebra. Let [ a ] = a + O ( V ) (cf. Subsection 2.4). Lemma 7.4. (1)
The Zhu algebra A ( R (3) ) is isomorphic to a quotient of U ( gl (2)) . (2) In the Zhu algebra A ( R (3) ) the following relation holds: [ e ]([ ω ] + 23 −
32 [ j ] ) = 0 . Proof.
Since R (3) is a quotient of W k ( sl (4) , θ ), the first assertion followsfrom Proposition 4.3. Let us prove the second assertion. We notice that: E E :=( S ( α + β − δ ) + 6 S ( ( δ + ϕ )) + 92 ( α + β − δ ) ( − ( δ + ϕ ) ( − ) e α + β . : eω :=( − ( α ( − + β ( − ) β ( − + β ( − + 34 ( δ − − δ ( − − ϕ − )) e α + β . By direct calculation we get the following relation: : E E : +3 : eω :=( e ( − + ( h ( − e ( − − h ( − e ( − )+ (( ϕ − + ϕ ( − ) e ( − + ϕ ( − e ( − )) . We have E ◦ E = ( E − + E ) E = − e ( − ω + e ( − + 32 ( h ( − e ( − − h ( − e ( − ) + 92 ( ϕ − + ϕ ( − ) e ( − + 92 ϕ ( − e ( −
2) + 3 e ( − + 9 ϕ ( − e ( − . This gives the following relation in the Zhu algebra: − e ][ ω ] − e ] + 92 ([ j ] + [ j ] − [ j ])[ e ] = − e ][ ω ] − e ] + 92 [ j ] [ e ] = 0 . The claim follows. (cid:3)
Proposition 7.5. (1) R (3) is a simple vertex algebra. (2) R (3) ∼ = ( M ⊗ Π(0)) int .Proof.
By using the fact that R (3) is a subalgebra of M ⊗ Π(0), we concludethat if w sing is a singular vector for W k ( sl (4) , θ ) in R (3) , it must have gl (2)-weight ( nω , m ) for n ∈ Z ≥ and m ∈ Z . This means that h (0) w sing = nw sing , ϕ (0) w sing = mw sing . This leads to the relation L (0) w sing = ( 3 n ( n + 2)4 − m ) w sing . On the other hand, w sing generates a submodule whose lowest componentmust be a module the for Zhu algebra. Now Lemma 7.4 implies that U ( gl (2)) w sing is annihilated by [ e ]([ ω ] + − [ j ] ). If n >
0, we get3 n ( n + 2)4 − m − m = 3 n ( n + 2) − m − , which gives a contradiction since m ∈ Z . So n = 0. Then the fact thatconformal weight must be positive implies that m = 0. Therefore w sing must be proportional to the vacuum vector. We deduce that there are nonon-trivial singular vectors, and therefore R (3) is a simple vertex algebra.This proves (1). The proof of assertion (2) is completely analogous. (cid:3) Proof of Theorem 7.1.
Apply Propositions 7.3 and 7.5. (cid:3)
ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 39 d gl (2) -singular vectors in R (3) .Lemma 7.6. Let ℓ ∈ Z . (1) If ℓ ≥ , then for every j ≥ v ℓ,j = Q j e ℓ ( δ + ϕ )+3 jδ is a non-trivial singular vector in R (3) . (2) If ℓ ≤ , then for every j ≥ v ℓ,j = Q j − ℓ e − ℓ ( δ − ϕ )+3 jδ is a non-trivial singular vector in R (3) .In particular, the set { v ℓ,j | j ≥ } provides an infinite family of linearlyindependent [ gl (2) -singular vectors in the ℓ -eigenspace of ϕ (0) .Proof. The non-triviality of the singular vectors v ℓ.j is well known (cf. [5]).The assertions now follow from the fact v ℓ,j belongs to a maximal sl (2)-integral part of M ⊗ Π(0). (cid:3)
Proof of Theorem 6.5.
Since we have proved that W k ( sl (4) , θ ) isisomorphic to the simple vertex algebra R (3) , Lemma 7.6 shows that each W k ( sl (4) , θ ) ( i ) contains infinitely many linearly independent singular vec-tors. Remark 7.2.
Assertion (3) of Theorem 7.1 implies that W k ( sl (4) , θ ) = R (3) is an object of the category KL k +1 of V k +1 ( sl (2))-module. In particular,each W k ( sl (4) , θ ) ( i ) is an object this category. Since k + 1 = − / [ sl (2), and the category KL k +1 is semisimple (this followseasily from [34], we skip details), we have that W k ( sl (4) , θ ) ( i ) is completelyreducible. So we actually proved that each W k ( sl (4) , θ ) ( i ) is a direct sum ofinfinitely many irreducible V k +1 ( gl (2))-modules.8. Explicit decompositions from Theorem 6.4: g ♮ is a Liealgebra In Theorem 6.4 we proved a semisimplicity result for conformal embed-dings of V k ( g ♮ ) in W k ( g , θ ) where g = sl ( n ) or g = sl (2 | n ). But this semisim-plicity result does not identify highest weights of the components W k ( g , θ ) ( i ) .In this section we shall identify these components in certain cases and provethat then W k ( g , θ ) is a simple current extension of V k ( g ♮ ).Recall from Section 2.7 that F n denotes a rank one lattice vertex algebraand F in , i = 0 , · · · , n − , denote its irreducible modules. The following resultrefines Theorem 6.4. Theorem 8.1. (1) If g = sl (2 n ) and k = − n , n ≥ , then Com( V k +1 ( sl (2 n − , W k ( g , θ )) ∼ = F n ( n − . Moreover, we have the following decomposition of W k ( g , θ ) as a V k +1 ( sl (2 n − ⊗ F n ( n − -module: W k ( g , θ ) ∼ = n − M i =0 L sl (2 n − ( k Λ + Λ i ) ⊗ F in n ( n − . (8.1) (2) If g = sl (2 | n ) and k = − h ∨ / ∈ Z , then Com( V − k − ( sl ( n )) , W k ( g , θ )) ∼ = F n . Moreover, we have the following decomposition of W k ( g , θ ) as a V − k − ( sl ( n )) ⊗ F n -module: W k ( g , θ ) ∼ = n − M i =0 L sl ( n ) ( − ( k + 2)Λ + Λ i ) ⊗ F i n . (8.2) Proof. (1) Let α = 2 nJ { c } and note that V k ( g ♮ ) = V k +1 ( sl (2 n − ⊗ M α (4 n ( n − . By Theorem 6.4 we have that each W k ( g , θ ) ( i ) is an irreducible V k ( g ♮ )-module, and, by checking the action of J { c } (0) , we see that there is a weight Λsuch that W k ( g , θ ) ( i ) = L sl (2 n − (Λ) ⊗ M α (4 n ( n − , in ) . Since W k ( g , θ ) (1) ∼ = L sl (2 n − ( k Λ + Λ ) ⊗ M α (4 n ( n − , n ) ,W k ( g , θ ) ( − ∼ = L sl (2 n − ( k Λ + Λ n − ) ⊗ M α (4 n ( n − , − n ) , the fusion rules result from Proposition 5.1 and the fusion rules (2.5) implythat W k ( g , θ ) ( i ) ∼ = L sl (2 n − ( k Λ + Λ ¯ i ) ⊗ M α (4 n ( n − , in ) , (8.3)where ¯ i ∈ { , . . . , n − } , i ≡ ¯ i mod (2 n − . SinceCom( V k +1 ( sl (2 n − , W k ( g , θ )) = { v ∈ W k ( g , θ ) | J { u } ( n ) v = 0 , n ≥ , u ∈ g ♮ } , we get thatCom( V k +1 ( sl (2 n − , W k ( g , θ )) ∼ = M i ∈ Z M α (4 n ( n − , in ( n − M α (4 n ( n − V k +1 ( sl (2 n − , W k ( g , θ )) ∼ = F n ( n − . The decomposition (8.1) now easily follows from (8.3). This proves (1).The proof of (2) is based on Proposition 5.2 and it is completely analogousto the proof of assertion (1). (cid:3)
ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 41 Remark 8.1.
Decompositions (8.1) and (8.2), together with the fusion rulesresult from Propositions 5.1 and 5.2 imply that the minimal W -algebras fromTheorem 8.1 are finite simple current extensions of the tensor product of anadmissible affine vertex algebra with a rank one lattice vertex algebra. It isalso interesting to notice that the r.h.s. of (8.1) and (8.2) have sense for thecases g = sl ( n ), n odd, and g = sl (2 | n ), k = − h ∨ ∈ Z . But Corollary 8.3shows that most likely we won’t get lattice vertex subalgebra in these cases. Remark 8.2.
The computation of the explicit decompositions in Theorem6.4 when V k ( g ) does not contain an admissible vertex algebra of type A needs a subtler analysis. Our approach motivates the study of the followingnon-admissible affine vertex algebras: • V k ′ ( sl (2 n + 1)) for k ′ = − n , • V k ′ ( sl ( n )) for k ′ = − n +13 , • V k ′ ( sl (3 n + 2)) for k ′ = − n − • V k ′ ( sl ( n )) for k ′ = − n +12 , n ≥ V − ( sl (3)) (cf. [8]): Proposition 8.2. [8]
For s ∈ Z ≥ set U s = L sl (3) ( − (1 + s )Λ + s Λ ) , U − s = L sl (3) ( − (1 + s )Λ + s Λ ) . • The set { U s | s ∈ Z } provides a complete list of irreducible V − ( sl (3)) modules from the category KL − . • The following fusion rules hold in the category KL − : U s × U s = U s + s ( s , s ∈ Z ) . By using this proposition we get the following refinement of Theorem 6.4(1) for the case n = 5: Corollary 8.3.
We have the following isomorphism of V − , / ( g ♮ ) -modules: W − ( sl (5) , θ ) ∼ = M s ∈ Z U s ⊗ M (3 / , s ) . Proof.
Set α = J { c } . Then V − ( g ♮ ) = V − ( sl (3)) ⊗ M α (3 / W k ( g , θ ) ( i ) is an irreducible V k ( g ♮ )-module, and, bychecking the action of J { c } (0) , we see that there is a weight Λ such that W k ( g , θ ) ( i ) = L sl (3) (Λ) ⊗ M α (3 / , i ) . The assertion follows as in the proof of Theorem 8.1 from the fusion rulesresult from Proposition 8.2. (cid:3)
Remark 8.3.
In [28], the vertex algebra U and its modules U s from Propo-sition 8.2 are realized inside of the Weyl vertex algebra M of rank three. Itwas proved in [8] that M ∼ = M s ∈ Z U s ⊗ M ( − , s ) . Note that although W − ( sl (5) , θ ) admits an analogous decomposition, onecan easily see that this W -algebra is not isomorphic to any subalgebra of M .We also believe that the modules which appear in the decomposition of W k ( g , θ ) in Theorem 6.4 (3) are also simple currents, so one can also ex-pect the decomposition like in Corollary 8.3. Indeed, we can show thatsuch decomposition holds but instead of applying fusion rules (which wedon’t know yet), we will apply results from our previous papers [9] and[10]. In [9] we proved that the affine vertex algebra V − n +12 ( sl ( n + 1))( n ≥
4) is semisimple as a V − n +12 ( gl ( n ))-module and identified highestweights of all modules appearing in the decomposition. In [10] we provedthat V − n +12 ( sl ( n + 1)) ( n ≥ , n = 5) is embedded in the tensor productvertex algebra W k ( sl (2 | n ) , θ ) ⊗ F − . An application of these results will givethe branching rules.For s ∈ Z ≥ , we set U ( n ) s = L sl ( n ) ( − ( n +12 + s )Λ + s Λ ) , U ( n ) − s = L sl ( n ) ( − ( n +12 + s )Λ + s Λ n ) . Theorem 8.4.
Let g = sl (2 | n ) , k = − h ∨ , n = 4 or n ≥ . We have anisomorphism as V k ( g ♮ ) -modules: W k ( g , θ ) ∼ = M s ∈ Z U ( n ) s ⊗ M ( nn − , s ) . Proof.
We first consider the Heisenberg vertex algebra M α ( nn − ) ⊗ M ϕ ( − α = J { c } and ϕ such that[ α λ α ] = nn − λ, [ ϕ λ ϕ ] = − λ. Define ϕ = α + ϕ, b ϕ = 2 − n α + nn − ϕ ) . Then M α ( nn − ⊗ M ϕ ( −
1) = M b ϕ ( − n ⊗ M ϕ ( 2 n − . Theorem 6.4 (3) implies that W k ( g , θ ) ( s ) ∼ = L sl ( n ) (Λ ( s ) ) ⊗ M α ( nn − , s ) , ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 43 where L sl ( n ) (Λ ( s ) ) is an irreducible highest weight module from KL − n +12 . Itwas proved in [10, Theorem 5.6] that V − n +12 ( sl ( n + 1)) ⊗ M ϕ ( 2 n − M s ∈ Z W k ( g , θ ) ( s ) ⊗ M ϕ ( − , − s )= M s ∈ Z L sl ( n ) (Λ ( s ) ) ⊗ M α ( nn − , s ) ⊗ M ϕ ( − , − s ) . = M s ∈ Z L sl ( n ) (Λ ( s ) ) ⊗ M b ϕ ( − n , s ) ! ⊗ M ϕ ( 2 n − . This implies that V − n +12 ( sl ( n + 1)) ∼ = M s ∈ Z L sl ( n ) (Λ ( s ) ) ⊗ M b ϕ ( − n , s ) . Now results from [9] (see in particular [9, Theorem 2.4, Theorem 5.1 (2)])imply that L sl ( n ) (Λ ( s ) ) ∼ = U ( n ) s . The claim follows. (cid:3) Explicit decompositions from Theorem 6.4: g ♮ is not a Liealgebra In this section we describe the decomposition of W − ( sl ( n + 5 | n ) , θ ) as V k ( g ♮ )-module. We obtain, similarly to the results of Section 8, that W − ( sl ( n + 5 | n ) , θ ) is a simple current extension of V k ( g ♮ ). We expect thisto hold in general.9.1. Simple current V k ( sl ( m | n )) –modules. Let us first recall a few de-tails on simple current modules obtained by using the simple current oper-ator ∆( α, z ) . Let V be a conformal vertex algebra with conformal vector ω . Let α bean even vector in V such that ω n α = δ n, α, α ( n ) α = δ n, γ ( n ≥ , where γ is a complex number. Assume that α (0) acts semisimply on V witheigenvalues in Z . Let [19]∆( α, z ) = z α (0) exp ∞ X n =1 α ( n ) − n ( − z ) − n ! . Then [19] ( V ( α ) , Y α ( · , z )) := ( V, Y (∆( α, z ) · , z ))(9.1) is a V –module, called a simple current V –module, and Y α ( ω, z ) = Y ( ω, z ) + z − Y ( α, z ) + 1 / γz − . (9.2)When V is the simple affine vertex algebra associated to sl ( m | n ), we will usethis construction to produce simple current modules in a suitable category.Let g = sl ( m | n ) ( m = n ), k ∈ C . Let e i,j denote the standard matrixunits in sl ( m | n ); consider the following vector in V k ( sl ( m | n )): α m,n = 1 m − n ( ne , + · · · + ne m,m + me m +1 ,m +1 + · · · + me m + n,m + n ) ( − . Note that(9.3) g = g − + g + g , g i = { x ∈ g | [ α m,n , x ] = ix } . In particular, g ∼ = sl ( m ) × sl ( n ) × C α m,n is the even part of g and g − + g is its odd part.For i ∈ {− , , } and n ∈ Z set g i ( n ) = g i ⊗ t n . The decomposition (9.3) implies that α m,n (0) acts semi-simply on V k ( sl ( m | n ))with integral eigenvalues. Moreover[ α m,n λ α m,n ] = 1( m − n ) ( n mk − m nk ) λ = − nmk m − n λ. Set U m,ns = V k ( g ) ( sα m,n ) ( s ∈ Z ) . By definition (9.1), we see that U m,ns is obtained from V k ( g ) by applyingthe automorphism π s of b g (and V k ( g )) uniquely determined by π s ( x ± ( r ) ) = x ± ( r ∓ s ) ( x ± ∈ g ± ) , (9.4) π s ( x ( r ) ) = x ( r ) ( x ∈ sl ( m ) × sl ( n ) ⊂ g ) , (9.5) π s ( α m,n ( r ) ) = α m,n ( r ) − nmk m − n sδ r, , (9.6)where r, s ∈ Z . Note that in U m,ns we have g ± ( n ± s ) . = 0 ( n ≥ . Theorem 9.1.
Assume that m, n ≥ . We have:(1) U m,ns , s ∈ Z , are irreducible V k ( g ) –modules from the category KL k .(2) Let s = ± . Then the lowest graded component of U m,ns is, as a vectorspace, isomorphic to ^ ( g s (0)) . . It has conformal weight − nmk m − n .(3) U m,ns , s ∈ Z , are simple current V k ( g ) –modules in KL k and the fol-lowing fusion rules holds in KL k : U m,ns × U m,ns = U m,ns + s ( s , s ∈ Z ) . (9.7) ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 45 Proof. (1) Since U m,ns = π s ( V k ( g )), we get that U m,ns is irreducible V k ( g )–module. Relations (9.4) -(9.6) together with (9.2) imply that U m,ns belongsto KL k . In fact, the lowest graded component is contained in the vectorspace ^ (cid:0) g (0) + · · · + g ( s − (cid:1) . ( s ≥ ^ (cid:0) g − (0) + · · · + g − ( − s − (cid:1) . ( s ≤ − . For s = ± V k ( g )–module ( M, Y M ) from the category KL k one can show that( M s , e Y M ( · , z )) = ( M, Y M (∆( α m,n , z ) · , z ))(9.8)is also an irreducible V k ( g )–modules from the category KL k (this followsfrom the fact that M s is essentially obtained by applying the automorphism π s ). Then [36, Theorem 2.13] gives the fusion rules M × U m,ns = M s . In particular, this proves the fusion rules (9.7). (cid:3)
Remark 9.1.
Let us consider the case s = ±
1. Then the lowest weightcomponent U m,ns (0) of U m,ns is a irreducible (sub)quotient of the Kac module K sm,n ( k ) induced from the 1–dimensional ( g + g − s )–module C with action g − s . = 0 ,x. = 0 ( x ∈ sl ( m ) × sl ( n )) ,α m,n . = − s nmk m − n . As a vector space K sm,n ( k ) ∼ = V g s . If we take an odd coroot β = e i,i + e m + j,m + j , 1 ≤ i ≤ m , 1 ≤ j ≤ n , by direct calculation we get(9.9) β. = − sk . . This implies that K sm,n ( k ) is typical iff k / ∈ {− ( m − , . . . , n − } .Recall that a weight λ of a basic Lie superalgebra is said to be typical if( λ + ρ )( β ) = 0 for each isotropic odd root β . To derive the above conditionon k , we make computations in a distinguished set of positive roots; wehave ρ = − s m X i =1 ( m − n − i + 1) ǫ i + n X j =1 ( n + m − j + 1) δ j , and from (9.9) we deduce that( λ + ρ )( β ) = − s ( k + m − i − j + 1) , which is non-zero if k / ∈ {− ( m − , . . . , n − } . Under this hypothesis, thelowest graded component U m,ns (0) of U m,ns is isomorphic to K sm,n ( k ) as a g –module. Now we specialize the previous construction to g = sl (4 |
1) and k = − α = α , = 13 ( e , + · · · + e , + 4 e , ) ( − ∈ V − ( sl (4 | . and [ α λ α ] = 43 λ, U s = U , s = V − ( sl (4 | ( sα ) ( s ∈ Z ) . Corollary 9.2.
We have:(1) U s , s ∈ Z , are irreducible V − ( sl (4 | –modules from the category KL − .(2) The lowest graded component of U is isomorphic to C | and that of U − is isomorphic to ( C | ) ∗ .(3) U s is an irreducible, simple current V − ( sl (4 | –module and the follow-ing fusion rules hold: U s × U s = U s + s ( s , s ∈ Z ) . Proof.
Proof follows from Theorem 9.1 and the fact that top component of U (resp. U − ) has the same highest weight as the sl (4 | C | (resp.( C | ) ∗ . (cid:3) Modules U s are actually obtained from the vertex algebra V k ( sl (4 | π s of \ sl (4 |
1) which leaves [ sl (4)–invariant.9.2. The decomposition for W k ( sl (6 | , θ ) , k = − . We now consider theminimal W –algebra W k ( g , θ ) for Lie superalgebra g = sl (6 |
1) at the confor-mal, non-collapsing level k = −
2. We shall prove that each W k ( g , θ ) ( i ) is asimple current V k ( g ♮ )–module. In order to see this, essentially it suffices toprove that W k ( g , θ ) ( ± are the simple current modules described in previoussection. Note that g ♮ = sl (4 |
1) + C , and that V k ( g ) = V k +1 ( sl (4 | ⊗ M β ( h ∨ − h ∨ ( k + h ∨ / , where β = J { c } , [ β λ β ] = h ∨ − h ∨ ( k + h ∨ /
2) = . By the irreducibility statement from Theorem 6.4 we see that there areweights Λ ± such that W k ( g , θ ) ( ± ∼ = L sl (4 | (Λ ± ) ⊗ M β ( , ± . The lowest graded component of L sl (4 | (Λ + ) (resp. L sl (4 | (Λ + ) ) is isomor-phic as sl (4 | C | (resp. ( C | ) ∗ ) and it has conformal weight h = . By Corollary 9.2, we get that W k ( g , θ ) ( ± ∼ = U ± ⊗ M β ( , ± . Since U ± and M β ( , ±
1) are simple current modules we get that W k ( g , θ ) isa simple current extension. In this way we have proved the following result,which gives a super-analog of Corollary 8.3. (Arguments are essentiallythe same, only the proof that W k ( g , θ ) ( ± are simple current modules usesdifferent techniques). ONFORMAL EMBEDDINGS OF AFFINE VERTEX ALGEBRAS IN W -ALGEBRAS 47 Corollary 9.3.
Let g = sl (6 | . We have the following isomorphism of V − , / ( g ♮ ) -modules: W − ( g , θ ) ∼ = M s ∈ Z U s ⊗ M β (3 / , s ) . Remark 9.2.
By using similar arguments one can obtain analogous decom-positions for g = sl ( n +5 | n ) and conformal level k = −
2. For decompositionsin the case of other conformal levels we need more precise fusion rules analy-sis. This and related questions will be discussed in our forthcoming papers.
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V.K. : Department of Mathematics, MIT, 77 Mass. Ave, Cambridge, MA 02139; [email protected]
P.MF. : Politecnico di Milano, Polo regionale di Como, Via Valleggio 11, 22100 Como,Italy; [email protected]
P.P. : Dipartimento di Matematica, Sapienza Universit`a di Roma, P.le A. Moro 2, 00185,Roma, Italy; [email protected]
O.P. : Department of Mathematics, Faculty of Science, University of Zagreb, Bijeniˇcka30, 10 000 Zagreb, Croatia;: Department of Mathematics, Faculty of Science, University of Zagreb, Bijeniˇcka30, 10 000 Zagreb, Croatia;