aa r X i v : . [ m a t h . QA ] O c t SUSY LATTICE VERTEX ALGEBRAS
REIMUNDO HELUANI AND VICTOR G. KAC
Abstract.
We construct and study SUSY lattice vertex algebras. As a simpleexample, we obtain the simple vertex algebra associated to the vertex algebra V c ( N
3) of central charge c = 3 /
2, as the SUSY lattice vertex algebra associatedto Z with bilinear form ( a, b ) = 2 ab . Introduction
In our recent paper [HK07] we developped a Λ-bracket formalism for N K = N supersymmetric (SUSY) vertex algebras, which greatly simplifies calculations withquantum superfields. Recall [Kac96] that a vertex algebra V is called N K = N supersymmetric if there exist N odd operators S i , satisfying [ S i , S j ] = 2 δ ij T , where T is the translation operator, such that[ S i , Y ( a, z )] = Y ( S i a, z )for any state a ∈ V . The presence of the operators S i allows one to introducesuperfields, and in [HK07] we use the Λ-bracket formalism to perform computationswith them.In the present note we study N K = 1 SUSY lattice vertex algebras V super Q ,associated to an integral lattice Q . We give a simple characterization of theseSUSY vertex algebras, similar to that of “ordinary” lattice vertex algebas [Kac96],and prove that, as ordinary vertex algebras,(1.1) V superQ ≃ V Q ⊗ F (cid:0) Π h (cid:1) , where V Q is the “ordinary” lattice vertex algebra, associated to Q , and F (Π h ) isthe vertex algebra of free fermions, based on the space h = C ⊗ Z Q with reversedparity.We prove the existence of a canonical N = 1 conformal structure for any V superQ .Moreover, for each skewsymmetric operator A on h such that A = 1, we constructon V superQ an N = 2 conformal structure, and for each triple of such operators A i , i = 1 , ,
3, such that A i A j = √− ε ijk A k for i = j ( σ -matrices), we construct onV superQ a “little” N = 4 conformal structure.We write down the characters and supercharacters for all irreducible V superQ -modules and show that they span an SL ( Z )-invariant finite-dimensional space offunctions, holomorphic on the upper-half plane.One of the simplest examples of V superQ , when Q = Z with the bilinear form( a, b ) = 2 ab , turns out to be isomorphic to the simple vertex algebra associatedto the N = 3 superconfomral algebra V / ( N
3) with central charge c = 3 /
2. This
R. Heluani was supported by the Miller Institute for basic research in science.V. Kac was supported in part by NSF grant DMS-0501395. result elucidates the classification on unitary representations of the N = 3 super-conformal algebra at c = 3 /
2, obtained in [SS87].Note also that this result shows that the vertex algebra V / ( N
3) is rational (andeven semisimple). To the best of our knowledge, this is the only known non-zerovalue of c (i.e. c = 3 / V c ( N
3) is a rational vertex algebra.Throghout the paper, we use the notation and terminology of our paper [HK07]without further notice. 2.
Existence and Uniqueness
Let Q be an integral lattice of rank r with a non-degenerate symmetric bilinearform ( · , · ). Let W = Π C ⊗ Z Q be the associated complex vector space with reversedparity. Extend ( · , · ) to W by bilinearity. We have the N K = 1 SUSY vertex algebra V ( W ) as in [HK07, Example 5.9], that is, the vertex algebra generated by oddsuperfields α ∈ W satisfying the Lambda-brackets:(2.1) [ α Λ β ] = ( α, β ) χ. Let C ε [ Q ] be the group algebra of Q , twisted by the C × -valued 2-cocycle ε , that isthe unital associative algebra generated by symbols e α , α ∈ Q , with multiplicationrules: e α e β = ε ( α, β ) e α + β . We will construct an N K = 1 SUSY vertex algebra structure on the vector spaceV superQ := V ( W ) ⊗ C C ε [ Q ] such that we have the Lambda brackets (2.1) and(2.2) [ α Λ e β ] = ( α, β ) e β . Equation (2.2) corresponds to the following. For each h ∈ W , j ∈ Z and J = 0 , h ( j | J ) on V ( W ). We extend these operators to V superQ by: h ( j | J ) ( s ⊗ e α ) = h ( j | J ) ( s ) ⊗ e α , ( j | J ) = (0 | ,h (0 | ( s ⊗ e α ) = ( h, α ) s ⊗ e α . Note that this preserves the commutation relations between the operators h ( j | J ) ,therefore we still have [ h Λ h ′ ] = ( h, h ′ ) χ, h, h ′ ∈ W. Let e α denote the operator of multiplication by 1 ⊗ e α on V superQ , then it is easyto check that [ h ( j | J ) , e α ] = 0 , ( j | J ) = (0 | , [ h (0 | , e α ] = ( h, α ) e α . From these commutation relations, and denoting by Γ α the super-field correspond-ing to 1 ⊗ e α for α ∈ Q , we obtain the Lambda bracket as in (2.2):(2.3) [ h Λ Γ α ] = ( h, α )Γ α . This in turn gives: [ h ( j | J ) , Γ α ( Z )] = ( − J ( h, α ) Z j | J Γ α ( Z ) . USY LATTICE VERTEX ALGEBRAS 3
Lemma 2.1.
This last formula implies Γ α ( Z ) = e α X j ∈ Z Z − − j | α ( j | exp − X j< Z − j | j α ( j | ×× exp − X j> Z − j | j α ( j | A α ( Z ) , for some operators A α ( Z ) on V superQ commuting with all h ( j | J ) .Proof. Let X α = exp X j< Z − j | j α ( j | − X j ∈ Z Z − − j | α ( j | ( e α ) − Γ α ( Z ) ×× exp X j> Z − j | j α ( j | . For ( j | J ) = (0 |
0) we have[ h (0 | , X α ] = exp X j< Z − j | j α ( j | − X j ∈ Z Z − − j | α ( j | ×× ( − ( h, α ))( e α ) − Γ α ( Z ) exp X j> Z − j | j α ( j | ++ exp X j< Z − j | j α ( j | − X j ∈ Z Z − − j | α ( j | ×× ( e α ) − ( h, α )Γ α ( Z ) exp X j> Z − j | j α ( j | = 0 . The other cases are similar (somewhat long) computations. (cid:3)
As a corollary, note that ∀ h ∈ W, h (0 | A α ( Z ) s ⊗ e β = ( h, β ) A α ( Z ) s ⊗ e β , therefore we must have A α ( Z ) s ⊗ e β = a α,β ( Z ) s ⊗ e β for some a α,β ∈ C (( Z )). Inparticular we see that A α ( Z ) | i = | i + θd α | i + O ( Z ), where O ( Z ) denotes a powerseries which is a multiple of z , and d α is an odd constant if we use a Grassmannalgebra L as our base ring for the vertex algebra V Q instead of C .Applying Γ α ( Z ) to the vacuum vector we obtain then:(2.4) 1 ⊗ e α + θ (cid:0) d α (1 ⊗ e α ) + α ( − | ⊗ e α (cid:1) + O ( z ) . Now recall from [HK07, Thm 4.16 (3)] that on V superQ we must have Y ( Sa, Z ) = D Z Y ( a, Z ), where D Z = ( ∂ θ + θ∂ z ), therefore we obtain the identity: D Z Γ α ( Z ) = d α Γ α ( Z )+ : α ( Z )Γ α ( Z ) : REIMUNDO HELUANI AND VICTOR G. KAC
Replacing Γ α ( Z ) by its expression from Lemma 2.1 in this last equation, we obtaina differential equation for A α . Indeed, note that:(2.5) ∂ θ Γ α ( Z ) = e α X j ∈ Z Z − − j | α ( j | exp − X j< Z − j | j α ( j | ×× exp − X j> Z − j | j α ( j | A α ( Z ) + e α X j ∈ Z Z − − j | α ( j | ×× exp − X j< Z − j | j α ( j | exp − X j> Z − j | j α ( j | ∂ θ A α ( Z ) , and similarly(2.6) θ∂ z Γ α ( Z ) = e α X j< Z − − j | α ( j | exp − X j< Z − j | j α ( j | ×× exp − X j> Z − j | j α ( j | A α ( Z ) + e α exp − X j< Z − j | j α ( j | ×× X j> Z − − j | α ( j | exp − X j> Z − j | j α ( j | A α ( Z )++ e α exp − X j< Z − j | j α ( j | exp − X j> Z − j | j α ( j | θ∂ z A α ( Z ) . Combining these last two expressions and recalling the definition of α − ( Z ) in [HK07,3.2.4] we obtain after canceling the exponentials: D Z A α ( Z ) = d α A α ( Z ) + Z − | α (0 | A α ( Z ) . Multiplying both sides of this equation by z − α (0 | and defining B α ( Z ) = A α ( Z ) z − α (0 | ,we obtain the diferential equation: D Z B α ( Z ) = d α B α ( Z ) . All of its solutions are of the form B α = C α (1 + θd α ) for some even constant C α . Itfollows from (2.4) that C α = 1 for all α ∈ Q , therefore we arrive at the expression:(2.7) Γ α ( Z ) = e α Z α (0 | | Z | d α + X j ∈ Z Z − − j | α ( j | ×× exp − X j< Z − j | j α ( j | exp − X j> Z − j | j α ( j | . Remark . Note that the lattice has to be integral, otherwise this expressionwould not be a well defined superfield.
USY LATTICE VERTEX ALGEBRAS 5
Remark . It is convenient to write Γ α in a “normally ordered” form:(2.8) Γ α ( Z ) = e α Z α (0 | | (cid:16) Z | d α (cid:17) exp X j< Z − − j | α ( j | ×× exp − X j< Z − j | j α ( j | exp − X j> Z − j | j α ( j | exp X j ≥ Z − − j | α ( j | . Remark . Note that from (2.5) and (2.6) it follows that(2.9) ( ∂ θ − θ∂ z )Γ α ( z, θ ) = d α Γ α ( z, θ )+ : α ( z, − θ )Γ α ( z, θ ) : . We need to check the axioms of an N K = 1 SUSY vertex algebra. First let’scheck translation invariance. Note that we already know S (1 ⊗ e α ) = d α (1 ⊗ e α ) + α ( − | ⊗ e α . We must have then: S ( s ⊗ e α ) = S ( s ) ⊗ e α + ( − p ( s ) ( s ⊗ (cid:16) d α (1 ⊗ e α ) + α ( − | ⊗ e α (cid:17) . We need to check translation invariance for the fields Γ α ( Z ). For this we firstnote (recall e α is the operator of multiplication by 1 ⊗ e α on V superQ ):[ S, e α ]( s ⊗ e β ) = ε ( α, β ) S ( s ⊗ e α + β ) −− e α (cid:16) S ( s ) ⊗ e β + ( − p ( s ) ( s ⊗ d β (1 ⊗ e β ) + β ( − | ⊗ e β ) (cid:17) == ε ( α, β ) (cid:16) S ( s ) ⊗ e α + β + ( − p ( s ) ( s ⊗ d α + β (1 ⊗ e α + β ) + ( α + β ) ( − | ⊗ e α + β ) (cid:17) −− ε ( α, β ) (cid:16) S ( s ) ⊗ e α + β + ( − p ( s ) ( s ⊗ d β (1 ⊗ e α + β ) + β ( − | ⊗ e α + β ) (cid:17) == ε ( α, β )( − p ( s ) s (cid:0) d α + β − d β + α ( − | (cid:1) ⊗ e α + β . Letting B α ( e β ) := d α + β e β , we can write this as (note that the sign ( − p ( s ) isaccounted for since B α and α ( − | are odd operators):[ S, e α ] = e α ( α ( − | + B α − B ) . Note also that [
S, z α (0 | ] = 0. Indeed: Sz ( α,β ) s ⊗ e β −− z α (0 | (cid:16) S ( s ) ⊗ e β + ( − p ( s ) ( s ⊗ (cid:16) d β (1 ⊗ e β ) + β ( − | ⊗ e β (cid:17)(cid:17) = 0 . REIMUNDO HELUANI AND VICTOR G. KAC
Now we can compute [ S, Γ α ( Z )].[ S, Γ α ( Z )] = e α (cid:0) α ( − | + B α − B (cid:1) Z α (0 | | (cid:16) Z | d α (cid:17) X j< Z − − j | α ( j | ×× exp − X j< Z − j | j α ( j | exp − X j> Z − j | j α ( j | X j ≥ Z − − j | α ( j | −− e α Z α (0 | | (cid:16) Z | d α (cid:17) X j< Z − − j | α ( j | ×× exp − X j< Z − j | j α ( j | exp − X j> Z − j | j α ( j | X j ≥ Z − − j | α ( j | ++ e α Z α (0 | | (cid:16) Z | d α (cid:17) X j< Z − − j | α ( j | ×× X j< Z − j | α ( j − | exp − X j< Z − j | j α ( j | exp − X j> Z − j | j α ( j | ×× X j ≥ Z − − j | α ( j | + e α Z α (0 | | (cid:16) Z | d α (cid:17) X j< Z − − j | α ( j | ×× exp − X j< Z − j | j α ( j | X j> Z − j | α ( j − | exp − X j> Z − j | j α ( j | ×× X j ≥ Z − − j | α ( j | − e α Z α (0 | | (cid:16) Z | d α (cid:17) X j< Z − − j | α ( j | ×× exp − X j< Z − j | j α ( j | ) exp − X j> Z − j | j α ( j | X j ≥ Z − − j | α ( j | , where we used (4.6.1) of [HK07].We can commute the operators B α to the left using e α B β = B β − α e α , to obtain: [ S, Γ α ( z, θ )] = ( B − B − α )Γ α ( z, θ )+ : α ( z, − θ )Γ α ( z, θ ) : , and comparing with (2.9) we obtain that translation invariance for the operatorsΓ α ( Z ) holds if and only if B − B − α = d α , and this in turn holds iff d α is additive in α ∈ Q . From now on, we will assume forsimplicity that our ring of scalars is C , therefore d α = 0 for all α ∈ Q . USY LATTICE VERTEX ALGEBRAS 7
We are left only to check locality between the operators Γ α ( Z ). Note first that X j ≥ Z − − j | α ( j | X j< W − − j | β ( j | == X j< W − − j | β ( j | X j ≥ Z − − j | α ( j | (cid:18) − i z,w θζ ( α, β ) z − w (cid:19) . Using standard computations we find:(2.10) Γ α ( Z )Γ β ( W ) = ε ( α, β ) i z,w ( z − w ) ( α,β ) (cid:18) − i z,w θζ ( α, β ) z − w (cid:19) Γ α,β ( Z, W ) , whereΓ α,β ( Z, W ) = e α + β Z α (0 | | W β (0 | | exp X j< Z − − j | α ( j | + W − − j | β ( j | ×× exp − X j< Z − j | j α ( j | + W − j | j β j | exp − X j> Z − j | j α ( j | + W − j | j β ( j | ×× exp X j ≥ Z − − j | α ( j | + W − − j | β ( j | . And finally note that we can rewrite (2.10) as:(2.11) Γ α ( Z )Γ β ( W ) = ε ( α, β ) i z,w ( z − w − θζ ) ( α,β ) Γ α,β ( Z, W ) , and(2.12) Γ β ( W )Γ α ( Z ) = ( − ( α,β ) ε ( β, α ) i w,z ( z − w − θζ ) ( α,β ) Γ α,β ( Z, W ) . Therefore we are in the same position as in the non-SUSY case. Namely, we needto declare the parity of Γ α to be the parity of ( α, α ), and locality holds if and onlyif:(2.13) ε ( α, β ) = ( − ( α,β )+( α,α )( β,β ) ε ( β, α ) . Using Lemma 4.2 in [HK07], we find:(2.14)[Γ α ( Z ) , Γ β ( W )] = ( , ( α, β ) ≥ ε ( α, β ) (cid:16) D ( − − ( α,β ) | W δ ( Z, W ) (cid:17) Γ α,β ( Z, W ) , ( α, β ) < . Example 2.5.
Let Q = Z with the generator α = 1 and the usual bilinear form( α, α ) = 1. Let V super Z be the corresponding N K = 1 SUSY lattice vertex algebra,where we put ε = 1. Then we have(2.15) [Γ α ( Z ) , Γ − α ( W )] = − ( D W δ )Γ α, − α ( Z, W )= ( D Z δ )Γ α, − α ( Z, W )= D Z ( δ Γ α, − α ( Z, W )) + δD Z Γ α, − α ( Z, W )= D Z ( δ Γ α, − α ( W, W )) + δD Z Γ α, − α ( Z, W ) | Z = W = − ( D W δ )Γ α, − α ( W, W ) + δD Z Γ α, − α ( Z, W ) | Z = W . REIMUNDO HELUANI AND VICTOR G. KAC and noting that Γ α, − α ( W, W ) = 1, D Z Γ α, − α ( Z, W ) | Z = W = α ( W ), we obtain:(2.16) [Γ α ( Z ) , Γ − α ( W )] = − D W δ ( Z, W ) + δ ( Z, W ) α ( W ) . Therefore we have:(2.17) [ e α Λ e − α ] = α + χ. Expanding the fields Γ ± α ( Z ) = ψ ± ( z ) + θϕ ± ( z ), where ψ ± are odd and ϕ ± areeven, and the field α ( Z ) = ψ ( z ) + θα ( z ), where ψ is odd and α is even, we obtainthe following lambda brackets:(2.18) [ ψ λ ψ ] = 1 , [ α λ α ] = λ, [ α λ ψ ± ] = ± ψ ± , [ ψ λ ψ ± ] = 0 , [ ψ λ ϕ ± ] = ± ψ ± , [ α λ ϕ ± ] = ± ϕ ± , [ ψ + λ ψ − ] = 1 , [ ψ ± λ ϕ ∓ ] = ± ψ , [ ϕ + λ ϕ − ] = α + λ, and all other brackets are zero (except the ones given by skew-symmetry). Werecognize these lambda brackets as the commutation relations of the generators of V ( sl , super ).On the other hand, let us consider the algebra generated by the three fermions ψ , ψ ± with commutation relations:(2.19) [ ψ λ ψ ] = 1 , [ ψ + λ ψ − ] = 1 . This is the tensor product of a free fermion with the usual lattice vertex algebra V Z . Define the following fields:(2.20) α =: ψ + ψ − : , ϕ ± = ± : ψ ψ ± :It is straightforward to check that the commutation relations (2.18) are satisfied bythe fields α, ϕ ± , ψ and ψ ± .Define the following fields in this algebra of three free fermions: G =:: ψ + ψ − : ψ : , L =: ( T ψ ) ψ : + : (: ψ + ψ − :)(: ψ + ψ − :) : . We claim that these fields generate a Neveu Shwarz algebra of central charge 3 / N = 1 structure of the boson fermion system, for the boson: ψ + ψ − : and the fermion ψ . We know that ψ is primary of conformal weight 1 / ψ + λ L ] = − : ψ + : ψ + ψ − :: − :: ψ + ψ − : ψ + : + Z λ ψ + = ( λ − T ) ψ + , hence ψ + (and similarly ψ − ) are primary of conformal weight 1 / G λ ψ ± ] and [ G λ ψ ]. We find easily [ ψ λ G ] =: ψ + ψ − : and [ ψ ± λ G ] = ∓ : ψ ± ψ :. Therefore the superfields are: α ( Z ) := ψ ( z ) + θ : ψ + ( z ) ψ − ( z ) : , Γ ± ( Z ) = ψ ± ( z ) ± θ : ψ ( z ) ψ ± ( z ) : , and these super-fields satisfy the Lambda brackets of the superfields α ( Z ) andΓ ± α ( Z ) computed above. USY LATTICE VERTEX ALGEBRAS 9 Conformal structure
Following [HK07, Ex. 5.8] we define:(3.1) G = X i : ( Sα i ) α i :where { α i } and { α i } are dual bases of W . We know from [HK07, Ex. 5.8] thatthis is a Neveu-Schwarz vector with central charge c = dim W on V ( W ), andthat all superfields h ( Z ) , h ∈ W are primary of conformal weight 1 /
2. From (2.3)we obtain [Γ α Λ h ] = ( − ( α,α ) ( h, α )Γ α (recall that h is always odd in our setting,while the parity of Γ α is the parity of ( α, α )). Therefore we can compute using thenon-commutative Wick formula:[Γ α Λ G ] = − X ( α i , α ) : (cid:16) ( χ + S )Γ α (cid:17) α i : + X ( − ( α,α ) ( α i , α ) : ( Sα i )Γ α : −− Z Λ0 ( − ( α,α ) ( η − χ )( α i , α )( α i , α )Γ α = − : ( S Γ α ) α : − χ : Γ α α : +( − ( α,α ) : ( Sα )Γ α : − ( − ( α,α ) λ ( α, α )Γ α , and recalling that : α ( Z )Γ α ( Z ) := D Z Γ α ( Z ) we see immediately that : αS Γ α := 0(cf. Appendix A). Using: Γ α α : = ( − ( α,α ) : α Γ α : , : ( S Γ α ) α : = − ( − ( α,α ) : αS Γ α : + Z −∇ χ ( − ( α,α ) ( α, α )Γ α = − ( − ( α,α ) : αS Γ α : +( − ( α,α ) ( α, α ) T Γ α = ( − ( α,α ) ( α, α ) T Γ α , we obtain [Γ α Λ G ] = ( − ( α,α ) (: ( Sα )Γ α : − χS Γ α − ( α, α )( λ + T )Γ α ) . Therefore: [ G Λ Γ α ] =: ( Sα )Γ α : + : αS Γ α : +( χ + S ) S Γ α + ( α, α ) λ Γ α = (2 T + ( α, α ) λ + χS ) Γ α . Hence Γ α is a primary field of conformal weight ( α, α ) / G . Notethat this last equation together with the fact that G is a conformal vector on V ( W ) imply that G is a conformal vector on V superQ . Indeed, the commutationrelation of G with itself is not changed (as it involves only fields from the subalgebra V ( W ) ⊂ V superQ ) and from this last equation we can easily show that G (0 | = S on V superQ .When r is even, we can enlarge this conformal structure to an N = 2 structureas follows. Let A be an endormorphism of W satisfying:(3.2) ( Aα, α ′ ) = − ( α, Aα ′ ) , A = Id , where Id is the identity operator in W . We can construct then an even super-fieldof V superQ , primary of conformal weight 1, given by(3.3) J = 12 r X i,j =1 ( α i , Aα j ) : α i α j : . Proposition 3.1. (1)
The superfields G and J generate the N = 2 conformal vertex algebra ofcentral charge r viewed as an N K = 1 SUSY vertex algebra as in [HK07,Ex. 5.10] . (2) The super vector space V superQ carries a conformal N K = 2 SUSY vertexalgebra structure of central charge r , namely, the vector τ = √− J ( − | | i is an N K = 2 conformal vector.Proof. (1) We already know that G is a Neveu-Schwarz super-field of centralcharge r and that J is a primary super-field of conformal weight 1. In orderto compute [ J Λ J ] we first compute (here we sum over repeated indexes):[ α k Λ J ] = − ( α i , Aα k ) χα i , therefore [ J Λ α k ] = ( α i , Aα k )( χ + S ) α i . It follows then:(3.4) [ J Λ J ] = 12 ( α k , Aα m )( α i , Aα k ) : (cid:16) ( χ + S ) α i (cid:17) α m : −
12 ( α k , Aα m )( α j , Aα m ) : α k ( χ + S ) α j : +12 ( α k , Aα m )( α i , Aα k ) Z Λ0 ( η − χ )( α i , α m ) ηd Γ , = ( α k , Aα m )( α i , Aα k ) : (cid:16) ( χ + S ) α i (cid:17) α m : + 12 ( α k , Aα i )( α i , Aα k ) λχ, =: (cid:0) ( χ + S ) α i (cid:1) α i : + r λχ = G + c λχ. And according to [HK07, Ex. 5.10] these are the commutation relationsof the N = 2 super-vertex algebra (viewed as an N K = 1 SUSY vertexalgebra).(2) The superfields G and J allow us to construct a conformal N K = 2 SUSYvertex algebra structure (see [HK07, Def. 5.6]) on the space V superQ asfollows. We define the operators S = G (0 | , S = J (0 | , and it follows from the first part of the proposition that these operatorssatisfy: [ S i , S j ] = 2 δ ij T. With these operators now we can construct superfields for each a ∈ V superQ as Y ( a, z, θ , θ ) = Y ( a, z ) + θ Y ( S a, z ) + θ Y ( S a, z ) + θ θ Y ( a, z ) , where Y ( a, z ) is the usual field associated to a when we view V superQ as an“ordinary” vertex algebra. Defining the vector τ ∈ V superQ as τ = √− J ( − | | i , we obtain easily now that this vector satisfies the properties of [HK07, Def.5.6], namely, it is an N K = 2 conformal vector. (cid:3) USY LATTICE VERTEX ALGEBRAS 11
Definition 3.2.
Let V be an N K = 1 conformal SUSY vertex algebra of centralcharge c . A little N = 4 conformal structure of central charge c on V consist ofthree superfields { J i , i = 1 , , } , such that each pair { G, J i } defines an N K = 2conformal structure of central charge c on V as in Prop. 3.1 (2) and moreover, thefields { G, J i } , i = 1 , ,
3, satisfy the commutation relations of the N = 4 vertexalgebra of central charge c as in [HK07, Ex 5.10].We obtain easily then: Proposition 3.3.
Let A i , i = 1 , , , be endomorphisms of W satisfying (3.2) andin addition A i A j = √− ε ijk A k , i = j, where ε is the totally antisymmetric tensor. Define the superfields J i , i = 1 , , by(3.3) with A replaced by A i . Then the superfields G, J i , i = 1 , . . . , define a little N = 4 conformal structure of central charge c = r on V superQ . Example 3.4.
It follows from the computation in Example 2.5 that when ( α, β ) = − α + β = 0 we have(3.5) [Γ α Λ Γ β ] = ε ( α, β ) (: α Γ α + β : + χ Γ α + β ) . Suppose now that ( α, α ) = 2, then we see from (2.14) that we have (we put ε ( α, − α ) = 1):(3.6)[Γ α ( Z ) , Γ − α ( W )] = − ( D | W δ ( Z, W ))Γ α, − α ( Z, W )= − ( D | Z δ ( Z, W ))Γ α, − α ( Z, W )= − D | Z ( δ ( Z, W )Γ α, − α ( Z, W )) − ( ∂ z δ ( Z, W )) D Z Γ α, − α ( Z, W )++ ( D Z δ ( Z, W )) ∂ z Γ α, − α ( Z, W ) − δ ( Z, W ) D | Z Γ α, − α ( Z, W )= − D | Z ( δ ( Z, W )Γ α, − α ( Z, W )) − ∂ z ( δ ( Z, W ) D Z Γ α, − α ( Z, W ))++ δ ( Z, W ) D | Z Γ α, − α ( Z, W ) + D Z ( δ ( Z, W ) ∂ z Γ α, − α ( Z, W ))= − D | Z ( δ ( Z, W )Γ α, − α ( W, W )) − ∂ z ( δ ( Z, W ) D Z Γ α, − α ( Z, W ) | Z = W )++ δ ( Z, W ) D | Z Γ α, − α ( Z, W ) | Z = W + D Z ( δ ( Z, W ) ∂ z Γ α, − α ( Z, W ) | Z = W )= − ( D | Z δ ( Z, W ))Γ α, − α ( W, W ) − ( ∂ z δ ( Z, W )) D Z Γ α, − α ( Z, W ) | Z = W ++ δ ( Z, W ) D | Z Γ α, − α ( Z, W ) | Z = W + ( D Z δ ( Z, W )) ∂ z Γ α, − α ( Z, W ) | Z = W = − ( D | W δ ( Z, W ))Γ α, − α ( W, W ) + ( ∂ w δ ( Z, W )) D Z Γ α, − α ( Z, W ) | Z = W ++ δ ( Z, W ) D | Z Γ α, − α ( Z, W ) | Z = W − ( D W δ ( Z, W )) ∂ z Γ α, − α ( Z, W ) | Z = W . Note that D Z Γ α,β ( Z, W ) | Z = W =: α ( W )Γ α + β ( W ) : ,∂ z Γ α,β ( Z, W ) | Z = W =: ( D W α ( W ))Γ α + β ( W ) : − : α ( W ) : α ( W )Γ α + β ( W ) :: D | Z Γ α,β ( Z, W ) | Z = W =: ∂ w α ( W )Γ α + β ( W ) : ++ : α ( W ) : ( D W α ( W ))Γ α + β ( W ) :: − : α ( W ) : α ( W ) : α ( W )Γ α + β ( W ) ::: . We obtain that for ( α, α ) = 2 we have:(3.7) [Γ α Λ Γ − α ] = T α + : αSα : + χSα + λα + λχ, where we have used the fact that : αα := 0.Expanding the fields Γ ± α ( Z ) = e ± ( z ) + θψ ± ( z ) and α ( Z ) = ψ ( z ) + θh ( z ), wefind the commutation relations (cf. Appendix A):(3.8)[ e + λ e − ] = h + λ, [ e ± λ ψ ∓ ] = − : ψ h : ∓ λψ , [ ψ + λ ψ − ] = − T h − : hh : + : ψ T ψ : − λh − λ , [ h λ e ± ] = ± e ± , [ h λ ψ ± ] = ± ψ ± , [ ψ λ ψ ± ] = ± e ± , [ ψ λ e ± ] = 0 , [ h λ h ] = 2 λ, [ ψ λ ψ ] = 2 , [ h λ ψ ] = 0 . With respect to the natural conformal structure defined above, the field ψ hasconformal weight 1 /
2, the fields h, e ± have conformal weight 1 and generate thecurrent sl algebra at level 1, commuting with the fermion ψ , and the fields ψ ± have conformal weight 3 / J = h, J ± = e ± , Φ = 1 √ ψ , ˜ G = − √ ψ h : , ˜ L = 14 : hh : −
14 : ψ T ψ : , ˜ G − = − √ ψ − , ˜ G + = 1 √ ψ + . It follows that these fields satisfy the commutation relations of the N = 3 supervertex algebra as defined in [KW04, sec. 8.5], the central charge is ˜ c = 3 / osp (3 | k = − /
4. This is computed explicitly in Appendix A.
Example 3.5.
Consider the rank 2 lattice generated by vectors α ± such that( α ± , α ± ) = ± α ± , α ∓ ) = 0. We define ε following [Kac96, 5.5] ε ( α ± , α ± ) = ± , ε ( α ± , α ∓ ) = ± , and extend by bimultiplicativity to Q .We have the operators Γ ± α ± satisfying the commutation relations[Γ ± α + Λ Γ ± α + ] = 0 , [Γ α + Λ Γ − α + ] = α + + χ, [Γ ± α − Λ Γ ∓ α − ] = 0 , [Γ α − Λ Γ α − ] = − (cid:18) χ + 12 S (cid:19) Γ α − . Example 3.6.
In this example we let W be the Cartan algebra of sl (2 |
1) with itsnon-degenerate pairing. Namely, we will consider the rank 2 lattice generated bytwo elements α and β , and the bilinear form is such that ( α, α ) = ( β, β ) = 0 and( α, β ) = ( β, α ) = −
1. We let ε ( α, β ) = − ε ( β, α ) = 1. We have the corresponding USY LATTICE VERTEX ALGEBRAS 13 vertex operators Γ ± α and Γ ± β which are even, as well as the operator Γ α + β whichis also even. They satisfy the commutation relations:[Γ α Λ Γ β ] = χ Γ α + β + : α Γ α + β : . Note that Γ α + β does not commute with itself. Indeed, it satisfies[Γ α + β Λ Γ α + β ] = −
12 (
T S + χT + λS + 2 λχ )) Γ α + β ) − : ( α + β ) T Γ α + β ) : . Representation Theory
The representation theory of SUSY lattice vertex algebras parallels the non-SUSY case, here we sketch a guideline: • The definition of a module over a SUSY VA is verbatim the definition inthe usual case, namely a super vector space M together with an associationto any vector a ∈ V , a superfield Y M ( a, Z ) with values in End( M ). Suchthat to the vacuum vector we associate the identity in M and that ( j | J )-th products are preserved. The notion of positive energy follows through,namely if the SUSY VA V is conformal, we require the operator L M to actdiagonally, with eigenvalues bounded below, and with finite dimensionaleigenspaces. • For a lattice Q and its dual Q ∗ , we construct a C ε [ Q ]-module C ε ∗ [ Q ∗ ] inthe usual way, by defining ε ∗ ( α, µ + β ) = ε ( α, β ) , α, β ∈ Q, µ ∈ Q ∗ . • We can construct then a V superQ -module M as follows: as a vector spacewe declare M = V ( W ) ⊗ C C ε ∗ [ Q ∗ ]. The operators h M ( n | j ) act, in the usualway, on the first factor for ( n | j ) = (0 |
0) and h M (0 | ( s ⊗ e λ ) = ( h, λ ) s ⊗ e λ , λ ∈ Q ∗ . The vertex operators Γ Mα ( Z ) are defined as in (2.7), where e α acts on C ε ∗ [ Q ∗ ] and the operators α ( n | j ) are replaced by the corresponding op-erators α M ( n | j ) (recall also that d α = 0). • As in the usual case, M decomposes as a sum of irreducible modules V super µ + Q = V ( W ) ⊗ C C ε ∗ [ µ + Q ] , µ + Q ∈ Q ∗ /Q. The proof of irreducibility reduces to analizing the action of h M (0 | , namely,if u = P s i ⊗ e λ i with pairwise distinct λ i , belongs to a non-zero V µ + Q sub-module U , we act diagonally by h M (0 | and since the Vandermonde matrixis invertible, we see that each s i ⊗ e λ i ∈ U . Now V ( W ) is irreducible (re-call it is the usual boson-fermion system) therefore | i⊗ e λ i ∈ U . ApplyingΓ Mβ ( Z ) we obtain that ( | i ⊗ e λ + β ) ∈ U for all β ∈ Q , and these generate V µ + Q . • As a corollary, V superQ is simple. Given the examples above we see that V super Z where Z is generated by α with ( α, α ) = m is the (irred. quotient ofthe) super affine algebra for ˆ sl at level 2 when m = 1, and it is the (irred.quotient of the) N = 3 super vertex algebra at central charge 3 / m = 2. • Comparing the action of h M (0 | we see that all modules V super µ + Q are non-isomorphic, and if Q is a positive lattice we see that they are positiveenergy modules. • The proof that V Q is rational and that irreducible representations areparametrized by Q ∗ /Q now follows exactly like in the non-SUSY case.4.1. Character formulas.
Now let Q be an even, integral lattice. For a coset λ ∈ Q ∗ /Q , let V λ be the corresponding module over V superQ . If Q is of rank 1 then V λ has for basis monomials of the form:(4.1) α ( − j m | . . . α ( − j | α ( − k n − / | . . . α ( − k − / | | i ⊗ | γ >, γ ∈ λ + Q where 1 ≤ j ≤ · · · ≤ j m are integers, and 1 / ≤ k < · · · < k n are half-integers.The general case is similar by fixing a basis { α i } for W . Let τ be a coordinate ofthe upper half plane, q = e πiτ , let us assume that z ∈ W , and let u be a complexparameter. We define the full character of a module M over V superQ to be(4.2) χ M ( τ, z, u ) = e πiu tr M e πiz (0 | q L M − c/ , where L M is the energy operator (recall that V superQ is a conformal N K = 1 SUSYvertex algebra) and c = r is the central charge. Similarly, we define the superchar-acter of M replacing the trace by the supertrace in (4.2). Let p ( j ) be the numberof partitions of the integer j without repeated odd parts. Similarly, let sp ( j ) bethe number of partitions of the integer j , without repeated odd parts and an evennumber of odd parts, minus the number of such partitions with odd number of oddparts. It is easy to show that the generating functions for p ( j ) and sp ( j ) are givenby ∞ X j =1 q j p ( j ) = ∞ Y j =1 (1 + q j − )(1 − q j ) , ∞ X j =1 q j sp ( j ) = ∞ Y j =1 (1 − q j − )(1 − q j ) . Using this, we see that the characters χ λ and the supercharacters χ sλ correspondingto V λ , λ ∈ Q ∗ /Q are described by:(4.3) χ λ ( τ, z, u ) = (cid:18) η ( τ ) η (2 τ ) η ( τ / (cid:19) r Θ Qλ ( τ, z, u ) ,χ sλ ( τ, z, u ) = (cid:18) η ( τ / η ( τ ) (cid:19) r Θ Qλ ( τ, z, u ) , where Θ Qλ are the classical Θ functions defined asΘ Qλ ( τ, z, u ) = e πiu X γ ∈ λ + Q q ( γ,γ )2 e πi ( z,γ ) , and η ( τ ) is the Dedekind eta function: η ( τ ) = q / ∞ Y j =1 (1 − q j ) . USY LATTICE VERTEX ALGEBRAS 15
Recall that these functions satisfy the following modular transformation properties (we omit the superscript Q where no confusion can arise):Θ λ (cid:18) − τ , zτ , u − ( z, z )2 τ (cid:19) = ( − iτ ) r/ | Q ∗ /Q | − / X λ ′ ∈ Q ∗ /Q e − πi ( λ,λ ′ ) Θ λ ′ ( τ, z, u ) , Θ λ ( τ + 1 , z, u ) = e iπ ( λ,λ ) Θ λ ( τ, z, u ) , and η (cid:18) − τ (cid:19) = ( − iτ ) / η ( τ ) , η ( τ + 1) = e πi/ η ( τ ) . We note also: η (cid:18) τ (cid:19) = e πi/ η ( τ ) η (2 τ ) η ( τ / . It follows that the characters (4.3) satisfy: χ λ (cid:18) − τ , zτ , u − ( z, z )2 τ (cid:19) = | Q ∗ /Q | − / X λ ′ ∈ Q ∗ /Q e − πi ( λ,λ ′ ) χ λ ′ ( τ, z, u ) χ λ ( τ + 1 , z, u ) = e πi ( λ,λ ) − πir/ χ sλ ( τ, z, u ) χ sλ ( τ + 1 , z, u ) = e πi ( λ,λ ) − πir/ χ λ ( τ, z, u ) . To analyse the action of S = ( τ
7→ − /τ ) ∈ SL (2 , Z ) on the supercharacters, weneed to consider the Ramond sector for each of the modules V λ . For this considerthe the automorphism σ of V superQ given by v ( − p ( v ) v . For each module V λ ,let us call the corresponding σ -twisted module V tw λ . If r = 1, the bases for thesemodules are of the form:(4.4) α − j m | . . . α − j | α − k n | . . . α − k | | i ⊗ | γ >, γ ∈ λ + Q, where 1 ≤ j ≤ · · · ≤ j m and 0 ≤ k < · · · < k n are integers (note that wesupressed the parenthesis in the subscripts of the creation operators since these aretwisted fields). For lattices of general rank, the basis is computed similarly. Themonomial in (4.4) has parity ( − n . To compute the energy, first we note thatfor a free fermion system generated by odd fields [ ψ λ ψ ′ ] = ( ψ, ψ ′ ), we consider thecorresponding twisted fields ψ tw ( z ) = X m ∈ Z ψ tw m z − / − m , with commutation relations [ ψ tw m , ψ ′ tw n ] = ( ψ, ψ ′ ) δ m, − n . We have the twisted Virasoro L tw ( z ) = P i : h i ( z ) h i ( z ) : − : ψ i ( z ) ∂ z ψ i ( z ) : tw ,where { ψ i } and { ψ i } are dual bases of W and similarly { h i } and { h i } are dualbases of Π W . It is easy to show that this is equal to: L tw ( z ) = 12 r X i =1 : h i ( z ) h i ( z ) : − r X i =1 : ψ tw i ( z ) ∂ z ψ i tw ( z ) : − r z − . Note that σ is a vertex algebra automorphism but it is not a SUSY vertex algebraautomorphism. It follows that the monomial (4.4) has energy: r
16 + ( γ, γ )2 + n X i =1 k i + m X i =1 j i . The general even integral lattice case is calculated similarly. It follows that thecharacters of these modules (clearly the supercharacters vanish), are given by (wespecialize the Θ function to z = u = 0): χ tw λ ( τ ) = 2 r η (2 τ ) η ( τ ) Θ λ ( τ ) , the factor 2 r appears since all the creation operators α | preserve the energy. Weobtain therefore χ sλ ( − τ ) = | Q ∗ /Q | − / r/ X λ ′ ∈ Q ∗ /Q e − πi ( λ,λ ′ ) χ tw λ ′ ( τ ) . Finally we find χ tw λ ( τ + 1) = e πi ( λ,λ ) χ tw λ ( τ ) , and therefore we arrive to the main theorem of this section: Theorem 4.1.
For each λ ∈ Q ∗ /Q , the linear span of the characters χ λ ( τ ) , su-percharacters χ sλ ( τ ) and twisted characters χ tw λ ( τ ) is invariant under the action of SL (2 , Z ) . They satisfy: χ λ ( τ + 1) = e πi ( λ,λ ) − πir/ χ sλ ( τ ) ,χ sλ ( τ + 1) = e πi ( λ,λ ) − πir/ χ λ ( τ ) ,χ tw λ ( τ + 1) = e πi ( λ,λ ) χ tw λ ( τ ) , and χ λ (cid:18) − τ (cid:19) = | Q ∗ /Q | − / X λ ′ ∈ Q ∗ /Q e − πi ( λ,λ ′ ) χ λ ′ ( τ ) ,χ sλ ( − τ ) = | Q ∗ /Q | − / (2) r/ X λ ′ ∈ Q ∗ /Q e − πi ( λ,λ ′ ) χ tw λ ′ ( τ ) .
5. V superQ as a vertex algebra
From the character formulas above, we guess that these SUSY lattice vertexalgebras are just tensor products of the usual (non-SUSY) lattice vertex algebrasand free fermions. To study this phenomena, recall that given a SUSY vertexalgebra V , we can view V as a vertex algebra by putting θ = 0 in all superfields. Inparticular, the SUSY vertex algebra associated to an integral lattice Q is generated(as a usual vertex algebra) by the fermions ¯ h ( z ) = h ( z, h ( z ) =( Sh )( z,
0) for h ∈ W , and the fields Γ α ( z ) = Γ α ( z,
0) and ¯Γ α ( z ) = ( S Γ α )( z, α ∈ Q . Note first that putting θ = 0 in (2.8) we obtain the usual expression forthe vertex operator Γ α ( z ) (see for example [Kac96, (5.5.9)]). Putting θ = 0 in theidentity S Γ α =: α Γ α : , we obtain that(5.1) ¯Γ α ( z ) =: ¯ h ( z )Γ α ( z ) : . USY LATTICE VERTEX ALGEBRAS 17
In particular, if we let V Q be the vertex algebra (as opposed to SUSY) associatedto the integral lattice Q and F ( W ) be the free fermions generated by odd fields ¯ h , h ∈ W satisfying [¯ h λ ¯ h ′ ] = ( h, h ′ ) , we can construct a surjective morphism V Q ⊗ F ( W ) → V superQ . Indeed this morphismmaps h ∈ V Q h ∈ V superQ , ¯ h ∈ F ( W ) ¯ h ∈ V superQ and Γ α ∈ V Q Γ α ∈ V superQ .To check that this morphism preserves lambda brackets we see that in V superQ :[ h λ h ′ ] = ( h, h ′ ) λ and [¯ h λ ¯ h ′ ] = ( h, h ′ ) which follow from (2.1). Finally, it followsfrom (2.3) that in V superQ : [¯ h λ Γ α ] = 0. To check surjectivity, we only need to showthat the generators of V superQ lie in the image of this morphism. But we alreadyknow that h, ¯ h and Γ α are in the image. It follows from (5.1) that ¯Γ α is in theimage of this morphism. Since both algebras are simple, we obtain an isomorphism V Q ⊗ F ( W ) ≃ V superQ .As a corolary we see that V ( sl , super ) is isomorphic the tensor product of twocharged fermions and one free fermion, and similarly we find that the simple N = 3super vertex algebra at central charge c = 3 / V ( sl ) with a free fermion. Remark . In particular we see that the simple V c ( N = 3) super vertex algebra,at central charge c = 3 / c for which V c ( N = 3) isrational and the list of all of its modules and corresponding characters is known.We note also that taking the centralizer of the fermion Φ in the N = 3 vertexalgebra, we find that the (irred. quotient of the) quantum Hamiltonian reductionof the super Lie algebra osp (3 |
2) as in [KW04, Sec. 8.5] for k = − / V ( sl ). Appendix A. N = 3 vertex algebra at central charge c = 3 / V super Z where Z is generated by α with ( α, α ) = 2. We expand the fields Γ ± α ( Z ) = e ± ( z ) + θψ ± ( z ) and α ( Z ) = ψ ( z ) + θh ( z ). It follows from (2.3) that[ ψ ( z ) + θh ( z ) , e ± ( w ) + ζψ ± ( z )] = ± θ − ζ ) δ ( z, w )( e ± ( w ) + ζψ ± ( w )) . Collecting the terms with θ, ζ and θζ we see that this implies:(A.1a) [ ψ ( z ) , e ± ( w )] = 0 , [ h ( z ) , ψ ± ( w )] = ± δ ( z, w ) ψ ± ( w ) , [ ψ ( z ) , ψ ± ( w )] = ± δ ( z, w ) e ± ( w ) , [ h ( z ) , e ± ( w )] = ± δ ( z, w ) e ± ( w ) . Similarly, it follows from [ α Λ α ] = 2 χ that[ ψ ( z ) + θh ( z ) , ψ ( w ) + ζh ( w )] = 2 δ ( z, w ) + 2 θζ∂ w δ ( z, w ) , and collecting terms we obtain:(A.1b)[ ψ ( z ) , ψ ( w )] = 2 δ ( z, w ) , [ ψ ( z ) , h ( w )] = 0 , [ h ( z ) , h ( w )] = 2 ∂ w δ ( z, w ) . Note also that from [Γ ± α Λ Γ ± α ] = 0 it follows(A.1c) [ e ± ( z ) , e ± ( w )] = 0 , [ ψ ± ( z ) , ψ ± ( w )] = 0 . Finally, we obtain from (3.7)[ e + ( z )+ θψ + ( z ) , e − ( w )+ ζψ − ( w )] = ( θ − ζ ) δ ( z, w ) (cid:0) ∂ w ( ψ ( w )+ ζh ( w ))+ : ψ ( w ) h ( w ) : ++ ζ : h ( w ) h ( w ) : − ζ : ψ ( w ) ∂ w ψ ( w ) (cid:1) ++ ( δ ( z, w ) + θζ∂ w δ ( z, w ))( h ( w ) + ζ∂ w ψ ( w ))++ ( θ − ζ ) ∂ w δ ( z, w )( ψ ( w ) + ζh ( w )) + ( ∂ w δ ( z, w ) + θζ∂ w δ ( z, w )) , and collecting terms as before:(A.1d) [ e + ( z ) , e − ( w )] = δ ( z, w ) h ( w ) + ∂ w δ ( z, w ) , [ ψ + ( z ) , e − ( w )] = δ ( z, w ) (cid:16) ∂ w ψ ( w )+ : ψ ( w ) h ( w ) : (cid:17) + (cid:0) ∂ w δ ( z, w ) (cid:1) ψ ( w ) , [ e + ( z ) , ψ − ( w )] = − δ ( z, w ) : ψ ( w ) h ( w ) : − ( ∂ w δ ( z, w )) ψ ( w ) , [ ψ + ( z ) , ψ − ( w )] = − δ ( z, w ) (cid:0) ∂ w h ( w )+ : h ( w ) h ( w ) : − : ψ ( w ) ∂ w ψ ( w ) : (cid:1) − ( ∂ w δ ( z, w )) h ( w ) − ( ∂ w δ ( z, w )) h ( w ) − ∂ w δ ( z, w ) . Equations (A.1) easily imply equations (3.8).Define now the fields ˜ G , ˜ G ± , ˜ L , J , J ± and Φ as in (3.9). First we note that˜ L is the Virasoro field of central charge 3 / /
2, the fields J , J ± are primary of conformal weight 1, and the fields ˜ G , ˜ G ± are primary ofconformal weight 3 / J λ J ± ] = ± J ± , [ J λ J ] = 2 λ, [ J + λ J − ] = J + λ, [ J + λ ˜ G − ] = − G + 2 λ Φ , [ J − λ ˜ G + ] = ˜ G + λ Φ , [ ˜ G ± λ ˜ G ± ] = 0 , [ ˜ G + λ ˜ G ] = 14 ( T + 2 λ ) J + , [ ˜ G + λ Φ] = 14 J + , [ ˜ G − λ Φ] = 12 J − , (A.2a) [ ˜ G + λ ˜ G − ] = ˜ L + 14 ( T + 2 λ ) J + λ . (A.2b)We have also:(A.2c) [ J λ ˜ G ] = [ h λ − √ ψ h :] = − λ √ ψ = − λ Φ , and similarly(A.2d) [Φ λ ˜ G ] = −
18 [ ψ λ : ψ h :] = − h = − J . USY LATTICE VERTEX ALGEBRAS 19
On the other hand(A.2e) [ ˜ G + λ ˜ G ] = −
18 [ ψ + λ : ψ h :]= − e + h : +2 : ψ ψ + : +2 Z λ [ e + µ h ] dµ ! = − (cid:0) : e + h : + : ψ ψ + : − λe + (cid:1) = −
14 (: he + : + : ψ ψ + : − T + λ ) e + )= 14 ( T + 2 λ ) J + −
14 : ψ ψ + := 14 ( T + 2 λ ) J + , where we have used quasi-commutativity in the 4th line and the fact that : he + := T e + and : ψ ψ + := 0 in the last line. Both identities follow from S Γ α =: α Γ α :and : αα := 0. Indeed: αα := − : αα : + Z −∇ [ α Λ α ] d Λ = − : αα : , and therefore we obtain by quasi-associativity: αS Γ α :=: α : α Γ α ::=:: αα : Γ α := 0 . It follows now ∂ z e + ( z ) + θ∂ z ψ + ( z ) = T Γ α ( Z ) = S Γ α ( Z ) = S : α ( Z )Γ α ( Z ) :==: ( Sα )( Z )Γ α ( Z ) :=: h ( z ) e + ( z ) : + θ (cid:16) : ( ∂ z ψ ( z )) e + ( z ) : + : h ( z ) ψ + ( z ) : (cid:17) , from where T e + =: he + :. We also have0 =: α ( Z ) S Γ α ( Z ) :=: ψ ( z ) ψ + ( z ) : + θ (cid:16) : h ( z ) ψ + ( z ) : − : ψ ( z ) ∂ z e + ( z ) : (cid:17) , from where : ψ ψ + := 0.In a similar fashion:(A.2f) [ ˜ G − λ ˜ G ] = 14 [ ψ − λ : ψ h :]= − e − h : − ψ ψ − : +2 Z λ [ e − µ h ] dµ ! = − (cid:0) : he − : +2 T e − + 2 λe − (cid:1) = −
12 ( T + 2 λ ) J − . We need to compute [ ˜ G λ ˜ G ], for this we compute first [ ψ λ : ψ h :] = 2 h and[ h λ : ψ h :] = 2 λψ , from where(A.2g) [ ˜ G λ ˜ G ] = 18 hh : − ψ ( λ + T ) ψ : +2 Z λ [ h µ h ] dµ ! = 14 (cid:0) : hh : − : ψ T ψ : + λ (cid:1) = ˜ L + λ , and according to [KW04, pp. 41] equations (A.2) are the commutation relationsfor the generators of the N = 3 super vertex algebra at central charge c = 3 / References [HK07] R. Heluani and V.G. Kac. Supersymmetric vertex algebras.
Communications in mathe-matical physics , 271:103–178, 2007.[Kac96] V. G. Kac.
Vertex algebras for beginners , University Lecture Series, vol. 10. AmericanMathematical Society, 1996. Second edition 1998.[KW04] V. G. Kac and M. Wakimoto. Quatum reduction and representation theory of supercon-formal algebras.
Adv. Math. , 185(2):400–458, 2004.[SS87] A. Schwimmer and N. Seiberg. Comments on the n = 2 , , Phys. Lett. B , 184:191–196, 1987.
Department of Mathematics, UC Berkeley, Berkeley, CA 94720, USA
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