Bershadsky-Polyakov vertex algebras at positive integer levels and duality
aa r X i v : . [ m a t h . QA ] N ov BERSHADSKY-POLYAKOV VERTEX ALGEBRAS AT POSITIVEINTEGER LEVELS AND DUALITY
DRAˇZEN ADAMOVI´C AND ANA KONTREC
Abstract.
We study the simple Bershadsky-Polyakov algebra W k = W k ( sl , f θ )at positive integer levels and classify their irreducible modules. In this waywe confirm the conjecture from [9]. Next, we study the case k = 1. We dis-cover that this vertex algebra has a Kazama-Suzuki-type dual isomorphic tothe simple affine vertex superalgebra L k ′ ( osp (1 | k ′ = − /
4. Using thefree-field realization of L k ′ ( osp (1 | W k and their highest weight modules. In a sequel, we plan to study fusionrules for W k . Introduction
In the recent years, minimal affine W algebras have attracted a lot of interest.They are obtained using quantum hamiltonian reduction from affine vertex algebras,and they can be described using generators and relations (cf. [27, 29]).The Bershadsky-Polyakov algebra W k = W k ( sl , f θ ) ([14], [36]) is the simplestminimal affine W –algebra. T. Arakawa proved in [11] that W k is rational for k + 3 / ∈ Z ≥ , while in other cases it is a non-rational vertex algebra. Morerecently, for k admissible and non-integral, irreducible W k –modules were classifiedin [9] in some special cases, and in [24] in full generality. A realization of W k , when2 k + 3 / ∈ Z ≥ , and its relaxed modules is presented in [8], which gives a naturalgeneralization of the realization of the affine vertex algebra V k ( sl (2)) from [3]. Letus now mention certain problems for Bershadsky-Polyakov vertex algebras, whichremain unsolved in papers listed above. A. Classification of irreducible W k –modules for integer levels k , k +2 ∈ Z ≥ . In [24], authors classified irreducible W k –modules for k admissible, non-integral. They showed that every irreducible highest weight module for W k isobtained as an image of the admissible modules for L k ( sl (3)) (which are classifedby T. Arakawa in [12]). However, when we pass to integral k , the methods of [24]are no longer applicable, since in this case quantum hamiltonian reduction sends L k ( sl (3))–modules to zero.In [9], we began the study of the representation theory of W k for integer levels k , k + 2 ∈ Z ≥ . The starting point was explicit formulas for singular vectors in W k (which generalized those of Arakawa in [11]). We presented a conjecture on theclassification of irreducible W k –modules for k + 2 ∈ Z ≥ , which we proved in cases Date : November 20, 2020.2010
Mathematics Subject Classification.
Primary 17B69; Secondary 17B68.
Key words and phrases. vertex algebra, W-algebras, Bershadsky-Polyakov algebra, Zhu’salgebra. k = − , O ) using explicit realizationsof W k . One of the main results in this paper is the proof of this conjecture. B. Free-field realization of W k and their modules for k + 2 ∈ Z ≥ . In [8],the Bershadsky-Polyakov algebra W k is realized as a vertex subalgebra of Z k ⊗ Π(0)(where Z k = W k ( sl (3) , f princ ) is the Zamolodchikov W -algebra [40]), for 2 k + 3 / ∈ Z ≥ . A realization of W k , when 2 k + 3 ∈ Z ≥ , requires a different approach. In [9]we constructed a free-field realization of W , but the cases when k > W k = W k ( sl , f θ ) at positive integer levels k and completely solve problem(A) for k ≥
0. We also partially solve the problem (B) for k = 1 and find dualityrelation of W with the affine vertex superalgebra associated to osp (1 | Classification of irreducible representations.
In [9], we found a necessarycondition for W k –modules, parametrizing the highest weights as zeroes of certainpolynomial functions (cf. Proposition 4.5) h i ( x, y ) = 1 i ( g ( x, y ) + g ( x + 1 , y ) + ... + g ( x + i − , y ))= − i + ki − xi + 3 i − x − k + 2 kx + 6 x + ky + 3 y − , and conjectured that this provides the complete list of irreducible modules for theBershadsky-Polyakov algebra W k when k ∈ Z , k ≥ −
1. This conjecture was provedin cases k = − k = 0 in [9], using explicit realizations of W − and W as theHeisenberg vertex algebra and a subalgebra of lattice vertex algebra, respectively. Inthis paper, we prove this conjecture for k ∈ Z , k ≥
1, thus obtaining a classificationof irreducible modules for W k at positive integer levels. Theorem 1.1.
The set { L ( x, y ) | ( x, y ) ∈ S k } is the set of all irreducible ordinary W k –modules, where S k = (cid:8) ( x, y ) ∈ C |∃ i, ≤ i ≤ k + 2 , h i ( x, y ) = 0 (cid:9) . In order to prove Theorem 1.1, we need to show that L ( x, y ), ( x, y ) ∈ S k , areindeed W k –modules. Idea of the proof is to construct an infinite family of irreducible W k –modules L ( x, y ) such that h i ( x, y ) = 0 for arbitrary 1 ≤ i ≤ k +2, using spectralflow construction of W k –modules (cf. Section 4). • We first consider a family of simple-current W k –modules Ψ n ( W k ), n ∈ Z ≥ . • We show that they are highest weight W k –modules satisfyingΨ n ( W k ) = L ( x n , y n ) and h ( x n , y n ) = h k +2 ( x n +1 , y n +1 ) = 0 . • Using a certain version of algebraic continuation (based on the fact thathighest weights of modules for Zhu’s algebra must be zeros of finitelymany curves in C ), we conclude that L ( x, y ) are W k –modules whenever h ( x, y ) = 0 or h k +2 ( x, y ) = 0. • Next, for every 2 ≤ i ≤ k + 1, we find special points ( x i , y i ) such that h i ( x i , y i ) = h k +2 ( x i , y i ) = 0, and again apply the spectral-flow automor-phism Ψ n . In this way we are able to construct infinitely many highestweight W k –modules L ( x i n , y i n ) such that h i ( x i n , y i n ) = 0. • Again using the algebraic continuation, we conclude that L ( x, y ) are W k –modules for each point of the curve h i ( x, y ) = 0. ERSHADSKY-POLYAKOV ALGEBRA AT POSITIVE INTEGER LEVELS 3
Realization of W and duality with L − / ( osp (1 | . Next, we give an in-depthstudy of the case k = 1. First we show that the Bershadsky-Polyakov vertex algebra W can be embedded into the tensor product of the affine vertex superalgebra L k ′ ( osp (1 | k ′ = − / F . Theaffine vertex algebra V k ( osp (1 | osp (1 ,
2) wasrealized by the first named author in [3]. Using this result, and the fact that atlevel k ′ = − / L k ′ ( sl (2)) into L k ′ ( osp (1 | W (cf. Theorem 5.2).Let F − be the lattice vertex superalgebra associated to the negative definitelattice Z √−
1. We show that the simple affine vertex superalgebra L − / ( osp (1 | W ⊗ F − (cf. Theorem 5.4). Moreover, there is aduality between W and the affine vertex superalgebra L k ′ ( osp (1 | k ′ = − / W = Com ( M h ⊥ (1) , L k ′ ( osp (1 | ⊗ F ) ,L k ′ ( osp (1 | (cid:0) M h (1) , W ⊗ F − (cid:1) , where M h ⊥ (1) and M h (1) are Heisenberg vertex algebras defined in Section 5.In [3] it was proved that L − / ( osp (1 , F / ⊗ Π / (0), where Π / (0) is a lattice type vertex algebra, and F / is a Clifford vertex superalgebra (cf. Subsection 2.2). Using the fact that all irre-ducible L − / ( osp (1 | W -modules. Consequences of duality and future work.
The notion of Kazama-Suzukidual was first introduced in the context of the duality of the N = 2 superconformalalgebra and affine Lie algebra b s l (2) (cf. [20, 1, 2]). Later it was shown that analogousduality relations hold for some other affine vertex algebras and W –algebras (cf.[4, 5, 16, 15]). Our result shows that L k ′ ( osp (1 | W .Relaxed modules for L k ′ ( osp (1 | W –modules, forwhich one expects it is easier to obtain the tensor category structure and calculatethe fusion rules. Recent results [10, 15] show compelling evidence that fusion rulesand (vertex) tensor category structure can be transferred onto duals. We expectthat the duality between L k ′ ( osp (1 | W could be used to study fusion rulesin the category of relaxed modules for L k ′ ( osp (1 | Setup. • The universal Bershadsky-Polyakov algebra of level k will be denoted with W k ( sl , f θ ) or W k , and its unique simple quotient with W k ( sl , f θ ) or W k .The spectral flow automorphism of W k is denoted by Ψ. • The Zhu algebra associated to the vertex operator algebra V with the Vi-rasoro vector ω will be denoted with A ω ( V ). • The Smith algebra corresponding to the polynomial g ( x, y ) ∈ C [ x, y ] isdenoted with R ( g ). • F − is the lattice vertex superalgebra associated to the lattice Z √− • F / is the Clifford vertex superalgebra, also called the free fermion algebra(cf. Section 2.2). It has an automorphism σ F / of order two which is lifted DRAˇZEN ADAMOVI´C AND ANA KONTREC from the automorphism Φ( r )
7→ − Φ( r ) of the Clifford algebra Cl / . The σ F / –twiseted F / –modules are denoted by M ± F . • F is the Clifford vertex superalgebra, also called the charged fermion al-gebra or the bc system (cf. Section 2.3). It has an automorphism σ F of order two which is lifted from the automorphism Ψ + ( r )
7→ − Ψ + ( r ),Ψ − ( r )
7→ − Ψ − ( r ) of the Clifford algebra Cl . The σ F –twisted F module isdenoted by M twF . • L k ( osp (1 | osp (1 |
2) at level k . The spectral flow automorphism of L k ( osp (1 | ρ . 2. Preliminaries
In this section we review certain properties of Clifford vertex superalgebras (cf.[22], [21]) and a construction of twisted modules for vertex superalgebras by H. Li(cf. [33]). Twisted modules for Clifford vertex superalgebras (cf. [22]) will play akey role in the realization of the Bershadsky-Polyakov algebra W .2.1. Twisted modules for vertex superalgebras.
Let V = V ⊕ V be a vertexsuperalgebra (cf. [21], [39]), with the vertex operator structure given by Y : V −→ (End V )[[ z, z − ]] , Y ( v, z ) = X n ∈ Z v n z − n − , for v ∈ V , v n ∈ End V . Then any element in V (resp. V ) is said to be even (resp.odd). For any homogenuous element u , we define | u | = 0 if u ∈ V and | u | = 1 if u ∈ V .We say that a linear automorphism σ : V → V is a vertex superalgebra automor-phism if it holds that σY ( v, z ) σ − = Y ( σv, z )for v ∈ V . Then σV α ⊂ V α for α ∈ (cid:8) , (cid:9) .Let V be a vertex superalgebra and σ an automorphism of V with period k ∈ Z ≥ (that is, σ k = 1). Let us now recall a construction of σ -twisted V -modules (cf. [33]).Let h ∈ V be an even element such that(2.1) L ( n ) h = δ n, h, h n h = δ n, γ for n ∈ Z > , for fixed γ ∈ Q . Assume that h acts semisimply on V with rational eigenvalues.It follows that h n satisfy:[ L ( m ) , h n ] = − nh m + n , [ h m , h n ] = mγδ m + n, , for m, n ∈ Z . Set ∆( h, z ) = z h exp ∞ X k =1 h k − k ( − z ) − k ! . Note that e πih is an automorphism of V . Set σ h = e πih and assume that σ h isof finite order. The following was proved in [33]: Proposition 2.1 ([33]) . Let V be a vertex superalgebra and let h ∈ V be an evenelement such that (2.1) holds and h acts on V with rational eigenvalues. Let ( M, Y M ( · , z )) be a V -module. Then ( M, Y M (∆( h, z ) · , z )) carries the structure of a σ h -twisted V -module. ERSHADSKY-POLYAKOV ALGEBRA AT POSITIVE INTEGER LEVELS 5
Clifford vertex superalgebra F / and its twisted modules. Let Cl / be the Clifford algebra with generators Φ( r ) , r ∈ + Z and commutation relations { Φ( r ) , Φ( s ) } = δ r + s, , r, s ∈ + Z . The fields Φ( z ) = X n ∈ Z Φ( n + ) z − n − generate on F / = ^ (Φ( − n − / | n ∈ Z ≥ )a unique structure of a vertex superalgebra with conformal vector ω F / = Φ( − )Φ( − ) , of central charge c F / = 1 / z )is usually called neutral fermion field, and F / is called free-fermion theory inphysics literature.A basis of F / is given byΦ( − n − ) . . . Φ( − n r − ) , where n > · · · > n r ≥ F / has the automorphism σ F / of order two whichis lifted from the automorphism Φ( r )
7→ − Φ( r ) of the Clifford algebra. The fixedpoints of this automorphism is the Virasoro vertex algebra L V ir ( , F / = L V ir ( , ⊕ L V ir ( , ).We briefly recall the properties of twisted modules for Clifford vertex superalge-bras, while details can be found in [22].Define the twisted Clifford algebra Cl / tw generated by Φ( m ) , m ∈ Z and rela-tions { Φ( m ) , Φ( n ) } = δ m + n, , m, n ∈ Z . Let M ± F / = ∞ M n =0 M ± F / ( n )be the two irreducible modules for the Clifford algebra Cl / tw , such that Φ(0) actson the one-dimensional top component M ± F / (0) as ± √ Id.Let Φ tw ( z ) = X m ∈ Z Φ( m ) z − m − / , and Y (Φ( − n − ) . . . Φ( − n r − ) , z ) =: ∂ n Φ tw ( z ) . . . ∂ n r Φ tw ( z r ) : , and extend by linearity to all of F / .Define the twisted operator Y twF / ( v, z ) := Y ( e ∆ z v, z ) , where ∆ z = 12 X m,n ∈ Z ≥ C m,n Φ( m + )Φ( n + ) z − m − n − , and C m,n = 12 m − nm + n + 1 (cid:18) − / m (cid:19)(cid:18) − / n (cid:19) . DRAˇZEN ADAMOVI´C AND ANA KONTREC
It holds that (cf. [21], [22]) e ∆ z ω F / = ω F / + z − . Then ( M ± F / , Y twF / ) has the structure of a σ F / -twisted module for the vertexsuperalgebra F / .Recall also that as a L V ir ( , M ± F / ∼ = L V ir ( , ) . Clifford vertex superalgebra F and its twisted modules. Consider theClifford algebra Cl with generators Ψ ± ( r ) , r ∈ + Z and relations { Ψ + ( r ) , Ψ − ( s ) } = δ r + s, , { Ψ ± ( r ) , Ψ ± ( s ) } = 0 , r, s ∈ + Z . The fields Ψ ± ( z ) = X n ∈ Z Ψ ± ( n + ) z − n − generate on F = ^ (cid:0) Ψ ± ( − n − / | n ∈ Z > (cid:1) a unique structure of a simple vertex superalgebra. This vertex algebra is sometimescalled bc –system.Let α =: Ψ + Ψ − :. Then ω F = 12 : αα :is a conformal vector for F of central charge c F = 1.A basis of F is given byΨ + ( − n − ) . . . Ψ + ( − n r − )Ψ − ( − k − ) . . . Ψ − ( − k s − ) , where n i , k i ∈ Z ≥ , n > · · · > n r , k > · · · > n s .The vertex superalgebra F has an automorphism σ F of order two which is liftedfrom the automorphism Ψ + ( r )
7→ − Ψ + ( r ), Ψ − ( r )
7→ − Ψ − ( r ) of the Clifford alge-bra.Let ( M twF , Y tw ( · , z )) so that M twF = F as a vector space, and the vertex operatoris defined by Y twF ( v, z ) = Y (∆( α/ , z ) v, z ) . By Proposition 2.1 we have that ( M twF , Y twF ) has the structure of a σ F -twistedmodule for the vertex superalgebra F .2.4. Lattice vertex superalgebras F − . Consider rank one lattice L = Z ϕ , h ϕ, ϕ i = −
1. Let F − be the associated vertex algebra. This vertex superalgebra isused for a construction of the inverse of Kazama-Suzuki functor in the context ofduality between affine b sl (2) and N = 2 superconformal algebra (cf. [20], [1], [2]).As a vector space F − = C [ L ] ⊗ M ϕ (1), where C [ L ] is a group algebra of L ,and M ϕ (1) the Heisenberg vertex algebra generated by the Heisenberg field ϕ ( z ) = P n ∈ Z ϕ ( n ) z − n − such that [ ϕ ( n ) , ϕ ( m )] = − nδ n + m, . ERSHADSKY-POLYAKOV ALGEBRA AT POSITIVE INTEGER LEVELS 7 F − is generated by e ± ϕ . We shall need the relations e ± ϕn e ± ϕ = 0 for n ≥ ,e ± ϕ − m e ± ϕ = S m ( ± ϕ ) e ϕ for m ≥ ,e ϕn e − ϕ = 0 for n ≥ − ,e ϕ − m − e ϕ = S m ( ϕ ) for m ≥ , where S m ( ϕ ) := S m ( ϕ ( − , ϕ ( − , · · · ) is the m -th Schur polynomial in variables ϕ ( − , ϕ ( − , · · · .2.5. Kazama-Suzuki duality.
In this subsection, we will define a duality of ver-tex algebras which is motivated by the duality between N = 2 superconformalvertex algebra and affine vertex algebra L k ( sl (2)).Recall first that if S is a vertex subalgebra of V , we have the commutant subal-gebra of V (cf. [35]):Com( S, V ) := { v ∈ V | a n v = 0 , ∀ a ∈ S, ∀ n ∈ Z ≥ } . Assume that
U, V are vertex superalgebras. We say that V is the Kazama-Suzukidual of U if there exist injective homomorphisms of vertex superalgebras ϕ : V → U ⊗ F, ϕ : U → V ⊗ F − , so that V ∼ = Com (cid:0) H , U ⊗ F (cid:1) , U ∼ = Com (cid:0) H , V ⊗ F − (cid:1) , where H (resp. H ) is a rank one Heisenberg vertex subalgebra of U ⊗ F (resp. V ⊗ F − ). 3. Bershadsky-Polyakov algebra W k ( sl , f θ )Bershadsky-Polyakov vertex algebra W k (= W k ( sl , f θ )) is the minimal affine W -algebra associated to the minimal nilpotent element f θ (cf. [11], [25], [29], [27],[31]). The algebra W k is generated by four fields T, J, G + , G − , of conformal weights2 , , , and is a Z -graded VOA. Definition 3.1.
Universal Bershadsky-Polyakov vertex algebra W k is the vertexalgebra generated by fields T, J, G + , G − , which satisfy the following relations: J ( x ) J ( y ) ∼ k + 33 ( z − w ) − , G ± ( z ) G ± ( w ) ∼ ,J ( z ) G ± ( w ) ∼ ± G ± ( w )( z − w ) − ,T ( z ) T ( w ) ∼ − c k z − w ) − + 2 T ( w )( z − w ) − + DT ( w )( z − w ) − ,T ( z ) G ± ( w ) ∼ G ± ( w )( z − w ) − + DG ± ( w )( z − w ) − ,T ( z ) J ( w ) ∼ J ( w )( z − w ) − + DJ ( w )( z − w ) − ,G + ( z ) G − ( w ) ∼ ( k + 1)(2 k + 3)( z − w ) − + 3( k + 1) J ( w )( z − w ) − ++ (3 : J ( w ) J ( w ) : + 3( k + 1)2 DJ ( w ) − ( k + 3) T ( w ))( z − w ) − , where c k = − (3 k +1)(2 k +3) k +3 . DRAˇZEN ADAMOVI´C AND ANA KONTREC
Vertex algebra W k is called the universal Bershadsky-Polyakov vertex algebraof level k . For k = − W k has a unique simple quotient which is denoted by W k .Let T ( z ) = X n ∈ Z L n z − n − J ( z ) = X n ∈ Z J n z − n − ,G + ( z ) = X n ∈ Z G + n z − n − ,G − ( z ) = X n ∈ Z G − n z − n − . The following commutation relations hold:[ J m , J n ] = 2 k + 33 mδ m + n, , [ J m , G ± n ] = ± G ± m + n , [ L m , J n ] = − nJ m + n , [ L m , G ± n ] = ( 12 m − n + 12 ) G ± m + n , [ G + m , G − n ] = 3( J ) m + n − + 32 ( k + 1)( m − n ) J m + n − − ( k + 3) L m + n − ++ ( k + 1)(2 k + 3)( m − m δ m + n, . Structure of the Zhu algebra A ( W k ) . By applying results from [27] wesee that for every ( x, y ) ∈ C there exists an irreducible representation L x,y of W k generated by a highest weight vector v x,y such that J v x,y = xv x,y , J n v x,y = 0 for n > ,L v x,y = yv x,y , L n v x,y = 0 for n > ,G ± n v x,y = 0 for n ≥ . Let A ω ( V ) denote the Zhu algebra associated to the VOA V (cf. [41]) with theVirasoro vector ω , and let [ v ] be the image of v ∈ V under the mapping V A ω ( V ).For the Zhu algebra A ω ( W k ) it holds that: Proposition 3.2 ([9], Proposition 3.2.) . There exists a homomorphism
Φ : C [ x, y ] → A ω ( W k ) such that Φ( x ) = [ J ] , Φ( y ) = [ T ] . It can be shown that the homomorphism Φ : C [ x, y ] → A ω ( W k ) is in fact anisomorphism, i.e. that A ω ( W k ) ∼ = C [ x, y ].If we switch to a shifted Virasoro vector ω = ω + DJ , the vertex algebras W k and W k become Z ≥ –graded with respect to L (0) = ω . In this case the Zhualgebras are no longer commutative, and they carry more information about therepresentation theory. The Zhu algebra associated to W k is then realized as a quo-tient of another associative algebra, the so-called Smith algebra R ( g ) (introducedin [38], see also [19]). These algebras were used by T. Arakawa in the paper [11] toprove rationality of W k ( sl , f θ ) for k = p/ − p ≥ p odd. ERSHADSKY-POLYAKOV ALGEBRA AT POSITIVE INTEGER LEVELS 9
We expand the original definition of Smith algebras R ( f ) by adding a centralelement. Definition 3.3.
Let g ( x, y ) ∈ C [ x, y ] be an arbitrary polynomial. Associative al-gebra R ( g ) of Smith type is generated by { E, F, X, Y } such that Y is a centralelement and the following relations hold: XE − EX = E, XF − F X = − F, EF − F E = g ( X, Y ) . In fact, Zhu algebra associated to W k is a quotient of the Smith-type algebra R ( g ) for a certain polynomial g ( x, y ) ∈ C [ x, y ] . Proposition 3.4 ([9], Proposition 4.2.) . Zhu algebra A ω ( W k ) is a quotient of theSmith algebra R ( g ) for g ( x, y ) = − (3 x − (2 k + 3) x − ( k + 3) y ) . Vertex algebra W k for k + 2 ∈ Z ≥ In this section we study the representation theory of the Bershadsky-Polyakovalgebra W k at positive integer levels. In [9], we parametrized the highest weights ofirreducible W k –modules as zeroes of certain polynomial functions (cf. Proposition4.5), and conjectured that this provides the complete list of irreducible modules forthe Bershadsky-Polyakov algebra W k when k ∈ Z , k ≥ −
1. This conjecture wasproved in cases k = − k = 0 in [9]. In this paper, we will prove this conjecturefor k ∈ Z , k ≥ Setup.
Let us choose a new Virasoro field L ( z ) := T ( z ) + 12 DJ ( z ) . Then ω = ω + DJ is a conformal vector ω n +1 = L ( n ) with central charge c k = − k + 1)(2 k + 3) k + 3 . The fields
J, G + , G − have conformal weights 1 , , J ( n ) = J n , G + ( n ) = G + n , G − ( n ) = G − n +1 . We have L ( z ) = X n ∈ Z L ( n ) z − n − , G + ( z ) = X n ∈ Z G + ( n ) z − n − , G − ( z ) = X n ∈ Z G − ( n ) z − n − . This defines a Z ≥ -gradation on W k .Define ∆( − J, z ) = z − J (0) exp ∞ X k =1 ( − k +1 − J ( k ) kz k ! , and let X n ∈ Z ψ ( a n ) z − n − = Y (∆( − J, z ) a, z ) , for a ∈ W k .The operator ∆( h, z ) associates to every V -module M a new structure of anirreducible V -module (cf. [32]). Let us denote this new module (obtained using themapping a n ψ ( a n )) with ψ ( M ). As the ∆-operator acts bijectively on the set ofirreducible modules, there exists an inverse ψ − ( M ). Remark 4.1.
Operators ψ m are also called the spectral flow automorphisms of W k ,see [8, Section 2] and [24, Subsection 2.2] for more details. From the definition of ∆( − J, z ) we have that ψ ( J ( n )) = J ( n ) − k + 33 δ n, , ψ ( L ( n )) = L ( n ) − J ( n ) + 2 k + 33 δ n, ,ψ ( G + ( n )) = G + ( n − , ψ ( G − ( n )) = G − ( n + 1) . Let L ( x, y ) top = { v ∈ L ( x, y ) : L (0) v = yv } , and denote ˆ x i = x + i − − k + 33 , ˆ y i = y − x − i + 1 + 2 k + 33 . The following was proved in [11]:
Lemma 4.2 ([11], Proposition 2.3.) . Let dim L ( x, y ) top = i . Then it holds that ψ ( L ( x, y )) ∼ = L (ˆ x i , ˆ y i ) . Necessary condition for W k –modules. Starting point in the classificationof irreducible W k –modules for integer levels k is the following formula for singularvectors (cf. [9]). These generalize a construction of a family of singular vectors byT. Arakawa in [11], where he found a similar formula for singular vectors in W k atlevels k = p/ − p ≥ p odd. Lemma 4.3 ([9], Lemma 8.1.) . Vectors G + ( − n , G − ( − n are singular in W k for n = k + 2 , where k ∈ Z , k ≥ − . Let g ( x, y ) = − (3 x − (2 k + 3) x − ( k + 3) y ) ∈ C [ x, y ]and define polynomials h i ( x, y ), for i ∈ N (cf. [11]) as h i ( x, y ) = 1 i ( g ( x, y ) + g ( x + 1 , y ) + ... + g ( x + i − , y ))= − i + ki − xi + 3 i − x − k + 2 kx + 6 x + ky + 3 y − . The next technical lemma follows immediately from the definiton of h i ( x, y ) andˆ x i , ˆ y i . Lemma 4.4.
Assume that i + j = k + 3 . Then it holds that h i ( x, y ) = h j (ˆ x i , ˆ y i ) . Proof.
We have h k +3 − i (ˆ x i , ˆ y i ) = ( − x − x + 12 x + 6 x − x + 3 xi − kx − xi + 4 kx + 2 kx − kx )++ ( ky + 3 y ) + ( − i + 2 ki + +6 i − k − k − k + 3 k − ki + 3 i −− ki − i + 2 k + 12 k + 18 + 3 k + 9 − i − i + 4 ki ++ 12 i − k − k − − k + 2 ki − k − k + 6 i − − k −− − ki + k + k + k − i + 3 + 2 k + 1)= ( − x − ix + 2 kx + 6 x ) + ( ky + 3 y ) + ( − i + ki + 3 i − k − h i ( x, y ) . (cid:3) ERSHADSKY-POLYAKOV ALGEBRA AT POSITIVE INTEGER LEVELS 11
Define the set S k = (cid:8) ( x, y ) ∈ C |∃ i, ≤ i ≤ k + 2 , h i ( x, y ) = 0 (cid:9) . In [9] we proved that in order for L ( x, y ) to be an irreducible ordinary W k –module, ( x, y ) needs to belong to the set S k . Proposition 4.5 ([9]) . Let k ∈ Z , k ≥ − . Then we have:(1) The set of equivalency classes of irreducible ordinary W k –modules is con-tained in the set { L ( x, y ) | ( x, y ) ∈ S k } . (2) Every irreducible W k –module in the category O is an ordinary module. We stated the following conjecture (and proved it for k = − k = 0): Conjecture 4.6. [9]
The set { L ( x, y ) | ( x, y ) ∈ S k } is the set of all irreducibleordinary W k –modules. The proof of Conjecture 4.6 is reduced to showing that L ( x, y ), ( x, y ) ∈ S k , areindeed W k –modules. In what follows we shall prove Conjecture 4.6.4.3. Simple current W k –modules. First step in the proof is to construct an infi-nite family of irreducible W k –modules L ( x, y ) satisfying the conditions h ( x, y ) = 0or h k +2 ( x, y ) = 0. We have the following important lemma: Lemma 4.7.
Assume that n ∈ Z ≥ . Define x n = − n k + 33 ,x n +1 = − n − − ( n + 2) k ,y n = n k + ( k + 3) n ) ,y n +1 = n + 13 ( n ( k + 3) + 2 k + 3) . Then for each n ∈ Z ≥ we have • L ( x n , y n ) is irreducible W k –module. • Ψ n ( W k ) ∼ = L ( x n , y n ) . • h ( x n , y n ) = h k +2 ( x n +1 , y n +1 ) = 0 .Proof. By direct calculation we have h i ( x n , y n ) = 0 ⇐⇒ i ∈ { , k + 2 + n ( k + 3) } ,h i ( x n +1 , y n +1 ) = 0 ⇐⇒ i ∈ { k + 2 , k + 2) + n ( k + 3) } . We see that ( x n , y n ) is the unique solution of the following recursive relations: x = y = 0 ,x n +1 = x n − k + 33 , y n +1 = ˆ y n = y n − x n +1 ,x n +2 = ˆ x n +1 = x n +1 + k , y n +2 = ˆ y n +1 = y n +1 − x n +2 . Then for each n ∈ Z ≥ we have L ( x n , y n ) = Ψ n ( L (0 , n ( W k ) . So L ( x n , y n ) is an irreducible W k –module. (cid:3) Lemma 4.8.
Assume that n ∈ Z < . Define x n = − n k + 33 ,x n − = 1 − n − ( n − k ,y n = − n k − ( k + 3) n ) ,y n − = − n k + 3 − n ( k + 3)) . Then for each n ∈ Z < we have • L ( x n , y n ) is irreducible W k –module. • Ψ n ( W k ) ∼ = L ( x n , y n ) . • h k +2 ( x n , y n ) = h ( x n +1 , y n +1 ) = 0 .Proof. By direct calculation we have h i ( x n , y n ) = 0 ⇐⇒ i ∈ { k + 2 , n ( k + 3) } ,h i ( x n − , y n − ) = 0 ⇐⇒ i ∈ { , − − k + n ( k + 3) } . We see that ( x n , y n ) is the unique solution of the following recursive relations: x = y = 0 ,x n − = x n + 2 k + 33 , y n − = y n + x n ,x n − = x n − − k , y n − = y n − + x n − . We have Ψ − ( L (0 , − ( W k ) = L ( 2 k + 33 , . Then for each n ∈ Z ≤ we have L ( x n , y n ) = Ψ n ( L (0 , n ( W k ) . Hence L ( x n , y n ) is an irreducible W k –module. (cid:3) By using [32, Theorem 2.15] (see also [10, Proposition 3.1]) we get:
Corollary 4.9.
In the fusion algebra of W k , L ( x n , y n ) are simple-current modules,i.e., the following fusion rules hold: Ψ n ( W k ) × L ( x, y ) = L ( x n , y n ) × L ( x, y ) = Ψ n ( L ( x, y )) . Modules L ( x, y ) such that h ( x, y ) = 0 or h k +2 ( x, y ) = 0 .Theorem 4.10. Assume that h ( x, y ) = 0 or h k +2 ( x, y ) = 0 . Then L ( x, y ) is anirreducible ordinary W k –module. ERSHADSKY-POLYAKOV ALGEBRA AT POSITIVE INTEGER LEVELS 13
Proof.
The solution of the equation h ( x, y ) = 0 is y = g ( x ) = − x − kx + 3 x k ( x ∈ C ) . So we need to prove that L ( x, g ( x )) is an W k –module for every x ∈ C . On theother hand, the Zhu’s algebra A ω ( W k ) is isomorphic to a certain quotient of C [ x, y ]by an ideal I . Since C [ x, y ] is Noetherian, the ideal I is finitely generated by finitelymany polynomials, say P , . . . , P ℓ . Hence the highest weights ( x, y ) of irreducible W k –modules are solutions of the equations: P i ( x, y + x/
2) = 0 , i = 1 , . . . , ℓ.
It remains to prove that P i ( x, g ( x ) + x/
2) = 0 , ∀ x ∈ C . By Lemma 4.7, we have that L ( x, g ( x )) are W k –modules for x = x n = − n k +33 . Sowe have P i ( x n , g ( x n ) + x n /
2) = 0 , i = 1 , . . . , ℓ.
Since each P i ( x, g ( x )+ x/
2) is a polynomial in one variable and given that it alreadyhas infinitely many zeros, it follows that P i ( x, g ( x ) + x/ ≡
0. This proves that L ( x, g ( x )) is W k –module, for every x ∈ C .Applying Lemma 4.4, we get that L ( x, y ) is a W k –module for each solution ofthe equation h k +2 ( x, y ) = 0. (cid:3) Proof of Conjecture 4.6.
The proof of the conjecture is reduced to theexistence of irreducible W k modules L ( x, y ) such that h i ( x, y ) = 0 for arbitrary1 ≤ i ≤ k + 2. Lemma 4.11.
Let ≤ i ≤ k + 2 . There exists an irreducible W k –module L ( x i , y i ) with highest weight ( x i , y i ) = (cid:18) − i , ( − i )( − i − k )3(3 + k ) (cid:19) such that dim L ( x, y ) top = i .Proof. Note that h j ( x i , y i ) = ( i − j )( − j − k ) = 0 ⇐⇒ j ∈ { i, k + 2 } . By Theorem 4.10 we know that L ( x i , y i ) is an W k –module. But since h i ( x i , y i ) = 0and h j ( x i , y i ) = 0 for j < i , we conclude that dim L ( x i , y i ) top = i . The prooffollows. (cid:3) Now we shall continue as in the previous subsection. We will apply the auto-morphism Ψ and construct new infinite family of W k –modules. Lemma 4.12.
For each n ∈ Z ≥ , Ψ n ( L ( x i , y i )) = L ( x in , y in ) , where x i n = 1 − i − n k + 33 y i n = 1 − i + i + k − ik + 12 n − in + 10 kn − ikn + 2 k n + 9 n + 6 kn + k n k ) x i n +1 = − i − n − k (2 + n )3 y i n +1 = 16 − i + i + 12 k − ik + 2 k + 21 n − in + 16 kn − ikn + 3 k n + 9 n + 6 kn + k n k ) Proof.
Using direct calculation we get h j ( x n , y n ) = ( i − j )( − j − n − k (1 + n )) ,h j ( x n +1 , y n +1 ) = − ( − i + j − k )( − i + j − k − n − kn ) , which implies thatdim L ( x n , y n ) top = i, dim L ( x n +1 , y n +1 ) top = k + 3 − i. We see that ( x in , y in ) is the unique solution of the following recursive relations: x i = x i , y i = y i ,x i n +1 = x i n + ( i − − k + 33 , y i n +1 = y i n − x i n +1 ,x i n +2 = x i n +1 + k + 2 − i − k + 33 , y n +2 = y i n +1 − x i n +2 . This proves that for each n ∈ Z ≥ we have L ( x in , y in ) = Ψ n ( L ( x i , y i )) . Since L ( x i , y i ) is a W k –module, we have that L ( x in , y in ) is an irreducible W k –module. The proof follows. (cid:3) Theorem 4.13.
Assume that h i ( x, y ) = 0 for ≤ i ≤ k + 2 . Then L ( x, y ) is anirreducible ordinary W k –module.Proof. By Lemma 4.12 we have that L ( x i n , y i n ) are W k –modules for every n ∈ Z ≥ .Since h i ( x i n , y i n ) = 0, we conclude that there is an infinite-family of points ( x, y )of the curve h i ( x, y ) = 0 such that L ( x, y ) is a W k –module. Applying the sameargument as in the proof of Theorem 4.10, we get that L ( x, y ) is a W k –module forevery ( x, y ) such that h i ( x, y ) = 0. (cid:3) Theorem 4.13 concludes the proof of Conjecture 4.6.5.
The duality of W and L − / ( osp (1 | . In this section we construct an embedding of the Bershadsky-Polyakov vertexalgebra W into the tensor product of the affine vertex superalgebra L k ( osp (1 | k = − / F . The affine vertex su-peralgebra V k ( osp (1 | L − / ( osp (1 | W by proving that there is an embedding of W into F − , where F − is the vertexsuperalgebra associated to the lattice Z √− ERSHADSKY-POLYAKOV ALGEBRA AT POSITIVE INTEGER LEVELS 15
Affine vertex superalgebra V k ( osp (1 | . Recall that g = osp (1 ,
2) is thesimple complex Lie superalgebra with basis { e, f, h, x, y } such that the even part g = span C { e, f, h } and the odd part g = span C { x, y } . The anti-commutationrelations are given by [ e, f ] = h, [ h, e ] = 2 e, [ h, f ] = − f [ h, x ] = x, [ e, x ] = 0 , [ f, x ] = − y [ h, y ] = − y, [ e, y ] = − x [ f, y ] = 0 { x, x } = 2 e, { x, y } = h, { y, y } = − f. Choose the non-degenerate super-symmetric bilinear form ( · , · ) on g such that non-trivial products are given by( e, f ) = ( f, e ) = 1 , ( h, h ) = 2 , ( x, y ) = − ( y, x ) = 2 . Let b g = g ⊗ C [ t, t − ] + C K be the associated affine Lie superalgebra, and V k ( g )(resp. L k ( g )) the associated universal (resp. simple) affine vertex superalgebra. Asusual, we identify x ∈ g with x ( − .The Sugawara conformal vector of V k ( osp (1 , ω sug = 12 K + 3 (cid:18) : ef : + : f e : + 12 : hh : −
12 : xy : + 12 : yx : (cid:19) . The notion of Ramond–twisted modules is defined as usual in the case of affinevertex superalgebras [28] (see also [37]).Recall also the spectral flow automorphism ρ for d osp (1 | ρe ( n ) = e ( n − , ρx ( n ) = x ( n − , ρf ( n ) = f ( n + 2) ,ρy ( n ) = y ( n + 1) ρh ( n ) = h ( n ) − δ n, K. ( n ∈ Z ) . As in the case of the automorphism Ψ of the W k , one shows that for any L k ( osp (1 | M, Y M ( · , z )) and n ∈ Z , ρ n ( M ) is again a L k ( osp (1 | Y ρ n ( M ) ( · , z )) := Y M (∆( − nh, z ) · , z )) . (see also [4, Proposition 2.1] for the proof of similar statement for spectral-flowautomorphism of L k ( sl (2)), and [10, Proposition 3.1] in the case of β − γ system).5.2. Embedding of W into L − / ( osp (1 | ⊗ F . The free field realization of L k ( osp (1 | W can be embedded into L − / ( osp (1 | ⊗ F , where F is the Clifford vertex superalgebra, introduced in Section 2.3.Set τ + =: Ψ + x : τ − =: Ψ − y : . Let α =: Ψ + Ψ − :. Then ω F = 12 : αα : . Denote by M h ⊥ (1) the Heisenberg vertex algebra generated by h ⊥ = α − h . For s ∈ C , denote by M h ⊥ (1 , s ) the irreducible M h ⊥ (1)–module on which h ⊥ (0) actsas s Id. Note that h ⊥ (1) h ⊥ = − . The Virasoro vector of central charge c = 1 in M h ⊥ (1) is ω ⊥ = − h ⊥ ( − . The following lemma can be proved easily using results from [32, 34].
Lemma 5.1.
Consider the M h ⊥ (1) –module M h ⊥ (1 , s ) with the vertex operator Y s ( · , z ) . Then for every n ∈ Z ( ] M h ⊥ (1 , s ) , f Y s ( · , z )) := ( M h ⊥ (1 , s ) , Y s (∆( − n h ⊥ , z ) · , z )) is an irreducible M h ⊥ (1) –module isomorphic to M h ⊥ (1 , n + s ) . Theorem 5.2. (i) There is a non-trivial homomorphism of vertex algebras
Φ : W → L − / ( osp (1 | ⊗ F uniquely determined by G + = 2 τ + = 2 : Ψ + x : G − = 2 τ − = 2 : Ψ − y : J = 53 α − hT = ω sug + ω F − ω ⊥ . (ii) Im (Φ) is isomorphic to the simple vertex algebra W .(iii) As a W ⊗ M h ⊥ (1) -module: L − / ( osp (1 | ⊗ F ∼ = M n ∈ Z Ψ − n ( W ) ⊗ M h ⊥ (1 , n )= M n ∈ Z L ( x n , y n ) ⊗ M h ⊥ (1 , − n ) . (iv) We have: Com ( M h ⊥ (1) , L − / ( osp (1 | ⊗ F ) ∼ = W . Proof.
Let k = 1. Using the formula(Ψ + − x ) n = ∞ X i =0 (cid:0) Ψ + − − i x ( n + i ) − x ( n − i − + i (cid:1) we obtain τ +2 τ − = − k ′ ,τ +1 τ − = − k ′ α − h,τ +0 τ − = − αh − : xy : − k ′ : D Ψ + Ψ − : . From the above formulas and OPE relations for W k it follows that for k = 1, k ′ needs to be equal to − /
4. For level k ′ = − / L k ′ ( sl (2)) into L k ′ ( osp (1 | ω sug = ω sl (2) , where ω sl (2) = 12( k ′ + 2) (cid:18) : ef : + : f e : + 12 : hh : (cid:19) . We have G +2 G − = 4 τ +2 τ − = 10 = ( k + 1)(2 k + 3) ,G +1 G − = 4 τ +1 τ − = − − α + h ) = 10 α − h = 6 J = 3( k + 1) J,G +0 G − = 4 τ +0 τ − = − hα : − xy : +5 : αα : +5 Dα.
ERSHADSKY-POLYAKOV ALGEBRA AT POSITIVE INTEGER LEVELS 17
Using the realization in [3] and the fact that there is a conformal embedding of L − / ( sl (2)) into L − / ( osp (1 | ω sug = 12 (: xy : − yx :) =: xy : − Dh.
This implies that T = : xy : − Dh + 12 : αα : + 13 (: αα : + : hh : − αh :)= : xy : − Dh + 56 : αα : + 13 : hh : −
23 : αh :Hence G +0 G − = − hα : − xy : +5 : αα : +5 Dα = 3 : J : +3 : DJ : − T = 3 : J : + 3( k + 1)2 : DJ : − ( k + 3) T. This proves assertion (i).Let us prove that W = Im(Φ) is simple.Let W = Ker L − / ( osp (1 , ⊗ F h ⊥ (0). It is clear that W is a simple vertex algebrawhich contains W ⊗ M h ⊥ (1).The simplicity of W follows from the following claim: Claim 1 W is generated by G + , G − , J, T, h ⊥ .For completeness, we shall include a proof of the Claim 1.Let U be the vertex subalgebra of W generated by { G + , G − , J, T, h ⊥ } . Thenclearly U ∼ = W ⊗ M h ⊥ (1).Let U ( n ) be the U –module obtained by the simple current construction( U ( n ) , Y ( n ) ( · , z )) := ( U, Y (∆( nα, ) · , z )) . Note next that α = J − h ⊥ , which implies that∆( α, z ) = ∆( − h ⊥ , z )∆( J, z ) . (5.1) • For a W –module W , by applying operator ∆( nJ, z ) we get module Ψ − n ( W ). • Using Lemma 5.1 we see that by applying the operator ∆( − n h ⊥ , z ) on M h ⊥ (1), we get module M h ⊥ (1)–module M h ⊥ (1 , n ).We get U ( n ) = Ψ − n ( W ) ⊗ M h ⊥ (1 , n ) . (5.2)Since by the boson-fermion correspondence F is isomorphic to the lattice vertexsuperalgebra V Z α , we get that U ( n ) is realized inside of L − / ( osp (1 | ⊗ F : U ( n ) ∼ = U.e nα . Note that h ⊥ (0) ≡ n Id on U ( n ) . Using H. Li construction from [34] (see also [26])we get that U = M U ( n ) (5.3) is a vertex subalgebra of L − / ( osp (1 | ⊗ F . But one shows that U contains allgenerators of L − / ( osp (1 | ⊗ F , so U = L − / ( osp (1 | ⊗ F. (5.4)This proves that U = W ∼ = W ⊗ M h ⊥ (1). Since W is simple, we have that W issimple, and therefore isomorphic to W . This proves Claim 1.The decomposition (iii) follows from relations (5.2)-(5.4). The assertion (iv)follows directly from (iii). (cid:3) The proof of the following result is similar to the one given in [4, Theorem 6.2]and [10, Theorem 5.1].
Theorem 5.3.
Assume that N (resp. N tw ) is an irreducible, untwisted (resp.Ramond twisted) L − / ( osp (1 | –module such that h (0) acts semisimply on N and N tw : N = M s ∈ Z +∆ N s , h (0) | N s ≡ s Id (∆ ∈ C ) ,N tw = M s ∈ Z +∆ ′ ( N tw ) s , h (0) | ( N tw ) s ≡ s Id (∆ ′ ∈ C ) , Then N ⊗ F and N tw ⊗ M twF are completely reducible W –modules: N ⊗ F = M s ∈ Z L s ( N ) , L s ( N ) = { v ∈ N ⊗ F | h ⊥ (0) v = ( s + ∆) v } ,N tw ⊗ M twF = M s ∈ Z L s ( N tw ) , L s ( N tw ) = { v ∈ N tw ⊗ M twF | h ⊥ (0) v = ( s + ∆ ′ ) v } , and each L s ( N ) and L s ( N tw ) are irreducible W –modules. Embedding of L − / ( osp (1 | into W ⊗ F − . In this section we consider thetensor product of W with the lattice vertex superalgebra F − (defined in Section2.4). We will show that the simple affine vertex superalgebra L − / ( osp (1 | W ⊗ F − .Let h = J + ϕ . Then for n ≥ h ( n ) h = δ n, . Let M h (1) be theHeisenberg vertex algebra generated by h , and M h (1 , s ) the irreducible highestweight M h (1)–module on which h (0) ≡ s Id.
Theorem 5.4. (i) There is a homomorphism of vertex algebras Φ inv : L − / ( osp (1 | → W ⊗ F − uniquely determined by x = 12 : G + e ϕ : y = −
12 : G − e − ϕ : e = 18 : ( G + ) e ϕ : f = −
18 : ( G − ) e − ϕ : h = − J − ϕ. ERSHADSKY-POLYAKOV ALGEBRA AT POSITIVE INTEGER LEVELS 19 (ii) We have: W ⊗ F − ∼ = M n ∈ Z ρ n ( L − / ( osp (1 | ⊗ M h (1 , − n ) . (iii) L − / ( osp (1 | Com ( M h (1) , W ⊗ F − ) . Proof.
We need to show that the following relations hold for n ≥ h ( n ) x = δ n, x, e ( n ) x = 0 , x ( n ) f = δ n, y,h ( n ) y = − δ n, y, e ( n ) y = − δ n, x, f ( n ) y = 0 ,x ( n ) x = 2 δ n, e, y ( n ) y = − δ n, f, x (0) y = h, x (1) y = − ,h (0) e = 2 e, h (0) f = − f, e (0) f = h. Direct calculation shows that x (1) y = − G +2 G = − ( k + 1)(2 k + 3)4 = −
54 ( x, y ) = − .x (0) y = −
14 ( ϕ ( − G +2 G + G +1 G ) = − ( k + 1)(2 k + 3)4 ϕ − k + 1)4 J = − ϕ − J = h.x (0) x = 2 e y (0) y = − fx (0) e = 0 , y (0) f = 0(note that above we used relations ( G ± ) = 0) h (0) x = x, h (0) y = − yh (1) h = 94 J J + 254 ϕ (1) ϕ = ( 94 2 k + 33 −
254 ) = − = −
54 ( h, h ) .h (0) e = − J ( G + ) e ϕ −
516 ( G + ) ϕ (0) e ϕ = −
38 ( G + ) e ϕ + 58 ( G + ) e ϕ = 14 ( G + ) e ϕ = 2 e,h (0) f = 316 J ( G − ) e − ϕ + 516 ( G − ) ϕ (0) e − ϕ = −
38 ( G − ) e − ϕ + 58 ( G − ) e − ϕ = 14 ( G − ) e − ϕ = − f, We use the formula (cf. [9, Section 8.1]) G +2 ( G − ) n = 2 n ( k − ( n − k − ( n −
2) + n/ G − ) n − ,G +1 ( G − ) n = 3 n ( k − ( n − J − ( G − ) n − + n ( n − k − ( n − G −− ( G − ) n − . which implies that for n = 2 and k = 1 we get G +2 ( G − ) = (4 ∗ ∗ G − = 8 G − . We have x (0) f = − G +2 ( G − ) e − ϕ = −
116 8 G − e − ϕ = − G − e − ϕ = y,e (0) y = −
116 (2 G + − G +2 G − e ϕ + 2 G +0 G +1 G − e ϕ ) = −
116 (2( k + 1)(2 k + 3) G + e ϕ + 6( k + 1) G +0 J − e ϕ )= −
116 (20 G + e ϕ − G + e ϕ ) = − G + e ϕ = − x,e (0) f = −
164 (2 G +1 G +2 ( G − ) + ( G +2 ) ( G − ) ϕ ) = −
164 (16 G +1 G − + 16 G +2 G − ϕ )= −
164 (16 · k + 1) J + 16( k + 1)(2 k + 3) ϕ ) = − J − ϕ = h. The operator h (0) acts semi–simply on W ⊗ F − and we have the followingdecomposition: W ⊗ F − = M n ∈ Z Z ( n ) , Z ( n ) = { v ∈ W ⊗ F − | h (0) v = nv } . (5.5)We have that Z (0) is a simple vertex superalgebra and each Z ( n ) is a simple Z (0) –module. The rest of the proof follows from the following claim Claim 2. Z (0) = Ker W ⊗ F − h (0) is generated by { x, y, e, f, h, h } .The proof of the Claim 2 is completely analogous to the proof of Claim 1. Thesearguments are also similar to those in [4, Theorem 6.2], [6, Proposition 5.4], [17,Section 5] in a slightly different setting.The Claim 2 implies that Z (0) ∼ = L − / ( osp (1 | ⊗ M h (1) . As in the proof of Theorem 5.2, using formula∆( nϕ, z ) = ∆( − nh, z )∆( − n h, z ) , we get Z ( − n ) = ρ n ( L − / ( osp (1 | ⊗ M h (1 , − n ) . (5.6)The decomposition (ii) follows now from relations (5.5) and (5.6). The assertion(iii) is a direct consequence of (ii).This concludes the proof of the Theorem. (cid:3) From relaxed L − / ( osp (1 | –modules to ordinary W -modules The free-field realization of the affine vertex superalgebra L k ′ ( osp (1 | k ′ = − /
4, it holds that L k ′ ( osp (1 | F / ⊗ Π / (0), whereΠ / (0) is a certain lattice type vertex algebra (cf. [3, Theorem 11.3]). All irre-ducible L k ′ ( osp (1 | W -modules,using the fact that the Bershadsky-Polyakov algebra W can be embedded into L k ′ ( osp (1 | ⊗ F , where F is a Clifford vertex superalgebra (cf. Theorem 5.2), ERSHADSKY-POLYAKOV ALGEBRA AT POSITIVE INTEGER LEVELS 21
Relaxed L k ′ ( osp (1 | –modules. The vertex algebra Π / (0) was introducedin [3], where Π / (0) = M (1) ⊗ C [ Z c . It is closely related to the lattice type vertex algebra Π(0) from [13]. Here c := k ( µ − ν ), where µ, ν satisfy h µ, µ i = −h ν, ν i = k , h µ, ν i = h ν, µ i = 0. It is easy tosee that g = exp( πiµ (0)) is an automorphism of order two for Π / (0).We will need the following fact about Π / (0)-modules from [3]: Proposition 6.1. [3, Proposition 4.1]
Let λ ∈ C and g = exp( πiµ (0)) . Then Π / − ( λ ) := Π / (0) e − µ + λc is an irreducible g -twisted Π / (0) -module. Using the realization of W , we have that: Lemma 6.2.
Assume that U tw is any g –twisted Π / (0) -module. Then F ⊗ M ± F / ⊗ U tw and M twF ⊗ F / ⊗ U tw are W -modules.Proof. In [3, Corollary 13.1] it was proved that if U tw is any g –twisted Π / (0)-module, then M ± F / ⊗ U tw is an untwisted L − / ( osp (1 , F / ⊗ U tw is a Ramond twisted L − / ( osp (1 , W in Theorem 5.2. (cid:3) We will consider the following F / ⊗ Π(0)–modules: • σ F / ⊗ g –twisted module F λ := M ± F / ⊗ Π / − ( λ ). • g = 1 ⊗ g –twisted module E λ := F / ⊗ Π / − ( λ ).First we recall the result from [3]. Proposition 6.3. [3, Theorem 13.2] F λ is an untwisted, relaxed L − / ( osp (1 , –module. F λ is irreducible if and only of λ / ∈ + Z . Using irreducibility of relaxed L − / ( sl (2))–modules we get: Proposition 6.4.
We have:(1) E λ is a Ramond twisted L − / ( osp (1 , –module.(2) E λ = E λ ⊕ E λ , where as L − / ( sl (2)) –modules E λ = L V ir ( , ⊗ Π − ( λ ) M L V ir ( , ) ⊗ Π − ( λ + ) E λ = L V ir ( , ⊗ Π − ( λ + ) M L V ir ( , ) ⊗ Π − ( λ ) (3) E λ (resp. E λ ) is irreducible Ramond twisted L − / ( osp (1 , –modules if λ / ∈ Z ∪ ( − + Z ) (resp. λ + / ∈ Z ∪ ( − + Z ) ).Proof. Using Proposition 6.1 and Lemma 6.2 we see that E λ is Ramond twisted L − / ( osp (1 | L V ir ( ,
0) : F / = L V ir ( , ⊕ L V ir ( , ) , and as a Π(0)–module: Π / − ( λ ) = Π − ( λ ) ⊕ Π − ( λ + ) , we easily get the decomposition in (2). Using the irreducibility results from [3] and[30] we get that as L − / ( sl (2))–modules: • L V ir ( , ⊗ Π − ( λ ) is irreducible iff λ / ∈ − ± + Z , • L V ir ( , ⊗ Π − ( λ + ) is irreducible iff λ + ∈ − ± + Z , • L V ir ( , ) ⊗ Π − ( λ ) is irreducible iff λ / ∈ − ± + Z , • L V ir ( , ) ⊗ Π − ( λ + ) is irreducible iff λ + / ∈ − ± + Z .One easily see that all the modules appearing above are irreducible as L − / ( sl (2))–modules if and only if λ / ∈ Z . Using the decomposition L − / ( osp (1 | L − / ( sl (2)) ⊕ L − / ( ω ), we easily see that as (Ramond twisted) L − / ( osp (1 | • E λ is irreducible iff λ / ∈ Z ∪ ( − + Z ) • E λ is irreducible iff λ + / ∈ Z ∪ ( − + Z ).The proof follows. (cid:3) Explicit realization of W –modules. From Theorem 4.13 it follows thatthe set { L ( x, y ) | ( x, y ) ∈ C , h i ( x, y ) = 0 , ≤ i ≤ } is the set of all irreducible ordinary W -modules. Now we will construct explicitrealizations of these modules, using results from the previous subsection. Lemma 6.5.
The irreducible highest weight W modules T (2) := { L ( x, y ) | ( x, y ) ∈ C , h ( x, y ) = 0 } are realized as irreducible quotients of U (2) ( λ ) = W .E λ , λ ∈ C , where E λ = F ⊗ twF / ⊗ e − µ + λc are highest weight vectors for W of highest weight ( x λ , y λ ) := ( −
23 ( − k + 2 λ ) , −
14 + 13 ( − k + 2 λ ) + 13 ( − k + 2 λ )) . (6.1) Proof.
Consider the σ F / ⊗ g -twisted F ⊗ F / ⊗ Π / (0)-module F (2) ( λ ) := F ⊗ M ± F / ⊗ Π / − ( λ ). Then F (2) ( λ ) is an untwisted W -module. It holds that (cf. [3]) h ( n ) E λ = δ n, ( − k + 2 λ ) E λ ,L sug ( n ) E λ = − δ n, E λ , n ∈ Z ≥ . We have J (0) E λ = −
23 ( − k + 2 λ ) E λ ,L (0) E λ = (cid:18) −
14 + 13 ( − k + 2 λ ) + 13 ( − k + 2 λ ) (cid:19) E λ = 112 (2( − k + 2 λ ) + 3)(2( − k + 2 λ ) − E λ . Set x λ := − ( − k + 2 λ ) and y λ := (2( − k + 2 λ ) + 3)(2( − k + 2 λ ) − J (0) E λ = x λ E λ , L (0) E λ = y λ E λ . Since y = x − x − , the pair ( x λ , y λ ) ∈ C satisfies the relation h ( x, y ) = − x + 2 x + 1 + 4 y = 0 . ERSHADSKY-POLYAKOV ALGEBRA AT POSITIVE INTEGER LEVELS 23
Hence W has a family of highest weight modules U (2) ( λ ), λ ∈ C , with highestweights ( x, y ). In particular, their irreducible quotients L ( x, y ) are also modulesfor W . (cid:3) We have the following irreducibility result:
Proposition 6.6.
Assume that λ / ∈ + Z . Then F λ ⊗ F is a completely reducible W ⊗ M h ⊥ (1) –module: F λ ⊗ F ∼ = M n ∈ Z Ψ − n ( L ( x λ , y λ )) ⊗ M h ⊥ (1 , ∆ + n )(6.2) where ∆ = k − λ and weights ( x λ , y λ ) are given by (6.1). In particular, U (2) ( λ ) isan irreducible W -module and it holds that U (2) ( λ ) = L ( x λ , y λ ) . Proof.
Since F λ is an irreducible L − / ( osp (1 , λ / ∈ + Z . (cf.Proposition 6.3), applying Theorem 5.3 we see that F λ ⊗ F is a completely reducible W ⊗ M h ⊥ (1)–module: F λ ⊗ F = M n ∈ Z L n ( F λ )where L n ( F λ ) = { v ∈ F λ ⊗ F | h ⊥ (0) v = ( n + ∆) v } is an irreducible W ⊗ M h ⊥ (1)–module. By using Lemma 6.5 we see that L ( F λ )must be isomorphic to the irreducible highest weight module L ( x λ , y λ ) ⊗ M h ⊥ (1 , ∆).Since Ψ − n ( W ) ⊗ M h ⊥ (1 , n ) are simple-current W ⊗ M h ⊥ (1)–modules we get that L n ( F λ ) = (cid:0) Ψ − n ( W ) ⊗ M h ⊥ (1 , n ) (cid:1) × ( L ( x λ , y λ ) ⊗ M h ⊥ (1 , ∆))= Ψ − n ( L ( x λ , y λ )) ⊗ M h ⊥ (1 , ∆ + n )(6.3)for every n ∈ Z . (Here ” × ” denotes the fusion product in the category of W ⊗ M h ⊥ (1)–modules).The proof follows. (cid:3) Lemma 6.7. W has a family of irreducible highest weight modules T (1) := { L ( x, y ) | ( x, y ) ∈ C , h ( x, y ) = 0 } which are realized as irreducible quotients of U (1) ( λ ) = W .E λ , λ ∈ C , where E λ := twF ⊗ F / ⊗ e − µ + λc are highest weight vectors for W .Proof. Consider the σ F ⊗ g -twisted F ⊗ F / ⊗ Π / (0)-module F (1) ( λ ) := M twF ⊗ F / ⊗ Π / − ( λ ), λ ∈ C . Then F (1) ( λ ) is an untwisted W -module.Let Y twF ( ω, z ) = Y (∆( h, z ) ω, z ) and set h = α . We have∆( α , z ) ω = ω + α z − + 12 α α z − = ω + α z − + 18 z − , hence L (0) twF = 1 / twF .Similarly, ∆( α , z ) α = α + 12 α (1) α ( − z − = α + 12 z − , hence α (0) twF = 1 / twF . It holds that h ( n ) E λ = δ n, ( − k + 2 λ ) E λ ,L sug ( n ) E λ = − δ n, E λ , n ∈ Z ≥ . We have J (0) E λ = (cid:18) −
23 ( − k + 2 λ ) (cid:19) E λ ,L (0) E λ = (cid:18) −
516 + 18 + 13 ( 12 − ( − k + 2 λ )) −
512 + 13 ( − k + 2 λ ) (cid:19) E λ = 148 (4( − k + 2 λ ) − − k + 2 λ ) + 5) E λ Set x := x λ = 56 −
23 ( − k + 2 λ ) , (6.4) y := y λ = 148 (4( − k + 2 λ ) − − k + 2 λ ) + 5)(6.5)so that J (0) E λ = x λ E λ , L (0) E λ = y λ E λ . Since y = x − x , the pair ( x λ , y λ ) ∈ C satisfies the relation h ( x, y ) = − x + 5 x + 4 y = 0 . Hence W has a family of highest weight modules U (1) ( λ ), λ ∈ C , with highestweights ( x, y ). In particular, their irreducible quotients L ( x, y ) are also modulesfor W . (cid:3) Using a twisted variant of Theorem 5.3 and Proposition 6.4 we get the followingirreducibility result.
Proposition 6.8.
Assume that λ / ∈ Z ∪ ( − + Z ) . Then E λ ⊗ M twF is a completelyreducible W ⊗ M h ⊥ (1) –module: E λ ⊗ M twF ∼ = M n ∈ Z Ψ − n ( L ( x λ , y λ )) ⊗ M h ⊥ (1 , ∆ ′ + n )(6.6) where ∆ ′ = + k − λ and weights ( x λ , y λ ) are given by (6.4)- (6.5). In particular, U (1) ( λ ) is an irreducible W -module and it holds that U (1) ( λ ) = L ( x λ , y λ ) . Remark 6.9.
Theorem 4.13 implies that there exists another family of irreduciblehighest weight W -modules L ( x, y ) , for which it holds that h ( x, y ) = 0 . Indeed,these modules can be obtained from T (1) using the spectral flow automorphism ψ − as T (3) := { ψ − ( L (ˆ x , ˆ y )) | ( x, y ) ∈ C , h ( x, y ) = 0 } . From Lemma 4.4 it easily follows that T (3) = { L ( x, y ) | ( x, y ) ∈ C , h ( x, y ) = 0 } . These modules also appear in the decomoposition in Proposition 6.8.
ERSHADSKY-POLYAKOV ALGEBRA AT POSITIVE INTEGER LEVELS 25
Acknowledgment
We would like to thank D. Ridout for useful discussions. The authors are par-tially supported by the QuantiXLie Centre of Excellence, a project coffinanced bythe Croatian Government and European Union through the European RegionalDevelopment Fund - the Competitiveness and Cohesion Operational Programme(KK.01.1.1.01.0004).
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Department of Mathematics, University of Zagreb, Croatia
E-mail address: [email protected]
A. Kontrec,
Department of Mathematics, University of Zagreb, Croatia