On Zhu's algebra and C 2 --algebra for symplectic fermion vertex algebra SF(d ) +
OOn Zhu’s algebra and C –algebra for symplecticfermion vertex algebra SF ( d ) +Draˇzen Adamovi´c and Ante ˇCeperi´cMay 29, 2020 Abstract
In this paper, we study the family of vertex operator algebras SF ( d ) + ,known as symplectic fermions. This family is of a particular interestbecause these VOAs are irrational and C -cofinite. We determine theZhu’s algebra A ( SF ( d ) + ) and show that the equality of dimensions of A ( SF ( d ) + ) and the C –algebra P ( SF ( d ) + ) holds for d ≥ d = 1 was treated by T. Abe in [Abe07]). We use these results to provea conjecture by Y. Arike and K. Nagatomo ([AN13]) on the dimension ofthe space of one-point functions on SF ( d ) + . Contents SF ( d ) . . . . . . . . . . . . . 73.3 Representations of SF ( d ) + . . . . . . . . . . . . . . . . . . . . . 73.3.1 SF ( d ) + θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3.2 SF ( d ) − θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3.3 SF ( d ) + . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3.4 SF ( d ) − . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3.5 (cid:100) SF ( d ) + . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3.6 (cid:100) SF ( d ) − . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Epimorphism π : A ( SF ( d ) + ) → A d . . . . . . . . . . . . . . . . . 9 P ( SF ( d ) + ) P ( SF ( d ) + ) . . . . . . . . . . . . . . . . . . . . 114.2 Monomials of length 2 . . . . . . . . . . . . . . . . . . . . . . . . 134.3 Monomials of length greater than 2 . . . . . . . . . . . . . . . . . 154.4 Proof of the Technical Lemma . . . . . . . . . . . . . . . . . . . . 16 SF ( d ) + . . . . 195.2 Further properties of A ( SF ( d ) + ) and P ( SF ( d ) + ) . . . . . . . . . 201 a r X i v : . [ m a t h . QA ] M a y Introduction
Symplectic fermions appeared first in physics literature in the papers by H. G.Kausch [Ka00], H. G. Kausch and M. Gaberdiel [GK96], [GK99] and in themathematical literature in the paper by T. Abe [Abe07]. T. Abe proved thatsymplectic fermion VOAs SF ( d ) + ( d ≥
1) form a family of irrational C -cofiniteVOAs at central charge c = − d . Orbifolds of symplectic fermions are studiedin detail by T. Creutzig and A. Linshaw [CL17]. Some CFT aspects of thetheory of symplectic fermions, including connection with quantum groups, wereinvestigated in [DR17], [GR17], [R14].They are connected to another family of irrational C -cofinite VOAs, thefamily of triplet algebras W ( p ) ( p ≥
2) (introduced in [AM08] by D. Adamovi´cand A. Milas) by SF (1) + (cid:39) W (2) . T. Abe in [Abe07] proved the following properties of VOA SF ( d ) + : • SF ( d ) + is a C –cofinite irrational VOA; • SF ( d ) + has four irreducible modules, and it contains logarithmic modules; • A ( SF (1) + ) (cid:39) M ( C ) ⊕ M ( C ) ⊕ C [ x ] / ( x ) ⊕ C ; • dim A ( SF (1) + ) = dim P ( SF (1) + ) = 11.Here, A ( V ) denotes the Zhu’s algebra of a VOA V , and P ( V ) denotes the C –algebra (also known as the Poisson algebra) of V , both introduced in [Zhu96]by Y. Zhu.Although the paper [Abe07] contains many general structural results onZhu’s algebra A ( SF ( d ) + ), it does not contain a complete description of A ( SF ( d ) + )for general rank d . The main goal of our paper is to completely determine theZhu’s algebra A ( SF ( d ) + ) for general rank d .Our method of proof will use the well known fact that for a general VOA V we have dim C A ( V ) ≤ dim C P ( V ) . (1.1)We will show that for SF ( d ) + , the equality of dimensions in (1.1) holds.The general problem of determining for which VOAs the equality in (1.1) holdswas posed and explored by M. Gaberdiel and T. Gannon in [GG09].This problem was solved for the family of triplet algebras W ( p ) in [AM11],and considered for some subalgebras of W ( p ) in [ALM13] and [ALM14]. Itis also important to mention papers [FL10] and [FFL11], which consider thisproblem for some rational VOAs.To expand on our method, we need some notation: A d := M d ( C ) ⊕ M d ( C ) ⊕ Λ even ( V d ) ⊕ C n d := dim C A d = 2 d − + 8 d + 1 , where V d stands for the standard (2 d -dimensional) representation of the Liealgebra sp (2 d ). The main result of this article is: Theorem 1.1.
We have A ( SF ( d ) + ) (cid:39) A d and dim C A ( SF ( d ) + ) = dim C P ( SF ( d ) + ) = n d . SF ( d ) + , obtainthe epimorphism of algebras π : A ( SF ( d ) + ) → A d .2. Find the spanning set of P ( SF ( d ) + ) of cardinality n d .These two steps (together with the inequality (1.1)) give us: n d ≤ dim C A ( SF ( d ) + ) ≤ dim C P ( SF ( d ) + ) ≤ n d , from which we conclude the equality of dimensions together with the fact thatthe epimorphism π is actually an isomorphism, giving us the full description ofthe Zhu’s algebra A ( SF ( d ) + ) for general rank d .We also explore some consequences of having the full description of A ( SF ( d ) + ): • We prove the Arike-Nagatomo conjecture ([AN13]) about the dimensionof a vector space of the one-point functions on SF ( d ) + . • We show that the center of A ( SF ( d ) + ) is isomorphic to Λ even ( V d ) ⊕ C ⊕ C ⊕ C (cf. Corollary 5.3). • We prove that the invariant subalgebra A ( SF ( d ) + ) sp (2 d ) is generated onlyby [ ω ] (cf. Proposition 5.5). This is quite surprising, since by [KL19], theinvariant vertex algebra SF ( d ) sp (2 d ) is isomorphic to W − d − / ( sp (2 n ) , f prin ),and its structure is therefore more complicated. Future work
Symplectic fermions belong to a class of vertex operator algebras related to thehigher rank logarithmic CFT. Other higher rank analogues of the triplet algebrasare vertex algebras W ( p ) Q introduced in [FT10] and studied in [AM14], [S20].We believe that for W ( p ) Q we have the equality of dimensions of Zhu’s algebraand C –algebra.We should mention that it is not easy to find examples of logarithmic VOAswhen dim C A ( V ) < dim C P ( V ). It seems that such example is c = 0 tripletvertex algebra (cf. [AM10] and [AM11, Remark 4]), but this is still a conjecture.We hope to study this problem in our forthcoming publications.3 Main definitions
We recall the definitions of Zhu’s algebra and C –algebra of a vertex operatoralgebra ( V, Y, , ω ) (following [Zhu96]). We will write the mode expansion of avertex operator associated to a ∈ V as Y ( a, z ) = (cid:88) n ∈ Z a n z − n − . As usual, for the Virasoro element ω we will write Y ( ω, z ) = (cid:88) n ∈ Z ω n z − n − = (cid:88) n ∈ Z L ( n ) z − n − . Per definition of a vertex operator algebra, we have a L (0)-grading V = (cid:77) n ∈ Z ≥ V n . We will refer to a ∈ V n as homogeneous elements and write deg a = n .To give a definition of the Zhu’s algebra, we define two bilinear maps ∗ : V × V → V, ◦ : V × V → V in the following way: for homogeneous a, b ∈ V , weput a ∗ b = Res z (1 + z ) deg a z Y ( a, z ) b = (cid:88) k ≥ (cid:18) deg ak (cid:19) a − k − ba ◦ b = Res z (1 + z ) deg a z Y ( a, z ) b = (cid:88) k ≥ (cid:18) deg ak (cid:19) a − k − b. We can extend these operations bilinearly to V , and denote by O ( V ) ⊆ V thelinear span of elements of the form a ◦ b . Denote A ( V ) := V /O ( V ). By [Zhu96],this space has a unital associative algebra structure (with multiplication inducedby ∗ and the unit 1 + O ( V )). This algebra is called the Zhu’s algebra of V andwe will write [ a ] = a + O ( V ) . It is well known that we can equip A ( V ) with an increasing filtration F n A ( V ) = (cid:32) n (cid:77) k =0 V k (cid:33) ⊕ O ( V )that gives A ( V ) the filtered algebra structure.For C –algebra, we let C ( V ) = span C { a − b : a, b ∈ V } , P ( V ) = V /C ( V ) . We will denote a = a + C ( V ). By [Zhu96], this quotient space P ( V ) has astructure of a graded commutative Poisson algebra with a · b = a − b { a, b } = a b, L (0)-grading. It is well known (see Chapter 2 of[Abe07]) that we have a natural epimorphism of unital algebrasgr A ( V ) → P ( V ) , (2.1)which gives us the inequality of dimensionsdim C A ( V ) ≥ dim C P ( V ) . (2.2)It is also important to mention the connection of V -modules to the A ( V )-modules. Let M be a weak V -module and letΩ( M ) := { m ∈ M : a n m = 0 for a ∈ V k , n > k − . } For the admissible and simple M , one can show that Ω( M ) is exactly thetop component of M (see Proposition 3.4. in [DLM98]). Proposition 2.1. [Zhu96] Let V be a VOA.1. Let M be a weak V -module. The linear map o : V → End(Ω( M )) , o ( a ) = a deg a − induces a representation of A ( V ) on Ω( M ) .2. If M is an irreducible V -module, then Ω( M ) is irreducible A ( V ) -module.3. The map M → Ω( M ) is a bijection from the set of inequivalent simpleadmissible V -modules to the set of inequivalent simple A ( V ) -modules. VOA of symplectic fermions
The VOA of symplectic fermions was defined in [Abe07], and we follow thenotation given there. First, one considers the symplectic vector space h ofdimension 2 d with the canonical basis e , e , . . . , e d , f , f , . . . , f d such that (cid:104) e i , e j (cid:105) = (cid:104) f i , f j (cid:105) = 0 , and (cid:104) e i , f j (cid:105) = − δ i,j . The subgroup of GL ( h ) preserving the symplectic form is the symplectic group Sp (2 d ). One can construct a Lie superalgebra and a vertex operator superalge-bra from h in a manner that is similar to the construction of Heisenberg VOA.We denote this VOSA SF ( d ). The easiest way to describe it is by saying that it isstrongly and freely generated by the odd generators e , e , . . . , e d , f , f , . . . , f d ,with the λ -bracket [ e iλ e j ] = [ f iλ f j ] = 0 , and [ e iλ f j ] = − λδ i,j and the Virasoro vector given by ω = (cid:80) di =1 e i − f i . One can easily see that wehave L (0) e i = e i , L (0) f i = f i , and that we can identify SF ( d ) with h .Now, the VOA of symplectic fermions SF ( d ) + is the even part of the VOSA SF ( d ). In [Abe07] it is shown that both of these algebras are simple and C -cofinite, and that the following vectors (for 1 ≤ i, j ≤ d ): e ij = e i − e j ,f ij = f i − f j ,h ij = e i − f j ,E ij = 12 (cid:16) e i − e j + e j − e i (cid:17) ,F ij = 12 (cid:16) f i − f j + f j − f i (cid:17) ,H ij = 12 (cid:16) e i − f j + f j − e i (cid:17) strongly generate SF ( d ) + . Notice that we have e ij = − e ji , f ij = − f ji ,E ij = E ji , F ij = − F ji . In context of our work, it is also important to mention the following result.
Theorem 3.1 ([Abe07], [AM11]) . We have A ( SF (1) + ) (cid:39) M ( C ) ⊕ M ( C ) ⊕ C [ x ] / ( x ) ⊕ C . Moreover, dim A ( SF (1) + ) = dim P ( SF (1) + ) = 11 . To compare with Theorem 1.1, notice that for two-dimensional V we haveΛ even ( V ) (cid:39) C [ x ] / ( x ) . .2 Automorphisms and derivations of SF ( d ) In this subsection we recall Abe’s results on the automorphisms of SF ( d ). De-note by θ the parity operator on SF ( d ) (the operator acting as ± SF ( d ) ± ). Theorem 3.2 ([Abe07]) . The automorphism groups of SF ( d ) and SF ( d ) + are Sp (2 d, C ) and Sp (2 d, C ) / (cid:104) θ (cid:105) , respectively. The important automorphisms to mention are the permutation automor-phisms e i (cid:55)→ e σ ( i ) , f i (cid:55)→ f σ ( i ) , for σ ∈ S d ( S d is here denoting the symmetric group on d letters) and automorphisms τ i , i = 1 , . . . , d defined by τ i ( e i ) = − f i , τ i ( f i ) = e i , τ i ( e j ) = e j , τ i ( f j ) = f j , for j = 1 , . . . , d, j (cid:54) = i .Mostly, we will prefer to use the Lie algebra sp (2 d ), which acts on SF ( d )with derivations. We will need to use some specific elements of this Lie algebra,so we need to write down one particular basis of sp (2 d ). We will use the notationfrom [FH91]. Let 1 ≤ i, j ≤ d, i (cid:54) = j : H i e k = δ ik e i , H i f k = − δ ik f i X ij e k = δ jk e i , X ij f k = − δ ik f j ,Y ij e k = 0 , Y ij f k = δ jk e i + δ ik e j ,Z ij e k = δ jk f i + δ ik f j , Z ij f k = 0 ,U i e k = 0 , U i f k = δ ik e i ,V i e k = δ ik f i , V i f k = 0 . SF ( d ) + In this subsection, we will recall Abe’s results on the representations of SF ( d ) + .In [Abe07], the θ -twisted SF ( d )-module SF ( d ) θ is defined. Theorem 3.3 ([Abe07]) . The list { SF ( d ) ± , SF ( d ) ± θ } gives a complete list ofinequivalent irreducible SF ( d ) + modules. Abe also exhibits reducible and indecomposable extensions of SF ( d ) ± de-noted by (cid:92) SF ( d ) ± . From the existence of such modules it follows that SF ( d ) + can’t be rational.To get the epimorphism π : A ( SF ( d ) + ) → A d and to prove Proposition5.5, we will need to recall the action of A ( SF ( d ) + ) on top components of theserepresentations, described in [Abe07]. By a result from [Zhu96], the images ofstrong generators of SF ( d ) + are the generators of A ( SF ( d ) + ), so we need toconsider only the action of strong generators. SF ( d ) + θ The top component of SF ( d ) + θ is one-dimensional, spanned with the vacuumvector θ with conformal weight − d/
8. As for the action of the strong generators7f SF ( d ) + , we have o ( h ii ) θ = − θ , while the rest of the generators act as 0. SF ( d ) − θ The top component of SF ( d ) − θ is 2 d -dimensional, spanned by e k − θ , f k − θ , for1 ≤ k ≤ d , with conformal weight − d/ /
2. We will slightly abuse notationand write e k = e k − θ , f k = f k − θ , for simplicity. Let 1 ≤ i, j ≤ d, i (cid:54) = j . The small generators act as: o ( e ij ) e k = 0 , o ( e ij ) f k = 12 ( δ ik e j − δ jk e i ) o ( f ij ) e k = 12 ( δ jk f i − δ i,k f j ) , o ( f ij ) f k = 0 o ( h ij ) e k = 12 δ jk e i , o ( h ij ) f k = 12 δ ik f j , ( i (cid:54) = j ) o ( h ii ) e k = 12 δ ik e i − e k , o ( h i,i ) f k = 12 δ ik f i − f k . The large generators act as (now we allow i = j ): o ( E ij ) e k = 0 , o ( E ij ) f k = −
14 ( δ ik e j + δ jk e i ) o ( F ij ) e k = 14 ( δ jk f i + δ ik f j ) , o ( F ij ) f k = 0 o ( H ij ) e k = 14 δ jk e i , o ( H ij ) f k = − δ ik f j . . SF ( d ) + The top component of SF ( d ) + is 1-dimensional (spanned by ) of conformalweight 0. All generators act trivially. SF ( d ) − The top component of SF ( d ) − is 2 d -dimensional, spanned by e k , f k , ≤ k ≤ d with conformal weight 1. The small generators act as o ( e ij ) e k = 0 , o ( e ij ) f k = δ ik e j − δ jk e i o ( f ij ) e k = δ jk f i − δ ik f j , o ( f ij ) f k = 0 o ( h ij ) e k = δ jk e i , o ( h ij ) f k = δ ik f j , while the large generators act as o ( E ij ) e k = 0 , o ( E ij ) f k = − δ ik e j − δ jk e i o ( F ij ) e k = δ jk f i + δ ik f j , o ( F ij ) f k = 0 o ( H ij ) e k = δ jk e i , o ( H ij ) f k = − δ ik f j . .3.5 (cid:92) SF ( d ) + In the Chapter 5 of [Abe07], it was shown that Ω( (cid:92) SF ( d ) + ) = (cid:92) SF ( d ) +0 . The topcomponent (cid:92) SF ( d ) +0 is 2 d − -dimensional and isomorphic to the even part of anexterior algebra on generators e i , f i , ≤ i ≤ d . Its conformal weight is 0. Smallgenerators act as multiplication by monomials of length 2 in the following way: o ( e ij ) = e i e j , o ( h ij ) = e i f j , o ( f ij ) = f i f j , while large generators act as 0. Notice that o ( ω ) is a nilpotent operator of degree d + 1. (cid:92) SF ( d ) − In the Chapter 5 of [Abe07] it was shown thatΩ( (cid:92) SF ( d ) − ) = (cid:92) SF ( d ) − (0) ⊕ (cid:92) SF ( d )[2 d ] − (1) , where (cid:92) SF ( d )[2 d ] − (1) is isomorphic as an A ( SF ( d ) + )-module to the top compo-nent of SF ( d ) − . π : A ( SF ( d ) + ) → A d In this subsection we will show the existence of an algebra epimorphism π : A ( SF ( d ) + ) → A d . For this, we will use a following generalization of the Chineseremainder theorem (see [IR90]): Theorem 3.4.
Let R be a ring (with unity), and let I , I , . . . , I k be two-sidedideals in R . If these ideals are pairwise coprime, then there is an epimorphism π : R → R/I ⊕ R/I ⊕ . . . ⊕ R/I k x (cid:55)→ ( x + I , x + I , . . . , x + I k ) with ker π = I ∩ I ∩ . . . ∩ I k . We will denote the homomorphisms connected to the representations of A ( SF ( d ) + ) in the following way: ρ ± : A ( SF ( d ) + ) → End C (Ω( SF ( d ) ± )) ρ ± θ : A ( SF ( d ) + ) → End C (Ω( SF θ ( d ) ± )) (cid:99) ρ ± : A ( SF ( d ) + ) → End C ( (cid:92) Ω( SF ( d ) ± )) . We want to use Theorem 3.4 on the ideals ker( (cid:99) ρ + ) , ker( ρ − ) , ker( ρ ± θ ) - forthat, we need to check that those ideals are pairwise coprime and we need toknow the images of these homomorphisms. We get the coprimality by observingthe minimal polynomials of o ( ω ) on different representations. We have:[ ω ] d +1 ∈ ker( (cid:99) ρ + )[ ω ] − ∈ ker( ρ − )[ ω ] + d/ ∈ ker( ρ + θ )[ ω ] + ( d/ − / ∈ ker( ρ − θ ) . ω ] are pairwise coprime.By Jacobson density theorem we getim( ρ − ) (cid:39) im( ρ − θ ) (cid:39) M d ( C ) , im( ρ + θ ) (cid:39) C for irreducible representations. It’s clear from the previous subsection thatim( (cid:99) ρ + ) is isomorphic to Λ even ( V d ) (even part of exterior algebra on the standardrepresentation of sp (2 d )). Now, we can use Theorem 3.4 to get an epimorphism π : A ( SF ( d ) + ) → A d = M d ( C ) ⊕ M d ( C ) ⊕ Λ even ( V d ) ⊕ C . Spanning set for P ( SF ( d ) + ) The main result of this section is the existence of a spanning set for P ( SF ( d ) + )of cardinality n d = 2 d − + 8 d + 1. Let us describe it more precisely. Put x i = e i , x i + d = f i , for 1 ≤ i ≤ d . Denote B d the set of all even length monomialsobtained by ( − x i , and B d the set of the following vectors: x i − x j , ≤ i ≤ j ≤ dx i − x j , ≤ i < j ≤ dx i − x j , ≤ i ≤ j ≤ dx i − x j , ≤ i < j ≤ de − f . It is easy to see that | B d | = 2 d − and | B d | = 8 d + 1. The main Theorem ofthis section is: Theorem 4.1.
Image of B d = B d ∪ B d in P ( SF ( d ) + ) is a spanning set of P ( SF ( d ) + ) . Also, in this section we will write C := C ( SF ( d ) + ) for simplicity. P ( SF ( d ) + ) In this section we prove some lemmas that will be helpful in the proof of Theorem4.1. First, let us recall some definitions and relations from [Abe07]. In theChapter 3 of [Abe07] it is defined that: B m,n ( a, b ) = ( m − n − m + n − a − m b − n , (4.1)for a, b ∈ h . Following relations hold in P ( SF ( d ) + ): B m,n ( a, b ) = ( − n − B m + n − , ( a, b ) (4.2) B m,n ( a, b ) = ( − m + n − B m,n ( b, a ) . (4.3)These relations show that every monomial of length 2 in SF ( d ) + can be writtenmodulo C as a linear combination of the vectors x i − n x j , where n ∈ Z > is odd and 1 ≤ i < j ≤ d,x i − n x j , where n ∈ Z > is even and 1 ≤ i ≤ j ≤ d. Also, we will need some relations from the proof of Proposition 3.12 in[Abe07]:( e i − m f i ) · ( e i − k f i ) = mk (cid:18) m + 1 + 1 k + 1 (cid:19) (cid:18) m + kk (cid:19) e i − m − k − f i (4.4) e i − e i = 0 (4.5) e i − f i = 0 (4.6)Now, we are ready to prove an important Lemma.11 emma 4.2. In P ( SF ( d ) + ) we have ( h ii − h jj ) = 0 (4.7)( h ii + h jj ) = 12( h ii + h jj ) · h ii · h jj . (4.8) for ≤ i < j ≤ d .Proof. Because of the permutation automorphisms (cf. subsection 3.2, it issufficient to show this in case i = 1 , j = 2 (in both cases). Relation (4.7) willfollow from: ( h − h ) · h = 0 (4.9) h · h = 12 ( h − h ) (4.10)We calculate: h − h = e − f e − f = e − f h − h = − f − e e − f = − f − e = e − f mod C . The second one follows from ( h ii ) = 2 e i − f i and some direct calculation.Now we turn to the relation (4.8). Let’s denote Z := ( h ) − h = e − f − e − f . Then, we need to prove ω = 12 ωZ. We check that L − Z = (cid:88) sym e − f − e − f − , and calculate: h − h + h − h = L − Z + e − f − e − f − + e − f − e − f − e − f + f − e = − L − Z − e − f − e − f − − e − f − e − f − + (cid:88) ( e i − f i − f i − e i ) h − h + h − h = − L − Z − e − f − e − f − − e − f − e − f − + (cid:88) ( e i − f i − f i − e i ) .
12f we subtract the second and third equation from the first one we see3 ω · Z + (cid:88) sym e − e − f − f − = 2 (cid:88) ( e i − f i − f i − e i )Because of L − Z = 2 L − Z + 2 (cid:88) sym e − e − f − f − , we can simplify to ω · Z = (cid:88) ( e i − f i − f i − e i ) (4.3) = 2 (cid:88) ( e i − f i ) . Using the relation (4.4) we have: ωZ = 19 ( h + h ) . To conclude: ω = h + 3 h h + 3 h h + h = h + 3 ωZ + h = 12 ωZ. Corollary 4.3. In P ( SF ( d ) + ) for ≤ i, j ≤ d we have ( h ii ) = ( h jj ) , which is equivalent to e i − f i = e j − f j . Proof.
Equivalence follows from the formula (4.4). Let’s put z i := h ii . In thatnotation, previous two Lemmas say: p ( z i , z j ) := ( z i − z j ) = 0 q ( z i , z j ) := ( z i + z j ) − z i + z j ) z i z j = 0 . Now, notice that we have: z i − z j = 14 (5( z i + z j ) p ( z i , z j ) + ( z j − z i ) q ( z i , z j )) . The goal of this subsection is to prove the following:13 roposition 4.4.
All monomials of length 2 in SF ( d ) + can be written (modulo C ) as a linear combination of the following vectors: x i − x j , ≤ i < j ≤ dx i − x j , ≤ i ≤ j ≤ dx i − x j , ≤ i < j ≤ dx i − x j , ≤ i ≤ j ≤ dx i − x j , ≤ i < j ≤ de − f . Notice that the first row lies in B d and that other rows lie in B d . Aswe already noticed in the subsection 4.1, relations (4.2) and (4.3) imply thatevery monomial of length 2 in SF ( d ) + can be written (modulo C ) as a linearcombination of vectors x i − n x j , where n ∈ Z > is odd and 1 ≤ i < j ≤ d,x i − n x j , where n ∈ Z > is even and 1 ≤ i ≤ j ≤ d. We need to remove the monomials with higher n from the list. Lemma 4.5.
For ≤ i ≤ d we have: e i − k e i ≡ C , for k ≥ e i − k f i ≡ C , for k = 6 or k ≥ . Proof.
First, using the relation (4.2) we get: h ii − ( e i − k e i ) = ke i − k − e i + e i − k e i − ≡ (cid:18) k + (cid:18) k + 12 (cid:19)(cid:19) e i − k − e i mod C h ii − ( e i − k f i ) = ke i − k − f i + e i − k f i − ≡ (cid:18) k + (cid:18) k + 12 (cid:19)(cid:19) e i − k − f i mod C It follows that e i − k e i ∈ C = ⇒ e i − k − e i ∈ C (4.11) e i − k f i ∈ C = ⇒ e i − k − f i ∈ C (4.12)From relation (4.3) we know that e i − k − e i ∈ C , k ≥
0, and we have a relation(4.5) which says that e i − e i ∈ C . From (4.11) it follows that e i − k e i ∈ C , for k ≥ V i ∈ sp (2 d ) to get C (cid:51) V i ( e i − e i ) = f i − e i + e i − f i (4.3) ≡ e i − f i mod C , and then proceed in similar fashion (together with the relation (4.6)).Using sp (2 d ) on the relations from the previous Lemma gives us everythingbut the last row in Proposition 4.4. For the last row, we use the Corollary 4.3.14 .3 Monomials of length greater than 2 Following [Abe07], we define L SF ( d ) = C , and for r ∈ Z > L r SF ( d ) = span C { x i − n . . . x i s − n s : 1 ≤ i j ≤ d, n j ∈ Z > , s ≤ r } . We also put L r SF ( d ) + = L r SF ( d ) ∩ SF ( d ) + . In this section, we will only use L r SF ( d ) + , so we will write L r + := L r SF ( d ) + for simplicity.We can generalize Abe’s definition (4.1) to B m ,...,m k ( a , . . . , a k ) := ( m − m − . . . ( m k − (cid:80) m i − a − m a − m . . . a k − m k . (4.13)Similar to k = 2 case, we have L − B m ,...,m k ( a , . . . , a k ) = ( k (cid:88) i =1 m i ) · k (cid:88) i =1 B m ,...,m i +1 ,...,m k ( a , . . . , a k ) ∈ C Iterating this relation, we get the following Lemma:
Lemma 4.6.
Each element of SF + ( d ) can be written (modulo C ) as a linearcombination of monomials of the form x i − n . . . x i k − n k , k ≥ , n j > , ≤ i j ≤ d, where at least one of n j equals . We can improve on this Lemma:
Lemma 4.7.
Each element of SF + ( d ) can be written (modulo C ) as a linearcombination of monomials of the form x i − n x i − . . . x i k − , k ≥ , n > , ≤ i j ≤ d (4.14) Proof.
We use the induction on the monomial length 2 k . For k = 0 the resultis obvious, and for k = 1 it follows from the Proposition 4.4. Let k ≥ a = x i − n . . . x i k − − n k − x i k . We calculate, using the associativity formula (see [LL04]):( x i k − − n k − x i k ) − ( x i − n . . . x i k − − n k − )= (cid:88) j ≥ ( − j (cid:18) − n k − j (cid:19) (cid:16) x i k − − n k − − j x i k − j + ( − − n k − x i k − n k − − − j x i k − j (cid:17) ( x i − n . . . x i k − − n k − )= x i k − − n k − x i k − x i − n . . . x i k − − n k − + u, u ∈ L (2 k − = a + u, u ∈ L (2 k − . Now, we can use the inductive hypothesis on ( x i − n . . . x i k − − n k − ) and thefact that ( − C space to conclude that we canrestrict ourselves to the monomials of the form b = x i − n x i − n x i − x i − . . . x i k − − x i k .
15e can use a similar trick to write b = ( x i k − − x i k ) − ( x i − n x i − n x i − . . . x i k − ) + u, u ∈ L (2 k − and use the inductive hypothesis on ( x i − n x i − n x i − . . . x i k − ), proving our re-sult.Next step is to check that the monomials of the form (4.14) reside in span C B d + C . We will first have to prove this for monomials of length 4 and then proceedwith an inductive argument. Lemma 4.8 (Technical Lemma) . Monomials of length 4 are in span C B d + C .Proof. See subsection 4.4.Now, we can finish the proof of Theorem 4.1.
Proposition 4.9.
All monomials in SF ( d ) + of length greater or equal to 4 liein span C B d + C . Proof.
We prove this Proposition by the induction on the length of monomials.The base case (monomials of length 4) is proven in previous Lemmas. Now,assume that all monomials of length less than 2 k lie in span C B d + C . Becauseof the Lemma 4.7 we can reduce to monomials of the form x i − n x i − . . . x i k − , n > , ≤ i j ≤ d. We can write x i − n x i − . . . x i k − = ( x i k − − x i k ) − ( x i − n x i − . . . x i k − − ) + u, u ∈ L (2 k − . By the inductive hypothesis, x = x i − n x i − . . . x i k − − ∈ span C B d + C . This means that we can write x (modulo C ) as a linear combination of mono-mials from B d of length less than 2 k. Let m ∈ B d of length < k . Then,( x i k − − x i k ) − m will be a linear combination of a monomial from B d of length2 k and some monomials of length < k .Let m ∈ B d . Then ( x i k − − x i k ) − m will be a linear combination of mono-mials of length ≤
4, and that case is already covered.
In this section we prove the Technical Lemma 4.8, that is, all monomials oflength 4 lie in span C B d + C .Recall the automorphisms τ i and the permutation automorphisms from sub-section 3.2. If we take a monomial of the form (4.14) of length 4, we can usethem to write it as e − n x i − x i − x i . We split this problem into several cases, de-pending on how many of i , i , i equal 1. We immediately see that we can’thave a non-zero monomial where i = i = i = 1. So, we start with:16 ase 1: Exactly two of i , i , i equal e − n e − f − x i , where i (cid:54) = 1. Using the automorphisms, it is enough to considerthe monomial e − n e − f − e . Notice that e − h = e − e − e − f + u, u ∈ L e − h = ( e − e − + e − e − ) e − f + v, v ∈ L The first row shows that e − e − f − e ∈ C + L , and we know that L ⊆ span C B d + C by Proposition 4.4. Because of h e − e − e − f = e − e − e − f we can conclude from the second row that e − e − e − f ∈ span C B d + C . It iseasy to see we can continue using this trick for higher n . Case 2: Exactly one of i , i , i equals e − n e − x i − x i or e − n f − x i − x i for i , i (cid:54) = 1. There is also a question of whether or not i equals i . Using automorphisms, we can reduce to these four subcases: Subcase 2.1: e − n e − e − e We use a similar idea as in
Case 1 : e − e = e − e − e − e − e − e = ( e − e − + e − e − ) e − e − . The first row shows that e − e − e − e − ∈ C , and because of h e − e − e − e − = e − e − e − e − ∈ C the second row shows that e − e − e − e − ∈ C . We can continue using the sametrick for higher n . Subcase 2.2: e − n f − e − e Looking at e − ( e − f ) , e − ( e − f ) , e − ( e f ) we can conclude that e − f − e − e , e − f − e − e ∈ C + L ⊆ C + span C B d . (4.15)Now if we calculate e − ( e − f ) = ( e − e − + e − e − + e − e − ) e − f + u, u ∈ L , and use h on the relation (4.15), we get e − f − e − e ∈ C + span B d . We canuse the same strategy for higher n . Subcase 2.3: e − n e − e − f From the proof of the Proposition 4.4 we know that e − n e ∈ C for all n ∈ N except n = 2 ,
4. Because of h − ( e − n e ) = e − n e − e − f
17e only need to check for these n . For n = 2 we have e − ( e − f ) = e − e − e − f + u, u ∈ L . For n = 4 we need to calculate: H ( e − e e f ) = 3 e − e e f + 2 e − e − e f h ( e − e e f ) = 3 e − e e f + e − e − e f . Subcase 2.4: e − n f − e − f From the proof of the Proposition 4.4 we know that e − n f ∈ C , ∀ n ∈ N , n (cid:54) = 1 , , , , , . Because of e − n f − e − f = h − ( e − n f ) we can reduce to cases n = 2 , , , , n = 2 , subcase 2.3 together with the fact thatfor V ∈ sp (2 d ): V ( e − n e − e − f ) = f − n e − e − f + e − n f − e − f ≡ e − n f − e − f mod C For the n = 3 , , e − n f − e − f is equivalent to ( h ) n +12 · h for odd n . We can use the relations(4.2) and (4.3) (similar to the proof of Corollary 4.3) to show they are in C + L . Case 3: None of i , i , i equal i , i , i are different, by use of the automorphisms we can seethat it’s enough to consider the monomials e − n e − e − e − . If we look at e ij − e kl for different choices of i, j, k, l ∈ { , , , } , we get e − e − e − e − ∈ C SF ( d ) + . We can proceed using the relation h ( e − n e − e − e − ) = ne − n − e − e − e − . If some of i , i , i take the same value, we get the monomial of the form e − n e i − f i − e j or e − n e i − f i − f j for i (cid:54) = j . Now we can use the permutationautomorphism which exchanges 1 and i to put ourselves in one of the formercases. 18 Consequences SF ( d ) + Following [AN13], denote by C ( V ) a vector space of one-point functions on aVOA V . In [AN13] (Theorem 6.3.2), the authors have constructed 2 d − + 3linearly independent pseudo-trace functions, therefore showing thatdim C C ( SF ( d ) + ) ≥ d − + 3holds for general d . (We won’t give the definitions of one-point functions orpseudo-trace functions here, instead we refer the reader to [Miy04] and [AN13].)They also showed that the equality holds in the case of d = 1 (using Abe’scalculation of Zhu’s algebra of SF (1) + from [Abe07]), and conjectured it forgeneral d . Now, because of the Theorem 1.1, we can use the same methodto show that their conjecture holds. That method depends on calculating thedimension of the space of symmetric linear functions on the Zhu’s algebra.We recall the notion: let A be a finite-dimensional associative C –algebra. Alinear function ϕ : A → C is called a symmetric linear function if ϕ ( ab ) = ϕ ( ba ) , ∀ a, b ∈ A. Following [AN13] we denote the space of those functions by S A , and for a VOA V , we put S V := S A ( V ) . The main result connecting S V to C ( V ) is [AN13,Theorem 3.3.6]: Theorem 5.1 ([AN13]) . Let V be a C -cofinite VOA. Suppose that any simple V -module is infinite-dimensional. Then dim C C ( V ) ≤ dim C S V . Now, all is set for the following Proposition.
Proposition 5.2.
For all natural d , we have dim C C ( SF ( d ) + ) = 2 d − + 3 . Proof.
Using aforementioned results from [AN13], it is enough to show thatdim C S SF ( d ) + = 2 d − + 3. By Theorem 1.1 we know that A ( SF ( d ) + ) (cid:39) Λ even ( V d ) ⊕ M d ( C ) ⊕ M d ( C ) ⊕ C . We need to know the dimension of the space of symmetric linear functions oneach summand. For the commutative algebra Λ even ( V d ), each linear functionis symmetric, giving usdim C S Λ even ( V d ) = dim C Λ even ( V d ) = 2 d − . For matrix algebras, we havedim C S M n ( C ) = 1 , ∀ n ∈ N . The result follows. 19 .2 Further properties of A ( SF ( d ) + ) and P ( SF ( d ) + ) First, let’s record some easy consequences of the Theorem 1.1:
Corollary 5.3.
1. Image of B d in P ( SF ( d ) + ) is actually a basis for P ( SF ( d ) + ) .2. We have gr A ( SF ( d ) + ) (cid:39) P ( SF ( d ) + ) .
3. Minimal polynomial of [ ω ] ∈ A ( SF ( d ) + ) is m d ( x ) = x d +1 ( x − (cid:18) x + d (cid:19) (cid:18) x + d − (cid:19) .
4. The center of A ( SF ( d ) + ) is isomorphic to Λ even ( V d ) ⊕ C ⊕ C ⊕ C . Proof.
Part 1 follows from Theorem 4.1 and the equality of dimensions of A ( SF ( d ) + ) and P ( SF ( d ) + ). Equality of dimensions also gives us that the epi-morphism (2.1) is actually an isomorphism in this case, solving part 2. Part 3 isa direct consequence of A ( SF ( d ) + ) (cid:39) A d and the discussion in the subsection3.4. Part 4 is a direct consequence of the Theorem 1.1.Let s d be a degree of nilpotency of ω ∈ P ( SF ( d ) + ). Parts 2 and 3 of theCorollary combine to give us that s d ≤ deg m d = d + 4. Actually, we can dobetter: Proposition 5.4.
We have: s d = (cid:40) , d ≤ d + 1 , d ≥ . Proof.
Notice that the element of maximal conformal weight in B d is e − f for d ≤
4, and e − f − e − f − . . . e d − f d for d ≥
5. Also, that maximal conformalweight is exactly 2( s d − ω ) s d = 0in P ( SF ( d ) + ). It remains to show that ( ω ) s d − isn’t equal to zero in P ( SF ( d ) + ).For d ≤
4, we can check by the computer (using relations proven in the subsec-tion 4.1) that we have:( ω ) = ( h ) in P ( SF (1) + )( ω ) = 165 ( h ) in P ( SF (2) + )( ω ) = 375 ( h ) in P ( SF (3) + )( ω ) = 725 ( h ) + 24 e − f − . . . e − f − in P ( SF (4) + ) . By relation (4.4) we get that h = 360 e − f . ω ) as a nonzero linear combinationof basis elements (by the part 1 of the Corollary 5.3).For d ≥
5, one can show that( ω ) d = d ! e − f − . . . e d − f d in P ( SF ( d ) + ) . The strategy is to write:( ω ) d = (cid:88) k + k + ... + k m d ! k ! k ! . . . k m ! m (cid:89) i =1 ( h ii ) k i , and then (using relations from Lemma 4.2) show that we have m (cid:89) i =1 ( h ii ) k i = 0for all choices of k i except the k = k = . . . = k d = 1 . We see that for all d > s d < deg m d . We can explain this for d = 2. Take a sp (2 d )-invariant element of conformal weight 4: J = ( e − f − f − e ) + ( e − f − f − e ) . One can show that in A ( SF (2) + ) we have[ J ] = − ω ] + 24[ ω ] + 295 [ ω ] . (5.1)It follows that we can write [ ω ] as a linear combination of elements of lowerdegree, and by part 2 of the Corollary 5.3 we must have ( ω ) = 0.In [CL17, Section 3], the authors showed that the VOA SF ( d ) sp (2 d ) is a W -algebra of type W (2 , , . . . , d ), generated by a primary field of conformalweight 4. Later, it was proved in [KL19] that SF ( d ) sp (2 d ) is isomorphic tothe simple principal W –algebra W k ( sp (2 d ) , f prin ) for k = − d − /
2. Specially, SF (2) sp (4) is strongly generated by ω and J , and it follows that A ( SF (2) sp (4) )is generated by [ ω ] and [ J ]. We have the induced algebra homomorphism A ( SF (2) sp (4) ) → A ( SF (2) + ) , and it is easy to see that the image of this homomorphism is the invariantsubalgebra A ( SF (2) + ) sp (4) . Equation (5.1) says that A ( SF (2) + ) sp (4) is exactlythe subalgebra of A ( SF (2) + ) generated by [ ω ].We can show that this also happens to sp (2 d )-invariant elements in A ( SF ( d ) + )for all d . Proposition 5.5.
Invariant subalgebra A ( SF ( d ) + ) sp (2 d ) is generated by [ ω ] ,and thus isomorphic to C [ x ] / ( m d ( x )) .Proof. Let W be a subalgebra of A ( SF ( d ) + ) generated by [ ω ]. Because ω isinvariant under the action of sp (2 d ), we have W ≤ A ( SF ( d ) + ) sp (2 d ) . Also,because of the part 3 of Corollary 5.3, we have W (cid:39) C [ x ] / ( m d ( x )) , dim C W = d + 4 .
21e will prove that dim C A ( SF ( d ) + ) sp (2 d ) = d + 4, and the rest will follow. Wecan write A ( SF ( d ) + ) = A − d ⊕ A ⊕ A − d + ⊕ A , where A λ = { x ∈ A ( SF ( d ) + ) : ([ ω ] − λ [ ]) N x = 0 , for N large enough } . Theorem 1.1 says that we have A − d (cid:39) C , A (cid:39) Λ even ( V d ) , A − d + (cid:39) A (cid:39) M d ( C ) . We want to show that A λ are in fact sp (2 d )-submodules. By the general the-ory of associative algebras (for the reference, see [Lam01]), there are centralidempotent elements e λ ∈ A ( SF ( d ) + ) such that A λ = A ( SF ( d ) + ) e λ . It is easy to see that these central idempotent elements are polynomials in [ ω ]by looking at the decomposition C [ x ] / ( m d ( x )) (cid:39) C [ x ] / ( x d +1 ) ⊕ C [ x ] / ( x − ⊕ C [ x ] / ( x + d/ ⊕ C [ x ] / ( x + d/ − / . It follows that each A λ is a sp (2 d )-submodule.It is obvious that A − d is a trivial sp (2 d )-module. By looking at top compo-nents of SF ( d ) − and SF ( d ) − θ one can see that both A and A − d/ / have alinear basis of the form e λ ∗ [ x ] , λ = 1 , − d/ / x is a strong generator of SF ( d ) + . It is easy to check that the linear spanof large generators in SF ( d ) + is a sp (2 d )-submodule isomorphic to Sym ( V d ),and that the linear span of small generators is a sp (2 d )-submodule isomorphicto Λ ( V d ). Now, by [FH91] we have that Sym ( V d ) is an irreducible sp (2 d )-module, and that Λ ( V d ) decomposes asΛ ( V d ) (cid:39) U ⊕ C , where U is an irreducible sp (2 d )-module (the trivial part corresponds to C ω ).Now, we have that dim C A sp (2 d )1 = dim C A sp (2 d ) − d/ / = 1 . For A , it is easy to see that it has a linear basis of the form e ∗ [ x ] , x ∈ B d , and that we have A (cid:39) d (cid:77) k =0 Λ k ( V d ) . Next we use a Theorem in [Pro07, Chapter 11.6.7] which says that for 0 ≤ k ≤ d we have Λ k ( V d ) (cid:39) k (cid:77) i =0 V ( i )2 d . V (0)2 d we denote the trivial representation, and by V ( i )2 d the i -th fun-damental representation). It is a well-known fact that for d < k ≤ d wehave Λ k V d (cid:39) Λ d − k V d . To conclude, for 0 ≤ k ≤ d we havedim C (Λ k V sp (2 d ) d ) = 1 , which implies dim C ( A sp (2 d )0 ) = d + 1 . Summing up, we get dim C (( A ( SF ( d ) + ) sp (2 d ) ) = d + 4 . Remark 5.6.
Note that for d > , the invariant subalgebra ( A ( SF ( d ) + ) sp (2 d ) isa proper subalgebra of the center of A ( SF ( d ) + ) . In the case of the triplet vertexalgebra W ( p ) (recall that SF (1) + = W (2) ), the center of A ( W ( p )) coincides withthe invariant subalgebra ( A ( W ( p ))) sl (2) ) (cf. [AM08], [AM11]). In our opinion,the reason for this difference is in the fact that SF ( d ) + for d > contains alarge number of generators of conformal weight which can contribute to thecenter of ( A ( SF ( d ) + ) , while weight two space of W ( p ) is -dimensional.In the case of higher rank triplet vertex algebra W ( p ) Q we expect that centerof A ( W ( p ) Q ) conicides with invariants A ( W ( p ) Q ) g , where g is the simple Liealgebra having root system Q . Acknowledgment
The results of this paper were reported by A. ˇC. at the conference RepresentationTheory XVI, Dubrovnik, June 23-29.2019.The authors are partially supported by the QuantiXLie Centre of Excellence,a project cofinanced by the Croatian Government and European Union throughthe European Regional Development Fund - the Competitiveness and CohesionOperational Programme (KK.01.1.1.01.0004).
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D. Adamovi´c,
Department of Mathematics, Faculty of Science, University ofZagreb, Croatia
E-mail address: [email protected]
A. ˇCeperi´c,
Department of Mathematics, Faculty of Science, University of Za-greb, Croatia
E-mail address: [email protected]@math.hr