Intertwining operators among twisted modules associated to not-necessarily-commuting automorphisms
IIntertwining operators among twistedmodules associated to not-necessarily-commuting automorphisms
Yi-Zhi Huang
Abstract
We introduce intertwining operators among twisted modules or twisted intertwiningoperators associated to not-necessarily-commuting automorphisms of a vertex operatoralgebra. Let V be a vertex operator algebra and let g , g and g be automorphismsof V . We prove that for g -, g - and g -twisted V -modules W , W and W , respec-tively, such that the vertex operator map for W is injective, if there exists a twistedintertwining operator of type (cid:0) W W W (cid:1) such that the images of its component operatorsspan W , then g = g g . We also construct what we call the skew-symmetry andcontragredient isomorphisms between spaces of twisted intertwining operators amongtwisted modules of suitable types. The proofs of these results involve careful analysis ofthe analytic extensions corresponding to the actions of the not-necessarily-commutingautomorphisms of the vertex operator algebra. In the present paper, we initiate the study of intertwining operators among twisted modulesassociated to not-necessarily-commuting automorphisms of a vertex operator algebra. Hereby twisted modules we mean (generalized or logarithmic) twisted modules introduced in [H7].For simplicity and to avoid confusion, when twisted modules are not mentioned, we shallcall such intertwining operators “twisted intertwining operators,” although these intertwiningoperators are not twisted directly, but are twisted instead in a suitable sense through twistedmodules.Intertwining operators among (untwisted) modules for a vertex operator algebra were firstintroduced mathematically by Frenkel, Lepowsky and the author in [FHL] and correspondto chiral vertex operators in physics (see [MS]). They have been studied systematically inthe papers [HL], [H1]–[H6], [HLZ1]–[HLZ4], [Y], [Ch1]–[Ch2] and [Fi1]–[Fi2]. Intertwiningoperators give chiral three-point correlation functions in two-dimensional conformal fieldtheories and are the building blocks of multi-point correlation functions on Riemann surfaces1 a r X i v : . [ m a t h . QA ] S e p f arbitrary genus. They are the main objects of interest in the representation theory ofvertex operator algebras and two-dimensional conformal field theory. Almost all importantresults in these theories are in fact properties of intertwining operators.Intertwining operators among twisted modules associated to commuting automorphismsof finite orders appeared implicitly in the work [FFR] of Feingold, Frenkel and Ries andwere introduced explicitly by Xu in [X] in terms of a generalization of the Jacobi identityfor twisted modules. However, there is still no definition of intertwining operators amongtwisted modules associated to noncommuting automorphisms in the literature. To constructorbifold conformal field theories associated to a noncommutative group of automorphisms ofa vertex operator algebra, it is necessary to study these intertwining operators.In [H8], the author formulated the following conjecture: Conjecture 1.1 ([H8])
Assume that V is a simple vertex operator algebra satisfying thefollowing conditions:1. V (0) = C , V ( n ) = 0 for n < and the contragredient V (cid:48) , as a V -module, is equivalentto V .2. Every grading-restricted generalized V -module is completely reducible3. V is C -cofinite, that is, dim V /C ( V ) < ∞ , where C ( V ) is the subspace of V spannedby the elements of the form Res x x − Y ( u, x ) v for u, v ∈ V and Y : V ⊗ V → V [[ x, x − ]] is the vertex operator map for V .Let G be a finite group of automorphisms of V . Then the twisted intertwining operatorsamong the g -twisted V -modules for all g ∈ G satisfy the associativity, commutativity andmodular invariance properties. If this conjecture is proved, then we obtain the genus-zero and genus-one parts of the chiralorbifold conformal field theory associated with the vertex operator algebra V and the group G of automorphisms of V . One consequence of Conjecture 1.1 is that the category of g -twistedmodules for all all g ∈ G has a natural structure of G -crossed braided tensor categorysatisfying additional properties.To even formulate this conjecture precisely, we have to first introduce twisted intertwiningoperators or intertwining operators among twisted modules associated to general automor-phisms and study their basic properties. In this paper, we give a definition of such twistedintertwining operators. Let V be a vertex operator algebra and let g , g and g be au-tomorphisms of V . We prove that for g -, g - and g -twisted V -modules W , W and W ,respectively, such that the vertex operator map for W is injective, if there exists a twistedintertwining operator of type (cid:0) W W W (cid:1) such that the images of its component operators span W , then g = g g . We also construct what we call the skew-symmetry and contragredientisomorphisms between spaces of twisted intertwining operators among twisted modules ofsuitable types. The proofs of these results are much more subtle and delicate than those for2he corresponding results in [FHL], [HL], [X] and [HLZ1] because they involve careful analy-sis of the analytic extensions corresponding to the actions of the not-necessarily-commutingautomorphisms of the vertex operator algebra.The motivation for the study of twisted intertwining operators does not just come fromorbifold conformal field theories and their potential applications in geometry and physics. Itin fact also comes intrinsically from the study of the uniqueness conjecture of the moonshinemodule vertex operator algebra proposed by Frenkel, Lepowsky and Meurman [FLM]. Thisconjecture was obtained by using the analogy among the Golay code, the Leech lattice andthe moonshine module vertex operator algebra. In Conway’s proof of the uniqueness of theLeech lattice [Co], the 24-dimensional vector space R plays a fundamental role. For theuniqueness conjecture for the moonshine module vertex operator algebra, the first difficultyis that there is no analogue of R . But we still need a structure large enough such that allthe works can be done in this structure.If the vertex operator subalgebra fixed by the automorphism group of a vertex operatoralgebra satisfy suitable conditions (for example, the three conditions in Conjecture 1.1),then one possible choice of such a structure large enough for our purpose is the intertwiningoperator algebra formed by the modules and intertwining operators for the fixed-point vertexoperator subalgebra. However, the assumption that these suitable conditions hold is in facta main difficult conjecture that we have to prove first. Because of this, instead of assumingthese conditions, we have to develop a theory that will lead us to a proof of these conditionsand a construction of the intertwining operator algebra.Since in this area, conjectures are sometimes claimed to have been proved in some books,papers, preprints or unpublished manuscripts without the evidence that the proofs indeedexist, the author would like to comment that the mathematics for obtaining conjectures andthe mathematics for finding proofs are often very different. To obtain conjectures, one canassume what one believes to be true and derive the consequences. But to prove conjectures,the assumptions used to derive the conjectures are often themselves the most difficult partsof the conjectures. Thus one might need different mathematical approaches and theories toprove these assumptions first. The theory of twisted intertwining operators initiated in thispaper is what the author believes to be needed in the proofs of the conjectures mentionedabove, including Conjecture 1.1 and the conjecture that suitable conditions hold for thefixed-point vertex operator subalgebra. We expect that this theory will play an importantrole in the study of orbifold conformal field theory and, in particular, in the study of theuniqueness conjecture of the moonshine module vertex operator algebra.In this paper, we use the approach based on multivalued analytic functions with pre-ferred branches developed and used in [H1], [H3], [H6], [HLZ2]–[HLZ3] and [Ch1]–[Ch2].Intertwining operators among (untwisted) modules can be defined using either such multi-valued analytic functions or formal variables. But to study products and iterates of inter-twining operators, it is necessary to use the approach based on such multivalued analyticfunctions. For twisted intertwining operators introduced and studied in this paper, even forthe definition and the properties involving only one twisted intertwining operator, we needthe approach based on such multivalued analytic functions, because the vertex operators for3wisted modules contain nonintegral powers and logarithm of the variable and, more impor-tantly, because the automorphisms associated to different twisted modules do not necessarilycommute with each other.In Section 2, we discuss the notations and conventions used in this paper, especially thoseinvolving multivalued analytic functions with preferred branches. In Section 3, we recall thenotion of twisted module introduced in [H7]. We also discuss in this section the functorsassociated to automorphisms of the vertex operator algebra and the contragredient functoron the category of twisted modules. Twisted intertwining operators are introduced in Section4. In the same section, we prove the result that for g -, g - and g -twisted V -modules W , W and W , respectively, such that the vertex operator map for W is injective, if thereexists a twisted intertwining operator of type (cid:0) W W W (cid:1) such that the images of its componentoperators span W , then g = g g . The skew-symmetry isomorphisms and contragredientisomorphisms are constructed in Section 5 and Section 6, respectively. To study intertwining operators, we have to work with multivalued analytic functions withpreferred branches. The approach that we use in this paper is the same as the one used in[H1], [H3], [H6], [HLZ2]–[HLZ3] and [Ch1]–[Ch2]. In this section, we recall and introducesome notations and conventions.We shall use i to denote √−
1. For z ∈ C × , we choose the value arg z of the argumentof z to be the one satisfying 0 ≤ arg z < π . We shall not use log z to denote the particularvalue log | z | + (arg z ) i of the logarithm of z as in [H1], [H3], [HLZ2]–[HLZ3] and [Ch1]–[Ch2].Instead, we shall always use l p ( z ) to denote the value log | z | + (arg z + 2 pπ ) i of the logarithmof z for p ∈ Z .Intertwining operators defined using formal variables in fact give multivalued analyticfunctions with preferred branches. We shall use log z to denote the multivalued logarithmfunction of the variable z with the preferred branch l ( z ) = log | z | + (arg z ) i . For n ∈ C ,we shall use z n to denote the multivalued analytic function e n log z with the preferred branch e nl ( z ) . Multivalued analytic functions with preferred branch on a region form a commutativeassociative algebra and can also be divided by such functions on the same region to obtainsuch functions on possibly smaller regions. In particular, f ( z , z ) = N (cid:88) i,j,k,l,m,n =1 a ijklmn z r i z s j ( z − z ) t k (log z ) l (log z ) m (log( z − z )) n for a ijklmn , r i , s j , t k ∈ C is a multivalued analytic function with preferred branch on the regiongiven by z , z (cid:54) = 0, z (cid:54) = z . For p , p , p ∈ Z , we shall use f p ,p ,p ( z , z ) to denote thesingle-valued branch N (cid:88) i,j,k,l,m,n =1 a ijklmn e r i l p ( z ) e s j l p ( z ) e t k l p ( z − z ) ( l p ( z )) l ( l p ( z )) m ( l p ( z − z )) n f ( z , z ).For a C -graded vector space W = (cid:96) n ∈ C W [ n ] , let W (cid:48) = (cid:96) n ∈ C W ∗ [ n ] be the graded dual of W and W = (cid:81) n ∈ C W [ n ] the algebraic completion of W . For n ∈ C , we use π n to denote thethe projection from W or W to W [ n ] .Let W be a vector space and X ( x ) = K (cid:88) k =0 (cid:88) n ∈ C a n,k x n (log x ) k ∈ (End W ) { x } [log x ] . For z ∈ C × , we shall use X p ( z ) to denote the series X ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x n = e nlp ( z ) , log x = l p ( z ) = K (cid:88) k =0 (cid:88) n ∈ C a n,k e nl p ( z ) ( l p ( z )) k with terms in End W . When W = (cid:96) n ∈ C W [ n ] is a C -graded vector space and a n,k for different n are homogeneous operators of different degrees, X p ( z ) ∈ Hom(
W, W ). When z changes in C × , X p ( z ) can be viewed as a function on C × valued in the space of series in W . We callthis function X p ( z ) the p -th analytic branch of X ( x ). In this paper, we fix a vertex operator algebra (
V, Y V , V , ω V ). In fact, the results of thepresent paper hold for a grading-restricted M¨obius vertex algebra, that is, a Z -graded vertexalgebra V = (cid:96) n ∈ Z V ( n ) equipped with operators L V ( − L V (0) and L V (1) such that V ( n ) = 0when n is sufficiently negative, dim V ( n ) < ∞ for n ∈ Z , the usual sl (2 , C ) commutatorrelations for L V ( − L V (0) and L V (1) hold and the usual commutator relations between L V ( − L V (0) and L V (1) and vertex operators hold. But for what we want to prove in thefuture, it is necessary for V to be a vertex operator algebra with the additional data of aconformal element satisfying some additional conditions.Let g be an automorphism of V . We first recall the definition of generalized g -twisted V -module first introduced in [H7]. But for simplicity, we shall omit the word “generalized.”In particular, in this paper, the vertex operator map for a g -twisted V -module in generalcontain the logarithm of the variable and the operator L (0) in general does not have to actsemisimply. Definition 3.1 A g -twisted V -module is a C × C / Z -graded vector space W = (cid:96) n ∈ C ,α ∈ C / Z W [ α ][ n ] (graded by weights and g -weights ) equipped with a linear map Y gW : V ⊗ W → W { x } [log x ] ,v ⊗ w (cid:55)→ Y gW ( v, x ) w satisfying the following conditions: 5. The equivariance property : For p ∈ Z , z ∈ C × , v ∈ V and w ∈ W ,( Y gW ) p +1 ( gv, z ) w = ( Y gW ) p ( v, z ) w, where for p ∈ Z , ( Y gW ) p ( v, z ) is the p -th analytic branch of Y gW ( v, x ).2. The identity property : For w ∈ W , Y g ( , x ) w = w .3. The duality property : For any u, v ∈ V , w ∈ W and w (cid:48) ∈ W (cid:48) , there exists a multivaluedanalytic function with preferred branch of the form f ( z , z ) = N (cid:88) i,j,k,l =0 a ijkl z m i z n j (log z ) k (log z ) l ( z − z ) − t for N ∈ N , m , . . . , m N , n , . . . , n N ∈ C and t ∈ Z + , such that the series (cid:104) w (cid:48) , ( Y gW ) p ( u, z )( Y gW ) p ( v, z ) w (cid:105) = (cid:88) n ∈ C (cid:104) w (cid:48) , ( Y gW ) p ( u, z ) π n ( Y gW ) p ( v, z ) w (cid:105) , (cid:104) w (cid:48) , ( Y gW ) p ( v, z )( Y gW ) p ( u, z ) w (cid:105) = (cid:88) n ∈ C (cid:104) w (cid:48) , ( Y gW ) p ( v, z ) π n ( Y gW ) p ( u, z ) w (cid:105) , (cid:104) w (cid:48) , ( Y gW ) p ( Y V ( u, z − z ) v, z ) w (cid:105) = (cid:88) n ∈ C (cid:104) w (cid:48) , ( Y gW ) p ( π n Y V ( u, z − z ) v, z ) w (cid:105) are absolutely convergent in the regions | z | > | z | > | z | > | z | > | z | > | z − z | >
0, respectively, and their sums are equal to the branch f p,p ( z , z ) = N (cid:88) i,j,k,l =0 a ijkl e m i l p ( z ) e n j l p ( z ) l p ( z ) k l p ( z ) l ( z − z ) − t of f ( z , z ) in the region | z | > | z | >
0, the region | z | > | z | >
0, the region given by | z | > | z − z | > | arg z − arg z | < π , respectively.4. The L (0) -grading condition and g -grading condition : Let L gW (0) = Res x xY gW ( ω, x ).Then for n ∈ C and α ∈ C / Z , w ∈ W [ α ][ n ] , there exists K, Λ ∈ Z + such that ( L gW (0) − n ) K w = ( g − e παi ) Λ w = 0. Moreover, gY gW ( u, x ) v = Y gW ( gu, x ) gv .5. The L ( − derivative property : For v ∈ V , ddx Y gW ( v, x ) = Y gW ( L V ( − v, x ) . A lower-bounded generalized g -twisted V -module is a g -twisted V -module W such that foreach n ∈ C , W [ n + l ] = 0 for sufficiently negative real number l . A g -twisted V -module W issaid to be grading-restricted if it is lower bounded and for each n ∈ C , dim W [ n ] < ∞ .6e shall denote the g -twisted V -module just defined by ( W, Y gW ) or simply by W when Y gW is clear.Let ( W, Y gW ) be a g -twisted V -module. Using the notation introduced in Section 2, wehave the p -th analytic branch ( Y gW ) p ( ω, z ) of the formal series Y gW ( ω, x ) for p ∈ Z . Since gω = ω , ( Y gW ) p +1 ( ω, z ) = ( Y gW ) p +1 ( gω, z ) = ( Y gW ) p ( ω, z )for p ∈ Z . Thus Y gW ( ω, x ) involves only integral powers of x . Let Y gW ( ω, x ) = (cid:88) n ∈ Z L W ( n ) x − n − . Then the same argument deriving the Virasoro relations for (untwisted) modules from axiomsother than those for the Virasoro operators give[ L W ( m ) , L W ( n )] = ( m − n ) L W ( m + n ) + c
12 ( m − m ) δ m + n, for m, n ∈ Z , where c is the central charge of V .Let ( W, Y gW ) be a g -twisted V -module. Let h be an automorphism of V and let φ h ( Y g ) : V × W → W { x } [log x ] v ⊗ w (cid:55)→ φ h ( Y g )( v, x ) w be the linear map defined by φ h ( Y g )( v, x ) w = Y g ( h − v, x ) w. The following result can be proved by a straightforward use of the axioms:
Proposition 3.2
The pair ( W, φ h ( Y g )) is an hgh − -twisted V -module. We shall denote the hgh − -twisted V -module in the proposition above by φ h ( W ).We also need contragredient twisted V -modules. Let ( W, Y gW ) be a g -twisted V -modulerelative to G . Let W (cid:48) be the graded dual of W . Define a linear map( Y gW ) (cid:48) : V ⊗ W (cid:48) → W (cid:48) { x } [log x ] ,v ⊗ w (cid:48) (cid:55)→ ( Y gW ) (cid:48) ( v, x ) w (cid:48) by (cid:104) ( Y gW ) (cid:48) ( v, x ) w (cid:48) , w (cid:105) = (cid:104) w (cid:48) , Y gW ( e xL (1) ( − x − ) L (0) v, x − ) w (cid:105) for v ∈ V , w ∈ W and w (cid:48) ∈ W (cid:48) . Proposition 3.3
The pair ( W (cid:48) , ( Y gW ) (cid:48) ) is a g − -twisted V -module. The proof of this result is a special case of the proof of Theorem 6.1 in Section 6 with W = V , g = 1 V , g = g and W = W = W (see also Remark 4.2 in Section 4 below). Sincethe proof of Theorem 6.1 uses only the definition of ( Y gW ) (cid:48) , quoting the proof of Theorem 6.1to give a proof of Proposition 3.3 does not constitute circular reasoning.The g − -twisted V -module ( W (cid:48) , ( Y gW ) (cid:48) ) is called the contragredient twisted V -module of ( W, Y g ). 7 Twisted intertwining operators
We introduce the notion of twisted intertwining operator or intertwining operator amongtwisted modules in this section. Twisted intertwining operators in this paper in generalinvolve the logarithm of the variable. Such intertwining operators are usually called loga-rithmic intertwining operators. For simplicity, we omit the word “logarithm,” unless thereis a need to emphasize that the intertwining operator indeed involve the logarithm of thevariable.
Definition 4.1
Let g , g , g be automorphisms of V and let W , W and W be g -, g -and g -twisted V -modules, respectively. A twisted intertwining operator of type (cid:0) W W W (cid:1) is alinear map Y : W ⊗ W → W { x } [log x ] w ⊗ w (cid:55)→ Y ( w , x ) w = K (cid:88) k =0 (cid:88) n ∈ C Y n,k ( w ) w x − n − (log x ) k satisfying the following conditions:1. The lower truncation property : For w ∈ W and w ∈ W , n ∈ C and k = 0 , . . . , K , Y n + l,k ( w ) w = 0 for l ∈ N sufficiently large.2. The duality property : For u ∈ V , w ∈ W , w ∈ W and w (cid:48) ∈ W (cid:48) , there exists amultivalued analytic function with preferred branch f ( z , z ; u, w , w , w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a ijklmn z r i z s j ( z − z ) t k (log z ) l (log z ) m (log( z − z )) n for N ∈ N , r i , s j , t k , a ijklmn ∈ C , such that for p , p , p ∈ Z , the series (cid:104) w (cid:48) , ( Y g W ) p ( u, z ) Y p ( w , z ) w (cid:105) = (cid:88) n ∈ C (cid:104) w (cid:48) , ( Y g W ) p ( u, z ) π n Y p ( w , z ) w (cid:105) , (4.1) (cid:104) w (cid:48) , Y p ( w , z )( Y g W ) p ( u, z ) w (cid:105) = (cid:88) n ∈ C (cid:104) w (cid:48) , Y p ( w , z ) π n ( Y g W ) p ( u, z ) w (cid:105) , (4.2) (cid:104) w (cid:48) , Y p (( Y g W ) p ( u, z − z ) v, z ) w (cid:105) = (cid:88) n ∈ C (cid:104) w (cid:48) , Y p ( π n ( Y g W ) p ( u, z − z ) w , z ) w (cid:105) (4.3)are absolutely convergent in the regions | z | > | z | > | z | > | z | > | z | > | z − z | >
8, respectively. Moreover, their sums are equal to the branches f p ,p ,p ( z , z ; u, w , w , w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p ( z ) e s j l p ( z ) e t k l p ( z − z ) ( l p ( z )) l ( l p ( z )) m ( l p ( z − z )) n ,f p ,p ,p ( z , z ; u, w , w , w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p ( z ) e s j l p ( z ) e t k l p ( z − z ) ( l p ( z )) l ( l p ( z )) m ( l p ( z − z )) n ,f p ,p ,p ( z , z ; u, w , w , w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p ( z ) e s j l p ( z ) e t k l p ( z − z ) ( l p ( z )) l ( l p ( z )) m ( l p ( z − z )) n , respectively, of f ( z , z ; u, w , w , w (cid:48) ) (recall the notations and convention in Section2) in the region given by | z | > | z | > | arg( z − z ) − arg z | < π , the regiongiven by | z | > | z | > − π < arg( z − z ) − arg z < − π , the region given by | z | > | z − z | > | arg z − arg z | < π , respectively.3. The L ( − -derivative property : ddx Y ( w , x ) = Y ( L ( − w , x ) . Remark 4.2
Let (
W, Y gW ) be a g -twisted V -module. Then by definition, the vertex operatormap Y gW is a twisted intertwining operator of type (cid:0) WV W (cid:1) . Remark 4.3
Let g , g and g be automorphisms of V , let V (cid:104) g ,g ,g (cid:105) be the vertex operatorsubalgebra of V consisting of elements of V fixed under the actions of g , g and g . Let W , W and W be g -, g - and g -twisted V -modules and let Y be a twisted intertwiningoperator of type (cid:0) W W W (cid:1) . Then W , W and W are V (cid:104) g ,g ,g (cid:105) -modules and Y is an (untwisted)intertwining operator of type (cid:0) W W W (cid:1) when W , W and W are viewed as V (cid:104) g ,g ,g (cid:105) -modules.But in general, an (untwisted) intertwining operator of type (cid:0) W W W (cid:1) when W , W and W areviewed as V (cid:104) g ,g ,g (cid:105) -modules might not be a twisted intertwining operator of type (cid:0) W W W (cid:1) . Infact, an (untwisted) intertwining operator of type (cid:0) W W W (cid:1) when W , W and W are viewedas V (cid:104) g ,g ,g (cid:105) -modules is required to satisfy only the duality property for vertex operatorsassociated to elements in V (cid:104) g ,g ,g (cid:105) while a twisted intertwining operator of type (cid:0) W W W (cid:1) must satisfy the more restrictive duality property in Definition 4.1. This is the reason whyeven when W , W and W are known to be twisted V -modules, we still want to add theword “twisted” in front of “intertwining operator” to call an intertwining operator of type (cid:0) W W W (cid:1) in Definition 4.1 a twisted intertwining operator.9 emark 4.4 In the duality property in the definition above, we require that the sum of(4.2) is equal to f p ,p ,p ( z , z ; u, w , w , w (cid:48) ) in the region given by | z | > | z | > − π < arg( z − z ) − arg z < − π . The choice of this region in fact gives an order of W and W to be W first and W second. See Theorem 4.7 and especially its proof below. Theother region is the region given by | z | > | z | > π < arg( z − z ) − arg z < π . If werequire the sum of (4.2) is equal to f p ,p ,p ( z , z ; u, w , w , w (cid:48) ) in this region, then the orderof W and W is chosen to be W first and W second. We choose the more natural order.We shall need the following two lemmas: Lemma 4.5
In Definition 4.1, the requirement in the duality property that the sum of (4.2)be equal to f p ,p ,p ( z , z ; u, w , w , w (cid:48) ) in the region given by | z | > | z | > and − π < arg( z − z ) − arg z < − π can be replaced by the requirement that the sum of (4.2) be equalto f p ,p ,p − ( z , z ; u, w , w , w (cid:48) ) in the region given by | z | > | z | > and π < arg( z − z ) − arg z < π . In the same definition, the requirement in the duality property that thesum of (4.1) be equal to f p ,p ,p ( z , z ; u, w , w , w (cid:48) ) in the region given by | z | > | z | > and | arg( z − z ) − arg z | < π can be replaced by the requirement that the sum of (4.1)be equal to f p ,p ,p − ( z , z ; u, w , w , w (cid:48) ) in the region given by | z | > | z | > and − π < arg( z − z ) − arg z < − π and to f p ,p ,p +1 ( z , z ; u, w , w , w (cid:48) ) in the region given by | z | > | z | > and π < arg( z − z ) − arg z < π .Proof. We prove only the first part. The second part can be proved similarly.Assume that Y is a twisted intertwining operator satisfying Definition 4.1. We choosea path l in the region given by | z | > | z | > z (1)1 , z (0)2 ) in the subregiongiven by | z | > | z | > − π < arg( z − z ) − arg z < − π to a point ( z (2)1 , z (0)2 ) inthe subregion given by | z | > | z | > π < arg( z − z ) − arg z < π by letting z pass through the set given by arg( z − z ) = 0 in the counter clockwise direction for thevariable z − z but keeping arg z between 0 and 2 π and fixing z = z (0)2 . See Figure 1.The sum of the series (4.2) is an analytic function of z and z and thus we can analyticallyFigure 1: The path l extend its value at ( z (1)1 , z (0)2 ) through the path l to its value at ( z (2)1 , z (0)2 ). At ( z (1)1 , z (0)2 ), its10alue is given by f p ,p ,p ( z (1)1 , z (0)2 ; u, w , w , w (cid:48) ). When the path l pass the point at whicharg( z − z ) = 0, there is a jump of arg( z − z ) from 0 to 2 π . When arg( z − z ) = 0, its value(at z , z ) is still f p ,p ,p ( z , z ; u, w , w , w (cid:48) ). But after the jump, since the sum is analyticand in particular is continuous, its value at ( z , z ) must be f p ,p ,p − ( z , z ; u, w , w , w (cid:48) ).In particular, its value at the arbitrary point ( z (2)1 , z (0)2 ) in the region given by | z | > | z | > π < arg( z − z ) − arg z < π must be f p ,p ,p − ( z (2)1 , z (0)2 ; u, w , w , w (cid:48) ).If we assume that Y satisfies the requirement that the sum of (4.2) be equal to f p ,p ,p − ( z , z ; u, w , w , w (cid:48) )in the region given by | z | > | z | > π < arg( z − z ) − arg z < π and all theother axioms in Definition 4.1, a completely analogous argument shows that the sum of(4.2) be equal to f p ,p ,p ( z , z ; u, w , w , w (cid:48) ) in the region given by | z | > | z | > − π < arg( z − z ) − arg z < − π . Lemma 4.6
For p , p , p ∈ Z , u ∈ V , w ∈ W , w ∈ W and w (cid:48) ∈ W (cid:48) , we have f p ,p ,p +1 ( z , z ; g u, w , w , w (cid:48) ) = f p ,p ,p ( z − z , z ; u, w , w , w (cid:48) ) (4.4) and f p +1 ,p ,p ( z , z ; g u, w , w , w (cid:48) ) = f p ,p ,p ( z , z ; u, w , w , w (cid:48) ) , (4.5) where f p ,p ,p ( z , z ; u, w , w , w (cid:48) ) for p , p , p ∈ Z is the branch given by p , p , p of themultivalued analytic function f ( z , z ; u, w , w , w (cid:48) ) with preferred branch in Definition 4.1.Proof. By the duality property for Y , when | z | > | z − z | > | arg z − arg z | < π , (cid:104) w (cid:48) , Y p (( Y g W ) p +1 ( g u, z − z ) w , z ) w (cid:105) converges absolutely to f p ,p ,p +1 ( z , z ; g u, w , w , w (cid:48) ) and (cid:104) w (cid:48) , Y p (( Y g W ) p ( u, z − z ) w , z ) w (cid:105) converges absolutely to f p ,p ,p ( z , z ; u, w , w , w (cid:48) ). But (cid:104) w (cid:48) , Y p (( Y g W ) p +1 ( g u, z − z ) w , z ) w (cid:105) = (cid:104) w (cid:48) , Y p (( Y g W ) p ( u, z − z ) w , z ) w (cid:105) Thus we have f p ,p ,p +1 ( z , z ; g u, w , w , w (cid:48) ) = f p ,p ,p ( z , z ; u, w , w , w (cid:48) ) . For general p , p , p ∈ Z , we obtain (4.4) by analytic extensions.On the other hand, by the duality property for Y , when | z | > | z | > − π < arg z − arg( − z ) < − π , (cid:104) w (cid:48) , Y p ( w , z )( Y g W ) p +1 ( g u, z ) w (cid:105) f p +1 ,p ,p ( z , z ; g u, w , w , w (cid:48) ) and (cid:104) w (cid:48) , Y p ( w , z )( Y g W ) p ( u, z ) w (cid:105) converges absolutely to f p ,p ,p ( z , z ; u, w , w , w (cid:48) ). But (cid:104) w (cid:48) , Y p ( w , z )( Y g W ) p +1 ( g u, z ) w (cid:105) = (cid:104) w (cid:48) , Y p ( w , z )( Y g W ) p ( u, z ) w (cid:105) . Thus we have f p +1 ,p ,p ( z , z ; g u, w , w , w (cid:48) ) = f p ,p ,p ( z − z , − z ; u, w , w , w (cid:48) ) . For general p , p , p ∈ Z , we obtain (4.5) by analytic extensions.We now prove that under suitable minor conditions, g = g g for the twisted intertwiningoperator defined in Definition 4.1. Theorem 4.7
Let g , g , g be automorphisms of V and let W , W and W be g -, g - and g -twisted V -modules, respectively. Assume that the vertex operator map for W given by u (cid:55)→ Y g W ( u, x ) is injective. If there exists a twisted intertwining operator Y of type (cid:0) W W W (cid:1) such that the coefficients of the series Y ( w , x ) w for w ∈ W and w ∈ W span W , then g = g g .Proof. Let Y be a twisted intertwining operator of type (cid:0) W W W (cid:1) such that the coefficients of Y ( w , x ) w for w ∈ W and w ∈ W span W . For u ∈ V , w ∈ W , w ∈ W and w (cid:48) ∈ W (cid:48) ,consider the multivalued analytic function f ( z , z ; u, w , w , w (cid:48) ) with preferred branch forthe twisted intertwining operator Y (see Definition 4.1). Fix z to be a nonzero negative realnumber − a where a ∈ R + . Then for any p ∈ Z , we have an analytic function f p − a ( z ) = N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p ( z ) e s j l p ( − a ) e t k l p ( z + a ) ( l p ( z )) l ( l p ( − a )) m ( l p ( z + a )) n of z and can be analytically extended to a multivalued analytic function f − a ( z ) = N (cid:88) i,j,k,l,m,n =0 a ijklmn z r i e s j l p ( − a ) ( z + a ) t k (log z ) l ( l p ( − a )) m (log( z + a )) n of z with preferred branch. Let a ∈ R + such that a > a > a − a . Consider the loopΓ in the z plane based at z = − a in Figure 2. We consider the value f p − a ( − a ) ofthe multivalued analytic function f − a ( z ) at − a . By the definition of twisted intertwiningoperator above, f p − a ( − a ) = (cid:104) w (cid:48) , ( Y g W ) p ( u, − a ) Y p ( w , − a ) w (cid:105) . (4.6)But by definition, when z goes around the loop above, the right-hand side of (4.6) changesto (cid:104) w (cid:48) , ( Y g W ) p − ( u, − a ) Y p ( w , − a ) w (cid:105) . (4.7)12igure 2: The loop Γ By the equivariance property of the g -twisted module W , (4.7) is equal to (cid:104) w (cid:48) , ( Y g W ) p ( g u, − a ) Y p ( w , − a ) w (cid:105) . (4.8)We also consider another loop Γ in the z plane and based at z = − a given in the order l first, l second, l third and l last in Figure 3. The loop Γ is in fact homotopy equivalentFigure 3: The loop Γ to the loop Γ . Thus when z goes around Γ , the right-hand side of (4.6) also changes to(4.8).On the other hand, we look at how the function values change when z goes through l , l , l and l . When z goes from − a to − a (see Figure 3) through l , the right-hand sideof (4.6) changes to N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p ( − a ) e s j l p ( − a ) e t k l p ( − a + a ) ( l p ( − a )) l ( l p ( − a )) m ( l p ( − a + a )) n (4.9)13ote that arg( − a ) = π , arg( − a ) = π and arg( − a + a ) = 0. Hence arg( − a + a ) − arg( − a ) = − π . Since | − a | > | − a | > − π < arg( − a + a ) − arg( − a ) < − π , bythe duality property of the twisted intertwining operator Y , (4.9) is equal to (cid:104) w (cid:48) , Y p ( w , − a )( Y g W ) p ( u, − a ) w (cid:105) . (4.10)Next let z go around the loop l . Then (4.10) changes to (cid:104) w (cid:48) , Y p ( w , − a )( Y g W ) p − ( u, − a ) w (cid:105) . (4.11)By the equivariance property of the g -twisted module W , (4.11) is equal to (cid:104) w (cid:48) , Y p ( w , − a )( Y g W ) p ( g u, − a ) w (cid:105) . (4.12)Now let z go from − a to − a through l . Then by reversing the argument above on thechange of the values when z goes through l , we see that the values change from (4.12) to (cid:104) w (cid:48) , ( Y g W ) p ( g u, − a ) Y p ( w , − a ) w (cid:105) . (4.13)Since | − a | > | − a | > | − a − ( − a ) | > | arg( − a − ( a )) − arg( − a ) | = 0 < π and | arg( − a ) − arg( − a ) | = 0 < π , by the duality property of the twisted intertwining operator Y , (4.13) is equal to (cid:104) w (cid:48) , Y p (( Y g W ) p ( g u, − a + a ) w , − a ) w (cid:105) . (4.14)Finally let z go around the loop l . The value changes from (4.14) to (cid:104) w (cid:48) , Y p (( Y g W ) p − ( g u, − a + a ) w , − a ) w (cid:105) . (4.15)By the equivariance property of the g -twisted module W , (4.15) is equal to (cid:104) w (cid:48) , Y p (( Y g W ) p ( g g u, − a + a ) w , − a ) w (cid:105) . (4.16)Again since | − a | > | − a | > | − a − ( − a ) | > | arg( − a − ( a )) − arg( − a ) | = 0 < π and | arg( − a ) − arg( − a ) | = 0 < π , by the duality property of the twisted intertwining operator Y , (4.16) is equal to (cid:104) w (cid:48) , ( Y g W ) p ( g g u, − a ) Y p ( w , − a ) w (cid:105) . (4.17)From the discussions above, we see that when z goes around Γ , the right-hand side of(4.6) changes to (4.17). Thus we see that (4.8) and (4.17) are equal, that is (cid:104) w (cid:48) , ( Y g W ) p ( g u − g g u, − a ) Y p ( w , − a ) w (cid:105) = 0 . (4.18)Since w , w and w (cid:48) are arbitrary and the coefficients of Y ( w , x ) w for w ∈ W and w ∈ W span W , we obtain from (4.18)( Y g W ) p ( g u − g g u, − a ) = 0 . (4.19)14eplacing u in (4.19) by L V ( − m u for m ∈ N , using the fact that L V ( −
1) commutes with g , g and g and then using the L ( − Y g W , we obtain d m dx m Y g W ( g u − g g u, x ) (cid:12)(cid:12)(cid:12)(cid:12) x n = e nlp ( − a , log x = l p ( − a ) = 0 . Using the Taylor series expansion, we obtain Y g W ( g u − g g u, x ) (cid:12)(cid:12)(cid:12)(cid:12) x n = e nlp ( z ) , log x = l p ( z ) = 0 (4.20)for z ∈ C × . Thus Y g W ( g u − g g u, x ) = 0 . (4.21)Since the vertex operator map u (cid:55)→ Y g W ( u, x ) is injective, (4.21) implies g u − g g u = 0 or g u = g g u . Since u is also arbitrary, we obtain g = g g . Remark 4.8
The proof of Theorem 4.7 can be intuitively understood using two braidinggraphs. See Figure 4. Since the two braiding graphs are topologically equivalent (isotopic),Figure 4: The braiding graphs corresponding to Γ (left) and Γ (right)the corresponding algebraic objects are equal and thus we have g u = g g u . In fact, just likethe theory of braided tensor categories, the correspondence between algebraic and analyticcalculations and the braiding graphs can be made mathematically precise so that proofs suchas the one for Theorem 4.7 can be given using such graphs. Note that in these graphs, we havesuppressed the associativity for the twisted vertex operators and the twisted intertwining15perator, just as what people usually do in the graphs for braided tensor categories. We alsonote that these braiding graphs explain only those results that are topological in nature. Foranalytic results such as those we shall prove in the next two sections, these graphs are notvery useful.Because of Theorem 4.7, in the rest of this paper, we shall discuss only twisted intertwin-ing operators of type (cid:0) W W W (cid:1) with W , W and W being g -, g - and g g -twisted V -modules. In this section, we construct what we call the skew-symmetry isomorphisms between thespaces of twisted intertwining operators of suitable types. These linear isomorphisms corre-spond to braidings in the still-to-be-constructed G -crossed braided tensor category structureon the category of g -twisted V -modules for all g in a group G of automorphisms of V .Let g , g be automorphisms of V , W , W and W g -, g - and g g -twisted V -modulesand Y a twisted intertwining operator of type (cid:0) W W W (cid:1) . We define linear mapsΩ ± ( Y ) : W ⊗ W → W { x } [log x ] w ⊗ w (cid:55)→ Ω ± ( Y )( w , x ) w by Ω ± ( Y )( w , x ) w = e xL ( − Y ( w , y ) w (cid:12)(cid:12)(cid:12)(cid:12) y n = e ± nπ i x n , log y =log x ± π i (5.1)for w ∈ W and w ∈ W .From the definition (5.1), for p ∈ Z , w ∈ W , w ∈ W and z ∈ C × ,Ω ± ( Y ) p ( w , z ) w = Ω ± ( Y )( w , x ) w (cid:12)(cid:12)(cid:12)(cid:12) x n = e nlp ( z ) , log x = l p ( z ) = (cid:32) e xL ( − Y ( w , y ) w (cid:12)(cid:12)(cid:12)(cid:12) y n = e ± nπ i x n , log y =log x ± π i (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) x n = e nlp ( z ) , log x = l p ( z ) = e zL ( − Y ( w , y ) w (cid:12)(cid:12)(cid:12)(cid:12) y n = e n ( lp ( z ) ± π i ) , log y = l p ( z ) ± π i . When arg z < π and arg z ≥ π , arg( − z ) = arg z + π and arg( − z ) = arg z − π , respectively.Hence e zL ( − Y ( w , y ) w (cid:12)(cid:12)(cid:12)(cid:12) y n = e n ( lp ( z )+ π i ) , log y = l p ( z )+ π i = e zL ( − Y p ( w , − z ) w when arg z < π and e zL ( − Y ( w , y ) w (cid:12)(cid:12)(cid:12)(cid:12) y n = e n ( lp ( z ) − π i ) , log y = l p ( z ) − π i = e zL ( − Y p ( w , − z ) w z ≥ π . In particular, for w ∈ W , w ∈ W and z ∈ C × satisfying arg z < π andarg z ≥ π , we have Ω + ( Y ) p ( w , z ) w = e zL ( − Y p ( w , − z ) w . (5.2)and Ω − ( Y ) p ( w , z ) w = e zL ( − Y p ( w , − z ) w , (5.3)respectively. Theorem 5.1
The linear maps Ω + ( Y ) and Ω − ( Y ) are twisted intertwining operators of types (cid:0) W W φ g − ( W ) (cid:1) and (cid:0) W φ g ( W ) W (cid:1) , respectively (recall the definition of φ g for an automorphism g of V in Section 3).Proof. The lower-truncation property and the L ( − u ∈ V , w ∈ W , w ∈ W and w (cid:48) ∈ W (cid:48) . We first need to give the multival-ued analytic functions with preferred branches in the duality property. We shall denotethese multivalued analytic functions for Ω + ( Y ) and Ω − ( Y ) by g + ( z , z ; u, w , w , w (cid:48) ) and g − ( z , z ; u, w , w , w (cid:48) ), respectively. Let f ( z , z ; u, w , w , w (cid:48) ) be the multivalued analyticfunction with preferred branch in the duality property for the twisted intertwining operator Y . Then we can write f ( z − z , − z ; u, w , w , e z L (cid:48) (1) w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a ijklmn ( z − z ) r i ( − z ) s j z t k (log z ) l (log z ) m (log( z − z )) n , where L (cid:48) (1) is the adjoint operator of L ( −
1) on W and is equal to the coefficient of the x − term in ( Y gW ) (cid:48) ( ω, x ). Define g ± ( z , z ; u, w , w , w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 e ± s j π i a ijklmn ( z − z ) r i z s j z t k (log( z − z )) l (log z + π i ) m (log z ) n . (5.4)When | z | > | z | > | z − z | > | z | > | arg( z − z ) − arg z | < π and arg z < π (for Ω + ) or arg z ≥ π (for Ω − ), from (5.3), the L ( − Y g W and theduality property for Y , (cid:104) w (cid:48) , ( Y g W ) p ( u, z )Ω ± ( Y ) p ( w , z ) w (cid:105) = (cid:104) w (cid:48) , ( Y g W ) p ( u, z ) e z L ( − Y p ( w , − z ) w (cid:105) = (cid:104) e z L (cid:48) (1) w (cid:48) , ( Y g W ) p ( u, z − z ) Y p ( w , − z ) w (cid:105) (5.5)17onverges absolutely to f p ,p ,p ( z − z , − z ; u, w , w , e z L (cid:48) (1) w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p ( z − z ) e s j l p ( − z ) e t k l p ( z ) ( l p ( z − z )) l ( l p ( − z )) m ( l p ( z )) n . (5.6)Since arg z < π (for Ω + ) or arg z ≥ π (for Ω − ), arg( − z ) = arg z ± π (for Ω ± ). Hence theright-hand side of (5.6) is equal to N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p ( z − z ) e s j ( l p ( z ) ± π i ) e t k l p ( z ) ( l p ( z − z )) l ( l p ( z ) ± π i ) m ( l p ( z )) n = g p ,p ,p ± ( z , z ; u, w , w , w (cid:48) ) . (5.7)But the left-hand side of (5.7) can be expanded in the region | z | > | z | > e l p ( z ) and e l p ( z ) and in finitely many nonnegative integral powers of l p ( z ) and l p ( z ) such that the real parts of the powers of e l p ( z ) is bounded from above and the realparts of the powers of e l p ( z ) is bounded from below. So the left-hand side of (5.5) as aseries of the same form that converges to the left-hand side of (5.7) in a smaller region mustbe convergent absolutely in the larger region given by | z | > | z | > | z | > | z | > | arg( z − z ) − arg z | < π and arg z < π (for Ω + ) orarg z ≥ π (for Ω − ). Since both the left-hand side of (5.5) and the left-hand side of (5.7) aresingle-valued analytic function in z and z with cuts at z ∈ R + and z ∈ R + , the fact thatthey are equal when | z | > | z | > | arg( z − z ) − arg z | < π and arg z < π (for Ω + ) orarg z ≥ π (for Ω − ) means that they are equal when | z | > | z | > | arg( z − z ) − arg z | < π .Thus we have proved that when | z | > | z | > | arg( z − z ) − arg z | < π , the left-handside of (5.5) is equal to g p ,p ,p ± ( z , z ; u, w , w , w (cid:48) ).When | z | > | z | > z ≥ π , (cid:104) w (cid:48) , Ω − ( Y ) p ( w , z )( Y g W ) p ( u, z ) w (cid:105) = (cid:104) w (cid:48) , e z L ( − Y p (( Y g W ) p ( u, z ) w , − z ) w (cid:105) = (cid:104) e z L (cid:48) (1) w (cid:48) , Y p (( Y g W ) p ( u, z ) w , − z ) w (cid:105) (5.8)converges absolutely and if in addition, | arg( z − z ) − arg( − z ) | < π , its sum is equal to f p ,p ,p ( z − z , − z ; u, w , w , e z L (cid:48) (1) w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p ( z − z ) e s j l p ( − z ) e t k l p ( z ) ( l p ( z − z )) l ( l p ( − z )) m ( l p ( z )) n . (5.9)18hen arg z ≥ π , arg( − z ) = arg z − π . Hence the right-hand side of (5.9) is equal to N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p ( z − z ) e s j ( l p ( z ) − π i ) e t k l p ( z ) ( l p ( z − z )) l ( l p ( z ) − π i ) m ( l p ( z )) n = g p ,p ,p − ( z , z ; u, w , w , w (cid:48) ) . (5.10)The same argument as above shows that the left-hand side of (5.8) converges absolutely when | z | > | z | > | z | > | z | > | arg( z − z ) − arg( − z ) | < π and arg z ≥ π . But when arg z ≥ π , arg( − z ) = arg z − π . Hence in this case, the inequality | arg( z − z ) − arg( − z ) | < π becomes − π < arg( z − z ) − arg z < − π . Also both theleft-hand side of (5.8) and the left-hand side of (5.10) are single valued analytic functions in z and z with cuts at z ∈ R + and z ∈ R + . Thus the same argument as above shows thatwhen | z | > | z | > − π < arg( z − z ) − arg z < − π , the left-hand side of (5.8) isequal to g p ,p ,p − ( z , z ; u, w , w , w (cid:48) ).Next we discuss the iterate of Ω − ( Y ) and the twisted vertex operator map φ g ( Y g W ). ByLemma 4.5, when | z | > | z − z | > z ≥ π , (cid:104) w (cid:48) , Ω − ( Y ) p ( φ g ( Y g W ) p ( u, z − z ) w , z ) w (cid:105) = (cid:104) w (cid:48) , Ω − ( Y ) p (( Y g W ) p ( g − u, z − z ) w , z ) w (cid:105) = (cid:104) w (cid:48) , e z L ( − Y p ( w , − z )( Y g W ) p ( g − u, z − z ) w (cid:105) = (cid:104) e z L (cid:48) (1) w (cid:48) , Y p ( w , − z )( Y g W ) p ( g − u, z − z ) w (cid:105) (5.11)converges absolutely and if in addition, π < arg z − arg( − z ) < π , its sum is equal to f p ,p ,p − ( z − z , − z ; g − u, w , w , e z L (cid:48) (1) w (cid:48) ). By (4.4), we have f p ,p ,p − ( z − z , − z ; g − u, w , w , e z L (cid:48) (1) w (cid:48) )= f p ,p ,p ( z − z , − z ; u, w , w , e z L (cid:48) (1) w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p ( z − z ) e s j l p ( − z ) e t k l p ( z ) ( l p ( z − z )) l ( l p ( − z )) m ( l p ( z )) n . (5.12)When arg z ≥ π , arg( − z ) = arg z − π and hence the right-hand side of (5.12) is equal to N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p ( z − z ) e s j ( l p ( z ) − π i ) e t k l p ( z ) ( l p ( z − z )) l ( l p ( z ) − π i ) m ( l p ( z )) n = g p ,p ,p − ( z , z ; u, w , w , w (cid:48) ) . (5.13)Thus the left-hand side of (5.11) converges absolutely when | z | > | z − z | > g p ,p ,p − ( z , z ; u, w , w , w (cid:48) ) when | z | > | z − z | > π < arg z − arg( − z ) < π and arg z ≥ π . But when arg z ≥ π , arg( − z ) = arg z − π and thus the inequality π < arg z − arg( − z ) < π is equivalent to | arg z − arg( z ) | < π . Then by the same19rguments as in the cases above, we see that when | z | > | z − z | > | arg z − arg z | < π ,the sum of left-hand side of (5.11) is equal to g p ,p ,p − ( z , z ; u, w , w , w (cid:48) ).We now come back to discuss Ω + ( Y ). When | z | > | z | > z < π , (cid:104) w (cid:48) , Ω + ( Y ) p ( w , z ) φ g − ( Y g W ) p ( u, z ) w (cid:105) = (cid:104) w (cid:48) , e z L ( − Y p ( φ g − ( Y g W ) p ( u, z ) w , − z ) w (cid:105) = (cid:104) e z L (cid:48) (1) w (cid:48) , Y p (( Y g W ) p ( g u, z ) w , − z ) w (cid:105) (5.14)converges absolutely and if in addition, | arg( z − z ) − arg( − z ) | < π , its sum is equal to f p ,p ,p ( z − z , − z ; g u, w , w , e z L (cid:48) (1) w (cid:48) ). By (4.5), we have f p ,p ,p ( z − z , − z ; g u, w , w , e z L (cid:48) (1) w (cid:48) )= f p − ,p ,p ( z − z , − z ; u, w , w , e z L (cid:48) (1) w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p − ( z − z ) e s j l p ( − z ) e t k l p ( z ) ( l p − ( z − z )) l ( l p ( − z )) m ( l p ( z )) n . (5.15)When arg z < π , arg( − z ) = arg z + π and hence the right-hand side of (5.15) is equal to N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p − ( z − z ) e s j ( l p ( z )+ π i ) e t k l p ( z ) ( l p − ( z − z )) l ( l p ( z ) + π i ) m ( l p ( z )) n = g p ,p ,p − ( z , z ; u, w , w , w (cid:48) ) . (5.16)Thus we see that the left-hand side of (5.14) converges absolutely when | z | > | z | > g p ,p ,p − ( z , z ; u, w , w , w (cid:48) ) when | z | > | z | > | arg( z − z ) − arg( − z ) | < π and arg z < π . But when arg z < π , arg( − z ) = arg z + π . Hence in this case,the inequality | arg( z − z ) − arg( − z ) | < π becomes π < arg( z − z ) − arg z < π . Alsoboth the left-hand side of (5.14) and the left-hand side of (5.16) are single-valued analyticfunction in z and z with cuts at z ∈ R + and z ∈ R + . Thus the same argument as aboveshows that when | z | > | z | > π < arg( z − z ) − arg z < π , the left-hand sideof (5.14) converges absolutely to g p ,p ,p − ( z , z ; u, w , w , w (cid:48) ). Then by Lemma 4.5, when | z | > | z | > − π < arg( z − z ) − arg z < − π , the sum of left-hand side of (5.14) isequal to g p ,p ,p + ( z , z ; u, w , w , w (cid:48) ).Finally, we discuss the iterate of Ω + ( Y ) and the twisted vertex operator map Y g W . When | z | > | z − z | > z < π , (cid:104) w (cid:48) , Ω + ( Y ) p (( Y g W ) p ( u, z − z ) w , z ) w (cid:105) = (cid:104) w (cid:48) , e z L ( − Y p ( w , − z )( Y g W ) p ( u, z − z ) w (cid:105) = (cid:104) e z L (cid:48) (1) w (cid:48) , Y p ( w , − z )( Y g W ) p ( u, z − z ) w (cid:105) (5.17)20onverges absolutely and if in addition, − π < arg z − arg( − z ) < − π , its sum is equal to f p ,p ,p ( z − z , − z ; u, w , w , e z L (cid:48) (1) w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p ( z − z ) e s j l p ( − z ) e t k l p ( z ) ( l p ( z − z )) l ( l p ( − z )) m ( l p ( z )) n . (5.18)When arg z < π , arg( − z ) = arg z + π . Hence the right-hand side of (5.18) is equal to N (cid:88) i,j,k,l,m,n =0 a ijklmn e r i l p ( z − z ) e s j ( l p ( z )+ π i ) e t k l p ( z ) ( l p ( z − z )) l ( l p ( z ) + π i ) m ( l p ( z )) n = g p ,p ,p + ( z , z ; u, w , w , w (cid:48) ) . (5.19)The same argument as above shows that the left-hand side of (5.17) converges absolutelywhen | z | > | z − z | > | z | > | z − z | > − π < arg z − arg( − z ) < − π and arg z < π . But when arg z < π , arg( − z ) = arg z + π . Hence inthis case, the inequality − π < arg z − arg( − z ) < − π becomes | arg( z − z ) − arg( − z ) | < π .Also both the left-hand side of (5.17) and the left-hand side of (5.19) are single valued analyticfunction in z and z with cuts at z ∈ R + and z ∈ R + . Thus the same argument as aboveshows that when | z | > | z − z | > − π < arg z − arg( − z ) < − π , the sum of the left-handside of (5.8) is equal to g p ,p ,p + ( z , z ; u, w , w , w (cid:48) ).Let V W W W be the space of twisted intertwining operators of type (cid:0) W W W (cid:1) . Then we have: Corollary 5.2
The maps Ω + : V W W W → V W W φ g − ( W ) and Ω − : V W W W → V W φ g ( W ) W are lin-ear isomorphisms. In particular, V W W W , V W φ g ( W ) W and V W W φ g − ( W ) are linearly isomorphic.Proof. It is clear that Ω + and Ω − are inverse of each other.The linear isomorphisms Ω + and Ω − are called the skew-symmetry isomorphisms . In this section, we construct what we call the contragredient isomorphisms between thespaces of twisted intertwining operators of suitable types. These linear isomorphisms willplay an important role in the study of rigidity and other related properties of the still-to-be-constructed G -crossed braided tensor category structure on the category of g -twisted V -modules for all g in a group G of automorphisms of V .Let g , g be automorphisms of V , W , W and W g -, g - and g g -twisted V -modulesand Y a twisted intertwining operator of type (cid:0) W W W (cid:1) . We define linear maps A ± ( Y ) : W ⊗ W (cid:48) → W (cid:48) { x } [log x ] w ⊗ w (cid:48) (cid:55)→ A ± ( Y )( w , x ) w (cid:48) (cid:104) A ± ( Y )( w , x ) w (cid:48) , w (cid:105) = (cid:104) w (cid:48) , Y ( e xL (1) e ± π i L (0) ( x − L (0) ) w , x − ) w (cid:105) (6.1)for w ∈ W and w ∈ W and w (cid:48) ∈ W (cid:48) .Let ( W, Y gW ) be a g -twisted V -module. When W = V , W = W = W and Y = Y gW , bydefinition, A + ( Y gW ) = A − ( Y gW ) = ( Y gW ) (cid:48) (see Section 3).Let L sW (0) be the semisimple part of L W (0). From the definition (6.1), for p ∈ Z , w ∈ W , w ∈ W , w (cid:48) ∈ W (cid:48) and z ∈ C × , we have (cid:104) A ± ( Y ) p ( w , z ) w (cid:48) , w (cid:105) = (cid:104) A ± ( Y ) p ( w , x ) w (cid:48) , w (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) x n = e nlp ( z ) , log x = l p ( z ) = (cid:104) w (cid:48) , Y ( e xL W (1) e ± π i L W (0) ( x − L W (0) ) w , x − ) w (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) x n = e nlp ( z ) , log x = l p ( z ) = (cid:104) w (cid:48) , Y ( e xL W (1) e ± π i L W (0) ( x − L sW (0) ) ·· ( e − ( L W (0) − L sW (0)) log x ) w , x − ) w (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) x n = e nlp ( z ) , log x = l p ( z ) = (cid:104) w (cid:48) , Y ( e zL W (1) e ± π i L W (0) ( e − l p ( z ) L sW (0) ) ·· ( e − ( L W (0) − L sW (0)) l p ( z ) ) w , y ) w (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) y n = e − nlp ( z ) , log y = − l p ( z ) = (cid:104) w (cid:48) , Y ( e zL W (1) e ± π i L W (0) e − l p ( z ) L W (0) w , y ) w (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) y n = e − nlp ( z ) , log y = − l p ( z ) . (6.2)When arg z = 0, arg z − = arg z = 0 and − l p ( z ) = l − p ( z − ). When arg z (cid:54) = 0, arg z − = − arg z + 2 π and − l p ( z ) = l − p − ( z − ). Hence when arg z = 0, the right-hand side of (6.2) isequal to (cid:104) w (cid:48) , Y ( e zL W (1) e ± π i L W (0) e l − p ( z − ) L W (0) w , y ) w (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) y n = e nl − p ( z − , log y = l − p ( z − ) = (cid:104) w (cid:48) , Y − p ( e zL W (1) e ± π i L W (0) e l − p ( z − ) L W (0) w , z − ) w (cid:105) (6.3)and when arg z (cid:54) = 0, it is equal to (cid:104) w (cid:48) , Y ( e zL W (1) e ± π i L W (0) e l − p − ( z − ) L W (0) w , y ) w (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) y n = e nl − p − z − , log y = l − p − ( z − ) = (cid:104) w (cid:48) , Y − p − ( e zL W (1) e ± π i L W (0) e l − p − ( z − ) L W (0) w , z − ) w (cid:105) . (6.4)From (6.2)–(6.4), for w ∈ W , w ∈ W , w (cid:48) ∈ W (cid:48) and z ∈ C × , we have (cid:104) A ± ( Y ) p ( w , z ) w (cid:48) , w (cid:105) = (cid:104) w (cid:48) , Y − p ( e zL W (1) e ± π i L W (0) e l − p ( z − ) L W (0) w , z − ) w (cid:105) (6.5)22hen arg z = 0 and (cid:104) A ± ( Y ) p ( w , z ) w (cid:48) , w (cid:105) = (cid:104) w (cid:48) , Y − p − ( e zL W (1) e ± π i L W (0) e l − p − ( z − ) L W (0) w , z − ) w (cid:105) (6.6)when arg z (cid:54) = 0. Theorem 6.1
The linear maps A + ( Y ) and A − ( Y ) are twisted intertwining operators oftypes (cid:0) φ g ( W (cid:48) ) W W (cid:48) (cid:1) and (cid:0) W (cid:48) W φ g − ( W (cid:48) ) (cid:1) , respectively.Proof. Just as in the proof of Theorem 5.1, compared with the duality property, the lower-truncation property and the L W ( − A + ( Y ) and A − ( Y ) by h + ( z , z ; u, w , w , w (cid:48) ) and h − ( z , z ; u, w , w , w (cid:48) ), respectively. Let f ( z , z ; u, w , w , w (cid:48) )be the multivalued analytic function with preferred branch in the duality property for thetwisted intertwining operator Y . Then we can write f ( z , z ; e z − L V (1) ( − z ) L V (0) u, e z − L W (1) e ± π i L W (0) e z L W (0) w , w , w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a ± ijklmn z r i z s j ( z − z ) t k (log z ) l (log z ) m (log( z − z )) n . Define h ± ( z , z ; u, w , w , w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 e ± t k π i a ± ijklmn z − ( r i + t k )1 z − ( s j + t k )2 ( z − z ) t k ·· ( − log z ) l ( − log z ) m (log( z − z ) − log z − log z ± π i ) n . (6.7)Let u ∈ V , w ∈ W , w ∈ W and w (cid:48) ∈ W (cid:48) . We consider z , z ∈ C satifying | z − | > | z − | > | z | > | z | >
0) and arg z , arg z (cid:54) = 0. Since | z − | > | z − | > Y g W ) (cid:48) = A + ( Y g W ) and the duality property for Y , (cid:104) φ g (( Y g W ) (cid:48) ) p ( u, z ) A + ( Y ) p ( w , z ) w (cid:48) , w (cid:105) = (cid:104) (( Y g W ) (cid:48) ) p ( g − u, z ) A + ( Y ) p ( w , z ) w (cid:48) , w (cid:105) = (cid:104) w (cid:48) , Y − p − ( e z L W (1) e π i L W (0) e z − ) L W (0) w , z − ) ·· ( Y g W ) − p − ( e z L V (1) ( − z − ) L V (0) g − u, z − ) w (cid:105) (6.8)23onverges absolutely and if in addition, − π < arg( z − − z − ) − arg z − < − π , its sum isequal to f − p − , − p − , − p − ( z − , z − ; e z L V (1) ( − z − ) L V (0) g − u, e z L W (1) e π i L W (0) e z − ) L W (0) w , w , w (cid:48) )= f − p − , − p − , − p − ( z − , z − ; g − e z L V (1) ( − z − ) L V (0) u, e z L W (1) e π i L W (0) e z − ) L W (0) w , w , w (cid:48) ) . (6.9)By (4.4), (6.9) is equal to f − p − , − p − , − p ( z − , z − ; e z L V (1) ( − z − ) L V (0) u, e z L W (1) e π i L W (0) e z − ) L W (0) w , w , w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a + ijklmn e r i l − p − ( z − ) e s j l − p − ( z − ) e t k l − p ( z − − z − ) ·· ( l − p − ( z − )) l ( l − p − ( z − )) m ( l − p ( z − − z − )) n . (6.10)Since arg z , arg z (cid:54) = 0, arg z − = − arg z + 2 π , arg z − = − arg z + 2 π , l − p − ( z − ) = − l p ( z ) and l − p − ( z − ) = − l p ( z ). Thus we also havearg( z − − z − ) = arg (cid:18) z − z z ( − z ) (cid:19) . Since 0 ≤ arg z < π for any z ∈ C × and − π < arg( z − − z − ) − arg z − = arg (cid:18) z − z z ( − z ) (cid:19) − arg z − < − π , we have arg (cid:18) z − z z ( − z ) (cid:19) = arg( z − z ) − arg z − arg z + (2 q + 1) π (6.11)with q = − − π < arg( z − z ) − arg z < − π , with q = 0 when | arg( z − z ) − arg z | < π and with q = 1 when π < arg( z − z ) − arg z < π . From (6.11), we obtain l − p ( z − − z − )= l − p (cid:18) z − z z ( − z ) (cid:19) = log | z − z | − log | z | − log | z | + arg( z − z ) i − arg z i − arg z i + (2 q + 1) π i + 2( − p ) π i = l p + q ( z − z ) − l p ( z ) − l p ( z ) + π i . (6.12)24rom l − p − ( z − ) = − l p ( z ), l − p − ( z − ) = − l p ( z ) and (6.12), the right-hand side of (6.10)is equal to N (cid:88) i,j,k,l,m,n =0 e ± t k π i a ± ijklmn e − r i l p ( z ) e − s j l p ( z ) e t k ( l p q ( z − z ) − l p ( z ) − l p ( z )+ π i ) ·· ( − l p ( z )) l ( − l p ( z )) m ( l p + q ( z − z ) − l p ( z ) − l p ( z ) + π i ) n = h p ,p ,p + q + ( z , z ; u, w , w , w (cid:48) ) . (6.13)From (6.8)–(6.13), the left-hand side of (6.8) converges absolutely when | z | > | z | > z , arg z (cid:54) = 0 and its sum is equal to the branch h p ,p ,p + q + ( z , z ; u, w , w , w (cid:48) ) when | z | > | z | > − π < arg( z − − z − ) − arg z − < − π and arg z , arg z (cid:54) = 0. In the case thatarg z = 0 or arg z = 0, we can also prove similarly that when | z | > | z | >
0, the left-handside of (6.8) converges absolutely and if in addition, − π < arg( z − − z − ) − arg z − < − π ,its sum is equal to h p ,p ,p + q + ( z , z ; u, w , w , w (cid:48) ). The main difference, for example, in thecase arg z = 0 is that arg z − = arg z = 0 instead of arg z − = − arg z + 2 π .When q = −
1, since the sum of the left-hand side of (6.8) is equal to the branch h p ,p ,p − ( z , z ; u, w , w , w (cid:48) ) when | z | > | z | > − π < arg( z − z ) − arg z < − π , byLemma 4.5, the left-hand side of (6.8) converges absolutely to h p ,p ,p + ( z , z ; u, w , w , w (cid:48) )when | z | > | z | > < arg( z − z ) − arg z < π . When q = 0, the left-handside of (6.8) converges absolutely to h p ,p ,p + ( z , z ; u, w , w , w (cid:48) ) when | z | > | z | > | arg( z − z ) − arg z | < π . When q = 1, since the left-hand side of (6.8) converges absolutelyto h p ,p ,p +1+ ( z , z ; u, w , w , w (cid:48) ) when | z | > | z | > π < arg( z − z ) − arg z < π ,by Lemma 4.5, the sum of the left-hand side of (6.8) is equal to h p ,p ,p + ( z , z ; u, w , w , w (cid:48) )when | z | > | z | > − π < arg( z − z ) − arg z <
0. Thus we have proved that when | z | > | z | > | arg( z − z ) − arg z | < π , the sum of the left-hand side of (6.8) is alwaysequal to h p ,p ,p + ( z , z ; u, w , w , w (cid:48) ).Next we consider the product of A + ( Y ) and the twisted vertex operator ( Y g W ) (cid:48) . Let u , w , w and w (cid:48) be the same as above. When | z − | > | z − | > | z | > | z | > z , arg z (cid:54) = 0, (cid:104) A + ( Y ) p ( w , z )(( Y g W ) (cid:48) ) p ( u, z ) w (cid:48) , w (cid:105) = (cid:104) w (cid:48) , ( Y g W ) − p − ( e z L V (1) ( − z − ) L V (0) u, z − ) ·· Y − p − ( e z L W (1) e π i L W (0) e l − p − ( z − ) L W (0) w , z − ) w (cid:105) (6.14)converges absolutely and if in addition, | arg( z − − z − ) − arg z − | < π , its sum is equal to f − p − , − p − , − p − ( z − , z − ; e z L V (1) ( − z − ) L V (0) u, e z L W (1) e π i L W (0) e l − p − ( z − ) L W (0) w , w , w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a + ijklmn e r i l − p − ( z − ) e s j l − p − ( z − ) e t k l − p − ( z − − z − ) ·· ( l − p − ( z − )) l ( l − p − ( z − )) m ( l − p − ( z − − z − )) n . (6.15)25gain we have arg z − = − arg z + 2 π , arg z − = − arg z + 2 π , l − p − ( z − ) = − l p ( z )and l − p − ( z − ) = − l p ( z ). Since 0 ≤ arg z < π for any z ∈ C × and − π < arg( z − − z − ) − arg z − = arg (cid:18) z − z z ( − z ) (cid:19) − arg z − < π , we havearg( z − − z − ) = arg (cid:18) z − z z ( − z ) (cid:19) = arg( z − z ) − arg z − arg z + (2 q + 1) π (6.16)with q = 0 when π < arg( z − z ) − arg z < π and with q = 1 when − π < arg( z − z ) − arg z < − π . From (6.16) and by the same calculation as in (6.12), we obtain l − p − ( z − − z − ) = l p + q − ( z − z ) − l p ( z ) − l p ( z ) + π i . (6.17)From l − p − ( z − ) = − l p ( z ), l − p − ( z − ) = − l p ( z ) and (6.17), the right-hand side of (6.15)is equal to N (cid:88) i,j,k,l,m,n =0 a + ijklmn e − r i l p ( z ) e − s j l p ( z ) e t k ( l p q − ( z − z ) − l p ( z ) − l p ( z )+ π i ) ·· ( − l p ( z )) l ( − l p ( z )) m ( l p + q − ( z − z ) − l p ( z ) − l p ( z ) + π i ) n = h p ,p ,p + q − ( z , z ; u, w , w , w (cid:48) ) . (6.18)From (6.14)–(6.18), the left-hand side of (6.14) converges absolutely when | z | > | z | > z , arg z (cid:54) = 0 and if in addition, | arg( z − − z − ) − arg z − | < π , its sum isequal to h p ,p ,p + q − ( z , z ; u, w , w , w (cid:48) ). In the case that arg z = 0 or arg z = 0, wecan also prove similarly that when | z | > | z | >
0, the left-hand side of (6.14) convergesabsolutely and if in addition, − π < arg( z − − z − ) − arg z − < − π , its sum is equal to h p ,p ,p + q + ( z , z ; u, w , w , w (cid:48) ).When q = 0, since the sum of the left-hand side of (6.14) is equal to the branch h p ,p ,p − ( z , z ; u, w , w , w (cid:48) ) when | z | > | z | > π < arg( z − z ) − arg z < π , byLemma 4.5, the left-hand side of (6.14) converges absolutely to h p ,p ,p + ( z , z ; u, w , w , w (cid:48) )when | z | > | z | > − π < arg( z − z ) − arg z < − π . When q = 1, the left-handside of (6.14) converges absolutely to h p ,p ,p + ( z , z ; u, w , w , w (cid:48) ) when | z | > | z | > − π < arg( z − z ) − arg z < − π . Thus we have proved that when | z | > | z | >
0, the left-hand side of (6.14) converges absolutely and if in addition, , − π < arg( z − z ) − arg z < − π ,its sum is equal to h p ,p ,p + ( z , z ; u, w , w , w (cid:48) ).Now we discuss A − ( Y ). When | z − | > | z − | > | z | > | z | >
0) andarg z , arg z (cid:54) = 0, (cid:104) (( Y g W ) (cid:48) ) p ( u, z ) A − ( Y ) p ( w , z ) w (cid:48) , w (cid:105) = (cid:104) w (cid:48) , Y − p − ( e z L W (1) e − π i L W (0) e l − p − ( z − ) L W (0) w , z − ) ·· ( Y g W ) − p − ( e z L V (1) ( − z − ) L V (0) u, z − ) w (cid:105) (6.19)26onverges absolutely and if in addition, − π < arg( z − − z − ) − arg z − < − π , its sum isequal to f − p − , − p − , − p − ( z − , z − ; e z L V (1) ( − z − ) L V (0) u, e z L W (1) e − π i L W (0) e l − p − ( z − ) L W (0) w , w , w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a − ijklmn e r i l − p − ( z − ) e s j l − p − ( z − ) e t k l − p − ( z − − z − ) ·· ( l − p − ( z − )) l ( l − p − ( z − )) m ( l − p − ( z − − z − )) n . (6.20)As in the case for A + ( Y ) above, since arg z , arg z (cid:54) = 0, l − p − ( z − ) = − l p ( z ) and l − p − ( z − ) = − l p ( z ). Also, the same argument as in that case gives l − p − ( z − − z − ) = l p + q ( z − z ) − l p ( z ) − l p ( z ) − π i (6.21)with q = − − π < arg( z − z ) − arg z < − π , with q = 0 when | arg( z − z ) − arg z | < π and with q = 1 when π < arg( z − z ) − arg z < π . From l − p − ( z − ) = − l p ( z ), l − p − ( z − ) = − l p ( z ) and (6.21), the right-hand side of (6.20) is equal to N (cid:88) i,j,k,l,m,n =0 a − ijklmn e − r i l p ( z ) e − s j l p ( z ) e t k ( l p q ( z − z ) − l p ( z ) − l p ( z ) − π i ) ·· ( − l p ( z )) l ( − l p ( z )) m ( l p + q ( z − z ) − l p ( z ) − l p ( z ) − π i ) n = h p ,p ,p + q − ( z , z ; u, w , w , w (cid:48) ) . (6.22)From (6.19)–(6.22) and the same argument as in the case for A + ( Y ) above, the left-handside of (6.19) converges absolutely when | z | > | z | > h p ,p ,p + q − ( z , z ; u, w , w , w (cid:48) ) when | z | > | z | > | arg( z − z ) − arg z | < π .Now we consider the product of A − ( Y ) and the twisted vertex operator φ g − (( Y g W ) (cid:48) ).When | z − | > | z − | > | z | > | z | >
0) and arg z , arg z (cid:54) = 0, (cid:104) A − ( Y ) p ( w , z ) φ g − (( Y g W ) (cid:48) ) p ( u, z ) w (cid:48) , w (cid:105) = (cid:104) A − ( Y ) p ( w , z )( Y g W ) (cid:48) p ( g u, z ) w (cid:48) , w (cid:105) = (cid:104) w (cid:48) , ( Y g W ) − p − ( e z L V (1) ( − z − ) L V (0) g u, z − ) ·· Y − p − ( e z L W (1) e − π i L W (0) e l − p − ( z − ) L W (0) w , z − ) w (cid:105) (6.23)converges absolutely and if in addition, | arg( z − − z − ) − arg z − | < π , its sum is equal to f − p − , − p − , − p − ( z − , z − ; e z L V (1) ( − z − ) L V (0) g u, e z L W (1) e − π i L W (0) e l − p − ( z − ) L W (0) w , w , w (cid:48) )= f − p − , − p − , − p − ( z − , z − ; g e z L V (1) ( − z − ) L V (0) u, e z L W (1) e − π i L W (0) e l − p − ( z − ) L W (0) w , w , w (cid:48) ) . (6.24)27y (4.4), (6.24) is equal to f − p − , − p − , − p − ( z − , z − ; e z L V (1) ( − z − ) L V (0) u, e z L W (1) e − π i L W (0) e l − p − ( z − ) L W (0) w , w , w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a − ijklmn e r i l − p − ( z − ) e s j l − p − ( z − ) e t k l − p − ( z − − z − ) ·· ( l − p − ( z − )) l ( l − p − ( z − )) m ( l − p − ( z − − z − )) n . (6.25)As in the case for A + ( Y ) above, since arg z , arg z (cid:54) = 0, l − p − ( z − ) = − l p ( z ) and l − p − ( z − ) = − l p ( z ). Also, the same argument as in that case gives l − p − ( z − − z − ) = l p + q − ( z − z ) − l p ( z ) − l p ( z ) − π i (6.26)with q = 0 when π < arg( z − z ) − arg z < π and with q = 1 when − π < arg( z − z ) − arg z < − π . From l − p − ( z − ) = − l p ( z ), l − p − ( z − ) = − l p ( z ) and (6.26), the right-handside of (6.25) is equal to N (cid:88) i,j,k,l,m,n =0 a − ijklmn e − r i l p ( z ) e − s j l p ( z ) e t k ( l p q − ( z − z ) − l p ( z ) − l p ( z ) − π i ) ·· ( − l p ( z )) l ( − l p ( z )) m ( l p + q − ( z − z ) − l p ( z ) − l p ( z ) − π i ) n = h p ,p ,p + q − − ( z , z ; u, w , w , w (cid:48) ) . (6.27)From (6.23)–(6.27) and the same argument as in the case for A + ( Y ) above, the left-handside of (6.23) converges absolutely when | z | > | z | > h p ,p ,p + q − ( z , z ; u, w , w , w (cid:48) ) when | z | > | z | > − π < arg( z − z ) − arg z < − π .Finally we study the iterate of A ± ( Y ) and the twisted vertex operator Y g W . Whenarg z (cid:54) = 0, from (6.6), we have (cid:104) A ± ( Y ) p (( Y g W ) p ( u, z − z ) w , z ) w (cid:48) , w (cid:105) = (cid:104) w (cid:48) , Y − p − ( e z L W (1) e ± π i L W (0) e l − p − ( z − ) L W (0) ( Y g W ) p ( u, z − z ) w , z − ) w (cid:105) . (6.28)Note that (3.61) and (3.62) in [HLZ1] with Y replaced by Y g W still holds since Y g W is a(logarithmic) intertwining operator among V (cid:104) g ,g ,g g (cid:105) -modules. Using these formulas, weobtain the formulas for Y g W ( u, x ) ( u ∈ V ) conjugated by e x L W (1) and y L W (0) . Using theseformulas, we obtain e x L W (1) y L W (0) Y g W ( u, x )= e x L W (1) Y g W ( y L V (0) u, yx ) y L W (0) = Y g W ( e x (1 − x yx ) L V (1) (1 − x yx ) − L V (0) y L V (0) u, yx (1 − x yx ) − ) e x L W (1) y L W (0) . (6.29)28ubstituting e ± nπ i x − n and ± π i − x for y n and log y , respectively, in (6.29), we obtain e x L W (1) e ± π i L W (0) ( x − L W (0)2 ) Y g W ( u, x )= Y g W (cid:18) e ( x + x ) L V (1) ( − ( x + x ) − ) L V (0) u, xx ( x + x ) x (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) x n = e ± nπ i , log x = ± π i ·· e x L W (1) e ± π i L W (0) ( x − L W (0)2 ) (6.30)Substituting e nl p ( z − z ) , l p ( z − z ), e nl p ( z ) and l p ( z ) for x n , log x , x n and log x , re-spectively, in (6.30) and then applying the resulting equality to w , we obtain in the region | z | > | z − z | > e z L W (1) e ± π i L W (0) e l − p − ( z − ) L W (0) Y g ,p W ( u, z − z ) w = Y g W (cid:18) e z L V (1) ( − z − ) L V (0) u, xx ( x + x ) x (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) x n = e nlp z − z , log x = l p ( z − z ) , x n = e nlp z log x = l p ( z ) , x n = e ± nπ i , log x = ± π i ·· e z L W (1) e ± π i L W (0) e l − p − ( z − ) L W (0) w . (6.31)In the region | z | > | z − z | >
0, either arg(1 + z − z z ) < π or π < arg(1 + z − z z ). Hencewhen | z | > | z − z | >
0, the expansion of (1 + z − z z ) m for m ∈ C as a power series in z − z z is absolutely convergent to e m log | z − z z | e m (arg(1+ z − z z ) i +2 qπ i ) = e m log | z | e − m log | z | e m (arg(1+ z − z z ) i +2 qπ i ) , (6.32)where q = 0 when arg(1 + z − z z ) < π and q = − π < arg(1 + z − z z ). Also, when | z | > | z − z | > | arg z − arg z | < π ,arg z = arg z + arg (cid:18) z − z z (cid:19) + 2 qπ, (6.33)where q = 0 when arg(1 + z − z z ) < π and q = − π < arg(1 + z − z z ). By (6.32) and(6.33), we obtain x m (cid:18) x x (cid:19) m (cid:12)(cid:12)(cid:12)(cid:12) x n = e nlp z − z , x n = e nlp z = e ml p ( z ) e m log | z | e − m log | z | e m ((arg(1+ z − z z )) i +2 qπ i ) = e m ((arg z ) i +2 p π i ) e m log | z | e m ((arg(1+ z − z z ) i )+2 qπ i ) = e m ((arg z ) i − (arg(1+ z − z z )) i − qπ i +2 p π i ) e m log | z | e m ((arg(1+ z − z z )) i +2 qπ i ) = e ml p ( z ) (6.34)29or m ∈ C . Using (6.34), we obtain (cid:18) xx ( x + x ) x (cid:19) m (cid:12)(cid:12)(cid:12)(cid:12) x n = e nlp z − z , x n = e nlp z , x n = e ± nπ i = e ± mπ i x m (cid:18) x m (cid:18) x x (cid:19) m (cid:19) x m (cid:12)(cid:12)(cid:12)(cid:12) x n = e nlp z − z , x n = e nlp z = e ± mπ i e ml p ( z − z ) e − ml p ( z ) e − ml p ( z ) = e m ( l p ( z − z ) − l p ( z ) − l p ( z ) ± π i ) . (6.35)Similarly, when | z | > | z − z | >
0, the expansion of log(1 + z − z z ) as a power series in z − z z is absolutely convergent tolog (cid:12)(cid:12)(cid:12)(cid:12) z − z z (cid:12)(cid:12)(cid:12)(cid:12) +arg (cid:18) z − z z (cid:19) i +2 qπ i = log | z |− log | z | +arg (cid:18) z − z z (cid:19) i +2 qπ i , (6.36)where q = 0 when arg(1 + z − z z ) < π and q = − π < arg(1 + z − z z ). By (6.33) and(6.36), we obtainlog (cid:18) xx ( x + x ) x (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) log x = l p ( z − z ) , log x = l p ( z ) , log x = ± π i = (cid:18) log x + log x − log x − log (cid:18) x x (cid:19) − log x (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) log x = l p ( z − z ) , log x = l p ( z ) , log x = ± π i = l p ( z − z ) − l p ( z ) − l p ( z ) ± π i . (6.37)On the other hand, for any z , z ∈ C × such that z (cid:54) = z , there exists m ∈ Z such that l p ( z − z ) − l p ( z ) − l p ( z ) ± π i = log | z − z | + (arg( z − z )) i − log | z | − (arg z ) i − log | z | − (arg z ) i + 2( p − p ) π i ± π i = log (cid:12)(cid:12)(cid:12)(cid:12) z − z z ( − z ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:18) arg (cid:18) z − z z ( − z ) (cid:19)(cid:19) i + 2( p − p ) π i + (2 m + 1) π i ± π i = log | z − − z − | + (arg( z − − z − )) i + 2 (cid:18) p − p + m + 1 ± (cid:19) π i = l p − p + m + ± ( z − − z − ) . (6.38)From (6.31) and (6.35)–(6.38), we see that when | z | > | z − z | > | arg z − arg z | < π , the right-hand side of (6.28) is equal to (cid:104) w (cid:48) , Y − p − (( Y g W ) p − p + m + ± ( e z L V (1) ( − z − ) L V (0) u, z − − z − ) ·· e z L W (1) e ± π i L W (0) e l − p − ( z − ) L W (0) w , z − ) w (cid:105) , (6.39)30hich, by the duality property for Y , converges absolutely to f − p − , − p − ,p − p + m + ± ( z − , z − ; e z L V (1) ( − z − ) L V (0) u,e z L W (1) e ± π i L W (0) e l − p − ( z − ) L W (0) w , w , w (cid:48) )= N (cid:88) i,j,k,l,m,n =0 a ± ijklmn e r i l − p − ( z − ) e s j l − p − ( z − ) e t k l p − p m + 1 ± ( z − − z − ) ·· ( l − p − ( z − )) l ( l − p − ( z − )) m ( l p − p + m + ± ( z − − z − )) n (6.40)when | z − | > | z − − z − | > | arg z − − arg z − | < π . In the case that arg z , arg z (cid:54) = 0,arg z − = − arg z + 2 π , arg z − = − arg z + 2 π , l − p − ( z − ) = − l p ( z ) and l − p − ( z − ) = − l p ( z ). Using these and (6.38), we see that the right-hand side of (6.40) is equal to N (cid:88) i,j,k,l,m,n =0 a ± ijklmn e − r i l p ( z ) e − s j l p ( z ) e t k ( l p ( z − z ) − l p ( z ) − l p ( z ) ± π i ) ·· ( − l p ( z )) l ( − l p ( z )) m ( l p ( z − z ) − l p ( z ) − l p ( z ) ± π i ) n = h p ,p ,p ± ( z , z ; u, w , w , w (cid:48) ) . (6.41)Note that | z − | > | z − − z − | > | z | > | z − z | > | arg z − − arg z − | < π is equivalent to | arg z − arg z | < π . Thus we have proved that the right-handside of (6.28) is absolutely convergent to h p ,p ,p ± ( z , z ; u, w , w , w (cid:48) ) in the region given by | z | , | z | > | z − z | > | arg z − arg z | < π and arg z , arg z (cid:54) = 0 ( | z | > | z − z | > | z | > | z − z | > e l p ( z − z ) and e l p ( z ) and in nonnegative integral powersof l p ( z − z ) and l p ( z ) with finitely many negative powers in e l p ( z − z ) and finitely manypositive powers of e l p ( z ) . Since h p ,p ,p ± ( z , z ; u, w , w , w (cid:48) ) can be expanded uniquely assuch a series in the region | z | > | z − z | >
0, the left-hand side of (6.28) must be absolutelyconvergent when | z | > | z − z | > | arg z − arg z | < π , its sum is equalto h p ,p ,p ± ( z , z ; u, w , w , w (cid:48) ).Just as in the skew-symmetry case, we have the following immediate consequence: Corollary 6.2
The maps A + : V W W W → V φ g ( W (cid:48) ) W W (cid:48) and A − : V W W W → V W (cid:48) W φ g − ( W (cid:48) ) are linearisomorphisms. In particular, V W W W , V φ g ( W (cid:48) ) W W (cid:48) and V W (cid:48) W φ g − ( W (cid:48) ) are linearly isomorphic.Proof. It is clear that A + and A − are inverse of each other.The linear isomorphisms A + and A − are called the contragredient isomorphisms .31 eferences [Ch1] L. Chen, On axiomatic approaches to intertwining operator algebras, Comm.Contemp. Math. (2016), 1550051.[Ch2] L. Chen, An S -symmetry of the Jacobi identity for intertwining operator alge-bras, New York J. Math (2015), 657-698.[Co] J. H. Conway, A characterisation of Leechs lattice, Invent. Math. (1969), 137–142.[FFR] A. Feingold, I. Frenkel and J. Ries, Spinor construction of vertex operator alge-bras, triality, and E (1)8 , Contemporary Mathematics, Vol. 121, Amer. Math. Soc.,Providence, 1991.[Fi1] F. Fiordalisi, Logarithmic Intertwining Operators and Genus-One CorrelationFunctions, Ph.D. Thesis, Rutgers University, 2015.[Fi2] F. Fiordalisi, Logarithmic Intertwining Operators and Genus-One CorrelationFunctions, Comm. Contemp. Math. (2016), 1650026.[FLM] I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and theMonster, Pure and Applied Math.,
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Comm. Contemp. Math. (2016), 1650009. Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscat-away, NJ 08854-8019