Intertwining operators among modules for affine Lie algebra and lattice vertex operator algebras which respect integral forms
aa r X i v : . [ m a t h . QA ] O c t Intertwining operators among modules for affine Liealgebra and lattice vertex operator algebras whichrespect integral forms
Robert McRae
Abstract
We define an integral intertwining operator among modules for a vertex operator al-gebra to be an intertwining operator which respects integral forms in the modules,and we show that an intertwining operator is integral if it is integral when restrictedto generators of the integral forms in the modules. We apply this result to classifyintegral intertwining operators which respect certain natural integral forms in modulesfor affine Lie algebra and lattice vertex operator algebras.
Integral forms for lattice vertex algebras were first introduced by Borcherds in [B] and werealso studied in [P]. They have been used in the modular moonshine program initiated byBorcherds and Ryba ([R], [BR1], [BR2], [GL]). Recently, integral forms in vertex algebrasrelated to lattice vertex algebras, including automorphism fixed points and the moonshinemodule V ♮ , have been constructed in [DG] and [GL], and their automorphism groups havebeen studied. The representation theory of integral vertex algebras based on affine Liealgebras and lattices was studied in [M2]: in particular, integral forms were constructed inboth vertex algebras and their modules. It was also determined that the graded Z -duals ofintegral forms in modules for these algebras form natural integral forms in the contragredientmodules.In the present paper, we continue to study the representation theory of integral vertexoperator algebras in general, and in particular the representation theory of integral affineLie algebra and lattice vertex operator algebras, by showing when intertwining operatorsamong modules respect integral forms. Intertwining operators play an essential role in therepresentation theory of vertex operator algebras. For instance associativity of intertwin-ing operators ([H1], [HLZ3]) and modular invariance for traces of products of intertwiningoperators ([H2]) are crucial for showing that modules for certain vertex operator algebrasform modular tensor categories ([H3]; see also the review article [HL2]). Thus it is naturalto consider intertwining operators among modules for vertex algebras over Z .Since also vertex algebras over Z lead to vertex algebras over finite fields (and moregenerally over arbitrary fields of prime characteristic) through reducing structure constantsmod a prime p , intertwining operators among modules for vertex algebras over Z lead to1ntertwining operators among modules for vertex algebras over fields of characteristic p .Intertwining operators among modules for the Virasoro vertex operator algebra L ( ,
0) overfields of odd prime characteristic have already been studied in [DR]. The present paper willhave immediate application to intertwining operators among modules for affine Lie algebraand lattice vertex operator algebras in prime characteristic.If V is a vertex operator algebra and W ( i ) for i = 1 , , V -modules, thenan intertwining operator of type (cid:0) W (3) W (1) W (2) (cid:1) is a linear map Y : W (1) ⊗ W (2) → W (3) { x } that satisfies a lower truncation axiom, an L ( − W (3) { x } represents formal series with arbitrary complex powers and coefficients in W (3) . If V and the modules W ( i ) have integral forms V Z and W ( i ) Z , respectively, then it isnatural to consider which intertwining operators among these modules respect the integralforms. In particular, it is natural to say that an intertwining operator of type (cid:0) W (3) W (1) W (2) (cid:1) isintegral with respect to the integral forms W ( i ) Z if Y ( w (1) , x ) w (2) ∈ W (3) Z { x } (1.1)for w (1) ∈ W (1) Z and w (2) ∈ W (2) Z .The main general result of this paper is that in order to check that an intertwiningoperator of type (cid:0) W (3) W (1) W (2) (cid:1) is integral, it is enough to check that (1.1) holds when w (1) and w (2) come from generating sets of W (1) Z and W (2) Z , respectively. This result fits with thegeneral philosopy of [M2] that integral forms of vertex operator algebras and modules areoften best studied using generating sets. In the remainder of the paper, we apply the generaltheorem to vertex operator algebras coming from affine Lie algebras and lattices; naturalintegral forms in modules for these algebras were obtained in [M2].Interestingly, the examples studied in this paper suggest another notion of integralityfor intertwining operators different from (1.1). Specifically, suppose that V is an affine Liealgebra or lattice vertex operator algebra with modules W ( i ) for i = 1 , ,
3, and supposethat V Z and W ( i ) Z are the integral forms in V and W ( i ) , respectively, that were constructedin [M2]. Then we are able to find and classify non-zero intertwining operators which areintegral with respect to W (1) Z , W (2) Z , and (( W (3) ) ′ Z ) ′ . Here (( W (3) ) ′ Z ) ′ is the graded Z -dual ofthe natural integral form in the contragredient module ( W (3) ) ′ ; this is an integral form of W (3) which is often larger than W (3) Z . In other words, we are able to classify the intertwiningoperators of type (cid:0) W (3) W (1) W (2) (cid:1) which satisfy h w ′ (3) , Y ( w (1) , x ) w (2) i ∈ Z { x } (1.2)when w (1) , w (2) , and w ′ (3) come from the natural integral forms of W (1) , W (2) , and ( W (3) ) ′ ,respectively. This suggests that (1.2) is a more fundamental integrality condition than (1.1),a notion supported by the fact that (1.2) corresponds to an integrality condition on physicallyrelevant correlation functions in conformal field theory.The remainder of this paper is structured as follows. In the next section we recall thenotions of integral form in a vertex operator algebra and in a module for a vertex operator2lgebra from [M2] and recall some results on integral forms in contragredient modules. InSection 3, we recall the definition of intertwining operator among modules for a vertexoperator algebra and define what it means for an intertwining operator to be integral withrespect to integral forms in the modules. We also prove our general result that an intertwiningoperator is integral if it is integral when restricted to generators. Finally, we derive a wayto use intertwining operators to identify a module with its contragredient; this will be usedin the last section when we study integral intertwining operators for lattice vertex algebras.In Section 4 we recall the construction of the level ℓ generalized Verma module vertexoperator algebra V b g ( ℓ,
0) based on an affine Lie algebra b g , where g is a finite-dimensionalsimple complex Lie algebra. We also recall the classification of irreducible V b g ( ℓ, L b g ( ℓ,
0) of V b g ( ℓ, L b g ( ℓ, ℓ . In Section 5, we recall the natural integral forms in V b g ( ℓ, L b g ( ℓ, L b g ( ℓ, ℓ is anonnegative integer.In Section 6, we recall the construction of the vertex algebra V L and its modules from aneven nondegenerate lattice L and recall from [DL] the construction of intertwining operatorsamong V L -modules. In Section 7, we recall from [M2] the natural integral form in a V L -moduleand use nondegenerate invariant bilinear pairings to explicitly describe the graded Z -dualof the natural integral form of a V L -module. Using this, we classify integral intertwiningoperators among V L -modules. Acknowledgments
This paper is part of my thesis [M1], completed at Rutgers University.I am very grateful to my advisor James Lepowsky for all of his advice and encouragement,and to Yi-Zhi Huang for comments on this paper.
Throughout this paper, we will use standard terminology and notation from the theory ofvertex operator algebras, sometimes without comment. In particular, we will use the notionof vertex algebra as defined in [B] and the notion of vertex operator algebra as defined in[FLM] (see also [LL]). We will also use the notion of module for a vertex (operator) algebra asdefined in [LL] (where modules for a vertex operator algebra have a conformal weight gradingby C ). We recall that a vertex operator algebra and its modules admit representations ofthe Virasoro Lie algebra, and as usual we use L ( n ) to denote the action of Virasoro algebraoperators on a vertex operator algebra and its modules.Vertex algebras over Z make sense, and we refer to them as vertex rings. As in [M2], wedefine an integral form in a vertex operator algebra as follows: Definition 2.1.
An integral form of a vertex operator algebra V is a vertex subring V Z ⊆ V that is an integral form of V as a vector space and that is compatible with the conformal eight grading of V : V Z = a n ∈ Z V n ∩ V Z , where V n is the weight space with L (0) -eigenvalue n . Remark 2.2.
Equivalently, an integral form V Z of V is the Z -span of a basis for V whichcontains the vacuum , is closed under vertex algebra products, and is compatible with theconformal weight gradation. Remark 2.3.
In [DG], the definition of an integral form V Z of V is somewhat different:compatibility with the weight gradation is replaced by the requirement that an integralmultiple of the conformal vector ω be in V Z . In the examples studied in this paper, bothconditions hold.Once we have fixed an integral form V Z of a vertex operator algebra V , it is easy to definethe notion of an integral form in a V -module: Definition 2.4.
An integral form in a V -module W is a V Z -submodule W Z ⊆ W that isan integral form of W as a vector space and that is compatible with the conformal weightgrading of W : W Z = a h ∈ C W h ∩ W Z , where W h is the weight space with L (0) -eigenvalue h . Remark 2.5.
Equivalently, an integral form W Z of W is the Z -span of a basis for W which is preserved by vertex operators from V Z and is compatible with the conformal weightgradation. Remark 2.6.
Note that the notion of an integral form of W depends on the integral form V Z of V that is used.Here we note that our approach to studying integral forms in vertex operator algebrasand their modules uses generating sets. If V Z is an integral form of a vertex operator algebra V , we say that S ⊆ V Z generates V Z if V Z is the smallest vertex subring of V containing S .Similarly, if W is a V -module with integral form W Z , we say that T ⊆ W Z generates W Z if W Z is the smallest V Z -submodule of W containing T .For our study of intertwining operators in this paper, we will also need to use contra-gredients of modules for a vertex operator algebra V . Recall from [FHL] that if W is a V -module, the contragredient V -module W ′ is the graded dual W ′ = a h ∈ C W ∗ h equipped with the vertex operator defined by h Y W ′ ( v, x ) w ′ , w i = h w ′ , Y oW ( v, x ) w i v ∈ V , w ∈ W , and w ′ ∈ W ′ . Here Y oW denotes the opposite vertex operator Y oW ( v, x ) = Y ( e xL (1) ( − x − ) L (0) v, x − ) . (2.1)Recall also from [FHL] that for any V -module W , W ∼ = ( W ′ ) ′ , so that it makes sense to referto contragredient pairs of V -modules.If V Z is an integral form of V and W is a V -module with integral form W Z , then thegraded Z -dual of W Z defined by W ′ Z = { w ′ ∈ W ′ | h w ′ , w i ∈ Z for w ∈ W Z } is an integral form of W ′ as a vector space. The following two propositions from [M2] (seealso Lemma 6.1, Lemma 6.2, and Remark 6.3 in [DG]) show when W ′ Z is a V Z -module: Proposition 2.7.
Suppose V Z is preserved by L (1) n n ! for n ≥ . Then W ′ Z is preserved by theaction of V Z . Proposition 2.8. If V Z is generated by vectors v such that L (1) v = 0 , then V Z is preservedby L (1) n n ! for n ≥ . Remark 2.9.
If an integral form V Z is invariant under the operators L (1) n n ! for n ≥ V Z is invariant under the integral form of the universal enveloping algebra of sl = span { L ( − , L (0) , L (1) } with basis given by the ordered products L ( − k k ! · (cid:18) L (0) m (cid:19) · L (1) n n !where k, m, n ≥
0. Note that V Z is automatically invariant under the operators L ( − k k ! because e xL ( − v = Y ( v, x ) for any v ∈ V Z and V Z is invariant under the operaters (cid:0) L (0) m (cid:1) bycompatibility of V Z with the weight gradation. But note that we cannot expect all of theseoperators to preserve integral forms of V -modules because for instance V -modules may notbe graded by integers.Sometimes a vertex operator algebra V is equivalent as a V -module to its contragredient V ′ . From [FHL], this is the case if and only if V has a nondegenerate bilinear form ( · , · ) thatis invariant in the sense that ( Y ( u, x ) v, w ) = ( v, Y o ( u, x ) w )for all u, v, w ∈ V . From [Li1], we know that the space of (not necessarily nondegenerate)invariant bilinear forms on V is linearly isomorphic to V /L (1) V , where as usual V n for n ∈ Z represents the weight space with L (0)-eigenvalue n . If V is a simple vertex operatoralgebra, then any non-zero invariant bilinear form is necessarily nondegenerate.5 Integral intertwining operators
For any vector space or Z -module V , let V { x } denote the space of formal series with complexpowers and coefficients in V . We recall the definition of intertwining operator among a tripleof modules for a vertex operator algebra (see for instance [FHL]): Definition 3.1.
Suppose V is a vertex operator algebra and W (1) , W (2) and W (3) are V -modules. An intertwining operator of type (cid:0) W (3) W (1) W (2) (cid:1) is a linear map Y : W (1) ⊗ W (2) → W (3) { x } ,w (1) ⊗ w (2)
7→ Y ( w (1) , x ) w (2) = X n ∈ C ( w (1) ) n w (2) x − n − ∈ W (3) { x } satisfying the following conditions:1. Lower truncation: For any w (1) ∈ W (1) , w (2) ∈ W (2) and n ∈ C , ( w (1) ) n + m w (2) = 0 for m ∈ N sufficiently large.2. The Jacobi identity: x − δ (cid:18) x − x x (cid:19) Y W (3) ( v, x ) Y ( w (1) , x ) − x − δ (cid:18) x − x − x (cid:19) Y ( w (1) , x ) Y W (2) ( v, x )= x − δ (cid:18) x − x x (cid:19) Y ( Y W (1) ( v, x ) w (1) , x )(3.1) for v ∈ V and w (1) ∈ W (1) .3. The L ( − -derivative property: for any w (1) ∈ W (1) , Y ( L ( − w (1) , x ) = ddx Y ( w (1) , x ) . (3.2) Remark 3.2.
We use V W (3) W (1) W (2) to denote the vector space of intertwining operators of type (cid:0) W (3) W (1) W (2) (cid:1) , and the dimension of V W (3) W (1) W (2) is the fusion rule N W (3) W (1) W (2) Remark 3.3.
Note that if W is a V -module, the vertex operator Y W is an intertwiningoperator of type (cid:0) WV W (cid:1) . The Jacobi identity (3.1) with v = ω implies that for w (1) ∈ W (1) and w (2) ∈ W (2) both homogeneous,wt ( w (1) ) n w (2) = wt w (1) + wt w (2) − n − n ∈ C . 6 emark 3.4. If W (1) , W (2) and W (3) are irreducible V -modules, then there are complexnumbers h i for i = 1 , , W ( i ) are contained in h i + N for each i . If Y is an intertwining operator of type (cid:0) W (3) W (1) W (2) (cid:1) and we set h = h + h − h , then (3.3) implies that we can write Y ( w (1) , x ) w (2) = X n ∈ Z w (1) ( n ) w (2) x − n − h (3.4)for any w (1) ∈ W (1) and w (2) ∈ W (2) , where w (1) ( n ) = ( w (1) ) n + h − for n ∈ Z . In particular,for w (1) ∈ W (1) h and W (2) h , note that w (1) (0) w (2) ∈ W (3) h .The Jacobi identity for intertwining operators, like the Jacobi identity for vertex algebrasand modules, implies commutator and iterate formulas. We will in particular need the iterateformula: for any v ∈ V , w (1) ∈ W (1) , and n ∈ Z , Y ( v n w (1) , x ) = Res x ( x − x ) n Y W (3) ( v, x ) Y ( w (1) , x ) − Res x ( − x + x ) n Y ( w (1) , x ) Y W (2) ( v, x ) . (3.5)We will also need weak commutativity for intertwining operators, whose proof is exactly thesame as the proof of weak commutativity for algebras and modules (Propositions 3.2.1 and4.2.1 in [LL]): Proposition 3.5.
Suppose W (1) , W (2) and W (3) are V -modules and Y ∈ V W (3) W (1) W (2) . Thenfor any positive integer k such that v n w (1) = 0 for n ≥ k , ( x − x ) k Y W (3) ( v, x ) Y ( w (1) , x ) = ( x − x ) k Y ( w (1) , x ) Y W (2) ( v, x ) . Proof.
Multiply the Jacobi identity (3.1) by x k and extract the coefficient of x − , obtaining( x − x ) k Y W (3) ( v, x ) Y ( w (1) , x ) − ( x − x ) k Y ( w (1) , x ) Y W (2) ( v, x )= Res x x k x − δ (cid:18) x − x x (cid:19) Y ( Y W (1) ( v, x ) w (1) , x ) . Since v n w (1) = 0 for n ≥ k , the right side contains no negative powers of x , so the residueis 0.We will need the following proposition, which is similar to Proposition 4.5.8 in [LL] anduses essentially the same proof: Proposition 3.6.
Suppose W (1) , W (2) , and W (3) are V -modules and Y ∈ V W (3) W (1) W (2) . Thenfor any v ∈ V , w (1) ∈ W (1) , and w (2) ∈ W (2) , and for any p ∈ C and q ∈ Z , ( w (1) ) p v q w (2) isan integral linear combination of terms of the form v r ( w (1) ) s w (2) . Specifically, let k and m be nonnegative integers such that v n w (1) = 0 for n ≥ k and v n w (2) = 0 for n > m + q . Then ( w (1) ) p v q w (2) = m X i =0 k X j =0 ( − i + j (cid:18) − ki (cid:19)(cid:18) kj (cid:19) v q + i + j ( w (1) ) p − i − j w (2) . roof. From weak commutativity, we have( w (1) ) p v q w (2) = Res x Res x x q x p Y ( w (1) , x ) Y W (2) ( v, x ) w (2) = Res x Res x x q x p ( − x + x ) − k [( x − x ) k Y ( w (1) , x ) Y W (2) ( v, x ) w (2) ]= Res x Res x x q x p ( − x + x ) − k [( x − x ) k Y W (3) ( v, x ) Y ( w (1) , x ) w (2) ] . (3.6)These formal expressions are well defined because Y W (2) ( v, x ) w (2) is lower-truncated, but wecannot remove the brackets from the last expression in (3.6). We observe that the term inbrackets in (3.6) may be written explicitly as X j ∈ N , m ∈ Z , n ∈ C ( − j (cid:18) kj (cid:19) v m ( w (1) ) n w (2) x k − j − m − x j − n − . Meanwhile, only a finite truncation of x q x p ( − x + x ) − k contributes to the residue since Y W (2) ( v, x ) w (2) is lower-truncated. In particular, the lowest possible integral power in x q Y W (2) ( v, x ) w (2) with a non-zero coefficient is x − m − by definition of m . Thus we cantake m X i =0 ( − k + i (cid:18) − ki (cid:19) x q + i x p − k − i as our truncation of x q x p ( − x + x ) − k . Then( w (1) ) p v q w (2) = Res x ,x X i,j,m,n ( − k + i + j (cid:18) − ki (cid:19)(cid:18) kj (cid:19) v m ( w (1) ) n w (2) x q + k + i − j − m − x p − k − i + j − n − = m X i =0 k X j =0 ( − k + i + j (cid:18) − ki (cid:19)(cid:18) kj (cid:19) v q + k + i − j ( w (1) ) p − k − i + j w (2) = m X i =0 k X j =0 ( − i + j (cid:18) − ki (cid:19)(cid:18) kj (cid:19) v q + i + j ( w (1) ) p − i − j w (2) , where we have changed j to k − j in the last equality.The Jacobi identity for intertwining operators makes sense in the vertex ring contextfor the same reasons that the Jacobi identity for algebras and modules does. However, the L ( − V Z is a vertex ring without a confor-mal vector ω , then there may be no L ( −
1) operator on V Z -modules. Additionally, sinceintertwining operators involve complex powers of x , the coefficients of the derivative of anintertwining operator may not make sense as maps from a V Z -module into a V Z -module.However, suppose V is a vertex operator algebra with integral form V Z and W (1) , W (2) ,and W (3) are V -modules with integral forms W (1) Z , W (2) Z , and W (3) Z , respectively: Definition 3.7.
An intertwining operator
Y ∈ V W (3) W (1) W (2) is integral with respect to W (1) Z , W (2) Z , and W (3) Z if for any w (1) ∈ W (1) Z and w (2) ∈ W (2) Z , Y ( w (1) , x ) w (2) ∈ W (3) Z { x } . emark 3.8. Note that whether an intertwining operator is integral or not will generallydepend on the integral forms used for the three V -modules.The main result of this section is the following theorem which reduces the problem ofshowing that an intertwining operator is integral to the problem of showing that it is integralwhen restricted to generators of W (1) and W (2) : Theorem 3.9.
Suppose V is a vertex operator algebra with integral form V Z and W (1) , W (2) ,and W (3) are V -modules with integral forms W (1) Z , W (2) Z , and W (3) Z , respectively. Moreover,suppose T (1) and T (2) are generating sets for W (1) Z and W (2) Z , respectively. If Y ∈ V W (3) W (1) W (2) satisfies Y ( t (1) , x ) t (2) ∈ W (3) Z { x } for all t (1) ∈ T (1) , t (2) ∈ T (2) , then Y is integral with respect to W (1) Z , W (2) Z , and W (3) Z .Proof. First let U (2) Z be the sublattice of W (2) Z consisting of vectors u (2) such that Y ( t (1) , x ) u (2) ∈ W (3) Z { x } for all t (1) ∈ T (1) . Note that if u (2) ∈ U (2) Z , t (1) ∈ T (1) , and v ∈ V Z , Proposition 3.6 impliesthat for any p ∈ C and q ∈ Z , ( t (1) ) p v q u (2) is an integral linear combination of terms of theform v r ( t (1) ) s u (2) , and is thus in W (3) Z . This means that Y W (2) ( v, x ) u (2) ∈ U (2) Z [[ x, x − ]] , for any v ∈ V Z , u (2) ∈ U (2) Z , that is, U (2) Z is a V Z -module. Since by hypothesis U (2) Z contains T (2) which generates W (2) Z , U (2) Z = W (2) Z and so Y ( t (1) , x ) w (2) ∈ W (3) Z { x } (3.7)for any t (1) ∈ T (1) and w (2) ∈ W (2) Z .Now we define U (1) Z as the sublattice of W (1) Z consisting of all vectors u (1) ∈ W (1) Z suchthat Y ( u (1) , x ) w (2) ∈ W (3) Z { x } . for all w (2) ∈ W (2) Z . In this case, U (1) Z is a V Z -submodule of W (1) Z by the iterate formula forintertwining operators, (3.5). Since U (1) Z contains T (1) by (3.7), U (1) Z = W (1) Z , and we haveshown that Y ( w (1) , x ) w (2) ∈ W (3) Z { x } for any w (1) ∈ W (1) Z and w (2) ∈ W (2) Z , that is, Y is integral with respect to W (1) Z , W (2) Z , and W (3) Z . Remark 3.10.
Note the role of Proposition 3.6 in the proof of the theorem. In particular,the commutator formula for intertwining operators (obtained from the coefficient of x − inthe Jacobi identity (3.1)) is not enough by itself without additional assumptions on thegenerating set for W (1) Z . 9e will apply Theorem 3.9 to classify integral intertwining operators among modulesfor affine Lie algebra and lattice vertex operator algebras in the following sections. In bothexamples, the graded Z -dual of a module integral form will also play a role. In the case ofmodules for lattice vertex operator algebras in Section 7, we will need to realize the graded Z -dual explicitly using intertwining operators and Proposition 3.12 below.We need to recall the symmetries among spaces of intertwining operators from [FHL],[HL1], and [HLZ2]. First, if W (1) , W (2) and W (3) are V -modules and Y is an intertwiningoperator of type (cid:0) W (3) W (1) W (2) (cid:1) , then for any r ∈ Z , there is an intertwining operator Ω r ( Y ) oftype (cid:0) W (3) W (2) W (1) (cid:1) defined byΩ r ( Y )( w (2) , x ) w (1) = e xL ( − Y ( w (1) , e (2 r +1) πi x ) w (2) (3.8)for w (1) ∈ W (1) and w (2) ∈ W (2) . Moreover, for any r ∈ Z there is an intertwining operator A r ( Y ) of type (cid:0) ( W (2) ) ′ W (1) ( W (3) ) ′ (cid:1) defined by h A r ( Y )( w (1) , x ) w ′ (3) , w (2) i W (2) = h w ′ (3) , Y or ( w (1) , x ) w (2) i W (3) (3.9)for w (1) ∈ W (1) , w (2) ∈ W (2) , and w ′ (3) ∈ ( W (3) ) ′ , where Y or ( w (1) , x ) w (2) = Y ( e xL (1) e (2 r +1) πiL (0) ( x − L (0) ) w (1) , x − ) w (2) . Then we have (see for example Propositions 3.44 and 3.46 in [HLZ2]):
Proposition 3.11.
For any r ∈ Z , the map Ω r : V W (3) W (1) W (2) → V W (3) W (2) W (1) is a linear iso-morphism with inverse Ω − r − . Moreover, the map A r : V W (3) W (1) W (2) → V ( W (2) ) ′ W (1) ( W (3) ) ′ is a linearisomorphism with inverse A − r − for any r ∈ Z . We say that a bilinear pairing ( · , · ) between V -modules W (1) and W (2) is invariant if( Y W (1) ( v, x ) w (1) , w (2) ) = ( w (1) , Y oW (2) ( v, x ) w (2) )for v ∈ V , w (1) ∈ W (1) , and w (2) ∈ W (2) . It is clear that if there is a nondegenerate invariantbilinear pairing between W (1) and W (2) , then W (1) and W (2) form a contragredient pair. Thenext proposition, which is a minor generalization of Remark 2.9 in [Li1], shows how certainintertwining operators yield nondegenerate invariant pairings between modules: Proposition 3.12.
Suppose V is a vertex operator algebra equipped with a nondegenerateinvariant bilinear pairing ( · , · ) V , and suppose W (1) and W (2) are V -modules. If Y is anintertwining operator of type (cid:0) VW (1) W (2) (cid:1) , then the bilinear pairing ( · , · ) between W (1) and W (2) given by ( w (1) , w (2) ) = Res x ( , Y o ( w (1) , e πi x ) e xL (1) w (2) ) V (3.10) for w (1) ∈ W (1) and w (2) ∈ W (2) is invariant. Moreover, if W (1) and W (2) are irreducible and Y is non-zero, then the pairing is nondegenerate and W (1) and W (2) form a contragredientpair. roof. We consider the intertwining operator Y ′ = Ω ( A ( Y )) of type (cid:0) ( W (2) ) ′ V ∼ = V ′ W (1) (cid:1) . The L ( − Y ′ ( , x ) equals its constant term − . Moreover,the coefficient of x − in the Jacobi identity (3.1) implies that Y ( W (2) ) ′ ( v, x ) − − − Y W (1) ( v, x ) = 0since Y ( v, x ) has no negative powers of x . Thus − = ϕ Y is a V -homomorphism from W (1) to ( W (2) ) ′ and we obtain a bilinear pairing ( · , · ) between W (1) and W (2) given by( w (1) , w (2) ) = h ϕ Y ( w (1) ) , w (2) i for w (1) ∈ W (1) , w (2) ∈ W (2) , which is invariant because ϕ Y is a homomorphism.To show that the invariant pairing ( · , · ) is also given by (3.10), we calculate using thedefinitions of Ω and A from (3.8) and (3.9), and identifying V ∼ = V ′ via ( · , · ) V . For w (1) ∈ W (1) and w (2) ∈ W (2) ,( w (1) , w (2) ) = h ϕ Y ( w (1) ) , w (2) i = Res x x − h Ω ( A ( Y ))( , x ) w (1) , w (2) i = Res x x − h e xL ( − A ( Y )( w (1) , e πi x ) , w (2) i = Res x x − h A ( Y )( w (1) , e πi x ) , e xL (1) w (2) i = Res x x − ( , Y o ( w (1) , e πi x ) e xL (1) w (2) ) V . This proves the first assertion of the proposition.To prove the nondegeneracy of ( · , · ) when W (1) , W (2) are irreducible and Y is non-zero,it is enough to prove that ϕ Y : W (1) → ( W (2) ) ′ is an isomorphism. Since W (1) and W (2) areirreducible, it suffices to prove that ϕ Y is non-zero, that is, Y ′ ( , x ) = 0. But if Y ′ ( , x ) = 0,then the creation property for vertex operator algebras and the iterate formula (3.5) implythat for any v ∈ V , Y ′ ( v, x ) = Y ′ ( v − , x ) = 0 . This is a contradiction because by Proposition 3.11, Y ′ is non-zero if Y is. In this section, we shall recall what we need from the representation theory of vertex operatoralgebras based on affine Lie algebras. We fix a finite-dimensional complex simple Lie algebra g , and let h denote a Cartan subalgebra of g . There is a unique up to scale nondegenerateinvariant bilinear form h· , ·i on g , which is nondegenerate on h and induces a bilinear formon h ∗ . We shall normalize the form h· , ·i on g so that h α, α i = 2for long roots α ∈ h ∗ . Irreducible finite-dimensional g -modules are in one-to-one correspon-dence with dominant integral weights λ ∈ h ∗ .11e now recall the affine Lie algebra b g = g ⊗ C [ t, t − ] ⊕ C k where k is central and all other brackets are determined by[ a ⊗ t m , b ⊗ t n ] = [ a, b ] ⊗ t m + n + m h a, b i δ m + n, k for a, b ∈ g and m, n ∈ Z . Using the notation of [LL] Chapter 6, for any finite-dimensional g -module U and any level ℓ ∈ C , we have the generalized Verma module V b g ( ℓ, U ) on which k acts as the scalar ℓ . The generalized Verma module V b g ( ℓ, U ) has a unique irreduciblequotient L b g ( ℓ, U ). The b g -modules V b g ( ℓ, U ) and L b g ( ℓ, U ) are linearly spanned by vectors ofthe form a ( − n ) · · · a k ( − n k ) u (4.1)for a i ∈ g , n i >
0. Here we use a ( n ) for a ∈ g and n ∈ Z to denote the action of a ⊗ t n on a b g -module.In the case U = C , the trivial one-dimensional g -module, V b g ( ℓ, U ) = V b g ( ℓ,
0) is a vertexalgebra, as is its irreducible quotient L b g ( ℓ, Y ( a ( − , x ) = X n ∈ Z a ( n ) x − n − (4.2)for a ∈ g . When ℓ = − h where h is the dual Coxeter number of g , V b g ( ℓ,
0) and L b g ( ℓ,
0) arealso vertex operator algebras.Let L λ denote the irreducible g -module with highest weight λ . Then for any level ℓ ∈ C ,the irreducible V b g ( ℓ, L ( ℓ, L λ ) for λ a dominant integralweight. Moreover, suppose θ is the highest root of g ; if ℓ is a nonnegative integer, theirreducible L b g ( ℓ, L ( ℓ, L λ ) where λ is a dominant integral weightsatisfying h λ, θ i ≤ ℓ ([FZ]; see also [LL]).As is well known, the dual of an irreducible g -module L λ is also an irreducible g -modulewith the action given by h x · v ′ , v i = −h v ′ , x · v i for x ∈ g , v ∈ L λ , and v ′ ∈ L ∗ λ . So we have L ∗ λ ∼ = L λ ∗ for some dominant integral weight λ ∗ .In fact, λ ∗ = − w ( λ ) where w is the element in the Weyl group of g of maximal length (seefor example [Hu]). Then the contragredient of the V b g ( ℓ, L b g ( ℓ, L λ ) is isomorphic to L b g ( ℓ, L λ ∗ ) ([FZ]).From now on, we assume that ℓ is a nonnegative integer; suppose that L b g ( ℓ, L λ ), L b g ( ℓ, L λ ), and L b g ( ℓ, L λ ) are irreducible L b g ( ℓ, h λ i , θ i ≤ ℓ for i = 1 , , (cid:0) L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) (cid:1) . First, let A ( L b g ( ℓ, L b g ( ℓ,
0) (see [Z] for thedefinition); from [FZ], as an associative algebra, A ( L b g ( ℓ, ∼ = U ( g ) / h x ℓ +1 θ i , where x θ is a root vector for the longest root θ of g . We also need the A ( L b g ( ℓ, A ( L b g ( ℓ, L λ )), which from [FZ] is given by A ( L b g ( ℓ, L λ )) ∼ = ( L λ ⊗ U ( g )) / h v λ ⊗ x ℓ −h λ ,θ i +1 θ i , v λ is a highest weight vector of L λ and h v λ ⊗ x ℓ −h λ ,θ i +1 θ i indicates the subbimodulegenerated by the indicated element. The A ( L b g ( ℓ, ∼ = U ( g ) / h x ℓ +1 θ i -bimodule structureon ( L λ ⊗ U ( g )) / h v λ ⊗ x ℓ −h λ ,θ i +1 θ i is induced by the following U ( g )-bimodule structure on L λ ⊗ U ( g ): x · ( v ⊗ y ) = ( x · v ) ⊗ y + v ⊗ xy for x, y ∈ U ( g ), v ∈ L λ , and ( v ⊗ y ) · x = v ⊗ yx. We also recall from [FZ] that the lowest weight spaces L λ and L λ of L b g ( ℓ, L λ ) and L b g ( ℓ, L λ ), respectively, are (left) A ( L b g ( ℓ, (cid:0) L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) (cid:1) fromTheorem 2.11 and Corollary 2.13 in [Li2] (see also Theorems 1.5.2 and 1.5.3 in [FZ]) is asfollows: V L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) ∼ = Hom A ( L b g ( ℓ, ( A ( L b g ( ℓ, L λ )) ⊗ A ( L b g ( ℓ, L λ , L λ ) . This space of A ( L b g ( ℓ, Lemma 4.1.
As (left) modules for A ( L b g ( ℓ, ∼ = U ( g ) / h x ℓ +1 θ i , A ( L b g ( ℓ, L λ )) ⊗ A ( L b g ( ℓ, L λ ∼ = ( L λ ⊗ L λ ) /W, with W the U ( g ) / h x ℓ +1 θ i -module generated by all vectors of the form v λ ⊗ x ℓ −h λ ,θ i +1 θ · w for w ∈ L λ .Proof. We know that A ( L b g ( ℓ, λ )) ∼ = ( L λ ⊗ U ( g )) /W ′ , where W ′ is the subbimodule gener-ated by v λ ⊗ x ℓ −h λ ,θ i +1 θ . We first define a mapΦ : ( L λ ⊗ U ( g )) ⊗ U ( g ) L λ → ( L λ ⊗ L λ ) /W as follows: for u = v ⊗ x where v ∈ L λ and x ∈ U ( g ), and for w ∈ L λ , we defineΦ( u ⊗ w ) = v ⊗ x · w + W. By the left U ( g )-module structure on ( L λ ⊗ U ( g )) ⊗ U ( g ) L λ and the tensor product g -modulestructure on ( L λ ⊗ L λ ) /W , it is easy to see that Φ is a U ( g )-homomorphism. Moreover, Φinduces a U ( g ) / h x ℓ +1 θ i -module homomorphism ϕ : ( L λ ⊗ U ( g )) /W ′ ⊗ U ( g ) / h x ℓ +1 θ i L λ → ( L λ ⊗ L λ ) /W because for y, y ′ ∈ U ( g ) and w ∈ L λ ,Φ( y · ( v λ ⊗ x ℓ −h λ ,θ i +1 θ ) · y ′ ⊗ w )= Φ(( y · v λ ) ⊗ x ℓ −h λ ,θ i +1 θ y ′ ⊗ w + v λ ⊗ yx ℓ −h λ ,θ i +1 θ y ′ ⊗ w )= ( y · v λ ) ⊗ ( x ℓ −h λ ,θ i +1 θ y ′ ) · w + v λ ⊗ ( yx ℓ −h λ ,θ i +1 θ y ′ ) · w + W = y · ( v λ ⊗ ( x ℓ −h λ ,θ i +1 θ y ′ ) · w ) + W = 0 .
13o obtain an inverse homomorphism, we define a U ( g )-homomorphismΨ : L λ ⊗ L λ → ( L λ ⊗ U ( g )) /W ′ ⊗ U ( g ) / h x ℓ +1 θ i L λ by defining for u ∈ L λ and w ∈ L λ :Ψ( u ⊗ w ) = ( u ⊗ W ′ ) ⊗ w. The map Ψ induces a U ( g ) / h x ℓ +1 θ i -homomorphism ψ : ( L λ ⊗ L λ ) /W → ( L λ ⊗ U ( g )) /W ′ ⊗ U ( g ) / h x ℓ +1 θ i L λ because for x ∈ U ( g ) and w ∈ L λ ,Ψ( x · ( v λ ⊗ x ℓ −h λ ,θ i +1 θ · w )) = Ψ(( x · v λ ) ⊗ x ℓ −h λ ,θ i +1 θ · w + v λ ⊗ ( xx ℓ −h λ ,θ i +1 θ ) · w )= (( x · v λ ) ⊗ W ′ ) ⊗ x ℓ −h λ ,θ i +1 θ · w + ( v λ ⊗ W ′ ) ⊗ xx ℓ −h λ ,θ i +1 θ · w = (( x · v λ ) ⊗ x ℓ −h λ ,θ i +1 θ + W ′ ) ⊗ w + ( v λ ⊗ xx ℓ −h λ ,θ i +1 θ + W ′ ) ⊗ w = ( x · ( v λ ⊗ x ℓ −h λ ,θ i +1 θ ) + W ′ ) ⊗ w = 0 . From the definitions, it is easy to see that ϕ and ψ are inverses of each other, so they give A ( L b g ( ℓ, ∼ = U ( g ) / h x ℓ +1 θ i -module isomorphisms.From this lemma, we have (see Theorem 3.2.3 in [FZ]): V L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) ∼ = Hom g (( L λ ⊗ L λ ) /W, L λ ) . From [Li2] and [FZ], we can describe the isomorphism as follows. An intertwining operator Y of type (cid:0) L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) (cid:1) induces a g -homomorphism π ( Y ) : L λ ⊗ L λ → L λ given by π ( Y )( w (1) ⊗ w (2) ) = w (1) (0) w (2) (4.3)using the notation of (3.4), where w (1) ∈ L λ ⊆ L b g ( ℓ, L λ ) and w (2) ∈ L λ ⊆ L b g ( ℓ, L λ ). Thishomomorphism π ( Y ) must equal 0 on W , so it induces a homomorphism, which we also call π ( Y ), from ( L λ ⊗ L λ ) /W to L λ . Note that it is not trivial to show that Y 7→ π ( Y ) is alinear isomorphism; the most difficult part is to show that a g -homomorphism f : ( L λ ⊗ L λ ) /W → L λ extends to a (unique) intertwining operator of type (cid:0) L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) (cid:1) .14 Integral intertwining operators among modules foraffine Lie algebra vertex operator algebras
In this section we fix a nonnegative integral level ℓ . We will classify integral intertwiningoperators among L b g ( ℓ, U ( b g ) of b g has an integral form U Z ( b g ): it is the subringof U ( b g ) generated by the vectors ( x α ⊗ t n ) k /k ! for k ≥
0, where n ∈ Z and x α is a root vectorin a Chevalley basis of g ([G], see also [Mi] and [P]). We describe a basis of U Z ( b g ) given in[Mi]. Consider a Chevalley basis { x α , h i } of g , where x α is a root vector corresponding to theroot α and h i is the coroot corresponding to a simple root α i ; suppose that θ is the highestroot of g with respect to the simple roots { α i } . Then b g has an integral basis consisting ofthe vectors x α ⊗ t n , h i ⊗ t n , − h θ ⊗ t + k (5.1)where n ∈ Z . Given any order of this basis, a basis for U Z ( b g ) consists of ordered products ofelements of the following forms:( x α ⊗ t n ) m m ! , (cid:18) h i ⊗ t + m − m (cid:19) , (cid:18) − h θ ⊗ t + k + m − m (cid:19) (5.2)where n ∈ Z and m ≥
0, as well as coefficients of powers of x in series of the formexp X j ≥ ( h i ⊗ t nj ) x j j ! (5.3)for n ∈ Z \ { } .Suppose L λ is a finite-dimensional irreducible g -module where λ is a dominant integralweight of g , and suppose v λ is a highest weight vector of L λ . Then V b g ( ℓ, L λ ) and L b g ( ℓ, L λ )have integral forms V b g ( ℓ, L λ ) Z and L b g ( ℓ, L λ ) Z given by U Z ( b g ) · v λ . Given that L λ is includedin V b g ( ℓ, L λ ) and L b g ( ℓ, L λ ) as their lowest conformal weight spaces (recall (4.1)), we can define( L λ ) Z = L λ ∩ U Z ( b g ) · v λ , a sublattice of both V b g ( ℓ, L λ ) Z and L b g ( ℓ, L λ ) Z which, by (5.2), agrees with the classicalintegral form of a g -module constructed in, for example, [Hu] Chapter 7. In light of the basisfor U Z ( b g ) described above, we have Proposition 5.1.
The integral forms V b g ( ℓ, L λ ) Z and L b g ( ℓ, L λ ) Z are spanned by vectors ofthe form P · u , where u ∈ ( L λ ) Z and P is a product of operators of the form x α ( − n ) m /m ! ,for α a root of g and n, m > , and coefficients of powers of x in series of the form exp X j ≥ h i ( − nj ) x j j ! for n > . roof. This follows immediately from (5.2), (5.3), and an appropriate choice of order on thebasis (5.1).The Lie-algebraic integral forms V b g ( ℓ, L λ ) Z and L b g ( ℓ, L λ ) Z are also vertex algebraic inte-gral forms, as shown by the following two results proved in [M2]: Theorem 5.2.
The integral form V b g ( ℓ, Z is the vertex subring of V b g ( ℓ, generated by thevectors x α ( − k k ! where k ≥ and x α is the root vector corresponding to the root α in thechosen Chevalley basis of g . Moreover, if L λ is a finite-dimensional g -module with highestweight vector v λ and W is V b g ( ℓ, L λ ) or L b g ( ℓ, L λ ) , then W Z is the V b g ( ℓ, Z -module generatedby v λ . Corollary 5.3.
The integral form L b g ( ℓ, Z of L b g ( ℓ, is the integral form of L b g ( ℓ, as avertex algebra generated by the vectors x α ( − k k ! where α is a root and k ≥ . It was also shown in [M2] that the graded Z -dual of a V b g ( ℓ, Z - or L b g ( ℓ, Z -module isalso a V b g ( ℓ, Z - or L b g ( ℓ, Z -module. In particular, suppose that λ is a dominant integralweight of g satisfying h λ, θ i ≤ ℓ where θ is the highest root of g , so that L b g ( ℓ, L λ ) is an L b g ( ℓ, Z -dual of L b g ( ℓ, L λ ) Z is the L b g ( ℓ, Z -module L b g ( ℓ, L λ ) ′ Z ,which is an integral form of L b g ( ℓ, L λ ∗ ) that is generally different from L b g ( ℓ, L λ ∗ ) Z . Remark 5.4.
The precise identity of the integral form L b g ( ℓ, L λ ) ′ Z ⊆ L b g ( ℓ, L λ ∗ ) depends onthe choice of isomorphism L b g ( ℓ, L λ ) ′ ∼ = L b g ( ℓ, L λ ∗ ). Because L b g ( ℓ, L λ ) is irreducible, such anisomorphism amounts to a g -module isomorphism L ∗ λ ∼ = L λ ∗ , in which the Z -dual of ( L λ ) Z is identified with a lattice ( L λ ) ′ Z ⊆ L λ ∗ .We now come to our main theorem on integral intertwining operators among L b g ( ℓ, L b g ( ℓ, L b g ( ℓ, L λ i ) for i = 1 , ,
3. In the statement of the following theorem, we use U Z ( g ) to denote U Z ( b g ) ∩ U ( g ),which by (5.2) has as a basis ordered monomials in elements of the forms x mα m ! , (cid:18) h i + m − m (cid:19) where m ≥ x α = x α ⊗ t and h i = h i ⊗ t . Theorem 5.5.
The lattice of intertwining operators within V L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) which are inte-gral with respect to L b g ( ℓ, L λ ) Z , L b g ( ℓ, L λ ) Z , and L b g ( ℓ, L λ ∗ ) ′ Z is isomorphic to Hom U Z ( g ) ((( L λ ) Z ⊗ Z ( L λ ) Z ) /W Z , ( L λ ∗ ) ′ Z ) where W Z is the U Z ( g ) -submodule of ( L λ ) Z ⊗ Z ( L λ ) Z generated by vectors of the form v λ ⊗ x ℓ −h λ ,θ i +1 θ ( ℓ − h λ , θ i + 1)! · w. Here v λ is a highest weight vector generating ( L λ ) Z as a U Z ( g ) -module, and w ∈ ( L λ ) Z . emark 5.6. Note that ( L λ ∗ ) ′ Z is an integral form of L λ , the lowest conformal weight spaceof L b g ( ℓ, L λ ). Remark 5.7.
We use vectors of the form v λ ⊗ x ℓ −h λ ,θ i +1 θ ( ℓ −h λ ,θ i +1)! · w to generate W Z , ratherthan vectors of the form v λ ⊗ x ℓ −h λ ,θ i +1 θ · w , to avoid unnecessary torsion in the quotient(( L λ ) Z ⊗ Z ( L λ ) Z ) /W Z . Remark 5.8.
Note that this theorem determines which intertwining operators of type (cid:0) L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) L b g ( ℓ,L λ ) (cid:1) satisfy h w ′ (3) , Y ( w (1) , x ) w (2) i ∈ Z { x } for any w (1) ∈ L b g ( ℓ, L λ ) Z , w (2) ∈ L b g ( ℓ, L λ ) Z , and w ′ (3) ∈ L b g ( ℓ, L λ ∗ ) Z , rather than whichintertwining operators satisfy Y : L b g ( ℓ, L λ ) Z ⊗ L b g ( ℓ, L λ ) Z → L b g ( ℓ, L λ ) Z { x } . It is not clear that there will generally be any non-zero intertwining operators which satisfythe latter condition.
Proof.
First suppose an intertwining operator Y is integral with respect to L b g ( ℓ, L λ ) Z , L b g ( ℓ, L λ ) Z , and L b g ( ℓ, L λ ∗ ) ′ Z . Then the map π ( Y ) : w (1) ⊗ w (2) w (1) (0) w (2) given in (4.3) sends ( L λ ) Z ⊗ Z ( L λ ) Z into ( L λ ∗ ) ′ Z and equals zero on W Z ⊆ W = h v λ ⊗ x ℓ −h λ ,θ i +1 θ · w i .Conversely, suppose f : ( L λ ) Z ⊗ Z ( L λ ) Z → ( L λ ∗ ) ′ Z is a U Z ( g )-homomorphism which equals zero on W Z . Then f extends to a U ( g )-homomorphismfrom L λ ⊗ L λ to L λ which equals zero on W . Thus since π given in (4.3) is an isomorphism, f induces a unique intertwining operator Y f : L b g ( ℓ, L λ ) ⊗ L b g ( ℓ, L λ ) → L b g ( ℓ, L λ ) { x } which satisfies π ( Y f ) = f and h w ′ (3) , Y f ( w (1) , x ) w (2) i ∈ Z { x } (5.4)for w (1) ∈ ( L λ ) Z , w (2) ∈ ( L λ ) Z , and w ′ (3) ∈ ( L λ ∗ ) Z . To complete the proof, it is enough toshow that (5.4) holds for all w (1) ∈ L b g ( ℓ, L λ ) Z , w (2) ∈ L b g ( ℓ, L λ ) Z , and w ′ (3) ∈ L b g ( ℓ, L λ ∗ ) Z .Since ( L λ ) Z and ( L λ ) Z generate L b g ( ℓ, L λ ) Z and L b g ( ℓ, L λ ), respectively, as L b g ( ℓ, Z -modules, by Theorem 3.9 it is enough to show that (5.4) holds for w (1) ∈ ( L λ ) Z , w (2) ∈ ( L λ ) Z , and w ′ (3) ∈ L b g ( ℓ, L λ ∗ ) Z . Since (5.4) holds when w ′ (3) ∈ ( L λ ∗ ) Z , by Proposition 5.1 itis enough to show that if it holds for some particular w ′ (3) ∈ L b g ( ℓ, L λ ∗ ) Z , then it also holdsfor the coefficients of powers of y inexp ( x α ( − n ) y ) · w ′ (3) α is a root and n >
0, and for coefficients of powers of y inexp X j ≥ h i ( − nj ) y j j ! · w ′ (3) where n > a ∈ g , m ∈ Z ,and intertwining operator Y , which follows from the Jacobi identity (3.1) by setting v = a ( − and taking the coefficient of x − x − m − : for w (1) ∈ L λ ,[ a ( m ) , Y ( w (1) , x )] = x m Y ( a (0) w (1) , x ) . Then if also w (2) ∈ L λ and m > a ( m ) Y ( w (1) , x ) w (2) = x m Y ( a (0) w (1) , x ) w (2) . (5.5)We will also use the fact that for any L b g ( ℓ, W , w ∈ W , w ′ ∈ W ′ , a ∈ g , and n ∈ Z , h a ( n ) w ′ , w i = h w ′ , − a ( − n ) w i . This follows from (2.1) and (4.2).Now suppose (5.4) holds for w ′ (3) ∈ L b g ( ℓ, L λ ∗ ) Z and n >
0; then by (5.5), h exp( x α ( − n ) y ) · w ′ (3) , Y f ( w (1) , x ) w (2) i = h w ′ (3) , exp( − x α ( n ) y ) Y f ( w (1) , x ) w (2) i = h w ′ (3) , Y f (exp( − x α (0) x n y ) · w (1) , x ) w (2) ∈ Z { x, y } because for any m ≥ x α (0) m m ! · w (1) ∈ ( L λ ) Z . Additionally, for each n > * exp X j ≥ h i ( − nj ) y j j ! · w ′ (3) , Y f ( w (1) , x ) w (2) + = * w ′ (3) , exp X j ≥ − h i ( nj ) y j j ! Y f ( w (1) , x ) w (2) + = * w ′ (3) , Y f (exp − h i (0) X j ≥ ( − j ( − x n y ) j j ! · w (1) , x ) w (2) + = h w ′ (3) , Y f (exp( h i (0)log(1 − x n y )) · w (1) , x ) w (2) i = h w ′ (3) , Y f ((1 − x n y ) h i (0) · w (1) , x ) w (2) i ∈ Z { x, y } because (1 − x n y ) h i (0) = X k ≥ ( − k (cid:18) h i (0) k (cid:19) ( x n y ) k . Note that (cid:0) h i (0) k (cid:1) · w (1) ∈ ( L λ ) Z because λ is a dominant integral weight of g , and thus h i (0)acts on basis elements of ( L λ ) Z as integers. This completes the proof.18 Lattice vertex operator algebras, their modules, andintertwining operators
In this section we recall the construction of vertex operator algebras from even lattices, theirmodules, and intertwining operators among their modules; see [FLM], [DL], and [LL] formore details. We fix a nondegenerate even lattice L with symmetric bilinear form h· , ·i , andthen set h = C ⊗ Z L , an abelian Lie algebra. Thus we can form the Heisenberg vertexoperator algebra M (1) = V b h (1 , L ◦ , the dual lattice of L , which is constructedfrom a central extension of L ◦ by a finite cyclic group. For our purposes here, we may viewthe twisted group algebra C { L ◦ } as the associative algebra with basis vectors e β for β ∈ L ◦ with identity e and product e β e γ = ε ( β, γ ) e β + γ . Here ε maps each pair of lattice elements to a root of unity and may be chosen to satisfy ε ( α + β, γ ) = ε ( α, γ ) ε ( β, γ ) , ε ( α, β + γ ) = ε ( α, β ) ε ( α, γ )for all α, β, γ ∈ L ◦ . We will also use the commutator map c of our central extension, whichis defined by e β e γ = c ( β, γ ) e γ e β for β, γ ∈ L ◦ . Note that from the definitions, c ( β, γ ) = ε ( β, γ ) ε ( γ, β ) − (6.1)for all β, γ ∈ L ◦ . The commutator map c is required to satisfy c ( α, β ) = ( − h α,β i for α, β ∈ L . (Note that L ⊆ L ◦ since L is even, and thus integral.)We define an action of h = h ⊗ t on C { L ◦ } by h (0) e β = h h, β i e β for h ∈ h , β ∈ L ◦ . Also for β ∈ L ◦ and x a formal variable, we define a map x β on C { L ◦ } by x β e γ = e γ x h β,γ i for any γ ∈ L ◦ .For any subset S ⊆ L ◦ , we use C { S } to denote the subspace of C { L ◦ } spanned by thevectors e β for β ∈ S . Then as a vector space, the vertex operator algebra associated to L is V L = M (1) ⊗ C { L } . For α ∈ L , we use the notation ι ( e α ) to denote the element 1 ⊗ e α ∈ V L . The vacuum of V L is = ι ( e ), and the vertex algebra structure is generated by the vectors ι ( e α ) for α ∈ L ,whose vertex operators are Y ( ι ( e α ) , x ) = E − ( − α, x ) E + ( − α, x ) e α x α (6.2)19here E ± ( β, x ) = exp X n ∈± Z + β ( n ) n x − n for any β ∈ L ◦ , and e α denotes the left multiplication action on C { L } . Remark 6.1.
The vertex algebra V L is a vertex operator algebra in the sense of [FLM] onlywhen L is positive definite. If L is not positive definite, V L is a conformal vertex algebra inthe sense of [HLZ1]: there are infinitely many weight spaces of negative conformal weight,and the weight spaces are infinite dimensional. However, all the results of this paper stillhold in this generality.We now recall the irreducible modules for V L . The space V L ◦ = M (1) ⊗ C { L ◦ } is a V L -module, with vertex operators for the ι ( e α ) acting on V L ◦ the same as in (6.2).Moreover, the submodules V β + L = M (1) ⊗ C { β + L } , where β runs over coset representatives of L ◦ /L , exhaust the irreducible V L -modules up toequivalence.We now recall the intertwining operators among V L -modules from [DL]. For any β ∈ L ◦ ,we have an intertwining operator Y β : V β + L ⊗ V L ◦ → V L ◦ { x } , where for any γ ∈ β + L , Y β ( ι ( e γ ) , x ) = E − ( − γ, x ) E + ( − γ, x ) e γ x γ e πiβ c ( · , β ) . (6.3)Here, the operator e πiβ on V L ◦ is given by e πiβ · u ⊗ ι ( e γ ) = e πi h β,γ i u ⊗ ι ( e γ )for u ∈ M (1), γ ∈ L ◦ , and the operator c ( · , β ) is defined by c ( · , β ) · u ⊗ ι ( e γ ) = c ( γ, β ) u ⊗ ι ( e γ ) . Then for any γ ∈ L ◦ , Y β | V β + L ⊗ V γ + L is an intertwining operator of type (cid:0) V β + γ + L V β + L V γ + L (cid:1) . From [DL]we have: Proposition 6.2.
For any α, β, γ ∈ L ◦ , V αβ γ = V V α + L V β + L V γ + L = (cid:26) C Y β | V β + L ⊗ V γ + L if α = β + γ α = β + γ . Remark 6.3.
Note that since for any β ∈ L ◦ , Y β | V β + L ⊗ V − β + L is a non-zero intertwiningoperator of type (cid:0) V L V β + L V − β + L (cid:1) , Proposition 3.12 implies that V ′ β + L ∼ = V − β + L .20 Integral intertwining operators among modules forlattice vertex operator algebras
We continue to fix a nondegenerate even lattice L in this section. We will classify the integralintertwining operators among V L -modules, using the integral forms for V L -modules that wereintroduced in [M2]. (The integral form of V L itself was first introduced in [B].) We recallthese integral forms by stating the following theorem from [M2] (the algebra part was alsoproved in [P] and [DG]): Theorem 7.1.
The vertex subring V L, Z of V L generated by the vectors ι ( e α ) for α ∈ L is anintegral form of V L . Moreover, for any β ∈ L ◦ , the V L, Z -submodule V β + L, Z of V β + L generatedby ι ( e β ) is an integral form of V β + L . Remark 7.2.
For this theorem to hold, we must assume that ε satisfies ε ( α, β ) = ± α, β ∈ L . In fact the proof (although not the statement) of Lemma 4.1 in [M2] showsthat we can choose ε to satisfy the stronger condition ε ( α, β ) = ± α ∈ L or β ∈ L . Remark 7.3.
The proof of Theorem 7.1 that is given in [M2], using the weaker conditionin Remark 7.2, shows that for β ∈ L ◦ , V β + L, Z is the Z -span of coefficients of products of theform E − ( − α , x ) · · · E − ( − α k , x k ) ι ( e α e β )where α , . . . , α k , α ∈ L . If we assume the stronger condition in Remark 7.2, then for any β ∈ L ◦ , V β + L, Z is the Z -span of coefficients of products of the form E − ( − α , x ) · · · E − ( − α k , x k ) ι ( e γ )where α , . . . , α k ∈ L and γ ∈ β + L . Note that the proof of Theorem 7.1 which is given in[M2] incorrectly states that the second description of V β + L, Z follows from the weaker conditionin Remark 7.2. In the rest of this section, we will generally use the first description of V β + L, Z since some of the formulas we need are slightly simpler this way. Remark 7.4.
The integral form V β + L, Z depends on the choice of ε , but once an ε satisfyingthe stronger condition of Remark 7.2 is fixed, the second description of V β + L, Z in Remark 7.3shows that it does not depend on the choice of generator. That is, the V L, Z -module in V β + L generated by ι ( e β ) is the same as the V L, Z -module generated by ι ( e β ′ ) for any β ′ ∈ β + L .In order to obtain integral intertwining operators, we will need an explicit description,as in Remark 7.3, of the graded Z -dual of the integral form of a V L -module. Recalling thatthe contragredient of V β + L is V − β + L , we shall use an invariant bilinear pairing as in (3.10) toidentify V ′− β + L, Z as a sublattice of V β + L, Z . First we calculate (3.10) with Y = Y β and withthe form ( · , · ) V L on V L normalized so that ( , ) V L = 1. Thus for u ∈ V β + L and v ∈ V − β + L ,we have ( u, v ) = Res x ( , ( Y β ) o ( u, e πi x ) e xL (1) v ) V L .
21n particular, (6.3) implies that for γ ∈ β + L , γ ′ ∈ − β + L ,( ι ( e γ ) , ι ( e γ ′ ))= Res x x − ( , e − πi h γ,γ i / x −h γ,γ i E − ( − γ, − x − ) e γ ( e πi x ) −h γ,γ ′ i e πi h β,γ ′ i c ( γ ′ , β ) ι ( e γ ′ )) V L = Res x x − ( , e πi ( h β − γ,γ ′ i−h γ,γ i / x −h γ,γ + γ ′ i E − ( − γ, − x − ) c ( γ ′ , β ) ε ( γ, γ ′ ) ι ( e γ + γ ′ )) V L = e πi h γ − β,γ i / c ( γ, β ) − ε ( γ, γ ) − δ γ + γ ′ , . (7.1)Using (7.1), we see that for any α ∈ L ,( ι ( e α e β ) , ι ( e − α e − β )) = ε ( α, β ) ε ( − α, − β )( ι ( e α + β ) , ι ( e − α − β ))= ε ( α, β ) e πi h α − β,α + β i / c ( α + β, β ) − ε ( α + β, α + β ) − = e πi h α,α i / ε ( α, α ) − e − πi h β,β i / c ( β, β ) − ε ( β, β ) − , since c ( α, β ) = ε ( α, β ) ε ( β, α ) − from (6.1).If we now renormalize the invariant pairing ( · , · ) between V β + L and V − β + L by setting( · , · ) new = e πi h β,β i / c ( β, β ) ε ( β, β )( · , · ) old , we see that now ( ι ( e α e β ) , ι ( e α ′ e − β )) = ( − h α,α i / ε ( α, α ) − δ α + α ′ , = 0 or ± α, α ′ ∈ L , using Remark 7.2. We can use this new invariant bilinear pairing between V β + L and V − β + L , which is the form in (3.10) with Y = e πi h β,β i / c ( β, β ) ε ( β, β ) Y β | V β + L ⊗ V − β + L , to identify V ′− β + L, Z as a sublattice of V β + L, Z , for any β ∈ L ◦ : Proposition 7.5.
For any β ∈ L ◦ , the integral form V ′− β + L, Z is the integral form of V β + L integrally spanned by coefficients of products of the form E − ( − β , x ) · · · E − ( − β k , x k ) ι ( e α e β ) (7.2) where β i ∈ L ◦ and α ∈ L .Proof. Let e V β + L, Z be the integral form of V β + L integrally spanned by coefficients of productsas in (7.2). We first show that e V β + L, Z ⊆ V ′− β + L, Z . For β ∈ L ◦ and n ∈ Z , it is easy to seefrom (2.1) that the adjoint of the operator β ( n ) is − β ( − n ). Thus the adjoint of E − ( − β, x ) = exp X n< − β ( n ) n x − n ! is exp X n< β ( − n ) n x − n ! = exp X n> − β ( n ) n x n ! = E + ( − β, x − ) . E + ( β, x ) E − ( γ, x ) = E − ( γ, x ) E + ( β, x ) (cid:18) − x x (cid:19) h β,γ i for any β, γ ∈ L ◦ . Thus for any β , . . . , β k ∈ L ◦ and α , . . . , α k ′ , α, α ′ ∈ L , (cid:0) E − ( − β , x ) · · · E − ( − β k , x k ) ι ( e α e β ) , E − ( − α , y ) · · · E − ( − α k ′ , y k ′ ) ι ( e α ′ e − β ) (cid:1) = (cid:0) ι ( e α e β ) , E + ( − β , x − ) · · · E + ( − β k , x − k ) E − ( − α , y ) · · · E − ( − α k ′ , y k ′ ) , ι ( e α ′ e − β ) (cid:1) = (cid:0) ι ( e α e β ) , E − ( − α , y ) · · · E − ( − α k ′ , y k ′ ) ι ( e α ′ e − β ) (cid:1) k Y i =1 k ′ Y j =1 (1 − x i y j ) h β i ,α j i = ( − h α,α i / ε ( α, α ) − δ a + α ′ , k Y i =1 k ′ Y j =1 (1 − x i y j ) h β i ,α j i ∈ Z [[ { x i } , { y j } ]] (7.3)since each h β i , α j i ∈ Z . Since V − β + L, Z is the integral span of coefficients of products of theform E − ( − α , y ) · · · E − ( − α k ′ , y k ′ ) ι ( e α ′ e − β )for α , . . . , α k ′ , α ′ ∈ L by Remark 7.3, it follows that e V β + L, Z ⊆ V ′− β + L, Z .Now we must show the opposite inclusion V ′− β + L, Z ⊆ e V β + L, Z . For any β ∈ L ◦ and n > s β,n denote the coefficient of x n in E − ( − β, x ), and for any partition Λ = ( λ , . . . , λ k )where λ ≥ . . . ≥ λ k > h Λ ( β ) = s β,λ · · · s β,λ k . If { α (1) , . . . , α ( l ) } is a basis for L , then V − β + L, Z has a basis consisting of the vectors h Λ ( α (1) ) · · · h Λ l ( α ( l ) ) ι ( e − α e − β ) (7.4)where α ∈ L and Λ , . . . , Λ k run over all partitions (see [M1] and [DG]).Let { β (1) , . . . , β ( l ) } be the basis of L ◦ dual to { α (1) , . . . , α ( l ) } . Then in the basis of V ′− β + L, Z dual to (7.4), the basis vector corresponding to h Λ ( α ( i ) ) ι ( e − α e − β ) for any i , α ∈ L , andpartition Λ is given by ( − h α,α i / ε ( α, α ) m Λ ( α ( i ) ) ι ( e α e β )where m Λ ( α ( i ) ) is a polynomial in the operators β ( i ) ( − n ), n >
0. It follows that the basisvector dual to the vector (7.4) is( − h α,α i / ε ( α, α ) m Λ ( α (1) ) · · · m Λ k ( α ( l ) ) ι ( e α e β ) . Thus it is enough to show that for any i and partition Λ, m Λ ( α ( i ) ) is an integer linearcombination of operators h Λ ′ ( − β ( i ) ) for partitions Λ ′ .We will need to use the well-known monomial and complete symmetric polynomials (seefor example [Mac] Chapter 1, Section 2). Fix a positive integer n . Then for any parti-tion Λ = ( λ , . . . , λ k ) where k ≤ n , the monomial symmetric polynomial in n variables m Λ ( y , . . . , y n ) is the sum of all distinct permutations of the monomial y λ · · · y λ k k . The23olynomials m Λ ( y , . . . , y n ) as Λ runs over partitions with less than or equal to n partsform a basis of the (graded) ring of n -variable symmetric polynomials with integer coeffi-cients. On the other hand, for any m >
0, the complete homogeneous symmetric polynomial h m ( y , . . . , y n ) is the sum of all distinct degree m monomials in n variables; for any partitionΛ = ( λ , . . . , λ k ), we define the polynomial h Λ ( y , . . . , y n ) = h λ ( y , . . . , y n ) · · · h λ k ( y , . . . , y n ) . The polynomials h Λ ( y , . . . , y n ) as Λ runs over all partitions with parts less than or equal to n forms another basis for the ring of n -variable symmetric polynomials. In particular, forany partition Λ of n , m Λ ( y , . . . , y n ) = X k Λ ′ h Λ ′ ( y , . . . , y n ) (7.5)where k Λ ′ ∈ Z and Λ ′ runs over all partitions of n .In light of (7.5), we will show that for any partition Λ = ( λ , . . . , λ k ) of an integer n , any i , and α ∈ L , (cid:0) m Λ ( α ( i ) ) ι ( e α e β ) , E − ( − α ( i ) , y ) · · · E − ( − α ( i ) , y n ) ι ( e − α e − β ) (cid:1) = ( − h α,α i / ε ( α, α ) − m Λ ( y , . . . , y n ) (7.6)and (cid:0) h Λ ( − β ( i ) ) ι ( e α e β ) , E − ( − α ( i ) , y ) · · · E − ( − α ( i ) , y n ) ι ( e − α e − β ) (cid:1) = ( − h α,α i / ε ( α, α ) − h Λ ( y , . . . , y n ) . (7.7)Since m Λ ( α ( i ) ) and h Λ ( − β ( i ) ) are operators of degree n and are polynomials in the operators β ( i ) ( − m ) for m > V − β + L, Z is spanned by coefficients of monomials in E − ( − α ( i ) , y ) · · · E − ( − α ( i ) , y n ) ι ( e − α e − β ) (7.8)and by vectors v such that (cid:0) m Λ ( α ( i ) ) ι ( e α e β ) , v (cid:1) = (cid:0) h Λ ( − β ( i ) ) ι ( e α e β ) , v (cid:1) = 0 . Thus (7.5), (7.6), and (7.7) imply that m Λ ( α ( i ) ) ι ( e α e β ) = X k Λ ′ h Λ ′ ( − β ( i ) ) ι ( e α e β ) , and thus m Λ ( α ( i ) ) = X k Λ ′ h Λ ′ ( − β ( i ) ) , which will complete the proof.To verify (7.6), we observe that by the definition of m Λ ( α ( i ) ), a monomial on the left sideof (7.6) has non-zero coefficient (equal to ( − h α,α i / ε ( α, α ) − ) if and only if the correspondingcoefficient of (7.8) is h Λ ( α ( i ) ) ι ( e − α e − β ). Since the coefficient of a monomial in (7.8) equals h Λ ( α ( i ) ) ι ( e − α e − β ) if and only if the monomial is a permutation of y λ · · · y λ k k , (7.6) followsfrom the definition of m Λ ( y , . . . , y n ). 24o verify (7.7), we see from (7.3) that the left side of (7.7) is the coefficient of x λ · · · x λ k k in ( − h α,α i / ε ( α, α ) − k Y i =1 n Y j =1 (1 − x i y j ) − = ( − h α,α i / ε ( α, α ) − k Y i =1 n Y j =1 X m ≥ x mi y mj = ( − h α,α i / ε ( α, α ) − k Y i =1 X m ≥ X m ,...,m n ≥ m + ··· + m n = m y m · · · y m n n x mi = ( − h α,α i / ε ( α, α ) − k Y i =1 X m ≥ h m ( y , . . . , y n ) x mi . Thus the coefficient of x λ · · · x λ k k is( − h α,α i / ε ( α, α ) − h Λ ( y , . . . , y n )as desired.Note that since L ⊆ L ◦ , Proposition 7.5 shows that V ′− β + L, Z is generally larger than V β + L, Z . We record the case β = 0 of Proposition 7.5: Corollary 7.6.
The integral form V ′ L, Z is the integral span of coefficients of products of theform E − ( − β , x ) · · · E − ( − β k , x k ) ι ( e α ) where β i ∈ L ◦ and α ∈ L . Remark 7.7.
Corollary 7.6 provides an alternative to the description of V ′ L, Z given in Propo-sition 3.6 of [DG]. Remark 7.8.
Note that the precise identity of V ′− β + L, Z depends on the choice of normal-ization of the invariant bilinear pairing between V β + L and V − β + L . We shall take V ′− β + L, Z asdescribed in Proposition 7.5 as the official V ′− β + L, Z , since we have V β + L, Z ⊆ V ′− β + L, Z this way.Finally we can classify some integral intertwining operators: Theorem 7.9.
For any β, γ ∈ L ◦ , there is a rank one lattice of intertwining operators within V β + γβ γ integral with respect to V β + L, Z , V γ + L, Z , and V ′− β − γ + L, Z . Moreover, this lattice is spannedby Y β,γ, Z = e − πi h β,γ i ε ( γ, β ) − Y β | V β + L ⊗ V γ + L . Proof.
From the definition (6.3) of Y β , we have Y β ( ι ( e β ) , x ) ι ( e γ ) = E − ( − β, x ) E + ( − β, x ) x h β,γ i e πi h β,γ i c ( γ, β ) ι ( e β e γ )= x h β,γ i E − ( − β, x ) e πi h β,γ i ε ( γ, β ) ε ( β, γ ) − ε ( β, γ ) ι ( e β + γ )= x h β,γ i E − ( − β, x ) e πi h β,γ i ε ( γ, β ) ι ( e β + γ ) . Y β,γ, Z ( ι ( e β ) , x ) ι ( e γ ) = x h β,γ i E − ( − β, x ) ι ( e β + γ ) ∈ V ′− β − γ + L, Z { x } (7.9)by Proposition 7.5.Since ι ( e β ) and ι ( e γ ) generate V β + L, Z and V γ + L, Z , respectively, as V L, Z -modules, Theorem3.9 implies that Y β,γ, Z is integral with respect to V β + L, Z , V γ + L, Z , and V ′− β − γ + L, Z . Moreover,we see from Proposition 7.5 and (7.9) that for c ∈ C , c Y β,γ, Z ( ι ( e β ) , x ) ι ( e γ ) ∈ V ′− β − γ + L, Z { x } if and only if c ∈ Z . Thus Y β,γ, Z spans the lattice of intertwining operators in V β + γβ γ whichare integral with respect to V β + L, Z , V γ + L, Z , and V ′− β − γ + L, Z . Remark 7.10.
Note that to prove Theorem 7.9 we only need the “easy” half of Proposition7.5 which states that for any β ∈ L ◦ , the coefficients of products as in (7.2) are contained in V ′− β + L, Z . But the more difficult opposite inclusion gives the additional information that for β, γ ∈ L ◦ , the coefficients of Y β,γ, Z ( w β , x ) w γ for any w β ∈ V β + L, Z , w γ ∈ W γ + L, Z are integrallinear combinations of the coefficients of products as in (7.2) (with β replaced by β + γ ). Remark 7.11.
Note that (7.9) shows that for β, γ ∈ L ◦ , Y β,γ, Z ( ι ( e β ) , x ) ι ( e γ ) / ∈ V β + γ + L, Z { x } since in general β / ∈ L . It is not clear that any non-zero integer multiple of Y β,γ, Z is in generalintegral with respect to V β + L, Z , V γ + L, Z , and V β + γ + L, Z . References [B] R. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster,
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