aa r X i v : . [ m a t h . QA ] S e p A PROOF OF FUSION RULES FORMULA
JIANQI LIU
Abstract.
A new proof of the fusion rules formula in the context of vertex operatoralgebra is given. Some more general relations between the space of intertwining operatorsand A ( V ) bimodules are obtained. Introduction
This paper is dedicated to present another proof of the well known fusion rules formulain the context of vertex operator algebra, which was claimed by Zhu in his thesis and firstproved by Li [9] using the Verma modules of VOA constructed in [2].Our method here is different with Li’s, but similar with the one Zhu had used toconstruct a module over VOA from an simple module over his famous algebra A ( V ). Theidea is that one define first an analogue of n-point correlation functions that verify certainconditions and then use these funtions to define the vertex operators Y . Just as Zhu hadclaimed in the joint paper with Frankel [7] that this method would also work in derivingthe fusion rules formula. However, they didn’t actually write out the proof of it in [7].To better understand his method we carry out a proof in this paper, and we can see fromour proof that some more general theorems regarding the fusion rules hold true.We assume that the readers are familar with the concept of vertex operator algebras(VOA), modules over VOA and the A ( V ) theory, see [1, 5, 6, 10] for example. The fusionrules formula can be stated as the following thoerem: Theorem 1.1. [7]
Let V be a VOA with three modules M , M , M , where M , M areirreducible. The space of intertwining operators of type (cid:0) M M M (cid:1) can be idenfified with ( M (0) ∗ ⊗ A ( V ) A ( M ) ⊗ A ( V ) M (0)) ∗ . (1.1)This is the first version of the fusion rules formula. It was originally used to computethe fusion fules of the irreducible modules over affine vertex operator algebras [7]. By theadjunction property of hom-tensor functors, one can easily see that the linear space in(1.1) can be identified withHom A ( V ) ( A ( M ) ⊗ A ( V ) M (0) , M (0)) , (1.2)which was proved to be isomorphic with the space of intertwining operators I (cid:0) M M M (cid:1) byLi [9]. Let’s recall some basic constructions regarding this problem. Assume that the V mod-ules M i , i = 1 , , L (0) weight space decomposition: M i = ∞ M n =0 M in + h i where h i ∈ C are the comformal weight of modules M i , i = 1 , , M i = L ∞ n =0 M i ( n ) with a m M i ( n ) ⊂ M i ( wta − m − n ) , ∀ a ∈ V, m ∈ Z , n ∈ Z ≥ , see [2, 3] for the definition of admissible modules. In [7, 9] it wasshown that an intertwining operator I ∈ (cid:0) M M M (cid:1) can be written as power series I ( v, z ) = X n ∈ Z v ( n ) z − n − · z − h where h = h + h − h . Furthermore, if one denote deg v = n for v ∈ M ( n ), the actionof v ( n ) satisfies: v ( m ) M ( n ) ⊆ M (deg v − m − n ) , ∀ m ∈ Z , n ∈ Z ≥ . Denote v (deg v −
1) by o ( v ), then there exists a linear map: π : I M M , M ! → ( M (0) ∗ ⊗ A ( V ) A ( M ) ⊗ A ( V ) M (0)) ∗ (1.3)given by π ( I ) = f I , where f I ( v ′ ⊗ v ⊗ v ) = h v ′ , o ( v ) v i for v ∈ M , v ′ ∈ M (0) ∗ , v ∈ M (0). For convenience we will denote deg v by wt v in the rest of this paper.In fact, it is not very hard to prove that this map π is injective [9]. However thesurjectivity of this map requires a lot of work to carry out. Shortly speaking, it is notquite obvious to see how to rise a linear functional f on M (0) ∗ ⊗ A ( V ) A ( M ) ⊗ A ( V ) M (0)to an intertwining operator I that satisfies f I = f .1.1. Idea of our proof.
Given a linear functional f : M (0) ∗ ⊗ A ( V ) A ( M ) ⊗ A ( V ) M (0) → C , we will first define a collection of (n+3)-point rational functions: S ( v ′ , ( a , z ) ... ( a k , z k )( v, w )( a k +1 , z k +1 ) ... ( a n , z n ) v ) (1.4)where a i ∈ V, v ′ ∈ M (0) ′ , v ∈ M (0) , v ∈ M ; k = 0 , , ..., n , and show that thesefunctions satisfy certain nice properties. For instance, we will show S doesn’t depend onthe position of term ( v, w ), namely, we can swap ( v, w ) with any one of the term ( a i , z i )and still end up with the same function S . Then we will use these rational functions S to define the intertwining operator I .We are going to see that S is an analogue of the classical (n+3)-point rational function:( v ′ , Y ( a , z ) ...I ( v, w ) , ..., Y ( a n , z n ) v ) , (1.5)which is the limit of power series h v ′ , Y ( a , z ) ...I ( v, w ) , ..., Y ( a n , z n ) v i in n+1 variables z i , i = 1 , , ..., n and w that converges in the domain D = { ( z , ..., w, ..., z n ) ∈ C n +1 || z | > | z | > ... > | w | > ... > | z n |} PROOF OF FUSION RULES FORMULA 3
Recall that the limit of this power series is a rational function in z i , w, z i − z j and z k − w with only possible poles at z i = 0 , w = 0 , z i = z j and z k = w , see [5]. Moreover, thisrational function doesn’t depend on place of the term I ( v, w ). That is to say, if one swapthe terms I ( v, w ) and Y ( a i , z i ) in the power series h v ′ , Y ( a , z ) ...I ( v, w ) , ..., Y ( a n , z n ) v i then take the limit, it will end up to be the same rational function as the original rationalfunction ( v ′ , Y ( a , z ) ...I ( v, w ) , ..., Y ( a n , z n ) v ). The only difference between these twopower series is the domain of convergence. This property essentially reflects the ”locality”of vertex operators.Note that if we fix the varialbes z i , i = 1 , , ..., n and w ,then the rational function abovecan be regarded as a map out of n+3 vector spaces:( − , Y ( − , z ) ...I ( − , w ) , ..., Y ( − , z n )) : M (0) ′ × V × ... × M × ...V × M (0) → F , (1.6)where F is the set of rational functions in n+1 variables z , z , ..., z n , w with only possiblepoles at z i = 0 , w = 0 , z i = z j , z k = w . This observation justified the name ”(n+3)-pointfunction”.This paper is organized as follows. We define the n + 3 point functions S inductivelyin section 2, and prove that the analogus of locality holds for our funtion S . In section 3we follow the method in [10] to extend the first and last inputs spaces of S from M (0) ′ and M (0) to the whole modules M ′ and M respectively, and carry out a proof of thefusion rules formula. Our proof also derives a more general relation between the space ofintertwining operators I (cid:0) M M M (cid:1) and the space ( M (0) ∗ ⊗ A ( V ) A ( M ) ⊗ A ( V ) M (0)) ∗ .2. The ( n + 3) -Point Function S The way Zhu constructed a V module M out of a simple A ( V ) module W [10] was to de-fine first a (n+2)-point rational function S ( v ′ , ( a , z ) ... ( a , z n ) v ), then use it to constructthe vertex operators Y M . The function S was defined inductively by the formula: S ( v ′ , ( a , z )( a , z ) ... ( a n , z n ) v ) = z − wta S ( π ( a ) v ′ , ( a , z ) ... ( a n , z n ) v )+ n X k =2 X i ≥ F wta ,i ( z , z k ) S ( v ′ , ( a , z ) ... ( a ( i ) a k , z k ) ... ( a n , z n ) v ) , (2.1)where F wta,i ( z, w ) is a rational function defined by: ι z,w ( F n,i ( z, w )) = X j ≥ (cid:18) n + ji (cid:19) z − n − j − w n + j − i ; F n,i ( z, w ) = z − n i ! (cid:18) ddw (cid:19) i w n z − w (2.2) JIANQI LIU
We can view the formula (2.1) as an expansion of S with respect to the left most term( a , z ). This formula could partially give the definition of our (n+3)-point function S ( v ′ , ( a , z ) ... ( a k , z k )( v, w )( a k +1 , z k +1 ) ... ( a n , z n ) v ) , as long as ( v, w ) is not at the first place i.e. k = 0. However, if we wish to define S ( v ′ , ( v, w )( a , z ) ... ( a n , z n ) v )under the assumption that all the (n+2) point functions S ( v ′ , ( a , z ) ... ( a k , z k )( v, w )( a k +1 , z k +1 ) ... ( a n − , z n − ) v )are given, then the formula (2.1) would fail to give us a proper definition because theterm ( v ( i ) a k , z k ) would make no sense.To remedy this situation, we will introduce another recurrent formula that is similarwith (2.1) which allows us to expand S with respect to the right most term.2.1. Formulas that motivates the definition.
Let I ∈ I (cid:0) M M M (cid:1) , v ′ ∈ M (0) ∗ , v ∈ M (0) , v ∈ M , a ∈ V . Following the proof of lemma 2.2.1 in [10], it is easily seen thatthe 4-point function ( v ′ , Y ( a, z ) I ( v, w ) v ) could be reduced to a sum of 3-point functionsby the formula:( v ′ , Y ( a, z ) I ( v, w ) v ) = ( v ′ o ( a ) , I ( v, w ) v ) z − wta + X i ≥ F wta,i ( z, w )( v ′ , I ( a ( i ) v, w ) v ) , (2.3)where v ′ o ( a ) is given by the natural right module action of A ( V ) on M (0) ∗ : h v ′ o ( a ) , v i := h v ′ , o ( a ) v i . Similarly, we can derive another recurrent formula for the same rational function but with Y ( a, z ) on the right, that is, a formula for the rational function ( v ′ , I ( v, w ) Y ( a, z ) v ). Lemma 2.1.
As rational functions, we have: ( v ′ , I ( v, w ) Y ( a, z ) v ) = ( v ′ , I ( v, w ) o ( a ) v ) z − wta + X i ≥ G wta,i ( z, w )( v ′ , I ( a ( i ) v, w ) v ) , (2.4) where G wta,i ( z, w ) is a rational function defined by ι w,z ( G wta,i ( z, w )) = − X j ≥ (cid:18) wta − − ji (cid:19) w wta − j − − i · z − wta +1+ j ; G n,i ( z, w ) = z − n +1 i ! (cid:18) ddw (cid:19) i (cid:18) w n − z − w (cid:19) (2.5) PROOF OF FUSION RULES FORMULA 5
Proof.
Note that a ( n ) v = 0 if wta − n − <
0, hence as power series: h v ′ , I ( v, w ) Y ( a, z ) v i = h v ′ , I ( v, w ) o ( a ) v i z − wta + X wta − n − > h v ′ , I ( v, w ) a ( n ) v i z − n − Recall that by taking residues from Jacobi indentity, one has[ a ( n ) , I ( v, w )] = X i ≥ (cid:18) ni (cid:19) I ( a ( i ) v, w ) w n − i , h v ′ , a ( n ) u i = X i ≥ i ! ( − i h ( L ( i ) a )(2 wta − n − i − v ′ , u i , (2.6)and ( L ( i ) a )(2 wta − n − i − v ′ ∈ M ′ ( − wta + n + 1) = 0, if wta − n − >
0. Thus, X wta − n − > h v ′ , I ( v, w ) a ( n ) v i z − n − = X wta − n − > h v ′ , a ( n ) I ( v, w ) v i z − n − − X wta − n − > h v ′ , [ a ( n ) , I ( v, w )] v i z − n − = − X wta − n − > X i ≥ (cid:18) ni (cid:19) h v ′ , I ( a ( i ) v, w ) v i z − n − w n − i . Make a change of variable n = wta − − j in the last term above, since wta − n − > ⇔ wta − − n = j ≥
0, it becomes − X j ≥ X i ≥ (cid:18) wta − j − i (cid:19) z − wta + j +2 − w wta − j − − i h v ′ , I ( a ( i ) v, w ) v i = X i ≥ ι w,z ( G wta,i ( z, w )) h v ′ , I ( a ( i ) v, w ) v i Therefore, as power series, h v ′ , I ( v, w ) Y ( a, z ) i = h v ′ , I ( v, w ) o ( a ) v i z − wta + X i ≥ ι w,z ( G wta,i ( z, w )) h v ′ , I ( a ( i ) v, w ) v i Taking the limit on both sides of this equation yields the formula (2.4). (cid:3)
In contrast to (2.3), the formula (2.4) can be regarded as expansion with respect to theright most term Y ( a, z ) in the rational function ( v ′ , I ( v, w ) Y ( a, z ) v ). Moreover, we havethe following relation of F and G : F n,i ( z, w ) − G n,i ( z, w ) = z − n i ! (cid:18) ddw (cid:19) i (cid:18) w n z − w − zw n − z − w (cid:19) = z − n i ! (cid:18) ddw (cid:19) i ( − w n − )= − ( n − n − ... ( n − − i + 1) 1 i ! z − n w n − − i = − (cid:18) n − i (cid:19) z − wta w wta − − i . JIANQI LIU
In particular, we have F wta,i ( z, w ) − G wta,i ( z, w ) = − (cid:18) wta − i (cid:19) z − wta w wta − − i . (2.7)2.2. Construction of 4-point and 5-point functions.
From now on, we assume that f is linear functional on the vector space M (0) ∗ ⊗ A ( V ) A ( M ) ⊗ A ( V ) M (0) ,v ′ ∈ M (0) ∗ , v ∈ M , v ∈ M (0) , and a i ∈ V . Denote by F ( z , z , ..., z n , w ) the vectorspace of rational functions in variables z i , i = 1 , , ..., n and w with only possible polesat z i = 0 , w = 0 , z i = z j , z k = w . We will use the same symbol v to denote the image v + O ( M ) of v ∈ M in the A ( V ) bimodule A ( M ), and we will denote the degree of v ,deg v by wt v .Motivated by the fact that the left hand sides of (2.3) and (2.4) give rise to the samerational function, we define first a 3-point function S M out of f , then use that to constructtwo 4-point functions S LV M , S
RMV in two different ways, and then show that the two 4-pointfunctions so constructed are the same rational function.
Definition 2.2.
Define S M : M (0) ∗ × M × M (0) → F ( w ) by the formula: S M ( v ′ , ( v, w ) v ) := f ( v ′ ⊗ v ⊗ v ) w − wt v , (2.8) where on the right hand side we use the same symbol v for its image in A ( M ) .Define S LV M : M (0) ∗ × V × M × M (0) → F ( z, w ) by the formula: S LV M ( v ′ , ( a, z )( v, w ) v ) = S M ( v ′ o ( a ) , ( v, w ) v ) z − wta + X i ≥ F wta,i ( z, w ) S M ( v ′ , ( a ( i ) v, w ) v ) . (2.9) Finally, define S RMV : M (0) ∗ × M × V × M (0) → F ( z, w ) by the formula S RMV ( v ′ , ( v, w )( a, z ) v ) = S M ( v ′ , ( v, w ) o ( a ) v ) z − wta + X i ≥ G wta,i ( z, w ) S M ( v ′ , ( a ( i ) v, w ) v ) . (2.10)The upper index L (resp. R ) in the 4-point functions S indicate that we use the ex-pansion formulas with respect to the left (resp. right) most term, i.e (2.3) (resp.(2.4)) todefine our new S . In what follows, we will denote the 3-point function S M simply by S . Proposition 2.3.
As rational functions in F ( z, w ) , we have: S LV M ( v ′ , ( a, z )( v, w ) v ) = S RMV ( v ′ , ( v, w )( a, z ) v ) . Therefore, we may use the common symbol S to denote these two 4-point functions. PROOF OF FUSION RULES FORMULA 7
Proof.
From the definition of S LV M , S RMV and S we see that: S LV M ( v ′ , ( a, z )( v, w ) v ) = f ( v ′ o ( a ) ⊗ v ⊗ v ) w − wtv z − wta + X i ≥ F wta,i ( z, w ) f ( v ′ ⊗ a ( i ) v ⊗ v ) w − wta − wtv + i +1 and S RMV ( v ′ , ( v, w )( a, z ) v ) = f ( v ′ ⊗ v ⊗ o ( a ) v ) w − wtv z − wta + X i ≥ G wta,i ( z, w ) f ( v ′ ⊗ a ( i ) v ⊗ v ) w − wta − wtv + i +1 . Combining these two equations with (2.7), we have S LV M ( v ′ , ( a, z )( v, w ) v ) − S RMV ( v ′ , ( v, w )( a, z ) v )= f ( v ′ o ( a ) ⊗ v ⊗ v ) w − wtv z − wta − f ( v ′ ⊗ v ⊗ o ( a ) v ) w − wtv z − wta − X i ≥ (cid:18) wta − i (cid:19) f ( v ′ ⊗ a ( i ) v ⊗ v ) w − wtv − wta + i +1 z − wta w wta − − i = f ( v ′ ⊗ ( a ∗ v − v ∗ a ) ⊗ v ) w − wtv z − wta − X i ≥ (cid:18) wta − i (cid:19) f ( v ′ ⊗ a ( i ) v ⊗ v ) z − wta w − wtv Recall that in the A ( V ) bimodule A ( M ) one has the following formula [10]: a ∗ v − v ∗ a = Res z Y ( a, z ) v (1 + z ) wta − = X i ≥ (cid:18) wta − i (cid:19) a ( i ) v. Hence we have S LV M ( v ′ , ( a, z )( v, w ) v ) − S RMV ( v ′ , ( v, w )( a, z ) v ) = 0. (cid:3) It follows from the proposition that both of the 4-point functions S LV M and S RMV indefinition 2.2 give rise to a single 4-point function S which satisfies S ( v ′ , ( a, z )( v, w ) v ) = S ( v ′ , ( v, w )( a, z ) v ) , (2.11)and it is defined as a rational function obtained from 3-point function S by the formulasof expansion with respect to either the left most term as in (2.3) or the right most termas in (2.4).We adopt the same idea as above to define 5-point functions. At first glimpse, therecould be in total four of the 5-point functions, namely, S LV MV , S LV V M , S RV MV and S RMV V ,where the sub index indicates the position of M . We can define them using suitableformulas as in (2.3) or (2.4) based on their upper index L and R . But in the end we willshow that they all give rise to the same rational function no matter what the upper orlower indices they have, just as the case in 4-point functions. JIANQI LIU
Definition 2.4.
The 5-point functions with upper indices LS LV MV ( v ′ , ( a , z )( v, w )( a , z ) v ) ,S LV V M ( v ′ , ( a , z )( a , z )( v, w ) v ) are defined by the expansion with respect to the left most term ( a , z ) , which is given bythe common formula: S ( v ′ o ( a ) , ( v, w )( a , z ) v ) z − wta + X j ≥ F wta ,j ( z , w ) S ( v ′ , ( a ( j ) v, w )( a , z ) v )+ X j ≥ F wta ,j ( z , z ) S ( v ′ , ( v, w )( a ( j ) a , z ) v ) . (2.12) The 5-point functions with upper indices RS RV MV ( v ′ , ( a , z )( v, w )( a , z ) v ) ,S RMV V ( v ′ , ( v, w )( a , z )( a , z ) v ) are defined by expansion with respect to the right most term ( a , z ) , which is also givenby a common formula S ( v ′ , ( a , z )( v, w ) o ( a ) v ) z − wta + X j ≥ G wta ,j ( z , w ) S ( v ′ , ( a , z )( a ( j ) v, w ) v )+ X j ≥ G wta ,j ( z , z ) S ( v ′ , ( a ( j ) a , z )( v, w ) v ) . (2.13)Note that the function S in expressions (2.12) and (2.13) above are 4-point functions. Infact, formula (2.12) is just the expansion of S LV MV ( v ′ , ( a , z )( v, w )( a , z ) v ) with respectto ( a , z ). But it follows from (2.11) that: S ( v ′ o ( a ) , ( v, w )( a , z ) v ) = S ( v ′ o ( a ) , ( a , z )( v, w ) v ); S ( v ′ , ( a ( j ) v, w )( a , z ) v ) = S ( v ′ , ( a , z )( a ( j ) v, w ) v ); S ( v ′ , ( a , z )( a ( j ) v, w ) v ) = S ( v ′ , ( a ( j ) v, w )( a , z ) v ) . So (2.12) is also an expansion of S LV V M (( v ′ , ( a , z )( a , z )( v, w ) v )) with respect to ( a , z ),hence it makes sense to define S LV MV and S LV V M by the same formula, and similar for S RV MV , S
RMV V .Now we want to show that all the 5-point functions defined as above are the same ratio-nal function, no matter which upper symbols or lower symbols they have, just like propertythe 4-point functions (2.11). In other words, there is only one S ( v ′ , ( a , z )( v, w )( a , z ) v ),and one can permute the term ( a , z ), ( a , z ) and ( v, w ) arbitrarily and still end up withthe same function. This is not an easy task, because first we have to show that (2.12) and(2.13) are the same rational function, which is far from being obvious, and furthermore, PROOF OF FUSION RULES FORMULA 9 observe that by definition 2.4 the function S V MV ( v ′ , ( a , z )( v, w )( a , z ) v ) would havetwo definitions: S LV MV ( v ′ , ( a , z )( v, w )( a , z ) v ) and S RV MV ( v ′ , ( a , z )( v, w )( a , z ) v )with the first one defined by expansion with respect to left most term ( a , z ) as in (2.12)and second one defined by expansion with respect to right most term ( a , z ) as in (2.13).If all of the 5-point functions in definition 2.4 give rise to one single 5-point function, thenthese two definition ought to be the same. Nevertheless, we have the following fact: Proposition 2.5. If S LV MV ( v ′ , ( a , z )( v, w )( a , z ) v ) = S RV MV ( v ′ , ( a , z )( v, w )( a , z ) v ) i.e. if (2.12) = (2.13) , then it follows that: S LV MV ( v ′ , ( a , z )( v, w )( a , z ) v ) = S RV MV ( v ′ , ( a , z )( v, w )( a , z ) v ) Proof.
Note that (2.12) is essentially the formula Zhu had used to define his n+1 pointfunction S out of n-point functions, so the formula (2.2.11) in [10] would also hold truein our case, and the proof would be exactly the same, so we won’t write out a proof here.That formula allows our to swap ( a , z ) and ( a , z ) in S LV V M : S LV V M ( v ′ , ( a , z )( a , z )( v, w ) v ) = S LV V M ( v ′ , ( a , z )( a , z )( v, w ) v ) . (2.14)By assumption of the proposition, definition 2.4 and (2.14), we have: S LV MV ( v ′ , ( a , z )( v, w )( a , z ) v ) = S LV V M ( v ′ , ( a , z )( a , z )( v, w ) v )= S LV V M ( v ′ , ( a , z )( a , z )( v, w ) v )= S LV MV ( v ′ , ( a , z )( v, w )( a , z ) v )= S RV MV ( v ′ , ( a , z )( v, w )( a , z ) v ) , which is the equation as proposed. (cid:3) Next, we are going to show the assumption in proposition 2.5 holds true, namely,the rational function (2.12) is the same with the one in (2.13). We will compute thedifference of three summands in those functions separately, then show that the sum oftheir differences is equal to 0. In order to make our computational work looks moreelegant, we introduce symbols ♥ , ⋆ , △ to denote the difference of three summands in(2.12) and (2.13), namely: S ( v ′ o ( a ) , ( v, w )( a , z ) v ) z − wta − S ( v ′ , ( a , z )( v, w ) o ( a ) v ) z − wta ♥ X j ≥ F wta ,j ( z , w ) S ( v ′ , ( a ( j ) v, w )( a , z ) v ) − X j ≥ G wta ,j ( z , w ) S ( v ′ , ( a , z )( a ( j ) v, w ) v ) ⋆ X j ≥ F wta ,j ( z , z ) S ( v ′ , ( v, w )( a ( j ) a , z ) v ) − X j ≥ G wta ,j ( z , z ) S ( v ′ , ( a ( j ) a , z )( v, w ) v ) △ Now our goal is to show that ♥ + ⋆ + △ = 0.By the property of 4-point functions, we may use formulas (2.9) and (2.10) to expandboth of the terms in ♥ with respect to ( a , z ). S ( v ′ o ( a ) , ( v, w )( a , z ) v ) z − wta − S ( v ′ , ( a , z )( v, w ) o ( a ) v ) z − wta = S ( v ′ o ( a ) o ( a ) , ( v, w ) v ) z − wta z − wta + X i ≥ F wta ,i ( z , w ) S ( v ′ o ( a ) , ( a ( i ) v, w ) v ) z − wta − S ( v ′ o ( a ) , ( v, w ) o ( a ) v ) z − wta z − wta + X i ≥ F wta ,i ( z , w ) S ( v ′ , ( a ( i ) v, w ) o ( a ) v ) z − wta = f ( v ′ ⊗ a ∗ a ∗ v ⊗ v ) w − wtv z − wta z − wta − f ( v ′ ⊗ a ∗ v ∗ a ⊗ v ) w − wtv z − wta z − wta + X i ≥ F wta ,i ( z , w ) f ( v ′ ⊗ ( a ∗ ( a ( i ) v )) − ( a ( i ) v ) ∗ a ) ⊗ v ) w − wta − wtv + i +1 z − wta = ♥ (1) + ♥ (2) + ♥ (3) . For ⋆ we can apply the formula (2.9) and (2.11) to expand each of the summand in ⋆ with respect to ( a , z ) on the left: X j ≥ F wta ,j ( z , w ) S ( v ′ , ( a ( j ) v, w )( a , z ) v ) − X j ≥ G wta ,j ( z , w ) S ( v ′ , ( a , z )( a ( j ) v, w ) v )= X j ≥ F wta ,j ( z , w ) S ( v ′ , o ( a )( a ( j ) v, w ) v ) z − wta + X j ≥ X i ≥ F wta ,j ( z , w ) F wta ,i ( z , w ) S ( v ′ , ( a ( i ) a ( j ) v, w ) v ) − X j ≥ G wta ,j ( z , w ) S ( v ′ , o ( a )( a ( j ) v, w ) v ) z − wta − X j ≥ X ı ≥ G wta ,j ( z , w ) F wta ,i ( z , w ) S ( v ′ , ( a ( i ) a ( j ) v, w ) v )= X j ≥ − (cid:18) wta − j (cid:19) S ( v ′ o ( a ) , ( a ( j ) v, w ) v ) z − wta z − wta w wta − − j (1) + X j ≥ X i ≥ − (cid:18) wta − j (cid:19) z − wta w wta − − j F wta ,w ( z , w ) S ( v ′ , ( a ( i ) a ( j ) v, w ) v ) (2) = ⋆ (1) + ⋆ (2) . PROOF OF FUSION RULES FORMULA 11
Finally, for the function △ we expand each of its summand in terms of ( a ( j ) a , z ) onthe left: X j ≥ F wta ,j ( z , z ) S ( v ′ , ( v, w )( a ( j ) a , z ) v ) − X j ≥ G wta ,j ( z , z ) S ( v ′ , ( a ( j ) a , z )( v, w ) v )= X j ≥ F wta ,j ( z , z ) S ( v ′ o ( a ( j ) a ) , ( v, w ) v ) z − wta − wta + j +12 + X j ≥ X i ≥ F wta ,j ( z , z ) F wta + wta − j − ,i ( z , w ) S ( v ′ , (( a ( j ) a )( i ) v, w ) v ) − X j ≥ G wta ,j ( z , z ) S ( v ′ o ( a ( j ) a ) , ( v, w ) v ) z − wta − wta + j +12 + X j ≥ X i ≥ G wta ,j ( z , z ) F wta + wta − j − ,i ( z , w ) S ( v ′ , (( a ( j ) a )( i ) v, w ) v )= X j ≥ − (cid:18) wta − j (cid:19) z − wta z wta − − j S ( v ′ o ( a ( j ) a ) , ( v, w ) v ) z − wta − wta + j +12(1) + X j ≥ X i ≥ − (cid:18) wta − j (cid:19) z − wta z wta − − j F wta + wta − j − ,i ( z , w ) S ( v ′ , ( a ( j ) a )( i )( v, w ) v ) (2) = △ (1) + △ (2) . Now we need to show ♥ (1) + ♥ (2) + ♥ (3) + ⋆ (1) + ⋆ (2) + △ (1) + △ (2) = 0. Sincein A ( M ) one has a ∗ v − v ∗ a = Res z Y ( a, z ) v (1 + z ) wta − = P j ≥ (cid:0) wta − j (cid:1) a ( j ) v , we canrewrite ⋆ (1) and △ (1) as: ⋆ (1) = − X j ≥ (cid:18) wta − j (cid:19) w − wta − wtv + j +1 z wta z wta w wta − j − f ( v ′ o ( a ) ⊗ a ( j ) v ⊗ v )= − w − wtv z − wta z wta f ( v ′ ⊗ ( a ∗ a ∗ v − a ∗ v ∗ a ) ⊗ v ); △ (1) = − X j ≥ (cid:18) wta − j (cid:19) z − wta z − wta w − wtv f ( v ′ o ( a ( j ) a ) ⊗ v ⊗ v )= − z − wta z − wta w − wtv f ( v ′ ⊗ ( a ∗ a ∗ v − a ∗ a ∗ v ) ⊗ v )Now we see that: ♥ (1) + ♥ (2) + ⋆ (1) + △ (1)= f ( v ′ ⊗ a ∗ a ∗ v ⊗ v ) w − wtv z − wta z − wta + f ( v ′ ⊗ a ∗ v ∗ a ⊗ v ) w − wtv z − wta z − wta − w − wtv z − wta z wta f ( v ′ ⊗ ( a ∗ a ∗ v − a ∗ v ∗ a ) ⊗ v ) − z − wta z − wta w − wtv f ( v ′ ⊗ ( a ∗ a ∗ v − a ∗ a ∗ v ) ⊗ v )= 0 Moreover, we can also rewrite ♥ (3) as ♥ (3) = X i,j ≥ (cid:18) wta − j (cid:19) z − wta w − wta − wtv + i +1 F wta ,i ( z , w ) f ( v ′ ⊗ a ( j ) a ( i ) v ⊗ v )Since what we need to show was ♥ + ⋆ + △ = 0, the equation above implies that we onlyneed to show ♥ (3) + ⋆ (2) + △ (2) = 0. We shall use the following lemma to prove thisequality. Lemma 2.6.
Let M be a V module, a , a ∈ V , v ∈ M , n ∈ Z ≥ , we have: X i,j ≥ (cid:18) wta − j (cid:19)(cid:18) wta + ni (cid:19) ( a ( j ) a ( i ) v − a ( i ) a ( j ) v )= X i,j ≥ (cid:18) wta − j (cid:19)(cid:18) wta + wta − j − ni (cid:19) ( a ( j ) a )( i ) v (2.15) Proof.
Choose complex variables z , z in the domain | z | < , | z | < , | z − z | < | z | .By Jacobi identity in the residue form[10], the left hand side of (2.15) can be written as:Res z ,z X i,j ≥ (cid:18) wta − j (cid:19)(cid:18) wta + ni (cid:19) z j z i ( Y ( a , z ) Y ( a , z ) v − Y ( a , z ) Y ( a , z ) v )= Res z ,z (1 + z ) wta − (1 + z ) wta + n ( Y ( a , z ) Y ( a , z ) v − Y ( a , z ) Y ( a , z ) v )= Res z Res z − z (1 + z + ( z − z )) wta − (1 + z ) wta + n Y ( Y ( a , z − z ) a , z ) v = Res z Res z − z X j ≥ (cid:18) wta − j (cid:19) (1 + z ) wta − − j + wta + n ( z − z ) j Y ( Y ( a , z − z ) a , z ) v = X i,j ≥ (cid:18) wta − j (cid:19)(cid:18) wta + wta − j − ni (cid:19) ( a ( j ) a )( i ) v, which is equal to the right hand side of (2.15). (cid:3) We are now left to show the sum of following three terms is equal to 0: X i,j ≥ (cid:18) wta − j (cid:19) z − wta w − wta − wtv + i +1 F wta ,i ( z , w ) f ( v ′ ⊗ a ( j ) a ( i ) v ⊗ v ) ♥ (3) X i,j ≥ − (cid:18) wta − j (cid:19) z − wta w wta + i +1 − wtv F wta ,w ( z , w ) f ( v ′ ⊗ a ( i ) a ( j ) v ⊗ v ) ⋆ (2) X i,j ≥ − (cid:18) wta − j (cid:19) z − wta z wta − − j F wta + wta − j − ,i ( z , w ) w − wta − wta + j +1+ i +1 − wtv △ (2) · f ( v ′ ⊗ ( a ( j ) a )( i ) v ⊗ v ) PROOF OF FUSION RULES FORMULA 13
Since the map ι z ,w is injective[5], it follows that if we only need to show ι z ,w ( ♥ (3) + ⋆ (2) + △ (2)) = 0. Recall that by (2.2), ι z ,w ( F wta ,i ( z , w )) can be written as: ι z ,w ( F wta ,i ( z , w )) = ι z ,w (cid:18) z − wta i ! (cid:18) ddw (cid:19) i (cid:18) w wta z − w (cid:19)(cid:19) = X n ≥ (cid:18) wta + ni (cid:19) w wta + n − i z − wta − n − Let’s denote the term z wta w − wtv + n +1 z − wta − n − by γ , then it follows from lemma ?? that ι z ,w ( ♥ (3)) + ι z ,w ( ⋆ (2))= X i,j ≥ (cid:18) wta − j (cid:19) z wta w − wta − wtv + i +1 ( X n ≥ (cid:18) wta + ni (cid:19) w wta + n − i z − wta − n − ) · ( f ( v ′ ⊗ a ( j ) a ( i ) v ⊗ v ) − f ( v ′ ⊗ a ( i ) a ( j ) v ⊗ v ))= X i,j,n ≥ (cid:18) wta − j (cid:19)(cid:18) wta + ni (cid:19) γ · f ( v ′ ⊗ ( a ( j ) a ( i ) v − a ( i ) a ( j ) v ) ⊗ v )= X i,j,n ≥ (cid:18) wta − j (cid:19)(cid:18) wta + wta + n − j − i (cid:19) γ · f ( v ′ ⊗ ( a ( j ) a )( i ) v ⊗ v )= − ι z ,w ( △ (2))Now the proof of (2.12)=(2.13) is complete. Putting this together with proposition 2.5and (2.14), we see that our definition 2.4 for the 5-point functions give rise to one single5-point function S , which satisfies: S ( v ′ , ( a , z )( a , z )( v, w ) v ) = S ( v ′ , ( a , z )( a , z )( v, w ) v )= S ( v ′ , ( a , z )( v, w )( a , z ) v ) = S ( v ′ , ( a , z )( v, w )( a , z ) v )= S ( v ′ , ( v, w )( a , z )( a , z ) v ) = S ( v ′ , ( v, w )( a , z )( a , z ) v ) (2.16)This is to say that the terms ( a , z ) , ( a , z ) and ( v, w ) can be permutated arbitrarilywithin S . Moreover, S ( v ′ , ( a , z )( a , z )( v, w ) v ) can be either defined by expansion withrespect to ( a i , z i ) , i = 1 , a i , z i ) , i = 1 , v ′ , Y ( a , z ) Y ( a , z ) I ( v, w ) v ) , which is defined to be the limit of series h v ′ , Y ( a , z ) Y ( a , z ) I ( v, w ) v i .2.3. Construction of (n+3)-point function S . We will define the general (n+3)-pointfunction S by induction on n . Note that the base cases n = 1 , section. Now suppose all the (n+2)-point functions: S : M (0) ∗ × V × ... × M × ... × V × M (0) → F ( z , ..., z n − , w )are given, and they satisfy the following two properties: The first property is that: S ( v ′ , ( a , z )( a , z ) ... ( a n − , z n − )( v, w ) v ) = S ( v ′ , ( b , w )( b , w ) ... ( b n , w n ) v ) (1)where the finite set { ( b i , z i ) } ni =1 = { ( a , z ) , ..., ( a n − , z n − ) , ( v, w ) } .Denote by S L (resp. S R ) the expansion with respect to the left (resp.right) most term in S , see (2.12)(resp.(2.13)). The seond property is that for any indices i and jS ( v ′ , ( b , w )( b , w ) ... ( b n , w n ) v ) = S L ( v ′ , ( a i , z i ) v ) = S R ( v ′ , ( a j , z j ) v ) , (2)where the symbol in S L represents an arbitrary combination of ( v, w ), { ( a k , z k ) } k = i ,while the in S R represents an arbitrary combination of ( v, w ), { ( a k , z k ) } k = j .Note that these two properties are satisfied by 4 and 5-point functions, see (2.11) and(2.16). With (n+2)-point functions S in our hand we may define (n+3)-point functionsby similar formulas as in definition 2.2 and definition 2.4: Definition 2.7.
Assume the number of V in the subindex of S LV V...M ...V and S RV...M ...V V are both equal to n, and the M in S L can be placed anywhere in between the second andthe last place, while the M in S R can be placed anywhere in between the first and the nthplace. We define these functions by the following formulas: S LV V...M ...V ( v ′ , ( a , z ) ... ( v, w ) ...v )= S ( v ′ o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) v ) z − wta + n X k =2 X j ≥ F wta ,j ( z , z k ) · S ( v ′ , ( a , z ) ... ( a ( j ) a k , z k ) ... ( v, w ) v )+ X j ≥ F wta ,j ( z , w ) · S ( v ′ , ( a , z ) ... ( a n , z n )( a ( j ) v, w ) v ); (2.17) S RV...M ...V V ( v ′ , ... ( v, w ) ... ( a , z ) v )= S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) o ( a ) v ) z − wta + n X k =2 X j ≥ G wta ,j ( z , z k ) · S ( v ′ , ( a , z ) ... ( a ( j ) a k , z k ) ... ( v, w ) v )+ X j ≥ G wta ,j ( z , w ) · S ( v ′ , ( a , z ) ... ( a n , z n )( a ( j ) v, w ) v ) , (2.18) where the S on right hand sides of these equations are (n+2)-point functions. Note that the definition above implies that S LV MV...V = S LV V M...V = ... = S LV V...V M , whichis resaonable because the (n+2)-point functions S on the right hand side of (2.17) satisfy PROOF OF FUSION RULES FORMULA 15 property (1), and for the same reason one can expect that S RMV...V V = S RV M...V V = ... = S V...V MV . Our next step is to show: S LV...M...V ( v ′ , ( a , z ) ... ( v, w ) ... ( a , z ) v )= S RV...M...V ( v ′ , ( a , z ) ... ( v, w ) ... ( a , z ) v ) , (2.19)which will lead to the conclusion that all of the (n+3)-point functions S LV V...M...V , S
RV...M...V V give rise to one single (n+3)-point function S that satisfies the same properties (1) and(2) as (n+2)-point functions. In order to prove (2.19), we apply again formula (2.2.1) in[10] to our situation and get: S LV V...M...V ( v ′ , ( a , z )( a , z ) ... ( v, w ) ...v )= S LV V...M...V ( v ′ , ( a , z )( a , z ) ... ( v, w ) ...v ) . (2.20)Since the definition (2.17) of S L is essentially the same with (2.2.6) in [10], it is not hadto see that the proof of (2.20) would be exactly the same with the corresponding one(2.2.11) in [10], so we omit the proof of it. Proposition 2.8. If S LV V...M...V ( v ′ , ( a , z ) ...v ) = S RV...M...V V ( v ′ , ... ( a , z ) v ) , i.e. if theright hand side of (2.17) is equal to the right hand side of (2.18) , then (2.19) follows.Proof. The proof is similar with proposition 2.5. By (2.20) and assumption, we have: S LV...M...V ( v ′ , ( a , z ) ... ( v, w ) ... ( a , z ) v ) = S LV V...M...V ( v ′ , ( a , z )( a , z ) ... ( v, w ) ...v )= S LV V...M...V ( v ′ , ( a , z )( a , z ) ... ( v, w ) ...v )= S RV...M...V V ( v ′ , ( a , z ) ... ( v, w ) ... ( a , z ) v )as asserted. (cid:3) Now we are left to prove that S LV V...M...V ( v ′ , ( a , z ) ...v ) = S RV...M...V V ( v ′ , ... ( a , z ) v ).We again introduce the following symbols to denote the corresponding terms in right handside of the difference (2.17)-(2.18). S ( v ′ o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) v ) z − wta − S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) o ( a ) v ) z − wta ♥ X j ≥ F wta ,j ( z , z ) S ( v ′ , ( a ( j ) a , z ) ... ( v, w ) v ) − G wta ,j ( z , z ) S ( v ′ , ( a ( j ) a , z ) ... ( v, w ) v )) △ n X k =3 X j ≥ F wta ,j ( z , z k ) S ( v ′ , ( a , z ) ... ( a ( j ) a k , z k ) ... ( v, w ) v ) − n X k =3 X j ≥ G wta ,j ( z , z k ) S ( v ′ , ( a , z ) ... ( a ( j ) a k , z k ) ... ( v, w ) v ) (cid:4) X j ≥ ( F wta ,j ( z , w ) S ( v ′ , ( a , z ) ... ( a ( j ) v, w ) v ) − G wta ,j ( z , w ) S ( v ′ , ( a , z ) ... ( a ( j ) v, w ) v )) . ⋆ Then we need to show that ♥ + △ + (cid:4) + ⋆ = 0.Our strategy is to apply formula (2.17) to each one of the (n+2)-point S above, namely,we expand them with respect to the left most term, then add them up and show that thesum is equal to 0.Start with ♥ , note that S ( v ′ o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) v ) z − wta can be written as: S ( v ′ o ( a ) o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) v ) z − wta z − wta ( ∗ )+ n X t =3 X i ≥ F wta ,i ( z , z t ) S ( v ′ o ( a ) , ( a , z ) ... ( a ( i ) a t , z t ) ... ( a n , z n )( v, w ) v ) z − wta + X i ≥ F wta ,i ( z , w ) S ( v ′ o ( a ) , ( a , z ) ... ( a n , z n )( a ( i ) v, w ) v ) z − wta and S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) o ( a ) v ) z − wta can be written as S ( v ′ o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) o ( a ) v ) z − wta z − wta ( ∗∗ )+ n X t =3 X i ≥ F wta ,i ( z , z t ) S ( v ′ , ( a , z ) ... ( a ( i ) a t , z t ) ... ( a n , z n )( v, w ) o ( a ) v ) z − wta + X i ≥ F wta ,i ( z , w ) S ( v ′ , ( a , z ) ... ( a n , z n )( a ( i ) v, w ) o ( a ) v ) z − wta we denote the difference of first, second and third terms in ( ∗ ) and ( ∗∗ ) by ♥ (1), ♥ (2) and ♥ (3) respectively. In order to cancel ♥ with rest of the symbols, we need the followinglemma: Lemma 2.9.
As (n+1)-point functions, we have: S ( v ′ o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) v ) − S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) o ( a ) v )= n X k =3 X j ≥ (cid:18) wta − j (cid:19) z wta − j − k S ( v ′ , ( a , z ) ... ( a ( j ) a k , z k ) ... ( a n , z n )( v, w ) v )+ X j ≥ X j ≥ (cid:18) wta − j (cid:19) w wta − j − S ( v ′ , ( a , z ) ... ( a n , z n )( a ( j ) v, w ) v ) (2.21) Proof.
By the induction hypothesis on (n+2)-point functions and (2.7), we have:0 = S ( v ′ , ( a , z )( a , z ) ... ( a n , z n )( v, w ) v ) − S ( v ′ , ( a , z ) ... ( a n , z n )( a , z )( v, w ) v )= S ( v ′ o ( a )( a , z ) ... ( a n , z n )( v, w ) v ) z − wta − S ( v ′ ( a , z ) ... ( a n , z n )( v, w ) o ( a ) v ) z − wta + n X k =3 X j ≥ ( F wta ,j ( z , z k ) − G wta ,j ( z , z k )) S ( v ′ , ( a , z ) ... ( a ( j ) a k , z k ) ... ( a n , z n )( v, w ) v ) PROOF OF FUSION RULES FORMULA 17 + X j ≥ X j ≥ ( F wta ,j ( z , w ) − G wta ,j ( z , w )) S ( v ′ , ( a , z ) ... ( a n , z n )( a ( j ) v, w ) v )= S ( v ′ o ( a )( a , z ) ... ( a n , z n )( v, w ) v ) z − wta − S ( v ′ ( a , z ) ... ( a n , z n )( v, w ) o ( a ) v ) z − wta + n X k =3 X j ≥ − (cid:18) wta − j (cid:19) z wta − j − k S ( v ′ , ( a , z ) ... ( a ( j ) a k , z k ) ... ( a n , z n )( v, w ) v )+ X j ≥ X j ≥ − (cid:18) wta − j (cid:19) w wta − j − S ( v ′ , ( a , z ) ... ( a n , z n )( a ( j ) v, w ) v ) , so the lemma follows. (cid:3) It follows from the lemma that ♥ (2) and ♥ (3) can be written as n X t =3 n X k =3 ,k = t X i,j ≥ F wta ,i ( z , z t ) (cid:18) wta − j (cid:19) z − wta z wta − − jk · S ( v ′ , ( a , z ) ... ( a ( j ) a k , z k ) ... ( a ( i ) a t , z t ) ... ( a n , z n )( v, w ) v ) (21) + n X t =3 X i,j ≥ F wta ,i ( z , z t ) (cid:18) wta − j (cid:19) z − wta z wta − − jt · S ( v ′ , ( a , z ) ... ( a ( j ) a ( i ) a t , z t ) ... ( a n , z n )( v, w ) v ) (22) + n X t =3 X i,j ≥ F wta ,i ( z , w ) (cid:18) wta − j (cid:19) z − wta w wta − − j · S ( v ′ , ( a , z ) ... ( a ( i ) a t , z t ) ... ( a n , z n )( a ( j ) v, w ) v ) (23) = ♥ (21) + ♥ (22) + ♥ (23) ♥ (2) n X k =3 X i,j ≥ F wta ,i ( z , z k ) (cid:18) wta − j (cid:19) z − wta z wta − −− jk · S ( v ′ , ( a , z ) ... ( a ( j ) a k , z k ) ... ( a n , z n )( a ( i ) v, w ) v ) (31) + X i,j ≥ F wta ,i ( z , w ) (cid:18) wta − j (cid:19) z − wta w wta − −− j · S ( v ′ , ( a , z ) ... ( a n , z n )( a ( j ) a ( i ) v, w ) v ) (32) = ♥ (31) + ♥ (32) ♥ (3)So that ♥ = ♥ (1) + ♥ (21) + ♥ (22) + ♥ (23) + ♥ (31) + ♥ (32).Now by (2.7) and the formula for expansion with respect to ( a , z ) on the left (2.17),we may express the terms △ , (cid:4) and ⋆ as follows: We express △ as: X j ≥ − (cid:18) wta − j (cid:19) z − wta z − wta S ( v ′ o ( a ( j ) a ) , ( a , z ) ... ( a n , z n )( v, w ) v ) (1) + n X k =3 X i,j ≥ − (cid:18) wta − j (cid:19) z − wta z wta − − j F wta + wta − j − ,i ( z , z k ) · S ( v ′ , ( a , z ) ... (( a ( j ) a )( i ) a k , z k ) ... ( a n , z n )( v, w ) v ) (2) + X i,j ≥ − (cid:18) wta − j (cid:19) z − wta z − wta − − j F wta + wta − j − ,i ( z , w ) · S ( v ′ , ( a , z ) ... ( a n , z n )(( a ( j ) a )( i ) v, w ) v ) (3) = △ (1) + △ (2) + △ (3) . We express (cid:4) as: n X k =3 X j ≥ − (cid:18) wta − j (cid:19) z − wta z wta − − jk S ( v ′ o ( a ) , ( a , z ) ... ( a ( j ) a k , z k ) ... ( v, w ) v ) z − wta + n X k =3 X j,i ≥ − (cid:18) wta − j (cid:19) z − wta z wta − − jk F wta ,i ( z , w ) · S ( v ′ , ( a , z ) ... ( a ( j ) a k , z k ) ... ( a n , z n )( a ( i ) v, w ) v ) (2) + n X k =3 n X t =3 ,t = k X j,i ≥ − (cid:18) wta − j (cid:19) z − wta z wta − − jk F wta ,i ( z , z t ) · S ( v ′ , ( a , z ) ... ( a ( i ) a t , z t ) ... ( a ( j ) a k , z k ) ... ( a n , z n )( v, w ) v ) (3) + n X k =3 X j,i ≥ − (cid:18) wta − j (cid:19) z − wta z wta − − jk F wta ,i ( z , z k ) · S ( v ′ , ( a , z ) ... ( a ( i ) a ( j ) a k , z k ) ... ( a n , z n )( v, w ) v ) (4) = (cid:4) (1) + (cid:4) (2) + (cid:4) (3) + (cid:4) (4) . PROOF OF FUSION RULES FORMULA 19
We express ⋆ as: X j ≥ − (cid:18) wta − j (cid:19) z − wta w wta − − j S ( v ′ o ( a ) , ( a , z ) ... ( a n , z n )( a ( j ) v, w ) v ) z − wta + n X t =3 X j,i ≥ − (cid:18) wta − j (cid:19) z − wta w wta − − j F wta − ,i ( z , z k ) · S ( v ′ , ( a , z ) ... ( a ( i ) a k , z k ) ... ( a n , z n )( a ( j ) v, w ) v ) (2) + X j,i ≥ − (cid:18) wta − j (cid:19) z − wta w wta − − j F wta ,i ( z , w ) · S ( v ′ , ( a , z ) ... ( a n , z n )( a ( i ) a ( j ) v, w ) v ) (3) = ⋆ (1) + ⋆ (2) + ⋆ (3) . Now by lemma 2.6 and the formula (2.2) of ι z ,z t F n,i ( z , z t ), we have: X i,j ≥ (cid:18) wta − j (cid:19) F wta ,i ( z , z t ) a ( j ) a ( i ) a t + X i,j ≥ − (cid:18) wta − j (cid:19) F wta ,i ( z , z t ) a ( j ) a ( i ) a t + X i,j ≥ − (cid:18) wta − j (cid:19) F wta + wta − j − ,i ( z , z t )( a ( j ) a )( i ) a t = 0and the same equation holds if we replace z t by w and a t by v in the equation above.Therefore, by looking at the terms we expressed above it is easily seen that ♥ (22) + △ (2) + (cid:4) (4) = 0 and ♥ (32) + △ (3) + ⋆ (3) = 0. Moreover, it is also easy to see that ♥ (23) + ⋆ (2) = 0, ♥ (21) + (cid:4) (3) = 0 and ♥ (31) + (cid:4) (2) = 0.Now it remains to show ♥ (1) + △ (1) + (cid:4) (1) + ⋆ (1) = 0, or equivalently, S ( v ′ o ( a ) o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) v ) − S ( v ′ o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) o ( a ) v )= X j ≥ (cid:18) wta − j (cid:19) S ( v ′ o ( a ( j ) a ) , ( a , z ) ... ( a n , z n )( v, w ) v )+ n X k =3 X j ≥ (cid:18) wta − j (cid:19) z wta − − jk S ( v ′ o ( a ) , ( a , z ) ... ( a ( j ) a k , z k ) ... ( v, w ) v )+ X j ≥ (cid:18) wta − j (cid:19) w wta − − j S ( v ′ o ( a ) , ( a , z ) ... ( a n , z n )( a ( j ) v, w ) v ) , (2.22)but this is a consequence of lemma 2.9. In fact, L.H.S of (2.22)= S ( v ′ o ( a ) o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) v ) − S ( v ′ o ( a ) o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) v ) + S ( v ′ o ( a ) o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) v ) − S ( v ′ o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) o ( a ) v ) . Note that S is linear with respect to the place M (0) ∗ , so S ( v ′ o ( a ) o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) v ) − S ( v ′ o ( a ) o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) v )= S ( v ′ [ o ( a ) o ( a )] , ( a , z ) ... ( a n , z n )( v, w ) v )= X j ≥ (cid:18) wta − j (cid:19) S ( v ′ o ( a ( j ) a ) , ( a , z ) ... ( a n , z n )( v, w ) v ) , which is the first term on the right hand side of (2.22). On the other hand, by lemma 2.9, S ( v ′ o ( a ) o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) v ) − S ( v ′ o ( a ) , ( a , z ) ... ( a n , z n )( v, w ) o ( a ) v )= n X k =3 X j ≥ (cid:18) wta − j (cid:19) z wta − − jk S ( v ′ o ( a ) , ( a , z ) ... ( a ( j ) a k , z k ) ... ( v, w ) v )+ X j ≥ (cid:18) wta − j (cid:19) w wta − − j S ( v ′ o ( a ) , ( a , z ) ... ( a n , z n )( a ( j ) v, w ) v )which gives us the last two summands in (2.22). Therefore, (2.22) and hence (2.19) holdstrue.Now let’s make a conclusion about what we’ve proved so far. It follows from (2.19),(2.20)and proposition 2.8 that all (n+3)-point functions S LV V...M...V and S RV...M...V V dfined by(2.17) and (2.18) give rise to one single (n+3)-point function: S : M (0) ∗ × V × ... × M × ... × V → F ( z , ..., z n , w ) , (2.23)where M can be placed anywhere in between first V and last V . This function S satisfiesthe following properties. First, S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) v ) = S ( v ′ , ( b , w ) ... ( b n +1 , w n +1 ) v ) , (2.24)where { ( b , w ) , ( b , w ) , ..., ( b n +1 , w n +1 ) } is an arbitrary permutation of the set of pairs { ( a , z ) , ..., ( a n , z n ) , ( v, w ) } . Second, S satisfis the following recurrent formula: S ( v ′ , ( b , w )( b , w ) ... ( b n +1 , w n +1 ) v ) = S L ( v ′ , ( a i , z i ) v ) = S R ( v ′ , ( a j , z j ) v ) (2.25)where the symbol in S L represents an arbitrary combination of ( v, w ), { ( a k , z k ) } k = i and S L is given by (2.17),which means that one expand in terms of ( a i , z i ) from the left.Meanwhile, the in S R represents an arbitrary combination of ( v, w ), { ( a k , z k ) } k = j and S R is given by (2.18), i.e expand in terms of ( a j , z j ) from the right.Note that (2.24) and (2.25) are the same properties as (1), (2) of (n+2)-point functions,so the induction step is complete. Therefore, we conclude that for any positive integer n ,there exists a (n+3)-point function S as in (2.23) that satisfies the properties (2.24) and(2.25) (or properties (1) and (2)). PROOF OF FUSION RULES FORMULA 21 Extension of S Following the proof of (2.2.8) and (2.2.9) in [10], it is easy to verify the followingformulas for our (n+3)-point function S (2.23): S ( v ′ , ( L ( − a , z ) ... ( a n , z n )( v, w ) v ) = ddz S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) v ) S ( v ′ , ( L ( − v, w )( a , z ) ... ( a n , z n ) v ) = ddw S ( v ′ , ( v, w )( a , z ) ... ( a n , z n ) v ); (3.1) Z C S ( v ′ , ( a , z )( v, w ) ... ( a n , z n ) v )( z − w ) n dz = S ( v ′ , ( a ( k ) v, w ) ... ( a n , z n ) v ) Z C S ( v ′ , ( a , z )( a , z ) ... ( v, w ) v )( z − z ) n dz = S ( v ′ , ( a ( k ) a , z ) ... ( v, w ) v ) , (3.2)where in the first equation of (3.2) C is a contour of z surrounding w with z , ..., z n outside of C ; while in the second equation of (3.2) C is a contour of z surrounding z with z , ..., z n , w outside of C .We only present a proof for the second equation of (3.1) because the proof for rest ofthem will be exactly the same with corresponding ones in [10], so we omit them.Use induction on n . When n = 0, by definition we have: S ( v ′ , ( L ( − v, w ) v ) = f ( v ′ ⊗ L ( − v ⊗ v ) w − wtv − . But in A ( M ) one has: L ( − v + L (0) v ≡ O ( M ) [10]. This implies f ( v ′ ⊗ L ( − v ⊗ v ) w − wtv − = − wtv · f ( v ′ ⊗ v ⊗ v ) w − wtv − = ddw S ( v ′ , ( v, w ) v ) . Now assume that formula (3.1) holds true for (n+2)-point functions, then by the properties(2.24) and (2.25) we have: S ( v ′ , ( L ( − v, w )( a , z ) ... ( a n , z n ) v )= S L ( v ′ , ( a , z ) ... ( a n , z n )( L ( − v, w ) v )= S ( v ′ o ( a ) , ( a , z ) ... ( a n , z n )( L ( − v, w ) v ) z − wta + n X k =2 X j ≥ F wta ,j ( z , z k ) S ( v ′ , ( a , z ) ... ( a ( j ) a k , z k ) ... ( a n , z n )( L ( − v, w ) v )+ X j ≥ F wta ,j ( z , w ) S ( v ′ , ( a , z ) ... ( a n , z n )( a ( j ) L ( − v, w ) v ) . Note that we may apply induction hypothesis to the first two terms. Moreover, by somebasic communication formulas of VOA that involve L ( −
1) [8] one has: a ( j ) L ( − v = L ( − a ( j ) v − [ L ( − , a ( j )] v = L ( − a ( j ) v + ja ( j − v . It follows from the definition of F n,i ( z , w ) that X j ≥ F wta ,j ( z , w ) S ( v ′ , ( a , z ) ... ( a n , z n )( a ( j ) L ( − v, w ) v )= X j ≥ F wta ,j ( z , w ) ddw S ( v ′ , ( a , z ) ... ( a n , z n )( a ( j ) v, w ) v )+ X j ≥ z − wta ( j − (cid:18) ddw (cid:19) j (cid:18) w wta z − w (cid:19) S ( v ′ , ( a , z ) ... ( a n , z n )( a ( j − v, w ) v )= ddw X j ≥ F wta ,j ( z , w ) S ( v ′ , ( a , z ) ... ( a n , z n )( a ( j ) v, w ) v ) . This proves (3.1).3.1.
Extension of the last component.
Our ultimate goal is to define an intertwin-ing operator I ( v, w ) from S , and intuitively we want I to be given by the equation: h v ′ , I ( v, w ) v i = S ( v ′ , ( v, w ) v ). However in this equation, the input vectors v and v ′ need to be taken over the whole vector space M and M ′ , not just the bottom level M (0) and M (0) ∗ . Therefore, in order to properly define I , we have to extend the firstand last ”point” or input spaces of S from the bottom level to the whole space and get anew (n+3)-point function: S : M ′ × V × ... × M ... × V × M → F ( z , ..., z n , w ) , (3.3)then use this extended function to define the intertwining operator I .We will first extend the last input space of S from M (0) to M . The method we usehere is similar with the way of extension in [10]. Let ¯ M be the same vector space as in[10], which is a vector space spanned by symbols:( b , i )( b , i ) ... ( b m , i m ) v (3.4)where b i ∈ V, i k ∈ Z , v ∈ M (0), and ( b, i ) linear in b . Denote the vector in (3.4) by x .Extend S to M (0) ∗ × V × ... × M ... × V × ¯ M by letting: S ( v ′ , ( a , z ) ... ( a n , v n )( v, w ) x )= Z C ... Z C m S ( v ′ , ( a , z ) ... ( a n , v n )( v, w )( b , w ) ... ( b m , w m ) v ) w i ...w i m m dw ...dw m , (3.5)where C k is a contour of w k , C k contains C k +1 and C m contains 0; z , ..., z n and w areoutside C . Introduce a gradation on ¯ M by setting wt (( b , i )( b , i ) ... ( b m , i m ) v ) := m X k =1 ( wtb k − i k − , then ¯ M becomes a graded space: ¯ M = L n ∈ Z ¯ M ( n ) with M (0) ⊆ ¯ M (0). PROOF OF FUSION RULES FORMULA 23
Similar as in [10], we define the radical of S on ¯ M by Rad ( ¯ M ) := { x ∈ ¯ M | S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) x ) = 0 , ∀ a i ∈ V, v ∈ M , v ∈ M (0) ∗ } . (3.6)Even through our (n+3)-point function S is not exactly same with the correspondingone in [10], but the recurrent formula (2.17) which defines S are the same, and our S alsohas intrinsic properties (3.1) and (3.2). So it’s not hard to show that Rad ( ¯ M ) and thequotient space M = ¯ M /Rad ( ¯ M ) share the same properties as the corresponding ones in[10]. In particular, we will show M = ¯ M /Rad ( ¯ M ) is an irreducible admissible modulewith bottom level M (0). Proposition 3.1.
Let W be the subspace of ¯ M spanned by the following vectors: ∞ X i =0 (cid:18) mi (cid:19) ( a ( l + i ) b, m + n − i ) v − (cid:18) ∞ X i =0 ( − i (cid:18) li (cid:19) ( a, m + l − i )( b, n + i ) v − ∞ X i =0 ( − l + i (cid:18) li (cid:19) ( b, n + l − i )( a, m + i ) v (cid:19) (3.7) where a, b ∈ V, m, n, l ∈ Z , v ∈ M (0) . Then we have W ⊂ Rad ( ¯ M ) .Proof. Denote the element of (3.7) by y . We adopt the following notations in [6]. Let C iR be the circle of w i , i = 1 , R , and let C ǫ ( w ) be the circle of w centered at w with radius ǫ . Choose R, r, ρ >
R > ρ > r . In view of (3.5)and (2.24), we have: S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) y )= Z C ρ ∞ X i =0 (cid:18) mi (cid:19) S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( a ( l + i ) b, w ) v ) w m + n − i dw − Z C R Z C ρ ∞ X i =0 ( − i (cid:18) li (cid:19) S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( a, w )( b, w ) v ) w m + l − i w n + i dw dw + Z C ρ Z C r ∞ X i =0 ( − l + i (cid:18) li (cid:19) S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( b, w )( a, w ) v ) w m + i w n + l − i dw dw = Z C ρ ∞ X i =0 (cid:18) mi (cid:19) S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( a ( l + i ) b, w ) v ) w m + n − i dw − Z C R Z C ρ S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( a, w )( b, w ) v ) · ι w ,w (( w − w ) l ) w m w n dw dw + Z C ρ Z C r S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( b, w )( a, w ) v ) · ι w ,w (( − w + w ) l ) w m w n dw dw = Z C ρ ∞ X i =0 (cid:18) mi (cid:19) S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( a ( l + i ) b, w ) v ) w m + n − i dw − Z C ρ Z C R − C r S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( a, w )( b, w ) v )( w − w ) l w m w n dw dw . Since the only possible poles of the function S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( b, w )( a, w ) v )( w − w ) l w m w n in the area enclosed by C R − C r are at w = w , so by Cauchy’s integral theorem we have: Z C ρ Z C R − C r S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( a, w )( b, w ) v )( w − w ) l w m w n dw dw = Z C ρ Z C ǫ ( w ) S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( a, w )( b, w ) v )( w − w ) l w m w n dw dw . On the other hand, we may choose ǫ small enough so that when w is taken from C ǫ ( w ),one has | w − w | < | w | . Hence in the integral above, we may write:( w − w ) l w m w n = ( w − w ) l ( w + w − w ) m w n = ∞ X i =0 (cid:18) mi (cid:19) ( w − w ) l + i w m + n − i . Now by (3.2) we have: Z C ρ Z C ǫ ( w ) S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( a, w )( b, w ) v )( w − w ) l w m w n dw dw = Z C ρ Z C ǫ ( w ) ∞ X i =1 (cid:18) mi (cid:19) S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( a, w )( b, w ) v ) · ( w − w ) l + i w m + n − i dw dw = Z C ρ ∞ X i =1 (cid:18) mi (cid:19) S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( a ( l + i ) b, w ) v ) w m + n − i dw dw , which implies S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) y ) = 0. (cid:3) Remark 3.2. (1) The graph of contours in the proof above looks like: C ρ ρ C R RC ǫ ( z ) C r r PROOF OF FUSION RULES FORMULA 25 (2) By the definition (3.5) of our extended S , it is easy to see that if we replace v in (3.7) by an arbitrary element x ∈ ¯ M the conclusion remains valid. That is to say: forany x ∈ ¯ M , a, b ∈ V and m, n ∈ Z the element ∞ X i =0 (cid:18) mi (cid:19) ( a ( l + i ) b, m + n − i ) x − (cid:18) ∞ X i =0 ( − i (cid:18) li (cid:19) ( a, m + l − i )( b, n + i ) x − ∞ X i =0 ( − l + i (cid:18) li (cid:19) ( b, n + l − i )( a, m + i ) x (cid:19) (3.8) belongs to Rad ( ¯ M ) . Furthermore, we observe the following facts about
Rad ( ¯ M ). Lemma 3.3. (1) If x ∈ Rad ( ¯ M ) , then ( b, i ) x ∈ Rad ( ¯ M ) , ∀ b ∈ V, i ∈ Z ;(2) M (0) T Rad ( ¯ M ) = 0 ;(3) If n < one has ¯ M ( n ) ⊂ Rad ( ¯ M ) .Proof. (1) By definition (3.5) of S and definition of Rad ( ¯ M ), we see that S ( v ′ , ( a , z ) ... ( v, w )( b, i ) x ) = Z C S ( v ′ , ( a , z ) ... ( v, w )( b, w ) x ) w i dw = Z C · w i dw = 0where C is a contour of w with z , ..., z n , w outside. Thus, ( b, i ) x ∈ Rad ( ¯ M ).(2) Suppose there exists some v = 0 in M (0) T Rad ( ¯ M ), then by the definition of4-point functions in secion 2.2, we have for any a ∈ V :0 = ι w,z ( S ( v ′ , ( a, z )( v, w ) v ))= S ( v ′ , ( v, w ) o ( a ) v ) z − wta + X i ≥ ι w,z ( G wta,i ( z, w )) S ( v ′ , ( a ( i ) v, w ) v )= f ( v ′ ⊗ v ⊗ o ( a ) v ) z − wta w − wtw − X i,j ≥ (cid:18) wta − − ji (cid:19) w wtv − j − z − wta +1+ j f ( v ′ ⊗ a ( i ) v ⊗ v ) . By comparing the coefficients of z − wta w − wtw on both sides of this equation, we see that f ( v ′ ⊗ v ⊗ o ( a ) v ) = 0 for any a ∈ V, v ∈ A ( M ) , v ′ ∈ M (0) ∗ . Moreover, since M (0) isan irreducible A ( V ) module we have M (0) = A ( V ) .v = span { o ( a ) v | a ∈ V } . It followsthat f = 0, and this is a contradiction.(3) Let x = ( b m , i m ) ... ( b , i ) v with P mk =1 ( wtb k − i k − <
0. We use induction on thelength m of x to show x ∈ Rad ( ¯ M ). For base case, let x = ( b, t ) v with wtb − t − < then by (3.5) we have S ( v ′ , ( a , z ) ... ( v, w ) x ) = Z C S ( v ′ , ( a , z ) ... ( v, w )( b, z ) v ) z t dz = Z C S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) o ( b ) v ) z t − wtb dz + Z C n X k =1 X i ≥ G wtb,i ( z, z k ) S ( v ′ , ( a , z ) ... ( b ( i ) a k , z k ) ... ( v, w ) v ) z t dz + Z C X i ≥ G wtb,i ( z, w ) S ( v ′ , ( a , z ) ... ( b ( i ) v, w ) v ) z t dz, (3.9)where C is a contour of z surrounding 0 with all other variables lying outside C . Inparticular, | z | < | z k | for any k and | z | < | w | . Hence for any i ≥ Z C G wtb,i ( z, z k ) z t dz = Z C z − wtb +1+ t i ! (cid:18) ddz k (cid:19) i (cid:18) z wtb − k z − z k (cid:19) dz = 0 , (3.10)since − wtb + 1 + t > / ( z − z k ) is a sum of positive powers in z within the contour C . Therefore, since t − wt b is greater than −
1, all the integrals on the right hand side of(3.9) are equal to 0. The proof for base case is finished.For general x = ( b m , i m ) ... ( b , i ) v , by definition (3.5) we have: S ( v ′ , ( a , z ) ... ( v, w ) x )= Z C m ... Z C S ( v ′ , ( a , z ) ... ( v, w )( b m , w m ) ... ( b , w ) v ) w i m m ...w i dw ...dw m = Z C m ... Z C S ( v ′ , ( a , z ) ... ( v, w )( b m , w m ) ...o ( b ) v ) w i m m ...w − wtb + i dw ...dw m ♥ + Z C m ... Z C n X k =1 X i ≥ G wtb ,i ( w , z k ) S ( v ′ , ... ( b ( i ) a k , z k ) ... ( v, w ) ...v ) w i m m ...w i dw ...dw m ⋆ + Z C m ... Z C X i ≥ G wtb ,i ( w , w ) S ( v ′ , ... ( b ( i ) v, w )( b m , w m ) ...v ) w i m m ...w i dw ...dw m △ + Z C m ... Z C m X l =2 X i ≥ G wtb ,i ( w , w l ) S ( v ′ , ... ( v, w ) ... ( b ( i ) b l , w l ) ...v ) w i m m ...w i dw ...dw m (cid:4) = ♥ + ⋆ + △ + (cid:4) , where C is a contour of w surrounding 0 with all other variables lying outside. We needto show that the sum of these integrals is equal to 0, i.e. ♥ + ⋆ + △ + (cid:4) = 0.Case 1. wtb − i − < z : Z C G wtb ,i ( w , z ) w i dw = Z C w − wtb +1+ i i ! (cid:18) ddz (cid:19) i (cid:18) z wtb − w − z (cid:19) dw = 0 . PROOF OF FUSION RULES FORMULA 27
Thus, ⋆ = △ = (cid:4) = 0, and we also have ♥ = 0, since − wtb + i > − wtb − i − > − wtb + i < −
1, which implies ♥ = 0. Moreover, for any variable z we have the following evaluation: Z C G wtb ,i ( w , z ) w i dw = − Z C X j ≥ (cid:18) wtb − − ji (cid:19) z wtb − j − − i w − wtb +1+ j + i dw = Res w =0 (cid:18) X j ≥ (cid:18) wtb − − ji (cid:19) z wtb − j − − i w − wtb +1+ j + i (cid:19) = − (cid:18) i i (cid:19) z i − i . (3.11)Apply the formula (3.11) to ⋆ , △ , and (cid:4) , we have: ⋆ = − Z C m ... Z C n X k =1 X i ≥ (cid:18) i i (cid:19) z i − ik S ( v ′ , ... ( b ( i ) a k , z k ) ... ( v, w )( b m , w m ) ... ( b , w ) v )= − n X k =1 X i ≥ (cid:18) i i (cid:19) z i − ik S ( v ′ , ( a , z ) ... ( b ( i ) a k , z k ) ... ( a n , z n )( v, w ) y ) , where y = ( b m , i m ) ... ( b , i ) v . Note that wty = wtx − ( wtb − i − < y is m −
1, by induction hypothesis, S ( v ′ , ( a , z ) ... ( b ( i ) a k , z k ) ... ( a n , z n )( v, w ) y ) = 0for any i , then it follows that ⋆ = 0. Similarly, we have △ = 0. Finally, (cid:4) = Z C m ... Z C m X l =2 X i ≥ (cid:18) i i (cid:19) w i − il S ( v ′ , ... ( v, w ) ... ( b ( i ) b l , w l ) ...v ) w i m m ...w i dw ...dw m = m X l =2 X i ≥ (cid:18) i i (cid:19) S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) y l ) , where y l = ( b m , i m ) ... ( b ( i ) b l , i + i l − i ) ... ( b , i ) v . Note that wt ( b ( i ) b l , i + i l − i ) = wtb + wtb l − i − − i − i l + i − wt ( b , i ) + wt ( b l , i l ) . Thus, wty l = P mk =1 wt ( b k , i k ) = wtx < y l is m −
1. By induction, onehas S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) y l ) = 0 for any l , which implies (cid:4) = 0.Case 3. wtb − i − z : Z C G wtb ,i ( w , z ) w i dw = Z C w i ! (cid:18) ddz (cid:19) i (cid:18) z wtb − w − z (cid:19) dw = 0 . Hence ⋆ = △ = (cid:4) = 0. Finally, since − wtb + i = −
1, we have: ♥ = Z C m ... Z C Z C S ( v ′ , ( a , z ) ... ( v, w )( b m , w m ) ...o ( b ) v ) w i m m ...w − wtb + i dw ...dw m = Z C m ... Z C S ( v ′ , ( a , z ) ... ( v, w )( b m , w m ) ...o ( b ) v ) w i m m ...w i dw ...dw m = S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) y ) , where y = ( b m , i m ) ... ( b , i ) v with wty = wtx < m −
1. Again by inductionhypothesis, S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) y ) = 0, and so ♥ = 0 . (cid:3) We define the vertex operator Y M on the quotient space M = ¯ M /Rad ( ¯ M ) as in [10]: Y M ( a, z )( b , i ) ... ( b m , i m ) v := X n ∈ Z ( a, n )( b , i ) ... ( b m , i m ) v z − n − , (3.12)where a ∈ V, ( b , i ) ... ( b m , i m ) v ∈ M , and we use the same notation ( b , i ) ... ( b m , i m ) v for the equivalent class of this element in the quotient space M . We may also interpret(3.12) in component form as: a ( n ) . ( b , i ) ... ( b m , i m ) v = ( a, n )( b , i ) ... ( b m , i m ) v , (3.13)for all a ∈ V, n ∈ Z , and ( b , i ) ... ( b m , i m ) v ∈ M . By part (1) of the lemma 3.3, wesee that a ( n ) .Rad ( ¯ M ) ⊆ Rad ( ¯ M ) i.e. Y M is well-defined. Now we claim that for any x = ( b , i ) ... ( b m , i m ) v ∈ M one has ( − .x = x and ( n ) .x = 0 for all n = − ( j ) .a = 0 for any a ∈ V and j ≥
0, we have: S ( v ′ , ( a , z ) ... ( v, w ) x )= Z C Z C m ... Z C S ( v ′ , ( , w )( a , z ) ... ( v, w )( b , w ) ... ( b m , w m ) v ) · w n w i ...w i m m dw ...dw m dw = Z C Z C m ... Z C S ( v ′ o ( ) , ( a , z ) ... ( v, w )( b , w ) ... ( b m , w m ) v ) w n w i ...w i m m dw ...dw m dw . We note that o ( ) = Id , and Z C w n dw = n = −
11 if n = − , so our claim follows. Moreover, given x = ( b , i ) ... ( b m , i m ) v ∈ M and a ∈ V , we have wt ( a ( n ) .x ) = wta − n − wtx < n >>
0, so by part (3) of the lemma a ( n ) .x = 0 if n is large enough. Finally, by proposition 3.1 and (3.8), we have: ∞ X i =0 (cid:18) mi (cid:19) ( a ( l + i ) b )( m + n − i ) .x = ∞ X i =0 ( − i (cid:18) li (cid:19) a ( m + l − i ) b ( n + i ) .x − ∞ X i =0 ( − l + i (cid:18) li (cid:19) b ( n + l − i ) a ( m + i ) .x (3.14)for any m, n, l ∈ Z ; a, b ∈ V and x ∈ M . Therefore, Y M satisfies the Jacobi identity andall the other axioms for a weak V module, that is to say, ( M, Y M ) is a weak V module. PROOF OF FUSION RULES FORMULA 29
Furthermore, we can show the following:
Proposition 3.4. M has a gradation: M = L n ≥ M ( n ) with each M ( n ) an eigenspaceof L (0) and M (0) = M (0) .Proof. Let M ( n ) be the image of ¯ M ( n ) under the quotient map. By lemma 3.3 we have M = X n ≥ M ( n ) , and M (0) ⊆ M (0) . In order to show that the subspaces M ( n ) of M form a direct sum, we first show that L (0) acts semisimply on M (0), and to this end we claim first that on M (0): a ( wta − .v = o ( a ) v (3.15)In fact, it suffices to show that ( a, wta − v − o ( a ) v ∈ Rad ( ¯ M ). By definition we have: S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( a, wta − v )= Z C S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( a, w ) v ) w wta − dw = Z C S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) o ( a ) v ) w − wta w wta − dw + n X k =1 X i ≥ Z C G wta,i ( w , z k ) S ( v ′ , ( a , z ) ... ( a ( i ) a k , z k ) ... ( a n , z n )( v, w ) v ) w wta − dw + X i ≥ Z C G wta,i ( w , w ) S ( v ′ , ( a , z ) ... ( a n , z n )( a ( i ) v, w ) v ) w wta − dw , where C is a contour of w surrounding 0 with other variables outside of C . Thus for anyvariable z , one has Z C G wta,i ( w , z ) w wta − dw = Z C w wta − w − wta +11 i ! (cid:18) ddz (cid:19) i (cid:18) z wta − w − z (cid:19) dw = 0 , which implies that all the integrals involving G wta,i are equal to 0, while S ( v ′ , ( a , z ) ... ( a n , z n )( v, w )( a, wta − v ) = S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) o ( a ) v ) . Now the proof of (3.15) is complete.Recall that [ L (0) , a ( n )] = ( wta − n − a ( n ) [8]. Apply (3.15) to L (0) = ω ( wtω − L (0) , o ( a )] v = [ L (0) , a ( wta − v = 0 . Thus, L (0) is commutative with the action of A ( V ) on the irreducible module M (0).Since M (0) is of countable dimension, then by Schur’s lemma L (0) = λ · Id on M (0). For any x = ( b , i ) ... ( b m , i m ) v = b ( i ) ...b m ( i m ) v ∈ M ( n ), it follows that: L (0) x = ( m X k =1 ( wtb k − k −
1) + λ ) x = ( n + λ ) x. Therefore, each M ( n ) is an eigenspace of L (0) of eigenvalue n + λ for all n ∈ Z ≥ . Thisimplies that M = L n ≥ M ( n ).Finally, for any x = ( b m , i m ) ... ( b , i ) v ∈ M (0) we use induction on the length m of x to show x ∈ M (0). The base case m = 0 is clear. Consider general m >
0, if all( b j , i j ) have weight 0 then by (3.15) x = o ( b m ) ...o ( b ) v ∈ M (0), and so we are done.Otherwise, there must exist some ( b j , i j ) has negative weight because wtx = 0. Withoutloss of generality, we assume that wtb m − i m − <
0. Then we may express x as : x = ( b m − , i m − ) ... ( b , i )( b m , i m ) v + [( b m , i m ) , ( b m − , i m − ) ... ( b , i ) v ]= 0 + m − X j =1 ( b m − , i m − ) ... [( b m , i m ) , ( b j , i j )] ... ( b , i ) v . Note that each of the term ( b m − , i m − ) ... [( b m , i m ) , ( b j , i j )] ... ( b , i ) v in the sum has thesame weight as x but smaller length than x , so by induction hypothesis they all belongto M (0), therefore x ∈ M (0) and M (0) = M (0). (cid:3) Next, we show that M = ¯ M /Rad ( ¯ M ) is in fact an irreducible V module, and so M is isomorphic to M . Note that for any x ∈ M , S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) x ) is also arational funtion in z , ...z n , w , and it has Laurent series expansion S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) x )= S ( v ′ , ( v, w )( a , z ) ... ( a n , z n ) x )= X i ,...,i n ∈ Z Z C n ... Z C S ( v ′ , ( v, w )( a n , z n ) ... ( a , z ) x ) z i ...z i n n dz ...dz n z − i − ...z − i n − n = X i ,...,i n ∈ Z S ( v ′ , ( v, w ) a n ( i n ) ...a ( i ) x ) z − i − ...z − i n − n (3.16)on the domain D = { ( z , ...z n , w ) || w | > | z n | > ... > | z | > } . Lemma 3.5.
For any y ∈ M ( n ) with n > one has: S ( v ′ , ( v, w ) y ) = 0 for all v ′ ∈ M (0) ∗ and v ∈ M .Proof. It follows from an easy induction that y can be written as a sum of the terms( b m , i m ) ... ( b , i ) v with wtb j − i j − > j . Let y ′ = ( b m , i m ) ... ( b , i ) v , bydefinition and properties of the function S , we have S ( v ′ , ( v, w ) y ′ ) = Z C m ... Z C S ( v ′ , ( v, w )( b m , w m ) ... ( b , w ) v ) w i ...w i m m dw ...dw m PROOF OF FUSION RULES FORMULA 31 = Z C m ... Z C S ( v ′ , ( b , w )( v, w )( b m , w m ) ... ( b , w ) v ) w i ...w i m m dw ...dw m = Z C m ... Z C S ( v ′ o ( b ) , ( v, w )( b m , w m ) ... ( b , w ) v ) w − wtb + i ...w i m m dw ...dw m + Z C m ... Z C X i ≥ F wtb ,i ( w , w ) S ( v ′ , ( b ( i ) v, w )( b m , w m ) ... ( b , w ) v ) w i ...w i m m dw ...dw m + Z C m ... Z C m X k =2 X i ≥ F wtb ,i ( w , w k ) S ( v ′ , ( v, w ) ... ( b ( i ) b k , w k ) ... ( b , w ) v ) · w i ...w i m m dw ...dw m Since − wtb + i < −
1, we have for any variable z (with | z | > | w | ) and any i ≥ Z C F wtb ,i ( w , z ) w i dw = Z C w − wtb + i i ! (cid:18) ddz (cid:19) i (cid:18) z wtb w − z (cid:19) dw = 0 . It follows that S ( v ′ , ( v, w ) y ′ ) and hence S ( v ′ , ( v, w ) y ) is equal to 0. (cid:3) Proposition 3.6. M = ¯ M /Rad ( ¯ M ) is an irreducible V module with bottom level M (0) .Consequently, M is isomorphic to the irreducible V module M that we were starting with.Proof. Let N be a submodule of M . Since M is a direct sum of eigenspace for L (0) and L (0) .N ⊆ N , we have N = L n ≥ N ( n ). By (3.15), o ( a ) N (0) = a ( wta − N (0) ⊆ N (0),so N (0) is a A ( V ) submodule of M (0).Since M (0) is an irreducible A ( V ) module, and it generate M as V module, we have N = M if N (0) = M (0). On the other hand, if N (0) = M (0) then N (0) = 0, and weclaim that in this case N = 0.In fact, it suffices to show that for any x ∈ N we have S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) x ) = 0for any a i ∈ V and v ∈ M . By (3.16) we have on the domain D = { ( z , ...z n , w ) || w | > | z n | > ... > | z | > } the Laurent series expansion of S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) x ): S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) x ) = X i ,...,i n ∈ Z S ( v ′ , ( v, w ) a n ( i n ) ...a ( i ) x ) z − i − ...z − i n − n Since N is a submodule with N (0) = 0, we have y = a n ( i n ) ...a ( i ) x ∈ N and if y = 0then we have wt ( y ) >
0. Now the lemma 3.5 tells us that S ( v ′ , ( v, w ) y ) = 0 for any such y .Therefore, the Laurent series and hence the rational function S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) x )itself is equal to 0. (cid:3) In conclusion, we proved that the extended function S : M (0) ∗ × V × ... × M × ... × V × ¯ M → F ( z , ...z n , w ) can factor through M = ¯ M /Rad ( ¯ M ) ∼ = M and yield a well-defined(n+3)-point function: S : M (0) ∗ × V × ... × M × ... × V × M → F ( z , ...z n , w ) This S is defined by formula 3.5. Namely, if x = b ( i ) ...b m ( i m ) v ∈ M we have S ( v ′ , ( a , z ) ... ( a n , v n )( v, w ) x ):= Z C ... Z C m S ( v ′ , ( a , z ) ... ( a n , v n )( v, w )( b , w ) ... ( b m , w m ) v ) w i ...w i m m dw ...dw m , (3.17)where C k is a contour of w k , C k contains C k +1 and C m contains 0; z , ..., z n and w areoutside of C . Remark 3.7.
It seems we took a long way to extend the last input component of S from M (0) to M , but it is necessary. Because if we just use the formula (3.17) alone to extendour S and don’t include ¯ M and Rad ( ¯ M ) in our discussion, then we would have to checkthe well-definedness of S . Namely, if x ∈ M has two expressions: x = b ( i ) ...b m ( i m ) v = c ( j ) ...c k ( j k ) v ′ , we would have to show the right hand side of (3.17) yields the same valueon these two different expressions. This is certainly not an easy task because we don’t knowhow many relations are there in M . Extension of the first component.
There is a final preparation step before wecan construct an intertwining operator I out of S . We need to extend the first inputcomponent of S from bottom level M (0) ∗ to the whole irreducible module M ′ .Before we formally extend the first component of S , let’s make some observations first.These observations are actually the motivation of our definition for the new S . Let M bea V module, recall that on the contragradient module M ′ [5] the action is given by: h Y ′ ( a, z ) x ′ , x i = h x ′ , Y ( e zL (1) ( − z − ) L (0) a, z − ) x i , where x ∈ M, x ′ ∈ M ′ , and if we compare the coefficients of z − n − on both sides, it iseasy to see that: h a ( n ) x ′ , x i = h x ′ , X j ≥ j ! ( − wta ( L (1) j a )(2 wta − n − j − x i . Let’s denote by a ′ ( n ) the term P j ≥ j ! ( − wta ( L (1) j a )(2 wta − n − j − w =0 h a ( n ) x ′ , Y ( b, w ) x i w m , we have the following computation:Res w =0 h a ( n ) x ′ , Y ( b, w ) x i w m = Z C h a ( n ) x ′ , Y ( b, w ) x i w m dw = Z C Z C h Y ′ ( a, z ) x ′ , Y ( b, w ) x i z n w m dzdw = Z C Z C h x ′ , Y ( e zL (1) ( − z − ) L (0) a, z − ) Y ( b, w ) x i z n w m dzdw, PROOF OF FUSION RULES FORMULA 33 where C is a contour of z surrounding 0 with w outside, and C is a contour of w whichcontains C . Without loss of generality, we assume C , C are circles centered at 0 ofradius R , R respectively, R > R > R < z → /z in the integral above. Note that the parametrizationof 1 /z is (1 /R ) e − iθ which gives us clockwise orientation, and d (1 /z ) = − (1 /z ) dz . Thus,if we let C ′ be the circle centered at 0 with radius 1 /R (with counterclockwise orientation),then we have:Res w =0 h a ( n ) x ′ ,Y ( b, w ) x i w m = Z C Z C h x ′ , Y ( e zL (1) ( − z − ) L (0) a, z − ) Y ( b, w ) x i z n w m dzdw = Z C Z − C ′ h x ′ , Y ( e z − L (1) ( − z ) L (0) a, z ) Y ( b, w ) x i ( − z − n − w m dzdw = Z C Z C ′ h x ′ , Y ( e z − L (1) ( − z ) L (0) a, z ) Y ( b, w ) x i z − n − w m dzdw, (3.18)where C ′ is a circle with radius R ′ = 1 /R > /R > R (since R < C ′ is acontour of z that contains the contour C of the variable w .Therefore, on the domain D = { w || w | < } the rational function ( a ( n ) x ′ , Y ( b, w ) x ) hasLaurent series expansion:( a ( n ) x ′ , Y ( b, w ) x ) = X m ∈ Z ( Z C Z C ′ h x ′ , Y ( e z − L (1) ( − z ) L (0) a, z ) Y ( b, w ) x i z − n − w m dzdw ) z − m − , and the right hand side is the Laurent series expansion of the rational function Z C ′ h x ′ , Y ( e z − L (1) ( − z ) L (0) a, z ) Y ( b, w ) x i z − n − dz on the same domain D . Hence on D we have the following equality of rational functions:( a ( n ) x ′ , Y ( b, w ) x ) = Z C ′ h x ′ , Y ( e z − L (1) ( − z ) L (0) a, z ) Y ( b, w ) x i z − n − dz, (3.19)where C ′ is a contour of z with w inside.Moreover, if we replace a ( n ) in (3.19) by a ′ ( n ), then we will get a more elegant formula:( a ′ ( n ) x ′ , Y ( b, w ) x ) = Z C ′ h x ′ , Y ( a, z ) Y ( b, w ) x i z n dz. (3.20)In fact,( a ′ ( n ) x ′ ,Y ( b, w ) x ) = ( X j ≥ j ! ( − wta ( L (1) j a )(2 wta − n − j − x ′ , Y ( b, w ) x )= X j ≥ j ! ( − wta Z C ′ h x ′ , Y ( e z − L (1) ( − z ) L (0) ( L (1) j a ) , z ) Y ( b, w ) x i z − wta + n + j dz = Z C ′ h x ′ , Y ( e z − L (1) ( − z ) L (0) e zL (1) ( − z − ) L (0) a, z ) Y ( b, w ) x i z n = Z C ′ h ( Y ( e z − L (1) e − z − L (1) a, z ) Y ( b, w ) x i z n dz = Z C ′ h x ′ , Y ( a, z ) Y ( b, w ) x i z n dz. We introduce the following two notations which we will use later. F ′ ( z , ..., z n , w ) : = { f ( z , ..., z n , w ) ∈ F ( z , ..., z n , w ) || z i | < , ∀ i ; | w | < } ;( a, z ) ′ : = ( e z − L (1) ( − z ) L (0) a, z ) . Note that one may restrict the variables z , ..., z n , w in S ( v ′ , ( a , z ) ... ( a n , z n )( v, w ) x ) tothe domain D = { ( z , ..., z n , w ) || z i | < , ∀ i ; | w | < } and get a map: S : M (0) ∗ × V × ... × M × ... × V × M → F ′ ( z , ..., z n , w ) (3.21)From now on, we assume that S is the one given in ( ?? ), i.e. we restrict the variables of S in the domain D . Motivated by formulas (3.19) and (3.20), we extend the first component M (0) ∗ of S to M ′ in the following way:Proceed like the extension in section 3.2, we define ˜ M to be the vector space spannedby symbols of the following form: x = ( b , i ) ... ( b m , i m ) v ′ , where b j ∈ V, i j ∈ Z and v ′ ∈ M (0) ∗ and ( b, i ) is linear in b . Next, we extend the firstcomponent of S from M (0) ∗ to ˜ M by letting: S (( b , i ) ... ( b m , i m ) v ′ , ( a , z ) ... ( a n , z n )( v, w ) x )= Z C m ... Z C S ( v ′ , ( b m , w m ) ′ ... ( b , w ) ′ ( a , z ) ... ( a n , z n )( v, w ) x ) w − i − ...w − i m − m dw ...dw m , (3.22)where C k is a contour of w k , C k contains C k − and C contains all variables z , ..., z n , w .Similar as in section 3.2, one can show ˜ M /Rad ( ˜ M ) has a natural V module structuredefined by S , and one can show ˜ M /Rad ( ˜ M ) is an irreducible V module isomorphic to M ′ . Since S factor through ˜ M /Rad ( ˜ M ) ∼ = M ′ , it follows that we have a well-definedextended function S : S : M ′ × V × ... × M × ... × V × M → F ′ ( z , ..., z n , w )such that S ( b ( i ) ...b m ( i m ) v ′ , ( a , z ) ... ( a n , z n )( v, w ) x ) is given by (3.22): S ( b ( i ) ...b m ( i m ) v ′ , ( a , z ) ... ( a n , z n )( v, w ) x )= Z C m ... Z C S ( v ′ , ( b m , w m ) ′ ... ( b , w ) ′ ( a , z ) ... ( a n , z n )( v, w ) x ) w − i − ...w − i m − m dw ...dw m . (3.23) PROOF OF FUSION RULES FORMULA 35
Moreover, by a similar computation as the proof of formula (3.20), we have: S ( b ′ ( i ) ...b ′ m ( i m ) v ′ , ( a , z ) ... ( a n , z n )( v, w ) x )= Z C m ... Z C S ( v ′ , ( b m , w m ) ... ( b , w )( a , z ) ... ( a n , z n )( v, w ) x ) w i ...w i m m dw ...dw m , (3.24)where z , ..., z n , w are inside of C .Note that in the defining formula (3.17) and (3.24) of the extension of S , the terms( a i , z i ) i = 1 , ..., n and ( v, w ) in the integrals can be permuted arbitrarily, thanks to (2.24).Therefore, our extended S also satisfies the permutation invariant property: S ( x ′ , ( a , z ) ... ( a n , z n )( v, w ) x ) = S ( x ′ , ( b , w ) ... ( b n , w n )( b n +1 , w n +1 ) x ) , (3.25)where x ′ ∈ M ′ and x ∈ M , and (( b , w ) , ..., ( b n +1 , w n +1 )) is an arbitrary permutationof ( a i , z i ), i = 1 , ..., n and ( v, w ).Finally, it is easy to see from (3.25) that the formulas (3.1) and (3.2) also hold true forour extended S : M ′ × V × ... × M × ... × V × M → F ′ ( z , ..., z n , w ). In other words,we have: S ( x ′ , ( L ( − a , z ) ... ( a n , z n )( v, w ) x ) = ddz S ( x ′ , ( a , z ) ... ( a n , z n )( v, w ) x ); S ( x ′ , ( L ( − v, w )( a , z ) ... ( a n , z n ) x ) = ddw S ( x ′ , ( v, w )( a , z ) ... ( a n , z n ) x ) . (3.26) Z C S ( x , ( a , z )( v, w ) ... ( a n , z n ) x )( z − w ) n dz = S ( x ′ , ( a ( k ) v, w ) ... ( a n , z n ) x ); Z C S ( x ′ , ( a , z )( a , z ) ... ( v, w ) x )( z − z ) n dz = S ( x ′ , ( a ( k ) a , z ) ... ( v, w ) x ) . (3.27)3.3. Proof fusion rules formula.
With (n+3)-point function S : M ′ × V × ... × M × ... × V × M → F ′ ( z , ..., z n , w )in our hand we construct an intertwining operator I ∈ I (cid:0) M M M (cid:1) in the following way:Let v ∈ M , define v ( n ) : M → M (= M ′′ ) by the formula: h x ′ , v ( n ) x i := Z C S ( x ′ , ( v, w ) x ) w n dw, (3.28)where C is a contour of w surrounding 0. Define I ( v, w ) by I ( v, w ) := X n ∈ Z v ( n ) w − n − · w − h , where h = h + h − h and h i , i = 1 , , M i , i = 1 , , In other words, h x ′ , I ( v, w ) x i is defined as the Laurent series expansion of S ( x ′ , ( v, w ) x ) · w − h at w = 0. For simplicity, we omit the term w − h in I ( v, w ) for the rest of this sectionand write: I ( v, w ) = X n ∈ Z v ( n ) w − n − . In particular, if we denote by ( x ′ , I ( v, w ) x ) the limit of the Laurent series h x ′ , I ( v, w ) x i in | w | <
1, then we have: ( x ′ , I ( v, w ) x ) = S ( x ′ , ( v, w ) x )as rational functions. Now it follows from (3.26) that( x ′ , I ( L ( − v, w ) x ) = S ( x ′ , ( L ( − v, w ) x )= ddw S ( x ′ , ( v, w ) x )= ddw ( x ′ , I ( v, w ) x ) . Therefore, I ( L ( − v, w ) = ddw I ( v, w ). Moreover, I ( v, w ) satisfies the Jacobi identity: Proposition 3.8.
The operators v ( n ) defined in (3.28) satisfies the following: ∞ X i =0 (cid:18) mi (cid:19) ( a ( l + i ) v )( m + n − i ) x = ∞ X i =0 ( − i (cid:18) li (cid:19) a ( m + l − i ) v ( n + i ) x − ∞ X i =0 ( − l + i (cid:18) li (cid:19) v ( n + l − i ) a ( m + i ) x (3.29) In particular, I ( v, w ) is an intertwining operator of the type (cid:0) M M M (cid:1) [5] .Proof. By (3.28), (3.24) and the definition of contragradient module, we have h x ′ , ∞ X i =0 ( − i (cid:18) li (cid:19) a ( m + l − i ) v ( n + i ) x i = ∞ X i =0 ( − i (cid:18) li (cid:19) h a ′ ( m + l − i ) x ′ , v ( n + i ) x i = ∞ X i =0 ( − i (cid:18) li (cid:19) Z C ′ S ( a ′ ( m + l − i ) x ′ , ( v, w ) x ) w n + i dw = ∞ X i =0 ( − i (cid:18) li (cid:19) Z C ′ Z C ′ S ( x ′ , ( a, z )( v, w ) x ) z m + l − i w n + i dw, (3.30)where C ′ is a contour of w , C ′ is a contour of z which contains C ′ . PROOF OF FUSION RULES FORMULA 37
On the other hand, by (3.17) we have: h x ′ , ∞ X i =0 ( − l + i (cid:18) li (cid:19) v ( n + l − i ) a ( m + i ) x i = ∞ X i =0 ( − l + i (cid:18) li (cid:19) Z C S ( x ′ , ( v, w ) a ( m + i ) x ) w n + l − i dw = ∞ X i =0 ( − l + i (cid:18) li (cid:19) Z C Z C S ( x ′ , ( v, w )( a, z ) x ) z m + i w n + l − i dzdw, (3.31)where C is a contour of w , C is a contour of z that is contained in C .We adopt the notations in proposition 3.1, let C zα be a circle of variable z centered at0 with radius α , and let C wβ be a circle of variable w centered at 0 with radius β . Choose R, r, ρ > > R > ρ > r . Combining (3.30), (3.31), (3.25) and (3.27), we have: h x ′ , ∞ X i =0 ( − i (cid:18) li (cid:19) a ( m + l − i ) v ( n + i ) x − ∞ X i =0 ( − l + i (cid:18) li (cid:19) v ( n + l − i ) a ( m + i ) x i = ∞ X i =0 ( − i (cid:18) li (cid:19) Z C wρ Z C zR S ( x ′ , ( a, z )( v, w ) x ) z m + l − i w n + i dwdz − ∞ X i =0 ( − l + i (cid:18) li (cid:19) Z C wρ Z C zr S ( x ′ , ( v, w )( a, z ) x ) z m + i w n + l − i dzdw = Z C wρ Z C zR S ( x ′ , ( a, z )( v, w ) x ) ι z,w ( z − w ) l z m w n dwdz − Z C wρ Z C zr S ( x ′ , ( v, w )( a, z ) x ) ι w,z ( z − w ) l z m w n dzdw = Z C wρ Z C zǫ ( w ) S ( x ′ , ( a, z )( v, w ) x )( z − w ) l z m w n dzdw = Z C wρ Z C zǫ ( w ) S ( x ′ , ( a, z )( v, w ) x )( z − w ) l ι w,z − w ( w + ( z − w )) m w n dzdw = X i ≥ (cid:18) mi (cid:19) Z C wρ Z C zǫ ( w ) S ( x ′ , ( a, z )( v, w ) x )( z − w ) l + i w n + m − i dzdw = X i ≥ (cid:18) mi (cid:19) Z C wρ S ( x ′ , ( a ( l + i ) v, w ) x ) w m + n − i = X i ≥ (cid:18) mi (cid:19) h x ′ , ( a ( l + i ) v )( m + n − i ) x i (3.32)The graph of different contours in (3.32) is given as follows: C wρ ρ C zR RC zǫ ( w ) C zr r Since x ′ in (3.32) can be taken arbitraily, it follows that (3.29) and hence the Jacobiidentity of I ( v, w ) holds. (cid:3) Therefore, starting with f ∈ ( M (0) ∗ ⊗ A ( V ) A ( M ) ⊗ A ( V ) M (0)) ∗ we can construct a(n+3)-point rational function S : S : M ′ × V × ... × M × ... × V × M → F ′ ( z , ..., z n , w ) , and using this S we can define an intertwining operator I = I f of the type (cid:0) M M M (cid:1) .Finally, we claim that π ( I f ) = f , where π : I M M , M ! → ( M (0) ∗ ⊗ A ( V ) A ( M ) ⊗ A ( V ) M (0)) ∗ is given by π ( I )( v ′ ⊗ v ⊗ v ) = h v ′ , o ( v ) v i as in (1.3), where v ′ ∈ M (0) , v ∈ M (0) and v ∈ M .In fact, by (3.28) and the definition of S we have: π ( I f )( v ′ ⊗ v ⊗ v ) = h v ′ , v ( wtv − v i = Z C S ( v ′ , ( v, w ) v ) w wtv − dw = Z C f ( v ′ ⊗ v ⊗ v ) w − wtv w wtv − dw = f ( v ′ ⊗ v ⊗ v ) . (3.33)This shows π ( I f ) = f . Therefore, the map π is surjective. It is shown in [9] that thrmap π is injective provided M and M are irreducible V modules, hence the map π in(1.3) is an isomorphism of vector spaces. Now the proof of fusion rules formula is complete. PROOF OF FUSION RULES FORMULA 39
Some Generalizations.
Actually, based on our proof of the fusion rules formula wecan derive some more facts about A ( V ) bimodules and fusion rules space I (cid:0) M M M (cid:1) . Recallthat in section 3.2, we’ve proved that M ∼ = ¯ M /Rad ( ¯ M ). Now if we take any V module N with bottom level N (0) = M (0), then because M ∼ = ¯ M /Rad ( ¯ M ) is irreducible andthe bottom level of which is M (0), the universal property guarantees the existence of anepimorphism: N ։ ¯ M /Rad ( ¯ M ) . It follows that the last component of S can be extended to a bigger space N : S : M (0) ∗ × V × ... × M × ... × V × N → F ′ ( z , ..., z n , w )Similarly, if we take any V module N with bottom level N (0) = M (0), then the firstcomponent of S can also be extended to N ′ , and so we have a well-defined function: S : N ′ × V × ... × M × ... × V × N → F ′ ( z , ..., z n , w ) . (3.34)Now one can use this new S to construct an intertwining operator I ′ f of type (cid:0) N M N (cid:1) byletting: h x ′ , v ( n ) x i := Z C S ( x ′ , ( v, w ) x ) w n dw as in (3.28) , while in this case x ′ ∈ N ′ , x ∈ N instead.It is easy to see that the proof in section 3.3 goes through for this I ′ f , since in the proofwe’ve only used the properties (3.24), (3.26) and (3.27) of S , and these properties arealso satisfied by our new S in (3.34). Furthermore, it is also straightforward to see that π ( I ′ f ) = f as in (3.33). Thus, we have the following more general theorem: Theorem 3.9.
Let N , N be any V modules with bottom level M (0) , M (0) respec-tively, where M (0) , M (0) are irreducible A ( V ) modules. Then the map π : I N M , N ! → ( M (0) ∗ ⊗ A ( V ) A ( M ) ⊗ A ( V ) M (0)) ∗ given by (1.3) is surjective. Our proof can also be generalized from A n ( V ) perspective. Recall that in [3] a sequenceof associative algebras A n ( V ), where n ∈ Z ≥ are defined. They satisfy the following:(1). For any V modules M = L ∞ n =0 M ( n ) the first n level M (0) ⊕ M (1) ⊕ ... ⊕ M ( n )is a left A n ( V ) module with action given by: A n ( V ) → End n M i =0 M ( i ) ! : [ a ] o ( a ) = a ( wta − . In particular, the bottom level M (0) is a left A n ( V ) module for any n ≥ (2). For any V module M there is a well-defined A n ( V ) bimodule A n ( M ) [4] with theleft and right module action a ∗ n v and v ∗ n a satisfy: a ∗ n v − v ∗ n a = Res z Y ( a, z ) v (1 + z ) wta − (3.35)Now let g : M (0) ∗ ⊗ A n ( V ) A n ( M ) ⊗ A n ( V ) M (0) be a linear functional. We may use thesame formulas in section 2.2 to define a rational function S . i.e. we first define a 3-pointfunction S M : M (0) ∗ × M × M (0) → F ( w ) by S M ( v ′ , ( v, w ) v ) := g ( v ′ ⊗ v ⊗ v ) w − wtv asin (2.8), then use the left and right expansion formula to extend it to a n-point function.Recall that in the proof of the well-definedness and ”locality” of S in section 2.2 whenit comes to the A ( V ) bimodule A ( M ), the only essential property of this bimodule we’veused is the formula: a ∗ v − v ∗ a = Res z Y ( a, z ) v (1 + z ) wta − . See proposition 2.3 and the computation in page 11 for example. Since the same formula(3.35) holds for the A n ( V ) bimodule A n ( M ), all the computations in section 2 go throughfor the S defined from linear functional g , and so the S defined by g is the same S asin (2.23). Therefore, the right hand side of the fusion rules formula theorem 1.1 can begeneralized to A n ( V ) and A n ( M ): Theorem 3.10.
Let M , M and M be three V modules with M , M irreducible, thenthe map π : I N M , N ! → ( M (0) ∗ ⊗ A n ( V ) A n ( M ) ⊗ A n ( V ) M (0)) ∗ given by π ( I )( v ′ ⊗ v ⊗ v ) = h v ′ , o ( v ) v i is a linear isomorphism. References [1] R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster,
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