Fusion products of twisted modules in permutation orbifolds
aa r X i v : . [ m a t h . QA ] J un Fusion products of twisted modules in permutation orbifolds
Chongying DongDepartment of Mathematics, University of California, Santa Cruz, CA 95064 USAHaisheng LiDepartment of Mathematical Sciences, Rutgers University, Camden, NJ 08102 USAFeng XuDepartment of Mathematics, University of California, Riverside, CA 92521 USANina YuSchool of Mathematical Sciences, Xiamen University, Fujian 361005 CHINAJuly 2, 2019
Abstract
Abstract: Let V be a vertex operator algebra, k a positive integer and σ a permutation automorphism of thevertex operator algebra V ⊗ k . In this paper, we determine the fusion product of any V ⊗ k -module with any σ -twisted V ⊗ k -module. This paper is about permutation orbifold vertex operator algebras. The theory of permutation orbifolds studies therepresentations of the tensor product vertex operator algebra V ⊗ k with the natural action of the symmetric group S k as an automorphism group, where V is a vertex operator algebra and k is a positive integer. The study of permuta-tion orbifolds was initiated in [BHS], where the twisted sectors (modules), genus one characters and their modulartransformations, as well as the fusion rules for cyclic permutations for affine Lie algebras and the Virasoro algebrawere studied. The genus one characters and modular transformation properties of permutation orbifolds for a generalrational conformal field theory were given in [Ba].The study on permutation orbifolds in the context of vertex operator algebras was started by Barron, Dong andMason in [BDM], where for a general vertex operator algebra V with any positive integer k , an intrinsic connectionbetween twisted modules for tensor product vertex operator algebra V ⊗ k with respect to permutation automorphismsand V -modules was found. More specifically, let σ be a k -cycle which is viewed as an automorphism of V ⊗ k .Then, for any V -module ( W, Y W ( · , z )) , a canonical σ -twisted V ⊗ k -module structure on W was obtained, whichwas denoted by T σ ( W ) . Furthermore, it was proved therein that this gives rise to an isomorphism of the categoriesof weak, admissible and ordinary V -modules and the categories of weak, admissible and ordinary σ -twisted V ⊗ k -modules, respectively. The C -cofiniteness of permutation orbifolds and general cyclic orbifolds was established laterin [A1, A2, M1, M2]. On the other hand, an equivalence of two constructions (see [FLM], [Le], [BDM]) of twistedmodules for permutation orbifolds of lattice vertex operator algebras was given in [BHL]. The permutation orbifolds1f the lattice vertex operator algebras with k = 2 and k = 3 were extensively studied in [DXY1, DXY2, DXY3]. Inparticular, the fusion rules for the permutation orbifolds were determined. On the other hand, a study on permutationorbifolds in the context of conformal nets was given in [KLX], where irreducible representations of the cyclic orbifoldwere determined and fusion rules were given for k = 2 .Note that intertwining operator and fusion rule for modules were introduced in [FHL], while a theory of tensorproducts for modules for vertex operator algebras was developed and has been extensively studied by Huang andLepowsky (see [HL1], [HL2], [H1], [H2]). Intertwining operators among three twisted modules were studied by Xu(see [X]) and a notion of tensor product for twisted modules was introduced in [DLM1].In this paper, we study fusion products of V ⊗ k -modules with σ -twisted V ⊗ k -modules for any σ ∈ S k . Morespecifically, by using the explicit construction of the σ -twisted V ⊗ k -modules due to [BDM], we completely determinethe fusion products. To achieve this goal, we study fusion products of twisted modules and obtain the associativity ofthe fusion product of twisted modules by using a theorem of [KO] and [CKM]. We also generalize and use a resultof [Li2] and [Li3] on an analogue of Hom-functor. More specifically, given any σ -twisted modules W and W fora vertex operator algebra V with a finite order automorphism σ , we introduce the space H ( W , W ) of generalizedintertwining operators from W to W and prove that H ( W , W ) has a natural weak V -module structure. Further-more, we prove that for any V -module W , the space Hom V ( W , H ( W , W )) is naturally isomorphic to the space ofintertwining operators of type (cid:0) W W W (cid:1) . As one of the main results, we show that for any V -modules M, N, W , thereis a canonical linear isomorphism from the space I V (cid:0) WM N (cid:1) of intertwining operators of the indicated type to the space I V ⊗ k (cid:0) T σ ( W ) M T σ ( N ) (cid:1) of intertwining operators, where M = M ⊗ V ⊗ ( k − is viewed as a V ⊗ k -module.Now, we describe the contents of this paper with more details. Let V be any vertex operator algebra and let k be apositive integer. Recall from [BDM] that for σ = (12 · · · k ) ∈ Aut ( V ⊗ k ) , every σ -twisted V ⊗ k -module is isomorphicto T σ ( W ) for some V -module W and T σ ( W ) is irreducible if and only if W is irreducible. On the other hand, from[FHL], irreducible V ⊗ k -modules are classified as M ⊗ · · · ⊗ M k , where M , . . . , M k are irreducible V -modules.Our first goal is to determine the fusion product ( M ⊗ · · · ⊗ M k ) ⊠ T σ ( W ) . We first consider the special case where M j = V for ≤ j ≤ k . While it is natural to start with a special case,actually the main motivation comes from the following observation ( M ⊗ · · · ⊗ M k ) ≃ ⊠ ki =1 ( V ⊗ ( i − ⊗ M i ⊗ V ⊗ ( k − i ) ) . (1.1)The following is our first main result: Theorem A.
Let V be any vertex operator algebra and set σ = (12 · · · k ) ∈ Aut ( V ⊗ k ) with k a positive integer. Let M and N be any V -modules such that a tensor product M ⊠ N exists. Then (cid:16) M ⊗ V ⊗ ( k − (cid:17) ⊠ V ⊗ k T σ ( N ) ≃ T σ ( M ⊠ N ) . To prove this theorem, we generalize and use a result of [Li2] and [Li3] on an analogue of the classical Hom-functor. Let V be any vertex operator algebra with a finite order automorphism τ . Let W and W be τ -twisted V -modules. We first introduce the space H ( W , W ) of what are called generalized intertwining operators from W to W , where the definition of a generalized intertwining operator φ ( x ) from W to W captures the essentialproperties of I ( w, x ) for an intertwining operator I ( · , x ) of type (cid:0) W W W (cid:1) with W a V -module and with w ∈ W . Morespecifically, in addition to the lower truncation condition, φ ( x ) satisfies the conditions that [ L ( − , φ ( x )] = ddx φ ( x ) and that for every v ∈ V , there exists a nonnegative integer n such that ( x − x ) n Y M ( v, x ) φ ( x ) = ( x − x ) n φ ( x ) Y M ( v, x ) . We prove that H ( W , W ) has a natural weak V -module structure and that for any V -module W , the linear spaceHom V ( W , H ( W , W )) is naturally isomorphic to the space of intertwining operators of type (cid:0) W W W (cid:1) . Coming backto vertex operator algebra V ⊗ k , let M, N, W be V -modules. By exploiting certain techniques from [BDM] we obtain2 linear isomorphism between I V ⊗ k (cid:0) T σ ( W ) M T σ ( N ) (cid:1) and I V (cid:0) WM N (cid:1) , where M = M ⊗ V ⊗ ( k − . Furthermore, we usethis natural isomorphism to obtain the fusion product isomorphism relation of Theorem A.As it was indicated before, the main idea is to prove Theorem A first and then use the relation (1.1) and theassociativity of the tensor product to deal with the general case. Using Theorem A and a general result (Lemma 6.4),we show (cid:16) V ⊗ ( r − ⊗ M ⊗ V ⊗ ( k − r ) (cid:17) ⊠ V ⊗ k T σ ( N ) ≃ (cid:16) M ⊗ V ⊗ ( k − (cid:17) ⊠ V ⊗ k T σ ( N ) ≃ T σ ( M ⊠ N ) for ≤ r ≤ k .On the other hand, to carry out this plan we shall need the associativity of the tensor product for twisted modules.Let V be a regular and self-dual vertex operator algebra of CFT type and let G be a finite abelian automorphism groupof V. For any g -twisted V -module M and h -twisted V -module N with g, h ∈ G , we establish that a tensor product of M and N exists and the tensor product functor is associative.This result is achieved by using a theorem of [KO] and [CKM] and a theorem of [H2], where our approach isessentially the same as that in [CKM]. First, it was proved in [CM, M2] that V G is also a regular and self-dual vertexoperator algebra of CFT type. Then by [H2], the V G -module category C V G is a modular tensor category. On theother hand, a fusion category Rep( V ) was introduced in [KO] and [CKM] as a subcategory of C V G . This in particularimplies that the tensor product functor for Rep( V ) is associative. It turns out that for any finite order automorphism g of V, a g -twisted V -module is an object in Rep( V ) . Then it is shown that the tensor product of two twisted modulesin the sense of [X] and [DLM1] exists and is equivalent to the tensor product in the category
Rep( V ) . Consequently,we obtain the existence and the associativity of the tensor product in the sense of [X] and [DLM1].The following is the second main theorem of this paper:
Theorem B.
Let V be a regular and self-dual vertex operator algebra of CFT type, and let M , ..., M k , N be V -modules. Let σ be a k -cycle permutation. Then ( M ⊗ · · · ⊗ M k ) ⊠ V ⊗ k T σ ( N ) ≃ T σ ( M ⊠ V · · · ⊠ V M k ⊠ V N ) . Now, let σ ∈ S k be an arbitrary permutation. A classical fact is that the conjugacy class of σ is uniquely determinedby a partition κ of k , i.e., κ = ( k , . . . , k s ) is a sequence of positive integers such that k ≥ k ≥ · · · ≥ k s , k + k + · · · + k s = k. Define σ κ = σ · · · σ s , where σ = (12 · · · k ) , σ = (( k + 1) · · · ( k + k )) , and so on. Then σ κ = µσµ − forsome permutation µ . It is known that the categories of σ -twisted modules and σ κ -twisted modules are isomorphiccanonically. Using [BDM] and [FHL], we show that irreducible σ κ -twisted V ⊗ k -modules are classified as T σ κ ( N , . . . , N s ) := T σ ( N ) ⊗ · · · ⊗ T σ s ( N s ) , (1.2)where N , . . . , N s are irreducible V -modules. Furthermore, irreducible σ -twisted V ⊗ k -modules are classified as T σ ( N , . . . , N s ) := ( T σ κ ( N , . . . , N s )) µ , (1.3)where for any σ κ -twisted V ⊗ k -module ( W, Y W ) , ( W µ , Y τW ) is a σ -twisted V ⊗ k -module with W µ = W as a vectorspace and Y µW ( a, x ) = Y W ( µ ( a ) , x ) for a ∈ V ⊗ k . The following is the third main result of this paper:
Theorem C.
Assume that V is a rational and C -cofinite vertex operator algebra of CFT type and k is a positiveinteger. Let σ ∈ S k ⊂ Aut ( V ⊗ k ) . Suppose µσµ − = σ κ , where κ = ( k , . . . , k s ) is a partition of k and µ ∈ S k . Let M , . . . , M k and N , . . . , N s be irreducible V -modules. Then ( M ⊗ · · · ⊗ M k ) ⊠ T σ κ ( N , N , . . . , N s ) ≃ T σ κ ( M [ k ] ⊠ N , . . . , M [ k s ] ⊠ N s ) (1.4)as a σ κ -twisted V ⊗ k -module, and ( M ⊗ · · · ⊗ M k ) µ ⊠ ( T σ ( N , N , . . . , N s )) ≃ T σ ( M [ k ] ⊠ N , . . . , M [ k s ] ⊠ N s ) (1.5)as a σ -twisted V ⊗ k -module, where M [ k ] = ⊠ k i =1 M i , M [ k ] = ⊠ k + k i = k +1 M i , and so on, are V -modules.3his paper is organized as follows. In Section 2, we review some basics on vertex operator algebras, we alsopresent the analytic function version of the twisted Jacobi Identity. In Section 3, we prove that for a regular, self-dualvertex operator algebra V of CFT type and a finite abelian automorphism group G of V , the tensor product M ⊠ V N forany g -twisted V -module M and h -twisted V -module N exists and is associative for g, h ∈ G. In Section 4, we presentthe weak V -module H ( M , M ) of generalized intertwining operators from a σ -twisted V -module M to another σ -twisted V -module M . In Section 5, for any V -modules M, N, W , we construct a linear isomorphism between thespaces I V (cid:0) WM N (cid:1) and I V ⊗ k (cid:0) T σ ( W ) M T σ ( N ) (cid:1) . Furthermore, we prove (cid:0) M ⊗ V ⊗ ( k − (cid:1) ⊠ T σ ( N ) = T σ ( M ⊠ V N ) . Finally,in Section 6, by applying associativity of the tensor product ⊠ in Section 3, we obtain the second main theorem of thispaper. Notations:
For this paper, we use Z + for the nonnegative integers. Let ( V, Y, , ω ) be a vertex operator algebra in the sense of [FLM] and [FHL] (cf. [Bo]). First, we recall the definitionsof an automorphism of V and a g -twisted V -module for a finite order automorphism g of V (see [FLM], [DLM1]). Definition 2.1. An automorphism of a vertex operator algebra V is a linear isomorphism g of V such that g ( ω ) = ω and gY ( v, z ) g − = Y ( gv, z ) for any v ∈ V . Denote by Aut ( V ) the group of all automorphisms of V .A straightforward consequence is that every automorphism g of V preserves the vacuum vector , i.e., g ( ) = .On the other hand, a standard fact is that for any subgroup G ≤ Aut ( V ) , the set of G -fixed points V G := { v ∈ V | g ( v ) = v for g ∈ G } is a vertex operator subalgebra.Let g be a finite order automorphism of V with a period T in the sense that T is a positive integer such that g T = 1 .Then V = ⊕ T − r =0 V r , (2.1)where V r = (cid:8) v ∈ V | gv = e πir/T v (cid:9) for r ∈ Z . Note that for r, s ∈ Z , V r = V s if r ≡ s (mod T ) . Definition 2.2. A weak g -twisted V -module is a vector space M with a linear map Y M ( · , z ) : V → ( End M ) [[ z /T , z − /T ]] v Y M ( v, z ) = X n ∈ T Z v n z − n − ( v n ∈ End M ) , which satisfies the following conditions: For all u ∈ V r , v ∈ V , w ∈ M with ≤ r ≤ T − , Y M ( u, z ) = X n ∈ rT + Z u n z − n − ,u n w = 0 for n sufficiently large ,Y M ( , z ) = Id M ,z − δ (cid:18) z − z z (cid:19) Y M ( u, z ) Y M ( v, z ) − z − δ (cid:18) z − z − z (cid:19) Y M ( v, z ) Y M ( u, z )= z − (cid:18) z − z z (cid:19) − rT δ (cid:18) z − z z (cid:19) Y M ( Y ( u, z ) v, z ) , (2.2)where δ ( z ) = P n ∈ Z z n . 4otice that for any weak g -twisted V -module ( M, Y M ) , from definition we have z rT Y M ( u, z ) w ∈ W (( z )) for u ∈ V r , r ∈ Z , w ∈ M. (2.3)On the other hand, we have (see [DL]) z − (cid:18) z − z z (cid:19) − rT δ (cid:18) z − z z (cid:19) = z − (cid:18) z + z z (cid:19) rT δ (cid:18) z − z z (cid:19) . Remark 2.3.
It is well known that the twisted Jacobi identity (2.2) is equivalent to the following associativity relation ( z + z ) l + rT Y M ( u, z + z ) Y M ( v, z ) w = ( z + z ) l + rT Y M ( Y ( u, z ) v, z ) w (2.4)for w ∈ M , where l is a nonnegative integer such that z l + rT Y M ( u, z ) w involves only nonnegative integral powers of z , and commutator relation [ Y M ( u, z ) , Y M ( v, z )]= Res z z − (cid:18) z − z z (cid:19) − rT δ (cid:18) z − z z (cid:19) Y M ( Y ( u, z ) v, z ) . (2.5)From [DLMi] (cf. Definition 2.7 and Lemma 2.9 of [LTW]), (2.4) can be equivalently replaced with the propertythat for u, v ∈ V , there exists a nonnegative integer m such that ( z − z ) m Y M ( u, z ) Y M ( v, z ) ∈ Hom (cid:16)
M, M (( z T , z T )) (cid:17) and z m Y M ( Y ( u, z ) v, z ) = ( z − z ) m Y M ( u, z ) Y M ( v, z ) | z T =( z + z ) T . Definition 2.4. A g - twisted V -module is a weak g -twisted V -module M which carries a C -grading induced by thespectrum of L (0) where L (0) is the component operator of Y ( ω, z ) = P n ∈ Z L ( n ) z − n − . That is, we have M = L λ ∈ C M λ , where M λ = { w ∈ M | L (0) w = λw } . Moreover, it is required that dim M λ < ∞ for all λ and for anyfixed λ , M nT + λ = 0 for all small enough integers n. Definition 2.5. An admissible g -twisted V -module is a weak g -twisted module with a T Z + -grading M = ⊕ n ∈ T Z + M ( n ) such that u m M ( n ) ⊂ M ( wt u − m − n ) for homogeneous u ∈ V and m, n ∈ T Z . Note that if M = ⊕ n ∈ T Z + M ( n ) is an irreducible admissible g -twisted V -module, then there is a complex number λ M such that L (0) | M ( n ) = λ M + n for all n. As a convention, we assume M (0) = 0 , and λ M is called the weight orconformal weight of M. In case g = 1 , we recover the notions of weak, ordinary and admissible V -modules (see [DLM2]). Definition 2.6.
A vertex operator algebra V is said to be g -rational if the admissible g -twisted module category issemisimple. In particular, V is said to be rational if V is -rational.It was proved in [DLM2] that if V is a g -rational vertex operator algebra, then there are only finitely many ir-reducible admissible g -twisted V -modules up to isomorphism and any irreducible admissible g -twisted V -module isordinary. Definition 2.7.
A vertex operator algebra V is said to be regular if every weak V -module M is a direct sum ofirreducible ordinary V -modules. Definition 2.8.
A vertex operator algebra V is said to be C -cofinite if V /C ( V ) is finite dimensional, where C ( V ) =span { u − v | u, v ∈ V } . emark 2.9. We here recall some known results we shall use extensively.(1) It was proved in [DLM2, Li4] that if V is a C -cofinite vertex operator algebra, then V has only finitely manyirreducible admissible modules up to isomorphism.(2) Assume that V is rational and C -cofinite. It was proved in [ADJR] that V is g -rational for any finite automor-phism g , and on the other hand, it was proved in [DLM3] that λ M is a rational number for every irreducible g -twisted V -module M .A vertex operator algebra V = ⊕ n ∈ Z V n is said to be of CFT type if V n = 0 for all negative integers n and V = C .We shall need the following result from [CM, M2]: Theorem 2.10.
Assume that V is a regular and self-dual vertex operator algebra of CFT type. Then for any solvablesubgroup G of Aut( V ) , V G is a regular and self-dual vertex operator algebra of CFT type. Definition 2.11.
Let M = L n ∈ T Z + M ( n ) be an admissible g -twisted V -module. Set M ′ = M n ∈ T Z + M ( n ) ∗ , the restricted dual , where M ( n ) ∗ = Hom C ( M ( n ) , C ) . For v ∈ V , define a vertex operator Y M ′ ( v, z ) on M ′ via h Y M ′ ( v, z ) f, u i = h f, Y M ( e zL (1) ( − z − ) L (0) v, z − ) u i , where h f, w i = f ( w ) is the natural paring M ′ × M → C . On the other hand, if M = ⊕ λ ∈ C M λ is a g -twisted V -module, we define M ′ = ⊕ λ ∈ C M ∗ λ and define Y M ′ ( v, z ) for v ∈ V in the same way.The same argument of [FHL] gives: Proposition 2.12. If ( M, Y M ) is an admissible g -twisted V -module, then ( M ′ , Y M ′ ) carries the structure of an ad-missible g − -twisted V -module. On the other hand, if ( M, Y M ) is a g -twisted V -module, then ( M ′ , Y M ′ ) carries thestructure of a g − -twisted V -module. Moreover, M is irreducible if and only if M ′ is irreducible.Let g , g , g be mutually commuting automorphisms of V of periods T , T , T , respectively. In this case, V decomposes into the direct sum of common eigenspaces for g and g : V = M ≤ j 7→ Y ( w, z ) = X n ∈ C w n z − n − such that for any w ∈ M , w ∈ M and for any fixed c ∈ C , w c + n w = 0 for n ∈ Q sufficiently large , − (cid:18) z − z z (cid:19) j /T δ (cid:18) z − z z (cid:19) Y M ( u, z ) Y ( w, z ) − z − (cid:18) z − z − z (cid:19) j /T δ (cid:18) z − z − z (cid:19) Y ( w, z ) Y M ( u, z )= z − (cid:18) z − z z (cid:19) − j /T δ (cid:18) z − z z (cid:19) Y ( Y M ( u, z ) w, z ) (2.7)on M for u ∈ V ( j ,j ) with j , j ∈ Z and w ∈ M , and ddz Y ( w, z ) = Y ( L ( − w, z ) . All intertwining operators of type (cid:0) W M M (cid:1) form a vector space, which we denote by I V (cid:0) M M M (cid:1) . Set N M M M =dim I V (cid:0) M M M (cid:1) , which is called the fusion rule . Remark 2.14. As it was proved in [X], if there are weak g s -twisted modules ( M s , Y M s ) for s = 1 , , such that N M M M > , then g = g g . In view of this, we shall always assume that g = g g .By the same arguments in Chapter 7 of [DL] for formulas (7.24) and (7.11) we have: Proposition 2.15. The twisted Jacobi identity (2.7) is equivalent to the generalized commutativity: For u ∈ V ( j ,j ) and w ∈ M , there is a positive integer n such that ( z − z ) n + j T Y M ( u, z ) Y ( w, z ) = ( − z + z ) n + j T Y ( w, z ) Y M ( u, z ) for u ∈ V ( j ,j ) , w ∈ M and generalized associativity: For u ∈ V ( j ,j ) , w ∈ M and w ∈ M , there is a positiveinteger n depending on u and w only such that ( z + z ) n + j T Y M ( u, z ) Y ( w, z ) w = ( z + z ) n + j T Y ( Y M ( u, z ) w, z ) w . Let V and V be vertex operator algebras and let σ , σ be finite order automorphisms of V , V , respectively. Let W be a V -module, W , W σ -twisted V -modules, and let N be a V -module, N , N σ -twisted V -modules. Notethat W ⊗ N is a V ⊗ V -module, while W ⊗ N and W ⊗ N are σ -twisted V ⊗ V -modules with σ = σ ⊗ σ . Forany intertwining operator Y ( · , z ) of type (cid:0) W W W (cid:1) and for any intertwining operator Y ( · , z ) of type (cid:0) N N N (cid:1) , definea linear map ( Y ⊗ Y )( · , z ) : W ⊗ N → ( Hom ( W ⊗ N , W ⊗ N )) { z } by ( Y ⊗ Y )( w ⊗ n , z )( w ⊗ n ) = Y ( w , z ) w ⊗ Y ( n , z ) n (2.8)for w ∈ W , w ∈ W , n ∈ N , n ∈ N . As in [ADL], by using the generalized commutativity and associativity,it is straightforward to show that ( Y ⊗ Y )( · , z ) is an intertwining operator of type (cid:0) W ⊗ N W ⊗ N W ⊗ N (cid:1) . Then we have alinear map ψ : I V (cid:18) W W W (cid:19) ⊗ I V (cid:18) N N N (cid:19) → I V ⊗ V (cid:18) W ⊗ N W ⊗ N W ⊗ N (cid:19) ( Y , Y ) 7→ Y ⊗ Y . (2.9)The same arguments of [ADL] in proving Theorem 2.10 (with minor necessary modifications) give: Theorem 2.16. Let V , V , σ , σ , W , W , W , N , N , N be given as above. Then the linear map ψ is one-to-one.Furthermore, if either dim I V (cid:18) W W W (cid:19) < ∞ , or dim I V (cid:18) N N N (cid:19) < ∞ , then ψ is a linear isomorphism. 7et D = C \ { ( z, z ) , ( z, , (0 , z ) | z ∈ C } . Then we have: Proposition 2.17. Let V be rational and C -cofinite, and let G ⊂ Aut( V ) be a finite abelian group. Then in thepresence of other conditions in Definition 2.13, the Jacobi identity (2.7) is equivalent to the following properties for g , g ∈ G and v ∈ V ( j ,j ) , w ∈ M , w ∈ M , w ′ ∈ M ′ :(a) The formal series h w ′ , Y M ( v, z ) Y ( w, z ) w i ( z − z ) j /T z j /T converges to a unique multi-valued analytic function f ( z , z ) = P i h i ( z ,z ) z ri z r z s ( z − z ) t for some h i ( z , z ) ∈ C [ z , z ] ,r i ∈ Q and r, s, t ∈ Z in the domain | z | > | z | > . (b) The formal series h w ′ , Y ( w, z ) Y M ( v, z ) w i ( − z + z ) j /T z j /T converges to f ( z , z ) in the domain | z | > | z | > . (c) The formal series h w ′ , Y ( Y M ( v, z − z )( w, z ) w i ( z − z ) j /T z j /T converges to f ( z , z ) in the domain | z | > | z − z | > . Proof. Since V G is rational by Theorem 2.10, from [DLM3] each M i is a direct sum of finitely many irreducible V G -modules whose weights are rational. Without loss we can assume that w, w , w ′ are vectors in some irre-ducible V G -modules. Then there exists a rational number r such that the powers of z in the formal power series h w ′ , Y M ( v, z ) Y ( w, z ) w i lie in r + Z . The same is true for the corresponding terms in (b) and (c). Now resultsfollow from Proposition 2.15 and the proofs of Propositions 7.6, 7.8 and 7.9 of [DL].Now we recall a notion of tensor product (see [HL1], [Li3], [DLM1]). Definition 2.18. Let g , g be commuting finite order automorphisms of a vertex operator algebra V and let M i be a g i -twisted V -module for i = 1 , . A tensor product for the ordered pair ( M , M ) is a pair ( M, F ( · , z )) where M is aweak g g -twisted V -module and F ( · , z ) is an intertwining operator of type (cid:0) MM M (cid:1) such that the following universalproperty holds: For any weak g g -twisted V -module W and for any intertwining operator I ( · , z ) of type (cid:0) WM M (cid:1) , there exists a unique weak twisted V -module homomorphism ψ from M to W such that I ( · , z ) = ψ ◦ F ( · , z ) . We shall use M ⊠ V M to denote a generic tensor product module of M and M , provided that the existence isverified. Remark 2.19. We here recall Huang-Lepowsky’s tautological construction of a tensor product (see [HL1]). Supposethat V is a vertex operator algebra with a finite order automorphism τ such that V is τ -rational, namely, every τ -twisted V -module is completely reducible. Let W be a V -module and T a τ -twisted V -module such that dim I V (cid:0) PW T (cid:1) < ∞ for every irreducible τ -twisted V -module P . Let { P α | α ∈ A} be a complete set of equivalence class representativesof irreducible τ -twisted V -modules. Set M = ⊕ α ∈A I (cid:18) P α W T (cid:19) ∗ ⊗ P α = ⊕ α ∈A Hom (cid:18) I (cid:18) P α W T (cid:19) , P α (cid:19) , (2.10)a τ -twisted V -module. For α ∈ A , define Y α ( w, z ) t ∈ Hom (cid:16) I (cid:0) P α W T (cid:1) , P α (cid:17) { z } by ( Y α ( w, z ) t )( I ) = I ( w, z ) t for I ∈ I (cid:18) P α W T (cid:19) . Set Y ( · , z ) = P α ∈A Y α ( · , z ) . Then ( M, Y ) is a tensor product of W and T .The following is a simple property of fusion rules (numbers) (see [FHL], [HL1], [DLM1]: Lemma 2.20. Let g , g be commuting finite order automorphisms of a vertex operator algebra V and let M i be a g i -twisted V -module for i = 1 , , with g = g g . Then N M M M = N M M M . Tensor products of twisted modules Throughout this section, we assume that V is a regular and self-dual vertex operator algebra of CFT type and G isa finite abelian automorphism group of V . The main goal of this section is to prove that for any g, h ∈ G and forany g -twisted V -module M and h -twisted V -module N, a tensor product of M and N exists and associativity holds.These results will be used later to determine the tensor product of an untwisted module with a twisted module for thestudy on permutation orbifolds.Denote by C V the category of ordinary V -modules and by C V G the category of ordinary V G -modules. From [H2],both C V and C V G are modular tensor categories. Furthermore, from [KO] and [CKM], V is a commutative associativealgebra in C V G as V = ⊕ χ ∈ irr( G ) V χ , where irr( G ) is the set of irreducible characters of G and V χ are irreducible V G -modules (cf. [DJX]), i.e., simple objects in C V G . Recall the following definition from [KO] and [CKM] (see Proposition 3.46 of [CKM]): Definition 3.1. Denote by Rep( V ) the subcategory of C V G consisting of every V G -module W together with a V G -intertwining operator Y W ( · , z ) of type (cid:0) WV W (cid:1) such that the following conditions are satisfied:1. (Associativity) For any u, v ∈ V, w ∈ W and w ′ ∈ W ′ , the formal series h w ′ , Y W ( u, z ) Y W ( v, z ) w i and h w ′ , Y W ( Y ( u, z − z ) v, z )) w i converge on the domains | z | > | z | > and | z | > | z − z | > , respectively, to multivalued analytic functionswhich coincide on their common domain.2. (Unit) Y W ( , z ) = Id W . Recall from [KO] (see also [CKM]) that there is a categorical tensor product functor ⊠ V in the category of Rep( V ) ,which is associative. We denote by M ⊠ V N the tensor product of M and N in Rep( V ) . Then for any two V G -modules M, N in Rep( V ) , M ⊠ V N is a quotient of M ⊠ V G N. Moreover, Rep( V ) is fusion category. In particular, Rep( V ) is a semisimple category with finitely many inequivalent simple objects.From the generalized associativity for a twisted module (see Proposition 2.15), we immediately have: Lemma 3.2. If W is a g -twisted V -module with g ∈ G , then W is an object of Rep( V ) . Furthermore, if W i is a g i -twisted V -module with g i ∈ G for i = 1 , , then W and W are equivalent objects in Rep( V ) if and only if g = g and W ≃ W as twisted V -modules.On the other hand, we have: Lemma 3.3. If W is a simple object in Rep( V ) , then W is an irreducible g -twisted V -module for some g ∈ G. Proof. We need to use the Frobenius-Perron dimension in a fusion category (see [BK], [ENO], [DMNO]). Let C bea fusion category and let K ( C ) denote the Grothendieck ring of C . According to [ENO], there exists a unique ringhomomorphism FPdim C : K ( C ) → R such that FPdim C ( X ) > for all nonzero X ∈ C , where FPdim C ( X ) is called the Frobenius-Perron dimension of X ∈ C . Furthermore, one has the Frobenius-Perron dimension for thecategory C : FPdim ( C ) = X X ∈O ( C ) FPdim C ( X ) , where O ( C ) denotes the equivalence classes of the simple objects in C . Note that both C V G and Rep( V ) are fusioncategories. It follows from [DMNO] and [ENO] that (FPdim C V G V )(FPdim (Rep( V ))) = FPdim ( C V G ) . (3.1)We also need quantum dimensions and global dimensions from [DJX] and [DRX]. Let M be a g -twisted V -modulewith g ∈ G and set ch q M = tr | M q L (0) − c/ . Then ch q M converges to a holomorphic function χ M ( τ ) on the upperhalf plane with q = e πiτ (see [Z], [DLM3]). The quantum dimension of M over V is defined as qdim V M = lim y → χ M ( iy ) χ V ( iy ) . qdim V M = FPdim Rep( V ) M, and qdim V G W = FPdim C V G W for any V G -module W . Denote the equivalence classes of irreducible g -twisted V -modules by M ( g ) . The globaldimension of V is defined as glob( V ) = X M ∈M (1) (qdim V M ) . It follows that glob( V G ) = FPdim ( C V G ) . Recall from [DJX] and [DRX] that qdim V G V = o ( G ) , glob( V G ) = o ( G ) glob( V ) , glob( V ) = X M ∈ M( g ) (qdim V M ) for any g ∈ G. Then we have FPdim C V G V = qdim V G V = o ( G ) , FPdim ( C V G ) = glob( V G ) = o ( G ) glob( V ) . Combining these relations with equation (3.1) we obtain FPdim (Rep( V )) = o ( G )glob( V ) . Since for any g ∈ G , every irreducible g -twisted module is a simple object in Rep( V ) by Lemma 3.2, we get FPdim Rep( V ) ≥ X g ∈ G X M ∈ M( g ) (qdim V M ) = o ( G )glob( V ) = FPdim Rep( V ) . Thus FPdim Rep( V ) = X g ∈ G X M ∈ M( g ) (qdim V M )) , which implies that every simple object in Rep( V ) is an irreducible g -twisted V -module for some g ∈ G. Remark 3.4. Note that Lemma 3.3 holds for any (not necessarily abelian) finite automorphism group G as long as V G is rational, C -cofinite.Let g, h ∈ G . For any g -twisted module M and h -twisted module N , as they are objects in Rep( V ) by Lemma3.2, M ⊠ V N exists in Rep( V ) . We will show next that the tensor product in the sense of Definition 2.18 exists and isisomorphic to M ⊠ V N. We first prove that a categorical intertwining operator is the same as an intertwining operatorin the sense of Definition 2.13.The following is a generalization of Theorem 3.53 in [CKM] where g i = 1 for i = 1 , , with a similar proof: Theorem 3.5. Let g , g , g be commuting finite order automorphisms of V and let M i be a g i -twisted V -module for i = 1 , , . Assume Y ( · , z ) · : M ⊗ M −→ M { z } w ⊗ w ( w , z ) w = X n ∈ C ( w ) n w z − n − is a bilinear map satisfying the following conditions:1. Lower truncation: For any w ∈ M , w ∈ M and α ∈ C , ( w ) α + m w = 0 for sufficiently large m ∈ N .10. The L ( − -derivative formula : [ L ( − , Y ( w , z )] = ddx Y ( w , z ) = Y ( L ( − w , z ) for w ∈ M .3. The L (0) -bracket formula: [ L (0) , Y ( w , z )] = z ddz Y ( w , z ) + Y ( L (0) w , z ) for w ∈ M .4. For any w ′ ∈ M ′ , w ∈ M , v ∈ V , w ∈ M , the following series define multivalued analytic functions onthe indicated domains, which coincide on the respective intersections of their domains: h w ′ , Y M ( a, z ) Y ( w , z ) w i on | z | > | z | > , (3.2) h w ′ , Y ( Y M ( a, z − z ) w , z ) w i on | z | > | z − z | > , (3.3) h w ′ , Y ( w , z ) Y M ( a, z ) w i on | z | > | z | > . (3.4)Specifically, the principal branches (that is, z = e log z ) of the first and second multivalued functions yieldsingle-valued functions which coincide on the simply connected domain S := { ( z , z ) ∈ C | Re z > Re z > Re( z − z ) > , Im z > Im z > Im( z − z ) > } and the principal branches of the second and third multivalued functions yield single-valued functions whichcoincide on the simply connected domain S := { ( z , z ) ∈ C | Re z > Re z > Re( z − z ) > , Im z > Im z > Im( z − z ) > } . 5. There exists a multivalued analytic function g defined on { ( z , z ) ∈ C | z , z , z − z = 0 } that restricts tothe three multivalued functions above on their respective domains.Then Y is an intertwining operator of type (cid:0) M M M (cid:1) in the sense of Definition 2.13. Proof. The proof here is a slight modification of that of Theorem 3.53 in [CKM]. First, fix z ∈ C × with Re z , Im z > . Then f z ( z ) := g ( z , e log z ) z j /T ( z − z ) j /T yields a possibly multivalued analytic function of z definedon { z ∈ C × | z = z } . However, note that f z has single-valued restrictions to each of { z ∈ C × | | z | > | z |} , { z ∈ C × | | z | > | z − z | > } and { z ∈ C × | | z | > | z |} , and these restrictions coincide on certain simply-connected subsets of the intersections of these domains. Thus these restrictions define a single-valued analytic functionon the union of these domains. This implies that f z has a single-valued restriction e f z to C × \ { z } with possiblepoles at , z , ∞ . Thus e f z is a rational function. It follows from Proposition 2.17 that Y satisfies the Jacobi identityand hence it is an intertwining operator of type (cid:0) M M M (cid:1) in the sense of Definition 2.13. Theorem 3.6. Let g , g , g ∈ G and let M i be g i -twisted V -modules for i = 1 , , . Then the space of intertwiningoperators of type (cid:0) M M M (cid:1) in the sense of Definition 2.13 is linearly isomorphic to the space of the categorical Rep( V ) intertwining operators of type (cid:0) M M M (cid:1) . Furthermore, a tensor product module M ⊠ V M exists and and M ⊠ V M and M ⊠ V M are isomorphic g g -twisted V -modules. Proof. By Theorem 3.44 of [CKM], the space of the categorical Rep( V ) intertwining operators of type (cid:0) M M M (cid:1) isisomorphic to the space of bilinear maps Y ( · , z ) · : M ⊗ M −→ M { z } w ⊗ w ( w , z ) w = X n ∈ C ( w ) n w z − n − satisfying the conditions 1-5 in Theorem 3.5. By Proposition 2.17 and Theorem 3.5, the latter space is the same as thespace of intertwining operators of type (cid:0) M M M (cid:1) in the sense of Definition 2.13. It follows from Remark 2.14 that thespace of the categorical Rep( V ) intertwining operators of type (cid:0) M M M (cid:1) is zero if g g = g . So we assume g g = g for the rest of the proof. 11et { M , . . . , M r } be a complete set of equivalence class representatives of irreducible g -twisted V -modules.From Lemma 3.2, M , . . . , M r are inequivalent simple objects in Rep( V ) . From the first part, the fusion rules N M i M ,M of the indicated types in the sense of Definition 2.13 and in Rep( V ) are the same. As M ⊠ V M is anobject in Rep( V ) (consisting of some V G -modules), the fusion numbers N M i M ,M are finite. For ≤ i ≤ r , set N i = N M i M ,M for short. Set W = ⊕ ri =1 ( C N i ⊗ M i ) , which is a g -twisted V -module and hence an object in Rep( V ) . It follows that there is an isomorphism in the category Rep( V ) from M ⊠ V M to W .For ≤ i ≤ r , let Y i , ..., Y iN i be a basis of I V (cid:0) M i M M (cid:1) . Then we have intertwining operators e s ⊗ Y ij for ≤ i ≤ r, ≤ s, j ≤ N i of type (cid:0) WM M (cid:1) , where e , . . . , e N i are the standard basis vectors of C N i . Set F ( · , z ) = r X i =1 N i X s,j =1 e s ⊗ Y ij ( · , z ) . (3.5)We now prove that ( W, F ) is a tensor product of M and M in the sense of Definition 2.18. Let M be any g -twisted V -module and I ∈ I V (cid:0) MM M (cid:1) . We need to show that there is a unique V -homomorphism ψ : W → M such that I = ψ ◦ Y . Since V is g -rational, M is isomorphic to a direct sum of M i for ≤ i ≤ r . Thus M = Hom V ( M i , M ) ⊗ M i .It can be readily seen that I V (cid:18) MM M (cid:19) = ⊕ ri =1 Hom V ( M i , M ) ⊗ I V (cid:18) M i M M (cid:19) . Then I = P ri =1 P N i j =1 ψ i,j ⊗ Y ij where ψ i,j ∈ Hom V ( M i , M ) . For ≤ i ≤ r, ≤ s ≤ N i , let p i,s be theprojection of M onto e s ⊗ M i , a V -homomorphism, and let π i,s be the identification map e s ⊗ M i ≃ M i . Set ψ = P ri =1 P N i j,s =1 ψ i,j π i,s p i,s , a V -homomorphism from W to M . It is clear that I = ψ ◦ F. Thus ( W, F ) is a tensorproduct of M and M in the sense of Definition 2.18. Consequently, we have M ⊠ V M ≃ M ⊠ V M . Since the tensor product functor ⊠ V is associative, we have: Corollary 3.7. Let M i be a g i -twisted V -module with g i ∈ G for i = 1 , , . Then ( M ⊠ V M ) ⊠ V M ≃ M ⊠ V ( M ⊠ V M ) . V -module H ( M , M ) of generalized intertwining operators Let τ ∈ Aut ( V ) of period T and let M and M be weak τ -twisted V -modules. In this section, we introduce aspace H ( M , M ) which consists of what we call generalized intertwining operators from M to M and we define avertex operator map Y H ( · , z ) . We prove that ( H ( M , M ) , Y H ( · , z )) carries the structure of a weak V -module. Thenwe prove for any weak V -module M , giving an intertwining operator Y of type (cid:0) M M M (cid:1) is equivalent to giving a V -homomorphism from M to H ( M , M ) .Let V be a vertex operator algebra and let τ ∈ Aut ( V ) of period T . In this section, we first prove that for any τ -twisted V -modules M and M , there is a weak module structure on the space H ( M , M ) of generalized inter-twining operators. Furthermore, we prove that for any weak V -module M , the space Hom V ( M , H ( M , M )) of V -homomorphisms from M to H ( M , M ) is canonically isomorphic to I V (cid:0) M M M (cid:1) . (Hence giving a V -homomorphismfrom M to H ( M , M ) is equivalent to giving an intertwining operator of type (cid:0) M M M (cid:1) .)Let V be a vertex operator algebra and let τ be a finite order automorphism of V of period T , which are all fixedthroughout this subsection. As before, for r ∈ Z set V r = { v ∈ V | τ v = η rT v } , η T = e π √− /T , the principal primitive T -th root of unity. Definition 4.1. Let ( M , Y M ) and ( M , Y M ) be weak τ -twisted V -modules. A generalized intertwining operator from M to M is an element φ ( x ) = P α ∈ C φ α x − α − ∈ ( Hom ( M , M )) { x } satisfying the following conditions:(G1) For any α ∈ C , w ∈ M , φ α + n w = 0 for n ∈ Z sufficiently large;(G2) [ L ( − , φ ( x )] = ddx φ ( x ) ;(G3) For any v ∈ V , there exists a nonnegative integer k such that ( x − x ) k Y M ( v, x ) φ ( x ) = ( x − x ) k φ ( x ) Y M ( v, x ) . (4.1)Denote by H ( M , M ) the space of all generalized intertwining operators from M to M .Let φ ( x ) ∈ H ( M , M ) and let u ∈ V r with ≤ r < T . For each n ∈ Z , we define u H n φ ( x )= Res x X j ≥ (cid:18) − rT j (cid:19) x − rT − j x rT (cid:0) ( x − x ) n + j Y M ( u, x ) φ ( x ) − ( − x + x ) n + j φ ( x ) Y M ( u, x ) (cid:1) , (4.2)an element of ( Hom ( M , M )) { x } . Remark 4.2. Here are some explanations about this definition. First, recall that for any weak τ -twisted V -module ( W, Y W ) , we have z rT Y W ( u, z ) w ∈ W (( z )) for u ∈ V r , r ∈ Z , w ∈ W. In view of this, the sum in (4.2) involves only integer powers of x , so that applying Res x to the sum makes sense.Second, due to the weak commutativity (4.1), we see that for any n ∈ Z , the sum contains only finitely many nonzeroterms, and u H n φ ( x ) = 0 for n sufficiently large. Third, as φ ( x ) satisfies the condition (G1), it can be readily seen that u H n φ ( x ) also satisfies the condition (G1).Now, set Y H ( u, z ) φ ( x ) = X n ∈ Z u H n φ ( x ) z − n − . (4.3)In terms of this generating function, we have Y H ( u, z ) φ ( x )= Res x (cid:18) x + zx (cid:19) − rT (cid:18) z − δ (cid:18) x − xz (cid:19) Y M ( u, x ) φ ( x ) − z − δ (cid:18) x − x − z (cid:19) φ ( x ) Y M ( u, x ) (cid:19) . (4.4)Using linearity, we obtain a vertex operator map Y H ( · , z ) : V → ( End (Hom( M , M )) { x } ) [[ z, z − ]] u Y H ( u, z ) . (4.5)First, we present some technical results. Lemma 4.3. Let v ∈ V r with r ∈ Z and let φ ( x ) ∈ H ( M , M ) . Then z k ( x + z ) rT Y H ( u, z ) φ ( x ) = (cid:16) ( x − x ) k x rT Y M ( u, x ) φ ( x ) (cid:17) | x = x + z , (4.6)where k is any nonnegative integer such that (4.1) holds. Furthermore, we have x − δ (cid:18) x + zx (cid:19) (cid:18) x + zx (cid:19) rT Y H ( u, z ) φ ( x )= z − δ (cid:18) x − xz (cid:19) Y M ( u, x ) φ ( x ) − z − δ (cid:18) x − x − z (cid:19) φ ( x ) Y M ( u, x ) . (4.7)13 roof. Write r = qT + r for some q, r ∈ Z with ≤ r < T . As Y H ( u, z ) φ ( x ) involves only finitely many negativepowers of z , we can multiply both sides of (4.4) by ( x + z ) r T and by doing so we get z k ( x + z ) r T Y H ( u, z ) φ ( x )= Res x z k x r T (cid:18) z − δ (cid:18) x − xz (cid:19) Y M ( u, x ) φ ( x ) − z − δ (cid:18) x − x − z (cid:19) φ ( x ) Y M ( u, x ) (cid:19) = Res x z − δ (cid:18) x − xz (cid:19) (cid:16) ( x − x ) k x r T Y M ( u, x ) φ ( x ) (cid:17) − Res x z − δ (cid:18) x − x − z (cid:19) (cid:16) ( x − x ) k x r T φ ( x ) Y M ( u, x ) (cid:17) = Res x x − δ (cid:18) x + zx (cid:19) (cid:16) ( x − x ) k x r T Y M ( u, x ) φ ( x ) (cid:17) . Then z k ( x + z ) rT Y H ( u, z ) φ ( x )= z k ( x + z ) q ( x + z ) r T Y H ( u, z ) φ ( x )= Res x x − δ (cid:18) x + zx (cid:19) ( x + z ) q (cid:16) ( x − x ) k x r T Y M ( u, x ) φ ( x ) (cid:17) = Res x x − δ (cid:18) x + zx (cid:19) x q (cid:16) ( x − x ) k x r T Y M ( u, x ) φ ( x ) (cid:17) = Res x x − δ (cid:18) x + zx (cid:19) (cid:16) ( x − x ) k x rT Y M ( u, x ) φ ( x ) (cid:17) = (cid:16) ( x − x ) k x rT Y M ( u, x ) φ ( x ) (cid:17) | x = x + z . This proves the first assertion. Furthermore, we have z k x rT (cid:18) z − δ (cid:18) x − xz (cid:19) Y M ( u, x ) φ ( x ) − z − δ (cid:18) x − x − z (cid:19) φ ( x ) Y M ( u, x ) (cid:19) = z − δ (cid:18) x − xz (cid:19) (cid:16) x rT ( x − x ) k Y M ( u, x ) φ ( x ) (cid:17) − z − δ (cid:18) x − x − z (cid:19) (cid:16) x rT ( x − x ) k φ ( x ) Y M ( u, x ) (cid:17) = x − δ (cid:18) x + zx (cid:19) (cid:16) ( x − x ) k x rT Y M ( u, x ) φ ( x ) (cid:17) = x − δ (cid:18) x + zx (cid:19) (cid:16) ( x − x ) k x rT Y M ( u, x ) φ ( x ) (cid:17) | x = x + z = x − δ (cid:18) x + zx (cid:19) z k ( x + z ) rT Y H ( u, z ) φ ( x ) , which is equivalent to (4.7).Note that for any u ∈ V , we have u = u + u + · · · + u T − , where for ≤ r ≤ T − , u r = 1 T T − X j =0 η − jrT τ j ( u ) ∈ V r . Using this and Lemma 4.3, we immediately get the the following generalization: Proposition 4.4. Let v ∈ V and let φ ( x ) ∈ H ( M , M ) . Then z k Y H ( u, z ) φ ( x ) = (cid:0) ( x − x ) k Y M ( u, x ) φ ( x ) (cid:1) | x /T =( x + z ) /T , (4.8)14here k is any nonnegative integer such that (4.1) holds. Furthermore, we have T − X j =0 x − δ η − jT (cid:18) x + zx (cid:19) T ! Y H ( τ j u, z ) φ ( x )= z − δ (cid:18) x − xz (cid:19) Y M ( u, x ) φ ( x ) − z − δ (cid:18) x − x − z (cid:19) φ ( x ) Y M ( u, x ) . (4.9)We also have the following generalization of the first part of Lemma 4.3 (cf. [Li5], Lemma 2.15): Lemma 4.5. Let v ∈ V and let φ ( x ) , . . . , φ p ( x ) ∈ H ( M , M ) . If p X j =1 f j ( x − x ) Y M ( v, x ) φ j ( x ) = p X j =1 f j ( x − x ) φ j ( x ) Y M ( v, x ) , where f ( x ) , . . . , f p ( x ) ∈ C [ x ] , then p X j =1 f j ( x − x ) Y M ( v, x ) φ j ( x ) | x /T =( x + z ) /T = p X j =1 f j ( z ) Y H ( v, z ) φ j ( x ) . (4.10)On the other hand, let v , . . . , v p ∈ V and let φ ( x ) ∈ H ( M , M ) . If p X j =1 f j ( x − x ) Y M ( v j , x ) φ ( x ) = p X j =1 f j ( x − x ) φ ( x ) Y M ( v j , x ) , where f ( x ) , . . . , f p ( x ) ∈ C [ x ] , then p X j =1 f j ( x − x ) Y M ( v j , x ) φ ( x ) | x /T =( x + z ) /T = p X j =1 f j ( z ) Y H ( v j , z ) φ ( x ) . (4.11) Proof. As φ j ( x ) ∈ H ( M , M ) , there is a nonnegative integer k such that ( x − x ) k Y M ( v, x ) φ j ( x ) = ( x − x ) k φ j ( x ) Y M ( v, x ) for all ≤ j ≤ p . Then ( x − x ) k f j ( x − x ) Y M ( v, x ) φ j ( x ) = ( x − x ) k f j ( x − x ) φ j ( x ) Y M ( v, x ) . In view of Lemma 4.3, for ≤ j ≤ p , we have (cid:0) ( x − x ) k Y M ( v, x ) φ j ( x ) (cid:1) | x /T =( x + z ) /T = z k Y H ( v, z ) φ j ( x ) , so that (cid:0) ( x − x ) k f j ( x − x ) Y M ( v, x ) φ j ( x ) (cid:1) | x /T =( x + z ) /T = f j ( x − x ) | x = x + z · (cid:0) ( x − x ) k Y M ( v, x ) φ j ( x ) (cid:1) | x /T =( x + z ) /T = z k f j ( z ) Y H ( v, z ) φ j ( x ) . z k p X j =1 f j ( x − x ) Y M ( v, x ) φ j ( x ) | x /T =( x + z ) /T = ( x − x ) k p X j =1 f j ( x − x ) Y M ( v, x ) φ j ( x ) | x /T =( x + z ) /T = p X j =1 (cid:0) ( x − x ) k f j ( x − x ) Y M ( v, x ) φ j ( x ) (cid:1) | x /T =( x + z ) /T = z k p X j =1 f j ( z ) Y H ( v, z ) φ j ( x ) , which amounts to (4.10). The second assertion can be proved similarly.As the first main result of this section, we have: Theorem 4.6. Let τ ∈ Aut( V ) with o ( τ ) = T and let M and M be weak τ -twisted V -modules. Then ( H ( M , M ) , Y H ) carries the structure of a weak V -module. Proof. First, we prove that H ( M , M ) is closed under the action of V given by (4.4), i.e., v H n φ ( x ) ∈ H ( M , M ) for v ∈ V, n ∈ Z , φ ( x ) ∈ H ( M , M ) . Assume v ∈ V r with r ∈ Z . As it was pointed out before, v H n φ ( x ) satisfies condition (G1) for every n ∈ Z . We nowestablish condition (G2). Using (G2) for φ ( x ) and the corresponding property for Y M ( v, x ) , we have [ L ( − , Y M ( v, x ) φ ( x )] = (cid:18) ∂∂x + ∂∂x (cid:19) Y M ( v, x ) φ ( x ) , [ L ( − , φ ( x ) Y M ( v, x )] = (cid:18) ∂∂x + ∂∂x (cid:19) φ ( x ) Y M ( v, x ) . On the other hand, notice that (cid:18) ∂∂x + ∂∂x (cid:19) z − δ (cid:18) x − xz (cid:19) = 0 , (cid:18) ∂∂x + ∂∂x (cid:19) z − δ (cid:18) x − x − z (cid:19) = 0 , (cid:18) ∂∂x + ∂∂x (cid:19) x − δ (cid:18) x + zx (cid:19) (cid:18) x + zx (cid:19) rT ! = 0 . x − δ (cid:18) x + zx (cid:19) (cid:18) x + zx (cid:19) rT [ L ( − , Y H ( v, z ) φ ( x )]= z − δ (cid:18) x − xz (cid:19) [ L ( − , Y M ( v, x ) φ ( x )] − z − δ (cid:18) x − x − z (cid:19) [ L ( − , φ ( x ) Y M ( v, x )]= (cid:18) ∂∂x + ∂∂x (cid:19) (cid:18) z − δ (cid:18) x − xz (cid:19) Y M ( v, x ) φ ( x ) − z − δ (cid:18) x − x − z (cid:19) φ ( x ) Y M ( v, x ) (cid:19) = (cid:18) ∂∂x + ∂∂x (cid:19) x − δ (cid:18) x + zx (cid:19) (cid:18) x + zx (cid:19) rT Y H ( v, z ) φ ( x ) ! = x − δ (cid:18) x + zx (cid:19) (cid:18) x + zx (cid:19) rT (cid:18) ∂∂x + ∂∂x (cid:19) ( Y H ( v, z ) φ ( x ))= x − δ (cid:18) x + zx (cid:19) (cid:18) x + zx (cid:19) rT ∂∂x ( Y H ( v, z ) φ ( x )) . (4.12)As Y H ( v, z ) φ ( x ) involves only finitely many negative powers of z , by cancellation we obtain [ L ( − , Y H ( v, z ) φ ( x )] = ∂∂x Y H ( v, z ) φ ( x ) . This implies that v H n φ ( x ) satisfy condition (G2) for all n ∈ Z .Next, we consider condition (G3). Let u ∈ V . Then there exists a positive integer k such that ( x − x ) k Y M i ( u, x ) Y M i ( v, x ) = ( x − x ) k Y M i ( v, x ) Y M i ( u, x ) , ( x − x ) k Y M ( u, x ) φ ( x ) = ( x − x ) k φ ( x ) Y M ( u, x ) for i = 2 , . Using these and (4.7) we get x − δ (cid:18) x + zx (cid:19) (cid:18) x + zx (cid:19) rT ( y − x ) k ( y − x − z ) k Y M ( u, y )( Y H ( v, z ) φ ( x ))= x − δ (cid:18) x + zx (cid:19) (cid:18) x + zx (cid:19) rT ( y − x ) k ( y − x ) k Y M ( u, y )( Y H ( v, z ) φ ( x ))= x − δ (cid:18) x + zx (cid:19) (cid:18) x + zx (cid:19) rT ( y − x ) k ( y − x ) k ( Y H ( v, z ) φ ( x )) Y M ( u, y )= x − δ (cid:18) x + zx (cid:19) (cid:18) x + zx (cid:19) rT ( y − x ) k ( y − x − z ) k ( Y H ( v, z ) φ ( x )) Y M ( u, y ) . Again, as ( Y H ( v, z ) φ ( x )) involves only finitely many negative powers of z , by cancellation we obtain ( y − x ) k ( y − x − z ) k Y M ( u, y )( Y H ( v, z ) φ ( x )) = ( y − x ) k ( y − x − z ) k ( Y H ( v, z ) φ ( x )) Y M ( u, y ) . (4.13)Let n ∈ Z such that v H n φ ( x ) = 0 for all n ≥ n . For m ∈ N , applying Res z z n − m to both sides we get k X j =0 (cid:18) kj (cid:19) ( − j ( y − x ) k − j Y M ( u, y ) (cid:0) v H n − m + j φ ( x ) (cid:1) = k X j =0 (cid:18) kj (cid:19) ( − j ( y − x ) k − j (cid:0) v H n − m + j φ ( x ) (cid:1) Y M ( u, y ) . (4.14)17t then follows from this identity and induction on m that for every m ∈ N , there exists a positive integer p such that ( y − x ) p Y M ( u, y ) (cid:0) v H n − m φ ( x ) (cid:1) = ( y − x ) p (cid:0) v H n − m φ ( x ) (cid:1) Y M ( u, y ) . (Note that if k = 0 , it is clear, and if k ≥ , (4.14) gives a recursion formula.) Thus we have proved v H n φ ( x ) ∈H ( M , M ) for n ∈ Z . Now, we have proved that Y H ( · , z ) is a linear map from V to ( End ( H ( M , M ))) [[ z, z − ]] such that Y H ( u, z ) φ ( x ) ∈ H ( M , M )(( z )) for u ∈ V, φ ( x ) ∈ H ( M , M ) . It can be readily seen from (4.4) that Y H ( , z ) = 1 .To prove that H ( M , M ) is a weak V -module, it suffices to prove the weak associativity. Let u ∈ V r , v ∈ V s with r, s ∈ Z and let φ ( x ) ∈ H ( M , M ) . Then there exists a positive integer n such that ( x − x ) n Y M ( u, x ) φ ( x ) = ( x − x ) n φ ( x ) Y M ( u, x ) , ( x − x ) n Y M ( v, x ) φ ( x ) = ( x − x ) n φ ( x ) Y M ( v, x ) , ( x − x ) n Y M ( u, x ) Y M ( v, x ) = ( x − x ) n Y M ( v, x ) Y M ( u, x ) . From the twisted Jacobi identity for ( M , Y M ) we have ( y + z ) rT Y M ( Y ( u, z ) v, y )= Res y y rT (cid:18) x − δ (cid:18) y − y z (cid:19) Y M ( u, y ) Y M ( v, y ) − x − δ (cid:18) y − y − z (cid:19) Y M ( v, y ) Y M ( u, y ) (cid:19) . Using these properties and the standard delta-function substitutions, we get ( y + z ) rT ( y − x ) n ( y − x + z ) n Y M ( Y ( u, z ) v, y ) φ ( x )= ( y + z ) rT ( y − x ) n ( y − x + z ) n φ ( x ) Y M ( Y ( u, z ) v, y ) . Multiplying both sides by ( y + z ) − rT (noticing that we are allowed to do so since Y ( u, z ) v ∈ V (( z )) ), we obtain ( y − x ) n ( y − x + z ) n Y M ( Y ( u, z ) v, y ) φ ( x )= ( y − x ) n ( y − x + z ) n φ ( x ) Y M ( Y ( u, z ) v, y ) . (4.15)Noticing that u m v ∈ V r + s for m ∈ Z , using Lemma 4.5 we obtain z n ( z + z ) n ( x + z ) r + sT Y H ( Y ( u, z ) v, z ) φ ( x )= (cid:16) ( x − x ) n ( x − x + z ) n x r + sT Y M ( Y ( u, z ) v, x ) φ ( x ) (cid:17) | x = x + z . (4.16)On the other hand, we have ( x + z ) rT z n Y M ( Y ( u, z ) v, x ) = (cid:16) ( x − x ) n x rT Y M ( u, x ) Y M ( v, x ) (cid:17) | x = x + z , which gives z n Y M ( Y ( u, z ) v, x ) = ( x + z ) − rT (cid:16) ( x − x ) n x rT Y M ( u, x ) Y M ( v, x ) (cid:17) | x = x + z . Then z n z n ( z + z ) n ( x + z ) r + sT Y H ( Y ( u, z ) v, z ) φ ( x )= (cid:16) ( x − x ) n ( x − x + z ) n x r + sT z n Y M ( Y ( u, z ) v, x ) φ ( x ) (cid:17) | x = x + z = (cid:16) ( x − x ) n ( x − x + z ) n x r + sT ( x + z ) − rT (cid:16) ( x − x ) n x rT Y M ( u, x ) Y M ( v, x ) φ ( x ) (cid:17) | x = x + z (cid:17) | x = x + z = (cid:16) (1 + z /x ) − rT (cid:16) ( x − x ) n ( x − x ) n ( x − x ) n x rT x sT Y M ( u, x ) Y M ( v, x ) φ ( x ) (cid:17) | x = x + z (cid:17) | x = x + z = (cid:18) z x + z (cid:19) − rT (cid:16)(cid:16) ( x − x ) n ( x − x ) n ( x − x ) n x rT x sT Y M ( u, x ) Y M ( v, x ) φ ( x ) (cid:17) | x = x + z (cid:17) | x = x + z , z n z n ( z + z ) n ( x + z ) sT ( x + z + z ) rT Y H ( Y ( u, z ) v, z ) φ ( x )= (cid:16)(cid:16) ( x − x ) n ( x − x ) n ( x − x ) n x rT x sT Y M ( u, x ) Y M ( v, x ) φ ( x ) (cid:17) | x = x + z (cid:17) | x = x + z . (4.17)On the other hand, we consider Y H ( u, z ) Y H ( v, z ) φ ( x ) . With (4.13), by Lemma 4.5 we have z k ( z − z ) k ( x + z ) rT Y H ( u, z ) Y H ( v, z ) φ ( x )= (cid:16) ( x − x ) k ( x − x − z ) k x rT Y M ( u, x )( Y H ( v, z ) φ ( x )) (cid:17) | x = x + z . We also have z k ( x + z ) sT Y H ( v, z ) φ ( x ) = (cid:16) ( x − x ) k x sT Y M ( v, x ) φ ( x ) (cid:17) | x = x + z . Then we get z k ( x + z ) sT z k ( z − z ) k ( x + z ) rT Y H ( u, z ) Y H ( v, z ) φ ( x )= (cid:16) ( x − x ) k ( x − x − z ) k x rT Y M ( u, x ) (cid:16) ( x − x ) k x sT Y M ( v, x ) φ ( x ) (cid:17) | x = x + z (cid:17) | x = x + z = (cid:16)(cid:16) ( x − x ) k ( x − x ) k ( x − x ) k x rT x sT Y M ( u, x ) Y M ( v, x ) φ ( x ) (cid:17) | x = x + z (cid:17) | x = x + z , which gives z k ( x + z ) sT ( z + z ) k z k ( x + z + z ) rT Y H ( u, z + z ) Y H ( v, z ) φ ( x )= (cid:16)(cid:16) ( x − x ) k ( x − x ) k ( x − x ) k x rT x sT Y M ( u, x ) Y M ( v, x ) φ ( x ) (cid:17) | x = x + z (cid:17) | x = x + z + z . Combining this with (4.17) (replacing n with a larger one so that n ≥ k ) we obtain z n z n ( z + z ) n ( x + z ) sT ( x + z + z ) rT Y H ( Y ( u, z ) v, z ) φ ( x )= z n ( x + z ) sT ( z + z ) n z n ( x + z + z ) rT Y H ( u, z + z ) Y H ( v, z ) φ ( x ) , noticing that under the formal series expansion convention, for any m ∈ Z , ( x + z ) m | x = x + z = (( x + z ) + z ) m = ( x + ( z + z )) m = x m | x = x + z + z . Multiplying both sides by z − n z − n ( x + z ) − sT ( x + z + z ) − rT we get ( z + z ) n Y H ( Y ( u, z ) v, z ) φ ( x ) = ( z + z ) n Y H ( u, z + z ) Y H ( v, z ) φ ( x ) . This establishes the weak associativity. Therefore, ( H ( M , M ) , Y H ) carries the structure of a weak V -module.We end this section with the following technical result: Lemma 4.7. Let τ, T, M , M be given as before and let φ ( x ) ∈ ( Hom ( W , W )) { z } . Suppose that φ ( x ) satisfiesthe conditions (G1) and (G2) and suppose that the condition (G3) holds for any u ∈ U , where U be a subset of V suchthat ∪ k − j =0 τ j ( U ) generates V as a vertex algebra. Then φ ( x ) ∈ H ( W , W ) . Proof. Let K consist of all vectors a ∈ V such that ( x − x ) n Y M ( a, x ) φ ( x ) = ( x − x ) n φ ( x ) Y M ( a, x ) for some nonnegative integer n . We must prove K = V . As ∪ T − j =0 τ j ( U ) generates V as a vertex algebra, it suffices toprove that K is a vertex subalgebra that contains ∪ T − j =0 τ j ( U ) . It is clear that K is a subspace with ∈ K , and fromassumption we have U ⊂ K . Note that for any v ∈ V , we have Y M i ( τ ( v ) , z ) = lim z /T → η − k z /T Y M i ( v, z ) i = 2 , . It then follows that τ ( K ) ⊂ K . Thus ∪ T − j =0 τ j ( U ) ⊂ K . Now, it remains to prove that u m v ∈ K forany u, v ∈ K, m ∈ Z . As τ ( K ) ⊂ K , it suffices to consider u ∈ K ∩ V r with r ∈ Z . Note that ( z + z ) rT Y M i ( Y ( u, z ) v, z )= Res z z rT (cid:18) z − δ (cid:18) z − z z (cid:19) Y M i ( u, z ) Y M i ( v, z ) − z − δ (cid:18) z − z − z (cid:19) Y M i ( v, z ) Y M i ( u, z ) (cid:19) (4.18)holds on M i for i = 2 , . Let k be a positive integer such that ( z − z ) k Y M ( a, z ) φ ( z ) = ( z − z ) k φ ( z ) Y M ( a, z ) for a ∈ { u, v } . Using these relations and (4.18), we get ( z − z ) k ( z + z − z ) k Y M ( Y ( u, z ) v, z ) φ ( z ) = ( z − z ) k ( z + z − z ) k φ ( z ) Y M ( Y ( u, z ) v, z ) . Just as in the first part of the proof of Theorem 4.6, it follows from this relation and an induction that u m v ∈ K for all m ∈ Z . Therefore, we conclude K = V and hence φ ( x ) ∈ H ( W , W ) . Proposition 4.8. Let M , M be weak τ -twisted V -modules and let M be any weak V -module. Then for anyintertwining operator I ( · , x ) of type (cid:0) M M M (cid:1) , we have I ( w, x ) ∈ H ( M , M ) for w ∈ M , and the map f I defined by f I ( w ) = I ( w, x ) for w ∈ M is a homomorphism of weak V -modules. Furthermore, the linear map θ : I V (cid:0) M M M (cid:1) → Hom V ( M, H ( M , M )) defined by θ ( I ) = f I is a linear isomorphism. Proof. Let I ( · , x ) be any intertwining operator of type (cid:0) M M M (cid:1) . From definition, we see that for every w ∈ M , I ( w, x ) satisfies conditions (G1) and (G2). As the Jacobi identity for I ( · , x ) implies commutator formula which furthermoreimplies locality, I ( w, x ) also satisfies (G3). Thus I ( w, x ) ∈ H ( M , M ) . Then we have a linear map f I from M to H ( M , M ) defined by f I ( w ) = I ( w, x ) for w ∈ M . For any u ∈ V r , w ∈ M with r ∈ Z , we have f I ( Y ( u, x ) w )= I ( Y ( u, x ) w, x )= Res x (cid:18) x + x x (cid:19) − rT (cid:18) x − δ (cid:18) x − xx (cid:19) Y M ( u, x ) I ( w, x ) − x − δ (cid:18) x − x − x (cid:19) I ( w, x ) Y M ( u, x ) (cid:19) = Y H ( u, x ) I ( w, x )= Y H ( u, x ) f I ( w ) . This proves that f I is a homomorphism of weak V -modules.For the second assertion, it is clear that θ is a one-to-one linear map. To prove that it is onto, let f be a V -homomorphism from M to H ( M , M )) . For every w ∈ M , noticing that as an element of H ( M , M )) , f ( w ) is aformal series in a formal variable, we write f ( w ) as f ( w )( x ) , and then set I f ( w, x ) = f ( w )( x ) ∈ H ( M , M ) . Thisgives a linear map I f ( · , x ) from M to (Hom C ( M , M )) { x } . We have [ L ( − , I f ( w, z )] = ddz I f ( w, z ) . For any v ∈ V, w ∈ M , as f ( w )( x ) ∈ H ( W , W ) , by condition (G3) there exists a nonnegative integer k such that ( x − x ) k Y M ( v, x ) f ( w )( x ) = ( x − x ) k f ( w )( x ) Y M ( v, x ) , which amounts to ( x − x ) k Y M ( v, x ) I f ( w, x ) = ( x − x ) k I f ( w, x ) Y M ( v, x ) . (4.19)20n the other hand, for v ∈ V r with r ∈ Z and w ∈ M , with f a homomorphism of weak V -modules, we have ( z + z ) rT I f ( Y M ( v, z ) w, z )= ( z + z ) rT f ( Y M ( v, z ) w )( z )= ( z + z ) rT Y H ( v, z ) f ( w )( z )= Res z z rT (cid:18) z − δ (cid:18) z − z z (cid:19) Y M ( v, z ) f ( w )( z ) − z − δ (cid:18) z − z − z (cid:19) f ( w )( z ) Y M ( v, z ) (cid:19) = Res z z rT (cid:18) z − δ (cid:18) z − z z (cid:19) Y M ( v, z ) I f ( w, z ) − z − δ (cid:18) z − z − z (cid:19) I f ( w, z ) Y M ( v, z ) (cid:19) . This together with (4.19) gives the Jacobi identity. Therefore, I f ( · , x ) is an intertwining operator of type (cid:0) M M M (cid:1) . It isclear that θ ( I f ) = f . This proves that θ is onto, concluding the proof. In this section, we shall apply the results of Section 4 to the permutation orbifold model. I V (cid:0) WM N (cid:1) and I V ⊗ k (cid:0) T σ ( W ) M T σ ( N ) (cid:1) Let V be a vertex operator algebra and let k be a positive integer, which are all fixed throughout this section. Set σ = (1 2 · · · k ) , which is an automorphism of V ⊗ k . Let M , N , and W be irreducible V -modules. Set W = M ⊗ V ⊗ ( k − , which is a V ⊗ k -module. Let T σ ( N ) and T σ ( W ) be the σ -twisted V ⊗ k -modules constructed in[BDM]. In this section, we prove that there is a canonical linear isomorphism between the spaces I V (cid:0) WM N (cid:1) and I V ⊗ k (cid:0) T σ ( W ) M T σ ( N ) (cid:1) . That is, given an intertwining operator of type (cid:0) WM N (cid:1) , we can construct an intertwining operatorof type (cid:0) T σ ( W ) M T σ ( N ) (cid:1) . Conversely, given an intertwining operator of type (cid:0) T σ ( W ) M T σ ( N ) (cid:1) , we can construct an intertwiningoperator of type (cid:0) WM N (cid:1) . As one of the main results, we obtain the following tensor product relation: (cid:16) M ⊗ V ⊗ ( k − (cid:17) ⊠ V ⊗ k T σ ( N ) ≃ T σ ( M ⊠ V N ) . I V (cid:0) WM N (cid:1) to I V ⊗ k (cid:0) T σ ( W ) M T σ ( N ) (cid:1) Let V be a vertex operator algebra as before, and let k be a positive integer with k ≥ and set σ = (1 2 · · · k ) , whichis viewed as an automorphism of V ⊗ k of order k .For any u ∈ V , ≤ j ≤ k , we denote by u j the vector in V ⊗ k whose j th tensor factor is u and the other tensorfactors are : u j = ⊗ ( j − ⊗ u ⊗ ⊗ ( k − j ) . (5.1)We have Y ( u j , z ) = 1 ⊗ ( j − ⊗ Y ( u, z ) ⊗ ⊗ ( k − j ) . (5.2)Note that σu j = u j +1 for ≤ j ≤ k , where u k +1 = u by convention. For any weak V ⊗ k -module ( W, Y W ) , wehave (cid:2) Y W ( u i , z ) , Y W ( v j , z ) (cid:3) = Res z z − δ (cid:18) z − z z (cid:19) Y W ( Y ( u i , z ) v j , z ) for u, v ∈ V, ≤ i, j ≤ k . For n ≥ , as u n = 0 (in V ) we have ( u i ) n ( v j ) = 0 (in V ⊗ k ) if i = j . Then weimmediately have: 21 emma 5.1. Let ( W, Y W ) be any weak V ⊗ k -module. Then (cid:2) Y W ( u i , z ) , Y W ( v j , z ) (cid:3) = 0 for any u, v ∈ V , i, j ∈{ , . . . , k } with i = j .Let k be a positive integer. Recall from [BDM] ∆ k ( z ) = exp X n ≥ a n z − nk L ( n ) k − L (0) z (1 /k − L (0) , (5.3)where the coefficients a n for n ≥ are uniquely determined by exp X n ≥ − a n x n +1 ddx x = 1 k (1 + x ) k − k . (5.4)As in [BDM], we shall also use the expression ∆ k ( z k ) − . For convenience, we set Φ k ( z ) = ∆ k ( z k ) − = ( kz k − ) L (0) exp − X n ≥ a n z − n L ( n ) . (5.5)The following results were obtained in [BDM] (Proposition 2.2 and Corollary 2.5): Proposition 5.2. Let ( W, Y W ) be any V -module. Then ∆ k ( x ) Y W ( v, z ) w = Y W (cid:16) ∆ k ( x + z ) v, ( x + z ) /k − x /k (cid:17) ∆ k ( x ) w, (5.6) Φ k ( x ) Y W ( v, z ) w = Y W (cid:0) Φ k ( x + z ) v, ( x + z ) k − x k (cid:1) Φ k ( x ) w, (5.7) ∆ k ( z ) L ( − w − k z /k − L ( − k ( z ) w = ddz ∆ k ( z ) w, (5.8) ∆ k ( z ) ω = 1 k z k − ω + z − c 24 (1 − k − ) (5.9)for v ∈ V, w ∈ W , where ω is the conformal vector of V and c is the central charge.The following is one of the main results of [BDM]: Theorem 5.3. Let V be any vertex operator algebra and let σ = (12 · · · k ) ∈ Aut ( V ⊗ k ) . Then for any (weak,admissible) V -module ( W, Y W ) , we have a (weak, admissible) σ -twisted V ⊗ k -module ( T σ ( W ) , Y T σ ( W ) ) , where T σ ( W ) = W as a vector space and the vertex operator map Y T σ ( W ) ( · , z ) is uniquely determined by Y T σ ( W ) ( u , z ) = Y W (∆ k ( z ) u, z /k ) for u ∈ V, (5.10)where u = u ⊗ ⊗ ( k − ∈ V ⊗ k . Furthermore, every (weak, admissible) σ -twisted V ⊗ k -module is isomorphic to onein this form.Set η k = e π √− k , (5.11)the principal primitive k -th root of unity.We shall need the L ( − -operator for the vertex operator algebra V ⊗ k on the σ -twisted V ⊗ k -module T σ ( W ) .Note that the conformal vector of V ⊗ k is ω ( k ) := ω + · · · + ω k , ω denotes the conformal vector of V . Write Y W ( ω, z ) = X n ∈ Z L ( n ) z − n − , Y T σ ( W ) ( ω ( k ) , z ) = X n ∈ Z L T σ ( n ) z − n − . (5.12)Using the covariance property of σ -twisted modules and the expression of ∆ k ( z ) ω from Proposition 5.2, we have Y T σ ( W ) ( ω ( k ) , z ) = k − X j =0 Y T σ ( W ) ( σ j ω , z ) = k − X j =0 lim z /k → η − jk z /k Y T σ ( W ) ( ω , z )= k − X j =0 lim z /k → η − jk z /k Y W (∆ k ( z ) ω, z /k )= k − X j =0 k η − jk z /k − Y W ( ω, η − jk z /k ) + z − c k (1 − k − )= 1 k X m ∈ Z L ( mk ) z − m − + z − c 24 ( k − k − ) . Then L T σ ( n ) = ( k L ( nk ) for n = 0 k L (0) + c ( k − k − ) for n = 0 . (5.13)In particular, we have L T σ ( − 1) = 1 k L ( − k ) . (5.14)We have the following result which will be used later: Lemma 5.4. Let ( W, Y W ) be any V -module. Then ddz ∆ k ( z ) = 1 k X i ≥ (cid:18) − k + 1 i (cid:19) z − − ( i − /k L ( i − k ( z ) (5.15)holds on W. Proof. It is similar to that of Corollary 2.5 of [BDM]. As in [BDM] we set ∆ xk ( z ) = exp − X j ≥ a j z − j/k x j +1 ∂∂x ! k x ∂∂x z ( − /k +1) x ∂∂x , an element of (End C [ x, x − ])[[ z /k , z − /k ]] . As in the proof of Corollary 2.5 of [BDM], it suffices to show that k X i ≥ (cid:18) − k + 1 i (cid:19) (cid:18) − x i ∂∂x (cid:19) z − − ( i − /k ∆ xk ( z ) = ∂∂z ∆ xk ( z ) (5.16)in (End C [ x, x − ])[[ z /k , z − /k ]] . ∆ xk ( z ) x = exp − X j ≥ a j z − j/k x j +1 ∂∂x ! k x ∂∂x z ( − /k +1) x ∂∂x x = exp − X j ≥ a j z − j/k x j +1 ∂∂x ! kz − /k +1 x = kz exp − X j ≥ a j z − j/k x j +1 ∂∂x ! z − /k x = kz exp − X j ≥ a j ( z − /k x ) j +1 ∂∂z − /k x ! z − /k x = kz (cid:18) k (1 + z − /k x ) k − k (cid:19) = z (cid:16) (1 + z − /k x ) k − (cid:17) , where for the second last equality we are using equation (5.4). Then k X i ≥ (cid:18) − k + 1 i (cid:19) (cid:18) − x i ∂∂x (cid:19) z − − ( i − /k ∆ xk ( z ) x = 1 k X i ≥ (cid:18) − k + 1 i (cid:19) (cid:18) − x i ∂∂x (cid:19) z − − ( i − /k z (cid:16) (1 + z − /k x ) k − (cid:17) = − X i ≥ (cid:18) − k + 1 i (cid:19) x i z − i/k (1 + z − /k x ) k − = − X i ≥ (cid:18) − k + 1 i (cid:19) x i z − i/k (cid:16) (1 + z − /k x ) k − (cid:17) + (1 + z − /k x ) k − = − z − /k x ) k − = ∂∂z z (cid:16) (1 + z − /k x ) k − (cid:17) = ∂∂z ∆ xk ( z ) x. ∆ xk ( z ) x n = (∆ xk ( z ) x ) n for n ∈ Z . Thus k X i ≥ (cid:18) − k + 1 i (cid:19) (cid:18) − x i ∂∂x (cid:19) z − − ( i − /k ∆ xk ( z ) x n = 1 k X i ≥ (cid:18) − k + 1 i (cid:19) (cid:18) − x i ∂∂x (cid:19) z − − ( i − /k z n (cid:16) (1 + z − /k x ) k − (cid:17) n = − n X i ≥ (cid:18) − k + 1 i (cid:19) x i z − i/k (1 + z − /k x ) k − z n − (cid:16) (1 + z − /k x ) k − (cid:17) n − = − n X i ≥ (cid:18) − k + 1 i (cid:19) x i z − i/k (1 + z − /k x ) k − z n − (cid:16) (1 + z − /k x ) k − (cid:17) n − + n (1 + z − /k x ) k − z n − (cid:16) (1 + z − /k x ) k − (cid:17) n − = − nz n − (cid:16) (1 + z − /k x ) k − (cid:17) n − + n (1 + z − /k x ) k − z n − (cid:16) (1 + z − /k x ) k − (cid:17) n − = nz n − (cid:16) (1 + z − /k x ) k − (cid:17) n − (cid:16) − z − /k x ) k − (cid:17) = n (∆ xk ( z ) x ) n − ∂∂z ∆ xk ( z )= ∂∂z (∆ xk ( z ) x ) n = ∂∂z ∆ xk ( z ) x n . Therefore, equation (5.16) holds.As the first main result of this section we have: Theorem 5.5. Let M, N and W be weak V -modules and let σ = (1 2 · · · k ) ∈ Aut ( V ⊗ k ) . Let Y ( · , z ) be anintertwining operator of type (cid:0) WM N (cid:1) . For w ∈ W , set Y ( w, z ) = Y (∆ k ( z ) w, z /k ) ∈ ( Hom ( N, W )) { z } . (5.17)Then Y ( w, z ) ∈ H ( T σ ( N ) , T σ ( W )) for w ∈ W. (5.18)Furthermore, the linear map f : M ⊗ V ⊗ ( k − → H ( T σ ( N ) , T σ ( W )) , defined by f ( w ⊗ a ) = a H− Y ( w, z ) for w ∈ M, a ∈ V ⊗ ( k − , is a weak V ⊗ k -module homomorphism from M ⊗ V ⊗ ( k − to H ( T σ ( N ) , T σ ( W )) . Proof. First, we show that Y ( w, z ) ∈ H ( T σ ( N ) , T σ ( W )) for w ∈ M . It is clear that Y ( w, z ) satisfies the condition(G1). For condition (G2), note that the following relations hold for any w ′ ∈ W, m ∈ Z : Y ( L ( − w ′ , z ) = ddz Y ( w ′ , z ) , [ L ( m ) , Y ( w ′ , z )] = X i ≥ (cid:18) m + 1 i (cid:19) z m +1 − i Y ( L ( i − w ′ , z ) . [ L T σ ( − , Y ( w, z )]= 1 k h L ( − k ) , Y (∆ k ( z ) w, z /k ) i = 1 k X i ≥ (cid:18) − k + 1 i (cid:19) ( z /k ) − k − i Y ( L ( i − k ( z ) w, z /k )= 1 k z k − Y ( L ( − k ( z ) w, z /k ) + 1 k X i ≥ (cid:18) − k + 1 i (cid:19) z − − ( i − /k Y ( L ( i − k ( z ) w, z /k )= 1 k z k − (cid:18) ∂∂y /k Y (∆ k ( z ) w, y /k ) (cid:19) | y = z + 1 k X i ≥ (cid:18) − k + 1 i (cid:19) z − − ( i − /k Y ( L ( i − k ( z ) w, z /k )= (cid:18) ∂∂y Y (∆ k ( z ) w, y /k ) (cid:19) | y = z + Y (cid:18) ∂∂z ∆ k ( z ) w, z /k (cid:19) = ddz Y (∆ k ( z ) w, z /k )= ddz Y ( w, z ) . For condition (G3), let u ∈ V, w ∈ M . By Lemma 3.3 in [BDM], we have Y T σ ( W ) ( u , z ) Y ( w, z ) − Y ( w, z ) Y T σ ( N ) ( u , z )= Res z k z − δ ( z − z ) /k z /k ! Y ( Y M ( u, z ) w, z )= Res z k k − X r =0 z − δ (cid:18) z − z z (cid:19) (cid:18) z − z z (cid:19) − r/k Y ( Y M ( u, z ) w, z )= Res z k k − X r =0 z − δ (cid:18) z + z z (cid:19) (cid:18) z + z z (cid:19) r/k Y ( Y M ( u, z ) w, z ) . (5.19)It follows that there exists a nonnegative integer n such that ( z − z ) n Y T σ ( W ) ( u , z ) Y ( w, z ) = ( z − z ) n Y ( w, z ) Y T σ ( N ) ( u , z ) . (5.20)Since { σ j ( u ) | u ∈ V, ≤ j ≤ k } generates V ⊗ k as a vertex algebra, by Lemma 4.7 we have Y ( w, z ) ∈H ( T σ ( N ) , T σ ( W )) . Note that V ⊗ k as a vertex algebra is generated by the subset { u j | u ∈ V, ≤ j ≤ k } .Second, we show that f is a homomorphism of weak V -modules where H ( T σ ( N ) , T σ ( W )) is viewed as a V -module through the embedding π of V into V ⊗ k . Note that for ≤ i ≤ k , Y T σ ( W ) ( u i , z ) = Y T σ ( W ) ( σ i − u , z ) = lim z /k → η − ik z /k Y T σ ( W ) ( u , z ) . Then using (5.19) (by replacing z /k with η − ik z /k ), we get Y T σ ( W ) ( u i , z ) Y ( w, z ) − Y ( w, z ) Y T σ ( N ) ( u i , z )= Res z k k − X r =0 z − δ (cid:18) z + z z (cid:19) (cid:18) z + z z (cid:19) r/k η r ( i − k Y ( Y M ( u, z ) w, z ) . (5.21)26or u ∈ V , write u i = u i, + u i, + · · · + u i,k − , (5.22)where u i,r ∈ ( V ⊗ k ) r for ≤ r ≤ k − . As Y T σ ( W ) ( u i,r , z ) ∈ z − rT (End T σ ( W ))[[ z , z − ]] , Y T σ ( N ) ( u i,r , z ) ∈ z − rT (End T σ ( N ))[[ z , z − ]] , from (5.21) we get Y T σ ( W ) ( u i,r , z ) Y ( w, z ) − Y ( w, z ) Y T σ ( N ) ( u i,r , z )= Res z k z − η ( i − rk (cid:18) z + z z (cid:19) r/k δ (cid:18) z + z z (cid:19) Y ( Y M ( u, z ) w, z ) . (5.23)As Y ( · , x ) is an intertwining operator of type (cid:0) WM N (cid:1) , we have z − δ x /k − x /k z ! Y W (cid:16) ∆ k ( x ) v, x /k (cid:17) Y (cid:16) ∆ k ( x ) w, x /k (cid:17) − z − δ − x /k + x /k z ! Y (cid:16) ∆ k ( x ) w, x /k (cid:17) Y N (cid:16) ∆ k ( x ) v, x /k (cid:17) = x − /k δ x /k − z x /k ! Y (cid:16) Y M (∆ k ( x ) v, z )∆ k ( x ) w, x /k (cid:17) . (5.24)Let p be a nonnegative integer such that z p Y M (∆ k ( x ) v, z )∆ k ( x ) w involves only nonnegative (integer) powers of z . Then we have ( x − x ) p Y W (cid:16) ∆ k ( x ) v, x /k (cid:17) Y (cid:16) ∆ k ( x ) w, x /k (cid:17) = ( x − x ) p Y (cid:16) ∆ k ( x ) w, x /k (cid:17) Y N (cid:16) ∆ k ( x ) v, x /k (cid:17) , which is ( x − x ) p Y T σ ( W ) ( v , x ) Y ( w, x ) = ( x − x ) p Y ( w, x ) Y T σ ( N ) ( v , x ) . Then z p Y H ( v , z )( Y ( w, x )) = (cid:0) ( x − x ) p Y T σ ( W ) ( v , x ) Y ( w, x ) (cid:1) | x /k =( x + z ) /k . (5.25)On the other hand, from (5.24) we get x − /k δ x /k + z x /k ! (cid:0) ( x − x ) p Y T σ ( W ) ( v , x ) Y ( w , x ) (cid:1) = x − /k δ x /k + z x /k ! ( x − x ) p Y (cid:16) Y M (∆ k ( x ) v, z )∆ k ( x ) w, x /k (cid:17) . Taking the residue with respect to x /k , we obtain (cid:0) ( x − x ) p Y T σ ( W ) ( v , x ) Y ( w, x ) (cid:1) | x /k = x /k + z = (cid:16) ( x − x ) p Y (cid:16) Y M (∆ k ( x ) v, z )∆ k ( x ) w, x /k (cid:17)(cid:17) | x /k = x /k + z . (5.26)Notice that for any f ( x, x , z ) ∈ x α U (( x /T ))(( x /k , z )) α a complex number, U a vector space and T a positive integer, we have (cid:16) f ( x, x , z ) | x /k = x /k + z (cid:17) | z =( x + z ) /k − x /k = f ( x, x , ( x + z ) /k − x /k ) | x /k =( x + z ) /k . In view of this, applying substitution z = ( x + z ) /k − x /k to (5.26), we get (cid:0) ( x − x ) p Y T σ ( W ) ( v , x ) Y ( w, x ) (cid:1) | x /k =( x + z ) /k = (cid:16) ( x − x ) p Y (cid:16) Y M (∆ k ( x ) v, ( x + z ) /k − x /k )∆ k ( x ) w, x /k (cid:17)(cid:17) | x /k =( x + z ) /k = z p Y (cid:16) Y M (∆ k ( x + z ) v, ( x + z ) /k − x /k )∆ k ( x ) w, x /k (cid:17) . (5.27)Combining this with (5.25), and then using (5.6) we get Y H ( v , z )( Y ( w, x )) = Y (cid:16) Y M (∆ k ( x + z ) v, ( x + z ) /k − x /k )∆ k ( x ) w, x /k (cid:17) = Y (cid:16) ∆ k ( x ) Y M ( v, z ) w, x /k (cid:17) = Y ( Y M ( v, z ) w, x ) . (5.28)Furthermore, for any v ∈ V, a ∈ C ⊗ V ⊗ ( k − , we have f ( Y ( v, z ) w ⊗ a ) = a H− Y ( Y M ( v, z ) w, x ) = a H− Y H ( v , z )( Y ( w, x )) = Y H ( v , z ) a H− Y ( w, x ) = Y H ( v , z ) f ( w ⊗ a ) , noticing that [ a H m , ( v ) H n ] = 0 for m, n ∈ Z as ( v ) i a = 0 for all i ≥ . This proves that f is a homomorphism ofweak V -modules.Third, we show that f is a homomorphism of weak C ⊗ V ⊗ ( k − -modules. To this end, we first show that if ≤ i ≤ k , then ( u i ) H n Y ( w, x ) = 0 for all u ∈ V, w ∈ M, n ≥ . (5.29)Recall u i = u i, + u i, + · · · + u i,k − , where u i,r ∈ ( V ⊗ k ) r for ≤ r ≤ k − . By definition, we have (cid:0) u i,r (cid:1) H n Y ( w, x )= X j ≥ (cid:18) − rk j (cid:19) Res x x − rk − j x rT (cid:8) ( x − x ) n + j Y T σ ( W ) (cid:0) u i,r , x (cid:1) Y ( w, x ) − ( − x + x ) n + j Y ( w, x ) Y T σ ( N ) (cid:0) u i,r , x (cid:1)(cid:9) n ∈ Z . Assume n ≥ . Using (5.23) we get (cid:0) u i (cid:1) H n Y ( w, x )= k − X r =0 (cid:0) u i,r (cid:1) H n Y ( w, x )= k − X r =0 Res x X j ≥ (cid:18) − rk j (cid:19) x − rk − j x rk ( x − x ) n + j (cid:0) Y T σ ( W ) ( u i,r , x ) Y ( w, x ) − Y ( w, x ) Y T σ ( N ) (cid:0) u i,r , x (cid:1)(cid:1) = Res x Res x k − X r =0 X j ≥ (cid:18) − rk j (cid:19) x − rk − j x rk ( x − x ) n + j · k η (1 − i ) rk x − δ (cid:18) x + x x (cid:19) (cid:18) x + x x (cid:19) r/k Y ( Y M ( u, x ) w, x )= Res x Res x k − X r =0 X j ≥ (cid:18) − rk j (cid:19) x − rk − j x rk x n + j · k η ( i − rk x − δ (cid:18) x + x x (cid:19) (cid:18) x + x x (cid:19) r/k Y ( Y M ( u, x ) w, x )= Res x Res x k − X r =0 (cid:18) x + x x (cid:19) − r/k x n · k η ( i − rk x − δ (cid:18) x + x x (cid:19) (cid:18) x + x x (cid:19) r/k Y ( Y ( u, x ) w, x )= k k − X r =0 η ( i − rk ! Res x x n Y ( Y M ( u, x ) w, x )= δ i, Y ( u n w, x ) . This proves (5.29). Since V i for ≤ i ≤ k generate C ⊗ V ⊗ ( k − as a vertex algebra, from [LL], Y ( w, x ) for w ∈ M are vacuum-like vectors in H ( T σ ( N ) , T σ ( W )) viewed as a weak module for C ⊗ V ⊗ ( k − in the sense of [Li1]. Thenby ([LL]; Proposition 4.7.7, or Corollary 4.7.8), the linear map f from M ⊗ V ⊗ ( k − to H ( T σ ( N ) , T σ ( N )) , definedby f ( w ⊗ a ) = a H− Y ( w, x ) for w ∈ M, a ∈ V ⊗ ( k − , is a weak module homomorphism for C ⊗ V ⊗ ( k − .As V and C ⊗ V ⊗ ( k − generate V ⊗ k as a vertex algebra, it follows immediately that f is a weak V ⊗ k -modulehomomorphism. This completes the proof.Given V -modules M, N , and W , we have the space I V (cid:0) WM N (cid:1) of intertwining operators. Furthermore, we have σ -twisted V ⊗ k -modules T σ ( N ) and T σ ( W ) , and we have the space I V ⊗ k (cid:0) T σ ( W ) M T σ ( N ) (cid:1) of intertwining operators of type (cid:0) T σ ( W ) M T σ ( N ) (cid:1) . On the other hand, in view of Theorem 4.6, we have a weak V ⊗ k -module ( H ( T σ ( N ) , T σ ( W )) , Y H ( · , z )) .Combining Theorem 5.5 and Proposition 4.8, we immediately have: Corollary 5.6. For every intertwining operator Y ( · , z ) of type (cid:0) WM N (cid:1) , there exists an intertwining operator Y ( · , z ) oftype (cid:0) T σ ( W ) M T σ ( N ) (cid:1) , which is uniquely determined by Y ( w , z ) = Y (∆ k ( z ) w, z /k ) for w ∈ M. (5.30)In view of Corollary 5.6, we have a linear map π : I V (cid:18) WM N (cid:19) → I V ⊗ k (cid:18) T σ ( W ) M T σ ( N ) (cid:19) ; Y ( · , z ) 7→ Y ( · , z ) . (5.31)29 .1.2 From I V ⊗ k (cid:0) T σ ( W ) M T σ ( N ) (cid:1) to I V (cid:0) WM N (cid:1) In this section, we shall prove that the linear map π defined above from I V (cid:0) WM N (cid:1) to I V ⊗ k (cid:0) T σ ( W ) M T σ ( N ) (cid:1) is an isomor-phism by constructing its inverse map. The arguments here are similar to those in [BDM] (Section 4).Let V, k , and σ be given as before. On the other hand, let M, N , and W be V -modules. Set M = M ⊗ V ⊗ ( k − ,which is a V ⊗ k -module. Recall the σ -twisted V ⊗ k -modules T σ ( N ) and T σ ( W ) .Recall ∆ k ( z ) from (5.3). Here, we shall need its inverse ∆ k ( z ) − = z (1 − /k ) L (0) k L (0) exp − X j ≥ a j z − j/k L ( j ) . (5.32)As before, by convention we define k α = e α ln k and ( z k ) α = z kα for any α ∈ C . Furthermore, for any formal series A ( x ) = X α ∈ C u α x α ∈ U { x } with U a vector space, we define A ( z k ) = X α ∈ C u α z kα ∈ U { z } . In particular, if w is an L (0) -eigenvector in a V -module with eigenvalue α ∈ C , we have z (1 − /k ) L (0) k L (0) w = z (1 − /k ) α k α w. Notice that ∆ k ( z k ) − = .The following lemma follows from the proof of Lemma 4.1 in [BDM]: Lemma 5.7. Let Y ( · , z ) be an intertwining operator of type (cid:0) T σ ( W ) M T σ ( N ) (cid:1) . For w ∈ M , set Y ( w, z ) = Y (cid:16)(cid:0) ∆ k ( z k ) − w (cid:1) , z k (cid:17) , an element of ( Hom ( N, W )) { z } , where (cid:0) ∆ k ( z k ) − w (cid:1) = ∆ k ( z k ) − w ⊗ ⊗ ( k − ∈ M { z } . Then Y ( L ( − w, z ) = ddz Y ( w, z ) on N for any w ∈ M .Furthermore, we have the following commutator formula: Lemma 5.8. Let Y ( · , z ) be an intertwining operator of type (cid:0) T σ ( W ) M T σ ( N ) (cid:1) . Then Y W ( u, z ) Y ( w, z ) − Y ( w, z ) Y N ( u, z ) = Res z z − δ (cid:18) z − z z (cid:19) Y ( Y M ( u, z ) w, z ) for u ∈ V , w ∈ M , where Y ( w, z ) for w ∈ M is defined as in Lemma 5.7. Proof. Let u ∈ V, w ∈ M . Noticing that ( σ j u ) n w = ( u j +1 ) n w = 0 for all ≤ j ≤ k − , n ≥ , from the twisted Jacobi identity (2.7), we get Y T σ ( W ) ( u , z ) Y ( w , z ) − Y ( w , z ) Y T σ ( N ) ( u , z )= Res z k z − δ ( z − z ) /k z /k ! Y (cid:0) Y ( u , z ) w , z (cid:1) . Y W ( u, z ) Y ( w, z ) − Y ( w, z ) Y N ( u, z )= Y T σ ( W ) (cid:18)(cid:16) ∆ k (cid:0) z k (cid:1) − u (cid:17) , z k (cid:19) Y (cid:18)(cid:16) ∆ k (cid:0) z k (cid:1) − w (cid:17) , z k (cid:19) − Y (cid:18)(cid:16) ∆ k (cid:0) z k (cid:1) − w (cid:17) , z k (cid:19) Y T σ ( N ) (cid:16)(cid:0) ∆ k ( z k ) − u (cid:1) , z k (cid:17) = Res x k z − k δ (cid:0) z k − x (cid:1) /k z ! Y (cid:18) Y (cid:18)(cid:16) ∆ k (cid:0) z k (cid:1) − u (cid:17) , x (cid:19) (cid:0) ∆ k ( z k ) − w (cid:1) , z k (cid:19) . We are going to use the change-of-variable x = z k − ( z − z ) k . Note that for any n ∈ Z , (cid:0) z k − x (cid:1) n/k | x = z k − ( z − z ) k = ( z − z ) n , (5.33)where by convention ( z k − x ) n/k = P j ≥ (cid:0) n/kj (cid:1) ( − j z n − kj x j . Using (3.12) in [BDM], we have Y W ( u, z ) Y ( w, z ) − Y ( w, z ) Y N ( u, z )= Res z z − k ( z − z ) k − δ (cid:18) z − z z (cid:19) Y (cid:18) Y (cid:16)(cid:0) ∆ k ( z k ) − u (cid:1) , z k − ( z − z ) k (cid:17) (cid:16) ∆ k (cid:0) z k (cid:1) − w (cid:17) , z k (cid:19) = Res z z − δ (cid:18) z − z z (cid:19) Y (cid:16) Y (cid:16)(cid:0) ∆ k ( z k ) − u (cid:1) , z k − ( z − z ) k (cid:17) (cid:0) ∆ k ( z k ) − w (cid:1) , z k (cid:17) = Res z z − δ (cid:18) z − z z (cid:19) Y (cid:18) Y (cid:18) ∆ k (cid:16) ( z + z ) k (cid:17) − u, ( z + z ) k − z k (cid:19) ∆ k ( z k ) − w (cid:19) , z k ! . Then it suffices to prove Y (cid:16) ∆ k (cid:0) ( z + z ) k (cid:1) − u, ( z + z ) k − z k (cid:17) ∆ k ( z k ) − = ∆ k ( z k ) − Y ( u, z ) , (5.34)or equivalently ∆ k ( z k ) Y (cid:16) ∆ k (cid:0) ( z + z ) k (cid:1) − u, ( z + z ) k − z k (cid:17) ∆ k ( z k ) − = Y ( u, z ) . This last relation follows from (4.24) ([BDM]; Proposition 2.2) by replacing u, z and z with ∆ k (( z + z ) k ) − u , z k and ( z + z ) k − z , respectively. Now, the proof is complete.As the second main result of section we have: Theorem 5.9. Let M , N and W be V -modules and let σ = (1 2 · · · k ) ∈ Aut ( V ⊗ k ) . Set M = M ⊗ V ⊗ ( k − . Let Y ( · , z ) be any intertwining operator of type (cid:0) T σ ( W ) M T σ ( N ) (cid:1) . For w ∈ M , set Y ( w, z ) = Y (cid:16)(cid:0) ∆ k ( z k ) − w (cid:1) , z k (cid:17) , an element of ( Hom ( N, W )) { z } . Then Y ( · , z ) is an intertwining operator of type (cid:0) WM N (cid:1) . Proof. As the L ( − -property has been given in Lemma 5.7, it remains to prove the Jacobi identity which is equivalentto the commutator formula which was obtained in Lemma 5.8 and the weak associativity which states that for any u ∈ V, w ∈ M, a ∈ N, there exists a nonnegative integer n such that ( z + z ) n Y W ( u, z + z ) Y ( w, z ) a = ( z + z ) n Y ( Y M ( u, z ) w, z ) a. Let u ∈ V, w ∈ M . Recall u = u ⊗ ⊗ ( k − ∈ V ⊗ k , w = w ⊗ ⊗ ( k − ∈ M . 31e have u = P k − r =0 u ,r , where u i,r ∈ ( V ⊗ k ) r , i.e., σu ,r = η rk u ,r . By the twisted Jacobi identity, there exists apositive integer l such that for n ≥ l , ( x + x ) r/k + n Y T σ ( W ) ( u ,r , x + x ) Y ( w , x ) a = ( x + x ) r/k + n Y (cid:0) Y M ( u ,r , x ) w , x (cid:1) a for r = 0 , , . . . , k − . Using change-of-variables x = z k and x = ( z + z ) k − z k , we obtain ( z + z ) r + kn Y T σ ( W ) (cid:0) u ,r , ( z + z ) k (cid:1) Y (cid:0) w , z k (cid:1) a = ( z + z ) r + kn Y (cid:0) Y M (cid:0) u ,r , ( z + z ) k − z k (cid:1) w , z k (cid:1) a. Thus there exists a positive integer p such that ( z + z ) n Y T σ ( W ) (cid:0) u ,r , ( z + z ) k (cid:1) Y (cid:0) w , z k (cid:1) a = ( z + z ) n Y (cid:0) Y M (cid:0) u ,r , ( z + z ) k − z k (cid:1) w , z k (cid:1) a for n ≥ p, r = 0 , , . . . , k − . Consequently, we have ( z + z ) n Y T σ ( W ) (cid:0) u , ( z + z ) k (cid:1) Y (cid:0) w , z k (cid:1) a = ( z + z ) n Y (cid:0) Y M (cid:0) u , ( z + z ) k − z k (cid:1) w , z k (cid:1) a for n ≥ p .Notice that ∆ k ( z k ) − u ∈ V [ z, z − ] and ∆ k ( z k ) − w ∈ M [ z, z − ] . (Both of them are finite sums.) Then, thereexists a positive integer q such that ( z + z ) n Y T σ ( W ) (cid:0) (∆ k ( z + z ) k ) − u ) , ( z + z ) k (cid:1) Y (cid:0) (∆ k ( z k ) − w ) , z k (cid:1) a = ( z + z ) n Y (cid:0) Y M (cid:0) (∆ k ( z + z ) k ) − u ) , ( z + z ) k − z k (cid:1) (∆ k ( z k ) − w ) , z k (cid:1) a for n ≥ q . Therefore, we obtain ( z + z ) n Y W ( u, z + z ) Y ( w, z ) a = ( z + z ) n Y T σ ( W ) (cid:18) ∆ k (cid:16) ( z + z ) k (cid:17) − u (cid:19) , ( z + z ) k ! Y (cid:18)(cid:16) ∆ k (cid:0) z k (cid:1) − w (cid:17) , z k (cid:19) a = ( z + z ) n Y (cid:18) Y M (cid:18)(cid:16) ∆ k (cid:0) ( z + z ) k (cid:1) − u (cid:17) , ( z + z ) k − z k (cid:19) (cid:16) ∆ (cid:0) z k (cid:1) − w (cid:17) , z k (cid:19) a = ( z + z ) n Y (cid:18)(cid:16) ∆ k (cid:0) z k (cid:1) − Y ( u, z ) w (cid:17) , z k (cid:19) a = ( z + z ) n Y ( Y M ( u, z ) w, z ) a for n ≥ p , where we are using (5.34) for the second last equality. This completes the proof.To summarize, we have: Corollary 5.10. The linear map π defined in (5.31) is a linear isomorphism from I V (cid:0) WM N (cid:1) to I V ⊗ k (cid:0) T σ ( W ) M T σ ( N ) (cid:1) . V ⊗ k -modules with permutation automorphism twisted V ⊗ k -modules In this section, we study tensor product using the results of Section 5 from the k -cycle σ to an arbitrary permutation in S k and present the second main theorem of this paper. Some of the results in this section are valid for arbitrary vertexoperator algebras.Let V be a general vertex operator algebra for now. Recall Definition 2.18 for a tensor product of a g -twisted V -module M with a g -twisted V -module N . As a convention, we shall use M ⊠ N to denote a generic tensor productmodule, provided that its existence is justified. 32 emark 6.1. Let V be any vertex operator algebra. Note that for any V -module ( M, Y M ) , the pair ( M, Y M ) is atensor product of V and M . Furthermore, ( M, Y oM ) is a tensor product of M and V , where Y oM ( · , z ) is defined by Y oM ( w, z ) v = e zL ( − Y M ( v, − z ) w for v ∈ V, w ∈ M (see [FHL]). Thus M ⊠ V ≃ M ≃ V ⊠ M . Remark 6.2. Let U and V be vertex operator algebras and let ( W, Y W ) be a U -module and ( M, Y M ) a V -module.Then it is straightforward to show that W ⊗ M is a tensor product module for U ⊗ V -modules W ⊗ V and U ⊗ M ,where Y ( w ⊗ v, z )( u ⊗ m ) = e zL ( − Y W ( v, − z ) w ⊗ Y M ( v, z ) m (6.1)for w ∈ M, v ∈ V, u ∈ U, m ∈ M . That is, W ⊗ M ≃ ( W ⊗ V ) ⊠ ( U ⊗ M ) .Let k be a positive integer. Recall from [BDM] that for any V -module W , we have a σ -twisted V ⊗ k -module T σ ( W ) which equals W as a vector space and a V -module homomorphism from W to W is exactly the same as a σ -twisted V ⊗ k -module homomorphism from T σ ( W ) to T σ ( W ) .Let M and N be V -modules. Assume that ( M ⊠ N, Y ) is a tensor product of M and N . That is, M ⊠ N is a V -module and Y is an intertwining operator of type (cid:0) M ⊠ NM N (cid:1) , satisfying the universal property. From Theorem 5.5, wehave an intertwining operator Y of type (cid:0) T σ ( M ⊠ N ) M T σ ( N ) (cid:1) . It follows from Corollary 5.10 that ( T σ ( M ⊠ N ) , Y ) is a tensorproduct of M ⊗ V ⊗ ( k − with T σ ( N ) . Thus we have proved: Theorem 6.3. Let M and N be V -modules. Suppose that there exists a tensor product ( M ⊠ N, Y ) of M and N .Then a tensor product of M ⊗ V ⊗ ( k − with T σ ( N ) exists and ( M ⊗ V ⊗ ( k − ) ⊠ T σ ( N ) ≃ T σ ( M ⊠ N ) In particular, we have ( M ⊗ V ⊗ ( k − ) ⊠ T σ ( V ) ≃ T σ ( M ) and Theorem A (in Introduction) is true.Next, we shall generalize this result. First, we formulate the following lemma which is straightforward to prove: Lemma 6.4. Let U be a vertex operator algebra and let σ, τ be automorphisms of U with σ of finite order. Assumethat W , W are σ -twisted U -modules and W is a V -module on which τ acts such that τ Y W ( u, z ) w = Y W ( τ ( u ) , z ) τ w for u ∈ U, w ∈ W. (6.2)If I ( · , z ) is an intertwining operator of type (cid:0) W W W (cid:1) , then I τ − ( · , z ) is an intertwining operator of type (cid:0) W W τ W (cid:1) ,where W τ = W as a vector space, Y W τ ( u, z ) = Y W ( τ ( u ) , z ) for u ∈ U , and I τ − ( w, z ) = I ( τ − ( w ) , z ) for w ∈ W. Furthermore, if ( P, Y ) is a tensor product of W and W , then ( P, Y τ − ) is a tensor product of W τ and W . Inparticular, we have W ⊠ W ≃ W τ ⊠ W , provided that one of the two tensor products exists.We now in a position to prove Theorem B (in Introduction). Proof. First of all, combining Theorem 6.3 with Lemma 6.4, we have ( V ⊗ ( i − ⊗ M ⊗ V ⊗ ( k − i ) ) ⊠ T σ ( N ) ≃ T σ ( M ⊠ N ) for any V -module M and for ≤ i ≤ k . This shows that this theorem is true if for some ≤ r ≤ k , M i = V for all i = r , as M ⊠ · · · ⊠ M k ≃ M r . For the general case, notice that as M i ⊠ V ≃ M i ≃ V ⊠ M i for ≤ i ≤ k , we have M ⊗ · · · ⊗ M k ≃ ⊠ ki =1 ( V ⊗ ( i − ⊗ M i ⊗ V ⊗ ( k − i ) ) , where we are also using Remark 6.2. Then the general case follows from the special case and Theorem 3.7 (anassociativity). 33ow, we consider an arbitrary permutation σ ∈ S k . Recall that a partition of k is a sequence of positive integers k , k , . . . , k s such that k ≥ k ≥ · · · ≥ k s ≥ and k + · · · + k s = k. Associated to each partition κ = ( k , k , . . . , k s ) , we have a permutation σ κ := σ ◦ σ ◦ · · · ◦ σ s , (6.3)where σ = (12 · · · k ) , σ = (( k + 1) · · · ( k + k )) , and so on. A classical fact is that every permutation in S k isconjugate to σ κ for a uniquely determined partition κ of k . Remark 6.5. Let U be a vertex operator algebra and let σ, τ be finite order automorphisms of U such that τ = µσµ − for some automorphism µ of U . It is straightforward to show that a linear map Y W ( · , z ) : U → (End W ) { z } is a τ -twisted U -module structure on W if and only if Y µW ( · , z ) is a σ -twisted U -module structure on W , where Y µW ( u, z ) = Y W ( µ ( u ) , z ) for u ∈ U. If W is a τ -twisted U -module, we denote by W τ the corresponding σ -twisted U -module. On the other hand, if W , W are τ -twisted U -modules, a τ -twisted U -module homomorphism from W to W is exactly the same as a σ -twisted U -module homomorphism from W µ to W µ . Therefore, the category of τ -twisted U -modules is isomorphicto the category of σ -twisted U -modules canonically.Furthermore, by a straightforward argument we have: Lemma 6.6. Let V be any vertex operator algebra and let σ and µ be automorphisms of V with σ of finite order. Set τ = µσµ − . Assume that W is a V -module and T , T are τ -twisted V -modules. Then an intertwining operator oftype (cid:0) T W T (cid:1) is exactly the same as an intertwining operator of type (cid:0) T µ W µ T µ (cid:1) . Proposition 6.7. Let V be any vertex operator algebra and let σ and µ be automorphisms of V with σ of finite order.Set τ = µσµ − . Let W be a V -module and T a τ -twisted V -module. Assume that there exists a tensor product W µ − ⊠ T (a τ -twisted V -module). Then there exists a tensor product W ⊠ T µ (a σ -twisted V -module) and W ⊠ T µ ≃ ( W µ − ⊠ T ) µ . (6.4) Proof. Recall that T µ is a σ -twisted V -module and W µ − is a V -module with W as the underlying space, where thevertex operator map Y µ − W ( · , x ) is given by Y µ − W ( v, x ) = Y W ( µ − v, x ) for v ∈ V. From definition, we have an intertwining operator Y of type (cid:0) W µ − ⊠ TW µ − T (cid:1) , satisfying the universal property. Let P beany σ -twisted V -module and I ( · , z ) be any intertwining operator of type of (cid:0) PW T µ (cid:1) . By Lemma 6.6, I ( · , z ) is also anintertwining operator of type (cid:0) P µ − W µ − T (cid:1) . Then there exists a τ -twisted V -module homomorphism ψ from W µ − ⊠ T to P µ − such that I ( w, z ) t = ψ ( Y ( w, z ) t ) for w ∈ W, t ∈ T . Note that from Remark 6.5, ψ is also a σ -twisted V -module homomorphism from ( W µ − ⊠ T ) µ to P . This proves that (( W µ − ⊠ T ) µ , Y ) is a tensor product of W with T µ . Therefore, we have ( W µ − ⊠ T ) µ ≃ W ⊠ T µ , as desired. Remark 6.8. Let V and V be vertex operator algebras and let τ , τ be finite order automorphisms of V , V , re-spectively. Set τ = τ ⊗ τ , an automorphism of V ⊗ V . Then the same arguments of [FHL] show that if W i is anirreducible τ i -twisted V i -module for i = 1 , , then W ⊗ W is naturally an irreducible τ -twisted V ⊗ V -module andon the other hand, every irreducible τ -twisted V ⊗ V -module is isomorphic to one in this form. Furthermore, if V i is τ i -rational for i = 1 , , then V ⊗ V is τ -rational (see [BDM], Lemma 6.1 and Proposition 6.2).As before, we view the symmetric group S k as an automorphism group of V ⊗ k . From Remarks 6.5 and 6.8 andTheorem 5.3 (due to [BDM]) we immediately have: 34 roposition 6.9. Let V be any vertex operator algebra and σ ∈ S k be any permutation. Suppose that µσµ − = σ κ ,where µ ∈ S k and κ is the partition of k which is uniquely determined by σ . Then for any s irreducible V -modules N , ..., N s , we have an irreducible σ -twisted V ⊗ k -module T σ ( N , . . . , N s ) := ( T σ ( N ) ⊗ · · · ⊗ T σ s ( N s )) µ , (6.5)where σ , . . . , σ s are disjoint cycles as in (6.3), and on the other hand, every irreducible σ -twisted V ⊗ k -module isisomorphic to one in this form. Furthermore, for irreducible V -modules N , ..., N s and P , . . . , P s , T σ ( N , . . . , N s ) ≃ T σ ( P , . . . , P s ) if and only if N i ≃ P i for ≤ i ≤ s .To present our general result, we also need the following result: Proposition 6.10. Let V , . . . , V s be vertex operator algebras, let τ , . . . , τ s be finite order automorphisms of V , . . . , V s ,respectively. Assume that V i is τ i -rational for ≤ i ≤ s . Set τ = τ ⊗ · · · ⊗ τ s ∈ Aut ( V ⊗ · · · ⊗ V s ) . Assume that W i is a V i -module and T i is a τ i -twisted V i -module for i = 1 , . . . , s such that dim I V i (cid:0) P i W i T i (cid:1) < ∞ forevery irreducible τ i -twisted V i -module P i for ≤ i ≤ s . Then there exist tensor products W ⊠ T , . . . , W s ⊠ T s , ( W ⊗ · · · ⊗ W s ) ⊠ ( T ⊗ · · · ⊗ T s ) , and ( W ⊠ T ) ⊗ · · · ⊗ ( W s ⊠ T s ) ≃ ( W ⊗ · · · ⊗ W s ) ⊠ ( T ⊗ · · · ⊗ T s ) . (6.6) Proof. By induction, it suffices to prove that it is true for s = 2 . From Remark 2.19, there exist tensor products W ⊠ T and W ⊠ T . By definition, we have an intertwining operator Y of type (cid:0) W ⊠ T W T (cid:1) and an intertwiningoperator Y of type (cid:0) W ⊠ T W T (cid:1) , satisfying the universal property. Define a linear map Y ( · , z ) · : ( W ⊗ W ) ⊗ ( T ⊗ T ) → (( W ⊠ T ) ⊗ ( W ⊠ T )) { z } by Y ( w ⊗ w , z )( t ⊗ t ) = Y ( w , z ) t ⊗ Y ( w , z ) t (6.7)for w i ∈ W i , t i ∈ T i with i = 1 , . Then Y is an intertwining operator of type (cid:0) ( W ⊠ T ) ⊗ ( W ⊠ T ) W ⊗ W T ⊗ T (cid:1) (see the commentsright before Theorem 2.16). We claim that (( W ⊠ T ) ⊗ ( W ⊠ T ) , Y ) is a tensor product of W ⊗ W and T ⊗ T .Let T be any τ -twisted V ⊗ V -module and let I be any intertwining operator of type (cid:0) TW ⊗ W T ⊗ T (cid:1) . We must showthat there exists a τ -twisted V ⊗ V -module homomorphism ψ from ( W ⊠ T ) ⊗ ( W ⊠ T ) to T such that I = ψ ◦Y . As V ⊗ V is τ -rational, it suffices to consider the case with T an irreducible τ -twisted V ⊗ V -module. From Remark 6.8, T ≃ P ⊗ P , where P i is an irreducible τ i -twisted V i -module for i = 1 , . As dim I V i (cid:0) P i W i T i (cid:1) < ∞ by assumption,from Theorem 2.16 (due to [ADL]), I = P rj =1 I ,j ⊗ I ,j , where I ,j are intertwining operators of type (cid:0) P W T (cid:1) and I ,j are intertwining operators of type (cid:0) P W T (cid:1) . By the universal property of ( W i ⊠ T i , Y i ) for i = 1 , , there exist τ i -twisted V i -module homomorphisms ψ i,j : W i ⊠ T i → P i such that I i,j ( · , z ) = ψ i,j ◦ Y i ( · , z ) for i = 1 , , ≤ j ≤ r .Set ψ = P rj =1 ψ ,j ⊗ ψ ,j , which is a τ -twisted V ⊗ V -module homomorphism from ( W ⊠ T ) ⊗ ( W ⊠ T ) to T such that I = ψ ◦ Y . This proves that (( W ⊠ T ) ⊗ ( W ⊠ T ) , Y ) is a tensor product of ( W ⊗ W ) with T ⊗ T .Thus tensor product ( W ⊗ W ) ⊠ ( T ⊗ T ) exists and ( W ⊠ T ) ⊗ ( W ⊠ T ) ≃ ( W ⊗ W ) ⊠ ( T ⊗ T ) .We now prove Theorem C. Proof. Using Proposition 6.10 and Theorem B, we obtain ( M ⊗ · · · ⊗ M k ) ⊠ T σ κ ( N , N , . . . , N s )= ( M ⊗ · · · ⊗ M k ) ⊠ ( T σ ( N ) ⊗ · · · ⊗ T σ s ( N s )) ≃ (cid:16) M [ k ] ⊠ T σ ( N ) (cid:17) ⊗ · · · ⊗ (cid:16) M [ k s ] ⊠ T σ s ( N ) (cid:17) ≃ T σ (cid:16) M [ k ] ⊠ N (cid:17) ⊗ · · · ⊗ T σ s (cid:16) M [ k s ] ⊠ N s (cid:17) = T σ κ (cid:16) ( M [ k ] ⊠ N ) , . . . , ( M [ k s ] ⊠ N s ) (cid:17) , proving the first assertion. Then using Proposition 6.7 we get the second assertion.35 eferences [A1] T. Abe, C -cofiniteness of the -cycle permutation orbifold models of minimal Virasoro vertex operatoralgebras, Commun. Math. Phys. (2011), 825–844.[A2] T. Abe, C -cofiniteness of 2-cyclic permutation orbifold models, Commun. Math. Phys. (2013), 425–445.[ADL] T. 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Dong : Department of Mathematics, University of California Santa Cruz, CA 95064 USA; [email protected] H. Li : Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102 USA; [email protected] F. Xu : Department of Mathematics, University of California, Riverside, CA 92521 USA; [email protected] N. Yu : School of Mathematical Sciences, Xiamen University, Fujian, 361005, CHINA; [email protected]@xmu.edu.cn