On parafermion vertex algebras of \frak{sl}(2)_{-3/2} and \frak{sl}(3)_{-3/2}
aa r X i v : . [ m a t h . QA ] M a y ON PARAFERMION VERTEX ALGEBRAS OF sl (2) − / AND sl (3) − / DRAˇZEN ADAMOVI ´C, ANTUN MILAS, AND QING WANG
Abstract.
We study parafermion vertex algebras N − / ( sl (2)) and N − / ( sl (3)). Using theisomorphism between N − / ( sl (3)) and the logarithmic vertex algebra W (2) A from [2], we showthat these parafermion vertex algebras are infinite direct sums of irreducible modules for theZamolodchikov algebra W (2 ,
3) of central charge c = −
10, and that N − / ( sl (3)) is a direct sumof irreducible N − / ( sl (2))–modules. As a byproduct, we prove certain conjectures about thevertex algebra W ( p ) A . We also obtain a vertex-algebraic proof of the irreducibility of a familyof W (2 , c modules at c = − Introduction
Parafermion vertex algebras, or parafermionic cosets, are closely connected to the so-called Z -algebras [45, 46, 47], which played an important role in the development of the parafermion confor-mal field theory [38, 54] and in the vertex-operator-theoretic interpretation of Rogers-Ramanujantype partition identities [46, 47]. They are first defined in [26] as the subalgebras of the generalizedvertex algebras generated by Z -operators.In the past decades, the parafermion vertex operator algebras associated to rational affine vertexoperator algebras at the positive integer level were thoroughly studied and their structure is well-understood(see [16, 17, 27, 28, 31, 32, 33, 34, 43, 41, 52] etc.). At other levels, including genericlevels, their structure is largely unknown. Recently, these algebras have appeared in [23, 44] asquotients of certain universal vertex algebras constructed from non-linear conformal algebras.Among them, a very interesting problem is to determine fusion rules for parafermion vertexalgebras at rational level (cf. [18]).Our general goal is to study parafermion vertex algebra beyond rational case. It is naturalto start with certain examples when a parafermion vertex algebra belongs to a certain class of W -algebras. Let us mention that in the case of sl (2), parafermion algebra N k ( sl (2)) coincideswith singlet vertex algebra for k = − , − (cf. [53], [3]), with super-singlet (cf. [2], [18], [9] ) for k = − . In this paper, we want to explore the parafermion vertex algebra at a certain non-integraladmissible level, which belongs to the class of logarithmic vertex algebras. More specifically, westudy the following vertex algebras: Date : May 7, 2020.2010
Mathematics Subject Classification.
Primary 17B69; Secondary 17B20, 17B65.
Key words and phrases. vertex algebra, W -algebra, parafermion algebra. • The parafermion vertex algebras N k ( sl (2)) and N k ( sl (3)) at level k = − ; • Higher rank logarithmic vertex algebras W ( p ) A and W ( p ) A for p = 2; • The (universal) principal W –algebra W (2 , c = W k ( sl (3) , f pr ), also known as Zamolod-chikov’s algebra, at central charge c = − N − / ( sl (3)) ∼ = W (2) A , where W (2) A is a logarithmic vertex algebra(the so-called ”octuplet” algebra) constructed from the lattice vertex algebra V √ A , and where W (2) A is the zero charge subalgebra of W (2) A [37, 50].One of our main results is the following theorem. Theorem 1.1. (1) W (2) A and W (2) A are completely reducible N − / ( sl (2)) –modules. (2) W (2) A and W (2) A are completely reducible W (2 , c = − –modules. (3) N − / ( sl (2)) is a completely reducible W (2 , c = − –module. (4) N − / ( sl (3)) is generated by primary vectors of weights , , . Note that N − / ( sl (2)) is conformally embedded into W (2) A , which is a parafermionic analogof the conformal embedding gl (2) ֒ → sl (3) at k = − investigated in [10]. We prove in Proposition5.1 the result: • W (2) A = L s ∈ Z N − / (2 jω ) . Our next goal is to determine the decomposition of parafermion algebras as W (2 , W (2) A is known, and from its expression one can conjecture thedecomposition of W (2) A as W (2 , • It seems that the character formula for irreducible W (2 , • One needs to identify singular vectors for W (2) A and its subalgebras.Although the primary goal of the paper is not the study of the algebra W (2) A , we need to usesome elements of its representation theory to solve the above mentioned problems. It turns outthat the most efficient tool is to use sl (3)–action on W (2) A .A general Lie algebra action on W ( p ) Q was conjectured by Feigin-Tipunin [37], and recentlywas proved by S. Sugimoto [51]. In the case of the triplet vertex algebra, the sl (2)–action wasobtained in [5], [12]. So we need this action for sl (3) and p = 2. Next problem is that in general,it is still not proved that W ( p ) Q is a simple vertex algebra, so one can not prove a semi-simplicityresult by applying quantum Galois theory. But in our case we can prove that W (2) A is simple byapplying the explicit realization of L − / ( sl (3)) from [2] and using identification of the subalgebra W (2) A as a parafermion subalgebra N − / ( sl (3)) for which we know that it is simple. We get: Theorem 1.2. (1)
The vertex algebra W (2) A is simple. N PARAFERMION VERTEX ALGEBRAS 3 (2) W (2) A is a completely reducible sl (3) × W (2 , c = − –module and the following decompositionholds W (2) A = M λ ∈ P + ∩ Q V A ( λ ) ⊗ T κ =1 / λ, . (3) The W (2 , c = − –module T κ =1 / λ, is irreducible. We apply this result on the decomposition of N − / ( sl (2)) as W (2 , c = − –modules. First weshow in Proposition 5.1: • N − / ( sl (2)) = ( W (2) A ) gl (2) .As a consequence we get: Theorem 1.3. N − / ( sl (2)) = L λ ∈ P + ∩ Q T κ =1 / λ, . Acknowledgments:
This work was partially done during the visit of D.A. to Xiamen inJanuary 2019, and during the conference Representation Theory XVI in Dubrovnik in June 2019.D.A. is partially supported by the QuantiXLie Centre of Excellence, a project cofinanced bythe Croatian Government and European Union through the European Regional DevelopmentFund - the Competitiveness and Cohesion Operational Programme (KK.01.1.1.01.0004). A.M.was partially supported by the NSF Grant DMS-1601070. Q.W. is partially supported by ChinaNSF grants (Nos.11531004, 11622107). 2.
Preliminaries
Settings and known facts.
In this part we setup some notation and summarize facts weneed later. • Let g be the simple Lie algebra with Cartan subalgebra h and triangular decomposition g = n − + h + n + . • Let ˆ g be the associated affine Lie algebra, and ˆ h be the associated Heisenberg subalgebra. • Let V k ( g ) be the universal affine vertex operator algebra of level k associated to the simpleLie algebra g . • Let L k ( g ) be the simple quotient of V k ( g ). • Let N k ( g ) = { v ∈ V k ( g ) | h ( n ) v = 0 h ∈ h , n ∈ Z ≥ } be the parafermion subalgebra of V k ( g ). • Let N k ( g ) = { v ∈ L k ( g ) | h ( n ) v = 0 h ∈ h , n ∈ Z ≥ } be the parafermion subalgebra of L k ( g ). • For a λ ∈ P + , let V g ( µ ) be the irreducible finite-dimensional g –module with the highestweight λ , where P + denotes the set of dominant integral weights for g . • Let V k ( λ ) be the generalized Verma module for ˆ g –module induced from g –module V g ( λ ).Let L k ( λ ) be its simple quotient. ON PARAFERMION VERTEX ALGEBRAS • For λ, µ ∈ P + , let T k +3 λ,µ denotes the W k ( g , f pr )–module obtained as H DS ( V k ( λ − ( k + 3) µ )(cf. [15]). • For k = − p and g = sl (3), the universal affine vertex algebra V k ( sl (3)) is simple(cf. [40]), and therefore by [14] H DS ( V k ( sl (3)) = W k ( sl (3) , f pr ) is simple. In particular, W (2 , c = − is a simple vertex algebra.We shall need the following facts which are well-known. Let g = sl (2) with a Chevalley basis { e, f, h } and let k = − . Then we have: • V k ( g ) = L k ( g ). • V k ( jω ) = L k ( jω ), j ∈ Z ≥ , where ω is the fundamental dominant weight for sl (2). • N k ( j ) := N k ( jω ) = N k ( jω ), j ∈ Z ≥ , where N k ( j ) = { v ∈ V k ( jω ) | h ( n ) v = 0 , ∀ n ∈ Z ≥ } and N k ( j ) = { v ∈ L k ( jω ) | h ( n ) v = 0 , ∀ n ∈ Z ≥ } .We denote by ch[ M ]( q ) := tr M q L (0) the character of a V -module M ; from the context it shouldbe clear what the vertex algebra is. Also, for simplicity we suppressed the conformal anomaly − c .2.2. The vertex algebra W (2 , c . Let W (2 , c denotes the principal affine W –algebra W k ( sl (3) , f pr )of central charge c = c k = 2 − ( k +2) k +3 [14]. It is generated by the Virasoro field L ( z ) = P m ∈ Z L ( m ) z − m − and another field of conformal weight 3: W ( z ) = X m ∈ Z W ( m ) z − m − satisfying bracket relations[ L ( m ) , W ( n )] = (2 m − n ) W ( m + n )[ W ( m ) , W ( n )] = (22 + 5 c ) c · ·
5! ( m − m − mδ m + n, + 13 ( m − n )Λ m + n + (22 + 5 c )( m − n )48 ·
30 (2 m − mn + 2 n − L ( m + n ) , where Λ = L ( − − L ( − . Let L W ( c, h, h W ) denotes the irreducible highest weight W (2 , c –module of the highest weight ( h, h W ) with respect to ( L (0) , W (0)).Next we discuss characters of modules for the W (2 ,
3) vertex algebra at c = −
10. Out strategyis to first give an upper bound for graded dimensions of a family of irreducible W (2 , κ = 3 + k , where k is the level.Also, for λ ∈ P + , weight lattice of sl (3), we denote by V κ ( λ ) the Weyl module with the topdegree isomorphic to V sl (3) ( λ ). Then in [15], a family of modules for the universal W (2 , T κλ,µ = H DS ( V κ ( λ − κµ )). We only consider modules with µ = 0. For given centralcharacter determined by λ , we denote by β λ the W (0) eigenvalue on the highest weight vector. N PARAFERMION VERTEX ALGEBRAS 5
For two q -series f and h , we write f ( q ) ≤ h ( q ) ifCoeff q n f ( q ) ≤ Coeff q n h ( q ) , for every n . Then results from [15] give Lemma 2.1.
For every m, n ∈ Z ≥ , we have: ch[ L W ( − , m + 23 n + 23 mn + m + n, β m,n )]( q ) ≤ q m + n + mn + m + n (1 − q m +1 )(1 − q n +1 )(1 − q m + n +2 )( q ; q ) ∞ . Proof.
We apply formula (5.7) from [15] for the character formula of T κλ, , with κ = and g = sl (3).For λ = mω + nω , m, n ≥
0, their formula isch[ T κλ, ]( q ) = q ˜∆ κλ, P w ∈ W ( − ℓ ( w ) q −h w ( λ + ρ ) ,ρ i ( q ; q ) ∞ , where ˜∆ κλ, = 12 κ ( λ, λ + 2 ρ ) + ( ρ, ρ ) . Plugging in λ = mω + nω , and summing over the Weyl group W , givesch[ T κλ, ]( q ) = q m + n + mn + m + n (1 − q m +1 )(1 − q n +1 )(1 − q m + n +2 )( q ; q ) ∞ . Since T κλ, is not necessarily irreducible for κ rational (it is always irreducible for κ / ∈ Q [15]) theremight be non-trivial maximal submodule in T κλ, so we concludech[ L W ( − , m + 23 n + 23 mn + m + n, β m,n )]( q ) ≤ ch[ T κλ, ]( q )as claimed. (cid:3) The Vertex algebra W ( p ) A and its companions In this section we recall the definition of vertex algebras W ( p ) A and W ( p ) A (cf. [7], [50] )which are a higher analog of the triplet vertex algebra and singlet vertex algebra (cf. [1], [4] [5]).We shall also recall the result of [2] which identifies W ( p ) A for p = 2, as a parafermionic vertexalgebra N − / ( sl (3)).In this part we closely follow [2]. We consider the integral lattice √ pA = Z γ + Z γ , h γ , γ i = h γ , γ i = 2 p, h γ , γ i = − p, ON PARAFERMION VERTEX ALGEBRAS and the associated lattice vertex algebra V √ pA . Let M γ ,γ (1) be the Heisenberg vertex subalgebraof V √ pA generated by the Heisenberg fields γ ( z ) and γ ( z ). Let ω st = 13 p ( γ ( − + γ ( − γ ( −
1) + γ ( − )be the standard Virasoro vector in the lattice vertex algebra V √ pA of central charge 2. Define anew conformal vector ω = ω st + p − p ( γ ( −
2) + γ ( − . We equip V √ pA with the conformal structure coming from ω , which has central charge c p =2 − ( p − p , e.g. c = −
10 for p = 2.The vertex algebra W ( p ) A is defined (cf. [7], [50] ) as a subalgebra of the lattice vertex algebra V √ pA realized as W ( p ) A = Ker V √ pA e − γ /p \ Ker V √ pA e − γ /p . We also have its subalgebra: W ( p ) A = Ker M γ ,γ (1) e − γ /p \ Ker M γ ,γ (1) e − γ /p One can also construct the following extension of W ( p ) A which is a higher rank analog of thedoublet vertex algebra A ( p ) from [6].Define the lattice Γ = Z δ + Z δ ⊃ √ pA such that δ = 13 (2 γ + γ ) , δ = 13 ( γ + 2 γ ) . Clearly, Γ = √ pP , where P is a weight lattice of A . V Γ has the structure of a generalized vertexalgebra (cf. [26]) which contains the lattice vertex algebra V √ A . Note that γ , γ belongs to thedual lattice of Γ, so screening operators e − γ i /p are well defined on V Γ . Definition 3.1.
We define: Ω( p ) A = Ker V Γ e − γ /p \ Ker V Γ e − γ /p . Then Ω( p ) A is a generalized vertex algebra. It is a vertex algebra for p ≡ W (2 , c p ⊂ W ( p ) A ⊂ W ( p ) A ⊂ Ω( p ) A . N PARAFERMION VERTEX ALGEBRAS 7 W ( p ) A and W ( p ) A have vertex subalgebras isomorphic to the simple W (2 , c p which is generated by ω and w = 1 p ( γ ( − + 32 γ ( − γ ( − − γ ( − γ ( − − γ ( − ) − p − p (2 γ ( − γ ( −
2) + γ ( − γ ( − − γ ( − γ ( − − γ ( − γ ( − p − p ( γ ( − − γ ( − . The overall normalization of w is not important. For example, in order to get bracket relationsas in Section 2.2, for c = −
10, we would have to consider √ w .By direct calculation we get: Proposition 3.1.
Assume that λ = mω + nω . Let κ = p . Let v m,n = e − mδ − nδ whereThen E κ [ m, n ] := W (2 , c .v m,n is a highest weight W (2 , c = − –module with highest weight ( h ( p ) m,n , β ( p ) m,n ) where h ( p ) m,n = p m + n + mn ) + ( p − m + n )(3.1) β ( p ) m,n = ( m − n )( − p + 2 mp + np )( − p + mp + 2 np )2 p . (3.2)We also have the long screening operators E = e γ , E = e γ which commutes with both e − γ /p and e − γ /p [7, 37, 49]. Therefore operators E and E act asderivations on W ( p ) A . In particular, H := E E e − γ − γ , H := E E e − γ − γ ∈ W A ( p )and they are (non-zero) singular vectors of conformal weight 3 p − p = 2, this weight is4).Note also that for p = 2:[ E , E ] = ( e γ e γ ) = ( γ ( − e γ + γ ) = − ( γ ( − e γ + γ ) . Let N k ( g ) be the parafermion vertex subalgebra of L k ( g ). Theorem 3.1. [2]
For k = − and p = 2 we have N k ( sl (3)) = W ( p ) A . Now we shall relate N k ( sl (2)) and N k ( sl (3)). ON PARAFERMION VERTEX ALGEBRAS
Proposition 3.2.
For k = − and p = 2 we have: N k ( sl (2)) ∼ = W ( p ) A ∩ L k ( sl (2)) ⊂ N k ( sl (3)) . Theorefore, N k ( sl (2)) is a vertex subalgebra of W ( p ) A for p = 2 .Proof. In [2], we realized V √ pA inside the lattice vertex algebra V L , where L = Z α + Z β + Z δ with scalar products h α, α i = −h β, β i = h δ, δ i = 1(all other scalar products are zero).We used the following realization γ = − α, γ = α + β − δ, h = ( − β + δ )( − ∈ sl (2) . Note that(3.3) ( − β + δ, γ ) = ( − β + δ, γ ) = 0 . This easily implies N k ( sl (2)) ⊂ W ( p ) A , and that W ( p ) A ∩ L k ( sl (2)) = N k ( sl (2)) . The proof follows. (cid:3)
Proposition 3.3. N − / ( sl (2)) is conformally embedded in W (2) A .Proof. We consider conformal embeddings L − / ( gl (2)) into L − / ( sl (3)) from [10]. Then the Sug-awara Virasoro vector from L − / ( sl (3)) coincides with Sugawara Virasoro vector of L − / ( gl (2)).Since, Cartan subalgebras gl (2) and sl (3) have the same rank, we conclude that the parafermionalgebra N − / ( sl (2)) is conformally embedded in N − / ( sl (3)) = W (2) A . The proof follows. (cid:3) On the sl (3) –action on W ( p ) A and its applications A geometric proof of the Lie algebra action on W ( p ) Q is given by S. Sugimoto in [51]. In thissection we shall explore this action in the case Q = A , p = 2. We shall prove that W (2) A is asimple vertex algebra and by applying quantum Galois theory we shall prove semi-simplicity of W (2) A as sl (3) × W (2 , c = − –module. We shall also reconstruct explicit formulas for g = sl (3)–action.Our method is based on the following facts:(1) As in [51], the borel subalgebra action is given by the screening operators E i = e γ i , i = 1 , F i such that F i , E i give sl (2)-actions on W (2) A .(3) Using the realization from [2], and identification of W (2) A as a subalgebra of the vacuumspace of L − / ( sl (3)), we construct an automorphism Ψ acting on W (2) A such that F i = − Ψ E i Ψ − . N PARAFERMION VERTEX ALGEBRAS 9
In Section 5 we will use these explicit formulas to describe N − / ( sl (2) as gl (2) invariants of W (2) A .Recall that the vacuum space is defined asΩ k ( g ) = { v ∈ L k ( g ) | ( h ⊗ t C [ t ]) .v = 0 } . and it is a generalized vertex algebra [26]. Moreover, N k ( g ) ⊂ Ω k ( g ) . By using the realization from [2], we have that Ω − / ( g ) is a generalized vertex algebra whichcontains vertex subalgebra W ( p ) A for p = 2.We shall now use results of [8] to construct some derivations of W ( p ) A for p = 2. Let a = e − γ , a = e − γ . For i = 1 ,
2, we define F i = ∞ X j ∈ Z ,j =0 j : a i − j a ij : , F twi = ∞ X j ∈
12 + Z j : a i − j a ij : . Lemma 4.1.
For i = 1 , , we have: (1) F i is a derivation on any F i –invariant vertex subalgebra V ⊂ Ker a i . (2) F i is a derivation on Ω − / ( sl (3)) and W (2) A . (3) E i , F i , h i := γ i (0) generate an sl (2) –action on Ω − / ( sl (3)) and W (2) A .Proof. Assertion (1) follows directly from [8].Let V = Ω − / ( sl (3)) or V = W A (2). We claim that on V we have a F = F tw a (4.4) a F = F tw a . (4.5)Then assertion (2) would follow directly from relations (4.4)-(4.5).It remains to prove these relations. Let us prove (4.4). Let v ∈ V . Note that a belongs to atwisted V L –module, and now [8] implies Y ( F v, z ) a = [ F tw , Y ( v, z )] a . Since F tw a = 0, we get Y ( F v, z ) a = F tw Y ( v, z ) a . Now skew-symmetry we get a F v = Res z Y ( a , z ) F v = Res z e − zL ( − Y ( F v, − z ) a = Res z e − zL ( − F tw Y ( v, − z ) a = F tw Res z e − zL ( − Y ( v, − z ) a = F tw a v This proves (4.4). The proof of (4.5) is analogous.
The assertion (3) follows from a direct calculation as in [35, Section 4.1]. (cid:3)
Now we can reconstruct the sl (3) action on W (2) A (obtained in [51] by slightly differentmethods). We skip details. • Let Ψ be automorphism of the VOA L k ( sl (3)) lifted from the automorphism α
7→ − α , α
7→ − α of the root lattice A . • Since Ω k ( sl (3)) is Ψ–invariant, we conclude that Ψ = Ψ | Ω − / ( sl (3)) is an automorphismof the generalized vertex algebra Ω k ( sl (3)) for every k . In particular, Ψ is an automorphismof Ω − / ( sl (3) and of W (2) A . • Using realization, we show that F i = Ψ − E i Ψ and F , = Ψ − E , Ψ, where E , = [ E , E ]and F , = [ F , F ]. • Since Ψ( ω ) = ω and Ψ( w ) = − w , we see that all derivations above fix W (2 , c = − .In this way we get an alternative proof of the Sugimoto theorem for Q = A , p = 2: Theorem 4.1. [51]
The Lie algebra g = sl (3) acts on Ω − / ( sl (3)) and on W (2) A by derivations.Moreover, W (2 , c = − ⊂ W (2) g A . We have the following important consequence:
Corollary 4.1.
The group
P SL (3 , C ) and the compact Lie group P SU (3) act on W (2) A asautomorphism groups. The group action commutes with the action of the vertex algebra W (2 , c . We first need the following simplicity result.
Lemma 4.2. (1)
The vertex algebra W (2) A is simple. (2) The generalized vertex algebra Ω − / ( sl (3)) is simple.Proof. We have decomposition of W (2) A = ⊕ α ∈ A W (2) ( α ) A with respect to gradation in the rootlattice A . From [2] we know that W (2) (0) A = N − / ( sl (3)) is a simple VOA (as the parafermionicalgebra of L − / ( sl )) and all W (2) ( α ) A are also irreducible again, because they are realized asparafermionic modules (see [22] for detailed analysis of Heisenberg cosets). Moreover, since each W (2) ( α ) A belongs to a simple VOA L − / ( sl ), we conclude that W (2) ( α ) A ·W (2) ( β ) A = 0 and therefore W (2) ( α ) A · W (2) ( β ) A = W (2) ( α + β ) A . This proves the simplicity of W (2) A . N PARAFERMION VERTEX ALGEBRAS 11
The proof of simplicity of Ω − / is analogous. We have Ω − / ( sl (3)) = ⊕ µ ∈ P Ω ( µ ) − / with respectto gradation of the weight lattice of A . Modules Ω ( µ ) − / are simple W (0) (2) A –modules since theyare realized inside of L − / ( sl ) as irreducible modules for the parafermionic algebra. This easilyproves that Ω − / is simple. (cid:3) Remark 4.1.
A different proof of simplicity of Ω − / ( sl (3)) can be given using more generalresults of [48] . It is not hard to see that W (2) A is a Z -orbifold of Ω − / ( sl (3)) , implying also analternative proof of simplicity of W (2) A . Now by applying quantum Galois theory (cf. [30], [25], [29]) we get:
Proposition 4.1.
Let g = s l (3) . The vertex algebra W (2) A is a completely reducible g ×W (2) g A –module and W (2) A = M n,m ≥ ,n ≡ m mod (3) V A ( mω + nω ) ⊗ L [ m, n ] where L [ m, n ] is certain irreducible W (2) g A –module.Proof. Using [30], and more precisely [25, Remark 2.3], and Lemma 4.2, we first get a decompo-sition of W (2) A as an g × W (2) g A -module: M λ ∈ P ′ V sl (3) ( λ ) ⊗ L λ where P ′ is a set that contains P + ∩ Q and L λ are irreducible W (2) g A -modules. From the definitionof the g –action on W (2) A we easily see that any weight in P ′ is necessarily inside λ ∈ P + ∩ Q .We conclude that W (2) A decomposes as M λ ∈ P + ∩ Q V g ( nω + mω ) ⊗ L [ m, n ] . The proof follows (cid:3) W (2) A and N − / ( sl (2)) as invariant subalgebras In this section we shall identify W (2) A as the parafermion subalgebra of the N = 4 super-conformal vertex algebra L N =4 c = − = V (2) (cf. [2]). By using identification of L − / ( sl (2)) as the sl (2) invariant subalgebra of V (2) we shall prove that the parafermion algebra N − / ( sl (2)) is the gl (2)–invariant subalgebra of W (2) A .We shall consider two subalgebras of the Lie algebra g = sl (3) for which acts on W (2) A byderivations.Let g = span C { E , F , h } ⊂ g and g = g + C h ⊂ g . Clearly, g ∼ = sl (2) and g ∼ = gl (2). Note that g acts on the N = 4 superconformal vertex algebra V (2) (cf. [2], [11]) and L − / ( sl (2)) = (cid:0) V (2) (cid:1) g .The vertex superalgebra V (2) admits a Z –gradation: V (2) = M ℓ ∈ Z V (2) ℓ , such that V (2)0 is its subalgebra which decomposes as follows: V (2)0 = ∞ M s =0 L sl (2) (2 sω ) . (5.6)Let M h (1) is the Heisenberg vertex algebra generated by h . Proposition 5.1.
We have: (1) W (2) A = Com ( M h (1) , W ) , and the following decomposition holds W (2) A = ∞ M s =0 N sl (2) (2 s ) . (2) N − / ( sl (2)) ∼ = W (2) g A . Proof.
In the realization presented in [2], L − / ( sl (3)) is realised as a subalgebra of tensor product V (2) ⊗ F − , where where F − is a lattice vertex algebra. Moreover, W (2) A = N − / ( sl (3)) isexactly the subalgebra of V (2)0 on which h ( n ), n ≥
0, act trivially. Using (5.6) we get W (2) A = ∞ M s =0 N sl (2) (2 s ) . This proves assertion (1).Consider next the parafermion vertex algebra of V (2) U (2) = Com( M h (1) , V (2) ) = { v ∈ V (2) | h ( n ) v = 0 , n ≥ } ⊂ Ω − / . Since the action of g commutes with operators h ( n ) we conclude that g acts on U (2) and wehave the following decomposition of U (2) as a g × N − / ( sl (2))–module U (2) = M j ∈ Z ≥ ρ j ⊗ N − / (2 j )where ρ j denotes the irreducible j + 1–dimensional g –module. Moreover, we have N − / ( sl (2)) = (cid:16) U (2) (cid:17) g ⊂ W A (2) g . Since W (2) g A ⊂ W (2) A = Com( M h (1) , V (2)0 ) ⊂ U (2) we have N − / ( sl (2)) ∼ = W (2) g A . The proof follows. (cid:3) N PARAFERMION VERTEX ALGEBRAS 13 The character of the parafermion vertex algebra N k ( sl (2))Let ( x ; q ) n = Q n − i =0 (1 − xq i ) and ( q ) n = ( q ; q ) n . For m ≤
0, letΦ m ( q ) = ∞ X r =0 ( − r q r ( r +1)2 + mr , and for m >
0, Φ m ( q ) = Φ − m ( q ) denote unary false theta functions. By an identity of Andrews[13] (see also [19, 21]), the character of N k ( sl (2)) can be computed asch[ N k ( sl (2))]( q ) = CT x xq ; q ) ∞ ( x − q ; q ) ∞ = Φ ( q ) − Φ − ( q )( q ) ∞ , where CT x denotes the constant term inside the range | q | < | x | < q –character for the vertex algebra N k ( sl (2)) for k = − . Although thesame formula is valid for any generic level, this specific shape is convenient for k = . Lemma 6.1.
We have: ch[ N k ( sl (2))]( q ) = P ∞ m =1 q m ( m − (1 − q m )(1 − q m )(1 − q m )( q ) ∞ . Proof.
We have: ∞ X m =1 q m ( m − (1 − q m )(1 − q m )(1 − q m )= ∞ X m =1 q m ( m − (1 − q m + 2 q m − q m )= ∞ X m =1 ( q m ( m − − q m (2 m − + 2 q m (2 m +1) − q m ( m +1) )= ∞ X m =1 ( q m ( m − − q m ( m +1) ) + ∞ X m =1 ( − q m (2 m − + 2 q m (2 m +1) )= 1 − ∞ X m =1 ( q m (2 m − − q m (2 m +1) )= 1 + 2 ∞ X i =1 ( − i q i ( i +1)2 = ∞ X i =0 ( − i q i ( i +1)2 − ∞ X i =0 ( − i q i ( i +1)2 − i = Φ ( q ) − Φ − ( q ) . The proof follows. (cid:3)
Lemma 6.2.
We have: ch[ N k (2 s )]( q ) = q s ( s +1) CT ( x − s + · · · + 1 + · · · + x s )( xq ; q ) ∞ ( x − q ; q ) ∞ = q s ( s +1) (Φ ( q ) + Φ − ( q ) − − s − ( q ))( q ; q ) ∞ . The following lemma can be viewed as a generalization of Lemma 6.1.
Proposition 6.1.
We let F m,n = q m + n + mn − m − n (1 − q m )(1 − q n )(1 − q m + n ) . Then for every s ≥ , we have X m ≥ s +1 F m,m + X ≤ i ≤ s,m ≥ i ( F m,m +3( s +1 − i ) + F m +3( s +1 − i ) ,m ) | {z } := G s ( q ) = ( q ; q ) ∞ ch[ N k (2 s )]( q ) . Proof.
Follows by direct computation by induction. For s = 0 this is the statement of Lemma6.1. For the induction step we observe (for s ≥ q ; q ) ∞ CT (cid:0) x − s − + · · · + 1 + · · · + x s +1 (cid:1) ( xq ; q ) ∞ ( x − q ; q ) ∞ = ( q ; q ) ∞ CT ( x − s + · · · + 1 + · · · + x s )( xq ; q ) ∞ ( x − q ; q ) ∞ + 2(Φ − s − ( q ) − Φ − s − ( q )) . The rest follows via manipulation with q -series as in Lemma 6.1. (cid:3) Proposition 6.2. X m,n ≥ ,m ≡ n (3) min( m, n ) q m + n + mn − m − n (1 − q m )(1 − q n )(1 − q m + n ) = X s ≥ G s . Proof.
Directly from definition of G s and observation that every F m,n appears exactly min( m, n )-times inside G s as a summand.The formula also follows from Proposition 5.1. (cid:3) We also record a q -hypergeometric expression for the character. Proposition 6.3.
We have ch[ N k ( sl (2))]( q ) = X n ≥ q n ( q ) n . where ( q ) n = Q ni ≥ (1 − q i ) . N PARAFERMION VERTEX ALGEBRAS 15
Proof.
It follows from Euler’s identity 1( x ; q ) ∞ = X n ≥ q n x n ( q ) n and the fact that ch[ N k ( sl (2))]( q ) is the constant term of x ; q ) ∞ ( x − ; q ) ∞ . (cid:3) Lemma 6.1 and Lemma 2.1 suggest the following result whose proof is postponed for Section10.
Theorem 6.1.
Let k = − . As a W (2 , c = − –module, we have N k ( sl (2)) ∼ = ∞ M m =1 L W ( − , m ( m − , . (6.7) 7. The Character of W (2) A In this section we discuss two formulas for the character of W (2) A from [21] (see also [20]).Using the realization of W (2) A from [2], the character of W (2) A can be computed in anelegant form. Theorem 7.1. [21]
We have ch[ W (2) A ]( q )= X m,n ≥ m = n ( mod min( m + 1 , n + 1) q m + mn + n + m + n (1 − q m +1 )(1 − q n +1 )(1 − q m + n +2 )( q ) ∞ . Observe that if we sum over the subset m = n ∈ N above, we obtain the character of N k ( sl (2))as a summand. This important observation will be explained in Section 8.Results from Section 5 can be used to give a new representation-theoretic proof of the followingresult from [21]. Proposition 7.1. ch[ W (2) A ]( q ) = P n ≥ ,n ∈ Z sgn( n )( − n q n n + n n +2 n +2 n ( q ; q ) ∞ , where sgn( n ) = 1 , n ≥ and − for n < .Proof. Using Proposition 5.1 we get: W (2) A = ∞ M s =0 N − / (2 s ) . Applying the character formulas for the relevant d sl (2)-modules, we can writech[ W A (2)]( q ) = CT x P s ≥ x s +1 / − x − s − / x / − x − / q s ( s +1) ( xq ; q ) ∞ ( x − q ; q ) ∞ . The rest follows simply by extracting the constant term, using ( m ∈ Z )Coeff x m X s ≥ x s +1 / − x − s − / x / − x − / q s ( s +1) = X s ≥| m | q s ( s +1) and ( m ≥
0) Coeff x m xq ; q ) ∞ ( x − q ; q ) ∞ = 1( q ; q ) ∞ (Φ − m ( q ) − Φ − m − ( q )) , discussed in Section 6. Finally, we have to split the numerator in the character formula as X n ,n ≥ ( − n q n n + n n +2 n +2 n − X n ,n ≥ ( − n q n n − n ( n +1)+2 n +2 n . (cid:3) The decomposition of W (2) A as W (2 , c –module We also require next computational lemma.
Lemma 8.1.
For every m, n ≥ such that mω + nω ∈ P + ∩ Q , v m,n = e − mω − nω ∈ W (2) A .Also, L (0) · v m,n = h m,n v m,n , W (0) · v m,n = β m,n v m,n , where h m,n := h (2) m,n = m + n + mn + m + n , β m,n := β (2) m,n = ( m − n )(3+4 m +2 n )(3+2 m +4 n ) .Proof. From the definition, v m,n = e − mγ − nγ . Under the imposed condition on m and n , thesevectors are annihilated by the screening operators e − γ and e − γ and therefore v m,n ∈ W (2) A .Computation of highest weights follows from Proposition 3.1. (cid:3) Theorem 8.1.
Let g = sl (3) , we have: (1) W (2) g A ∼ = W (2 , c = − , (2) W (2) A ∼ = M n,m ≥ ,n ≡ m ( mod V sl (3) ( nω + mω ) ⊗ L W ( − , h m,n , β m,n ) , (3)(8.8) W (2) A ∼ = M n,m ≥ ,n ≡ m ( mod min( m + 1 , n + 1) L W ( − , h m,n , β m,n ) . N PARAFERMION VERTEX ALGEBRAS 17
Proof.
First we notice that W (2) g A ⊂ W (2) A , and therefore W (2) A is a completely reducible W (2) g A –module. The contribution of an irreducible sl (3)-module in W (2) A is controlled by theweight zero subspace whose dimension is given bydim( V sl (3) ( mω + nω ) ) = min( m + 1 , n + 1) , mω + nω ∈ Q ∩ P + . Thus we have W (2) A = M n,m ≥ ,n ≡ m mod (3) min( m + 1 , n + 1) L [ m, n ] . (8.9)From Theorem 7.1, we getch[ W (2) A ]( q ) = X n,m ≥ ,n ≡ m ( mod min( m + 1 , n + 1)ch[ T κmω + nω , ]( q ) , implying that ch[ T κmω + nω , ]( q ) is the q –character of certain irreducible W (2) g A –module. Since • W (2 , c = − ⊂ W (2) g A , • ch[ W (2 , c = − ]( q ) = ch[ T κ , ]( q ) < ch[ W (2) g A ]( q ) , we conclude that W (2) g A ∼ = W (2 , c = − . This proves the assertion (1). Therefore all L [ m, n ]are irreducible W (2 , c = − –modules. Lemma 8.1 implies that L [ m, n ] = L W ( − , h m,n , β m,n ) . Then assertion (2) follows from Proposition 4.1, and assertion (3) from the decomposition (8.9). (cid:3)
From the previous result we also easily get a proof of Theorem 6.1. Also, as a consequence, weobtain character formulas for irreducible W (2 , Corollary 8.1.
For m, n ≥ , we have ch[ L W ( − , m + 23 mn + 23 n + m + n, β m,n )]( q )= q m + mn + n + m + n (1 − q m +1 )(1 − q n +1 )(1 − q m + n +2 )( q ; q ) ∞ . Proof.
For m = n mod 3 this follows directly from Theorem 8.1 and Lemma 2.1. For m = n mod 3, this is a consequence of the same analysis applied to W (2) A -modules. We omit detailshere. (cid:3) Remark 8.1.
We should point out that irreducible W (2 , -modules and their characters werestudied in an old work of Koos and Driel [42] . In particular, they analyzed irreducible repre-sentations with central charge c = 50 − p − p , for p ∈ N , parametrized by dominant integralweights. For p = 1 (i.e. c = 2 ), the structure of such representations is well-known due to unitarityof the bosonic construction. For p ≥ , they proposed explicit formulas for characters based onsummation over the finite Weyl group, subject to conjectural embedding formulas among Vermamodules controlled by certain double cosets in the affine Weyl group of sl (3) . These embeddingsformulas are recently (rigorously) proven by by Dhillon [36] . His result supposedly clarifies thecharacter formulas used in [42] (and also in [37] for other higher rank algebras). However, for p = 2 (i.e. c = − ) the embedding structure among Verma modules [42, Table VIII] is differentcompared to p = 3 (i.e. c = − ); specifically [42, Table IX] . Even though Table VIII yieldscorrect formulas for ch[ L W ( − , , q ) and ch[ L W ( − , , q ) , we believe that our Corollary8.1 gives the first rigorous derivations of characters in all cases. The generators of W (2) A Theorem 9.1.
We have: (1)
The vertex algebra W (2) A is generated by N − / ( sl (2)) + N − / (2) . (2) The vertex algebra W (2) A is a simple W -algebra generated by the conformal vector and threeprimaries of weight , and .Proof. Consider again the subalgebra V (2)0 = ∞ M ℓ =0 W ℓ , W ℓ = L − / (2 ℓω )of the N = 4 superconformal algebra with central charge c = −
9. Using the sl (2) action, quantumGalois theory, and the same arguments as in the [25, Lemma 2.6], we get W · W ℓ = W ℓ +2 + W ℓ − ( ℓ ≥ , where the dot product is defined U · V = span { u m v : u ∈ U, v ∈ V, m ∈ Z } . But restriction to the parafermion algebra gives that N − / (2) · N − / (2 ℓ ) = N − / (2 ℓ + 2) + N − / (2 ℓ −
2) ( ℓ ≥ . This proves assertion (1).(2) By Part (1), W (2) A is generated by primary vectors of degree 2 , , , ,
5. However it iseasy to see that the weight 5 primary vector can be expressed using other primaries. (cid:3)
Remark 9.1.
We expect that W (2) A is of type (2 , , ) . N PARAFERMION VERTEX ALGEBRAS 19
The decomposition of N − / ( sl (2)) as a W (2 , c = − –module We also need the following result which easily follows from [39, Chapter 8].
Lemma 10.1.
For every m, n ∈ Z ≥ , we have dim V sl (3) ( nω + mω ) g = δ n,m . Theorem 10.1.
Let k = − . As a W (2 , c = − –module, we have N k ( sl (2)) ∼ = ∞ M m =0 L W ( − , m ( m + 1) , . (10.10) Proof.
Using Proposition 5.1, Lemma 10.1 and Theorem 8.1 we get: N − / ( sl (2)) = W A (2) g = M n,m ≥ ,n ≡ m ( mod V sl (3) ( nω + mω ) g ⊗ L W ( − , h m,n , β m,n )= M m ≥ L W ( − , h m,m , β m,m )= ∞ M m =0 L W ( − , m ( m + 1) , . The proof follows. (cid:3)
Remark 10.1.
One can also give decompositions of N − / ( sl (2)) -modules N − / (2 ℓ ) , ℓ ≥ . Therelevant irreducible W (2 , c = − -modules can be read off from Proposition 6.1. Remark 10.2.
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