Fusion rules for Z 2 -orbifolds of affine and parafermion vertex operator algebras
aa r X i v : . [ m a t h . QA ] A p r Fusion rules for Z -orbifolds of affine and parafermionvertex operator algebras Cuipo Jiang a and Qing Wang b a School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240,China b School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
Abstract
This paper is about the orbifold theory of affine and parafermion vertex operatoralgebras. It is known that the parafermion vertex operator algebra K ( sl , k ) asso-ciated to the integrable highest weight modules for the affine Kac-Moody algebra A (1)1 is the building block of the general parafermion vertex operator K ( g , k ) for anyfinite dimensional simple Lie algebra g and any positive integer k . We first classifythe irreducible modules of Z -orbifold of the simple affine vertex operator algebraof type A (1)1 and determine their fusion rules. Then we study the representationsof the Z -orbifold of the parafermion vertex operator algebra K ( sl , k ), we give thequantum dimensions, and more technically, fusion rules for the Z -orbifold of theparafermion vertex operator algebra K ( sl , k ) are completely determined. This paper is a continuation in a series of papers on the study of the orbifold theory ofaffine and parafermion vertex operator algebras. It is known that the parafermion vertexoperator algebra K ( g , k ) is the commutant of a Heisenberg vertex operator subalgebra inthe simple affine vertex operator algebra L ˆ g ( k, L ˆ g ( k,
0) is the integrable highestweight module with the positive integer level k for the affine Kac-Moody algebra ˆ g asso-ciated to a finite dimensional simple Lie algebra g over C . We denote K ( sl , k ) by K and L ˆ sl ( k,
0) by L ( k,
0) in this paper. Since parafermion vertex operator algebras canbe identified with W -algebras [17], the orbifold theory of the parafermion vertex algebrascorresponds to the orbifold theory of W -algebras. Some conjectures in the physics liter-ature about the orbifold W -algebras have been studied and solved in [4], [3], [30]. Theseresults about the orbifold W -algebras are mainly structural aspects. Our interest is tostudy the representation theory of the orbifold parafermion vertex operator algebra fromthe point of vertex algebras. From [17], we know that the full automorphism group of theparafermion vertex operator algebra K for k ≥ σ , which is determined by σ ( h ) = − h , σ ( e ) = f , σ ( f ) = e , where { h, e, f } is a standard Chevalley basis of sl with brackets [ h, e ] = 2 e , [ h, f ] = − f and [ e, f ] = h .We have classified the irreducible modules of the orbifold parafermion vertex operatoralgebra K σ in [28], where K σ is the fixed-point vertex operator subalgebra of K under Supported by China NSF grants No.11771281 and No.11531004. Supported by China NSF grants No.11622107 and No.11531004, Natural Science Foundation of FujianProvince No.2016J06002. . A natural problem next is to determine the fusion rules for K σ . Note that the vertexoperator algebra K σ can be viewed as a subalgebra of the orbifold affine vertex operatoralgebra L ( k, σ , where L ( k, σ is the fixed-point vertex operator subalgebra of L ( k, σ . In order to understand the representation theory of the orbifold parafermionvertex operator algebra K σ better, we should first understand the representation theoryof the orbifold affine vertex operator algebra L ( k, σ first. For this purpose, we classifythe irreducible modules of L ( k, σ and determine the fusion rules for L ( k, σ in Section3. We obtain Theorem 3.22 that there are two kinds of irreducible modules for L ( k, σ .One kind is the untwisted type modules coming from the irreducible L ( k, σ -twisted L ( k, L ( k, σ -modules in Theorem 3.25. These results together with the symmetric property of fusionrules imply that we only need to determine two kinds of fusion products, one is the fusionproduct between the untwisted type modules and the untwisted type modules, and theother is the fusion product between the untwisted type modules and the twisted typemodules. Our first step is to construct the intertwining operators among untwisted andtwisted L ( k, L ( k,
0) andthe intertwining operator constructed from the ∆-operator. Furthermore, by observingthe action of the automorphism σ on the ∆-operator, the fusion products between theuntwisted type modules and the untwisted type modules follow from the fusion productsbetween the untwisted type modules and the twisted type modules.The determination of the fusion rules for K σ is much more complicated. We firstdetermine the quantum dimensions of the irreducible K σ -modules, which can help us todetermine the fusion rules for K σ . However it is far from the complete determination ofthe fusion rules for K σ . Our strategy is to employ the lattice realization of the irreducible K -modules [17] and the lowest weights of the irreducible K σ -modules [28], together withthe decomposition of the irreducible L ( k, L ( k, i ) viewed as the modules of thelattice vertex operator subalgebra V Z γ ⊆ L ( k, ≤ i ≤ k . From the classificationresults of the irreducible modules of K σ , there are two families of untwisted type K σ -modules. One family is from the irreducible modules of K , which are not irreducibleas K σ -modules. We call it the untwisted module of type I . The other family is fromthe irreducible modules of K , which are also irreducible as K σ -modules. We call itthe untwisted module of type II . We would like to point out that the main difficulty todetermine the fusion products between the untwisted type modules and the untwisted typemodules of K σ is to find which one of the irreducible K σ -modules of type I can survivein the decomposition of the fusion product, and to distinguish the inequivalent modulesemerging in the decomposition of the fusion product. The fusion products between theuntwisted type modules and the twisted type modules of K σ are extremely complicatedin the case that the level k is even, because from [28], we know that in the level k , thereare two irreducible twisted modules of K , and the lowest weight vector can be in thegrade zero or in the grade of the σ -twisted module of K . Thus as the K σ -modules,there are four irreducible modules in the level k , when it emerges in the decomposition2f the fusion product between the untwisted type module and the twisted type module of K σ . We need to distinguish which one can survive for certain cases. The strategy is thatwe come back to the lattice realization of the irreducible K -modules M i,j for 0 ≤ i ≤ k ,0 ≤ j ≤ i [17], and we technically use another basis of the Lie algebra sl and apply theintertwining operator among the modules of the lattice vertex operator algebra, togetherwith the analysis of the lowest weights of the irreducible K σ -modules we obtained in [28].Furthermore, we determine the contragredient modules of all the irreducible K σ -modules,thus the fusion rules for K σ are completely determined.The paper is organized as follows. In Section 2, we recall some results about theparafermion vertex operator algebra K , its orbifold vertex operator subalgebra K σ andtheir irreducible modules. In Section 3, we classify the irreducible modules of the Z -orbifold L ( k, σ of the affine vertex operator algebra L ( k,
0) and determine the fusionrules for L ( k, σ . In Section 4, we give the quantum dimensions for irreducible K σ -modules. In Section 5, we determine the fusion rules for the Z -orbifold of parafermionvertex operator algebra K . In this section, we recall from [17], [19], [23], [5] and [28] some basic results on theparafermion vertex operator algebra associated to the irreducible highest weight modulefor the affine Kac-Moody algebra A (1)1 of level k with k being a positive integer and their Z -orbifolds. We first recall the notion of the parafermion vertex operator algebra.We are working in the setting of [17]. Let { h, e, f } be a standard Chevalley basis of sl with Lie brackets [ h, e ] = 2 e , [ h, f ] = − f , [ e, f ] = h and the normalized Killing form h h, h i = 2, h e, f i = 1, h h, e i = h h, f i = h e, e i = h f, f i = 0. Let b sl = sl ⊗ C [ t, t − ] ⊕ C C be the affine Lie algebra associated to sl . Let k ≥ V ( k,
0) = V b sl ( k,
0) = Ind b sl sl ⊗ C [ t ] ⊕ C C C be the induced b sl -module such that sl ⊗ C [ t ] acts as 0 and C acts as k on = 1. Then V ( k,
0) is a vertex operator algebra generated by a ( − for a ∈ sl such that Y ( a ( − , z ) = a ( z ) = X n ∈ Z a ( n ) z − n − where a ( n ) = a ⊗ t n , with the vacuum vector and the Virasoro vector ω aff = 12( k + 2) (cid:16) h ( − + e ( − f ( − + f ( − e ( − (cid:17) = 12( k + 2) (cid:16) − h ( − + 12 h ( − + 2 e ( − f ( − (cid:17) of central charge kk +2 (e.g. [27], [29], [34, Section 6.2]).3et M ( k ) be the vertex operator subalgebra of V ( k,
0) generated by h ( − with theVirasoro element ω γ = 14 k h ( − of central charge 1.The vertex operator algebra V ( k,
0) has a unique maximal ideal J , which is generatedby a weight k + 1 vector e ( − k +1 [29]. The quotient algebra L ( k,
0) = V ( k, / J is asimple, rational vertex operator algebra as k is a positive integer (cf. [27], [34]). Moreover,the image of M ( k ) in L ( k,
0) is isomorphic to M ( k ) and will be denoted by M ( k ) again.Set K ( sl , k ) = { v ∈ L ( k, | h ( m ) v = 0 for h ∈ h , m ≥ } . Then K ( sl , k ) which is the space of highest weight vectors with highest weight 0 for b h isthe commutant of M ( k ) in L ( k,
0) and is called the parafermion vertex operator algebraassociated to the irreducible highest weight module L ( k,
0) for c sl . The Virasoro elementof K ( sl , k ) is given by ω = ω aff − ω γ = 12 k ( k + 2) (cid:16) − kh ( − − h ( − + 2 ke ( − f ( − (cid:17) with central charge k − k +2 , where we still use ω aff , ω γ to denote their images in L ( k, K ( sl , k ) by K .Set W = k h ( − + 3 kh ( − h ( − + 2 h ( − − kh ( − e ( − f ( − + 3 k e ( − f ( − − k e ( − f ( − in V ( k, L ( k,
0) by W . It was proved in [17](cf.[19], [22])that the parafermion vertex operator algebra K is simple and is generated by ω and W . If k ≥
3, the parafermion vertex operator algebra K in fact is generated by W .The irreducible K -modules M i,j for 0 ≤ i ≤ k, ≤ j ≤ k − K = M , . It was also proved in [17, Theorem 4.4] that M i,j ∼ = M k − i,k − i + j as K -module . Theorem 8.2 in [5] showed that the k ( k +1)2 irreducible K -modules M i,j for1 ≤ i ≤ k, ≤ j ≤ i − K -modules. Moreover, K is C -cofinite [5] and rational [6] (see also [20]).Let L ( k, i ) for 0 ≤ i ≤ k be the irreducible modules for the rational vertex operatoralgebra L ( k,
0) with the top level U i = L ij =0 C v i,j which is an ( i + 1)-dimensional irre-ducible module of the simple Lie algebra C h (0) ⊕ C e (0) ⊕ C f (0) ∼ = sl . The top level of M i,j is a one dimensional space spanned by v i,j for 0 ≤ i ≤ k, ≤ j ≤ i [17]. The followingresult was due to [17]. Lemma 2.1.
The operator o ( ω ) = ω acts on v i,j , ≤ i ≤ k, ≤ j ≤ i as follows: o ( ω ) v i,j = 12 k ( k + 2) (cid:16) k ( i − j ) − ( i − j ) + 2 kj ( i − j + 1) (cid:17) v i,j . (2.1)4et σ be an automorphism of Lie algebra sl defined by σ ( h ) = − h, σ ( e ) = f, σ ( f ) = e . σ can be lifted to an automorphism σ of the vertex operator algebra V ( k,
0) of order2 in the following way: σ ( x ( − n ) · · · x s ( − n s ) ) = σ ( x )( − n ) · · · σ ( x s )( − n s ) for x i ∈ sl and n i >
0. Then σ induces an automorphism of L ( k,
0) as σ preservesthe unique maximal ideal J , and the Virasoro element ω γ is invariant under σ . Thus σ induces an automorphism of the parafermion vertex operator algebra K . In fact, σ ( ω ) = ω, σ ( W ) = − W . Lemma 2.2. [17] If k ≥ , the automorphism group Aut K = h σ i is of order 2. Remark 2.3. If k = 1 , K = C . If k = 2 , K is generated by ω . Thus the automorphismgroup Aut K = { } is trivial for k = 1 and k = 2 . Therefore, by Lemma 2.2, we only needto consider the orbifold of parafermion vertex operator algebra under the automorphism σ for k ≥ . Let K σ be the Z -orbifold vertex operator algebra, i.e., the fixed-point vertex op-erator subalgebra of K under the automorphism σ . The following theorem gives theclassification of the irreducible modules of K σ for k ≥ Theorem 2.4. [28] If k = 2 n + 1 , n ≥ , there are ( k +1)( k +7)4 inequivalent irreduciblemodules of K σ . If k = 2 n , n ≥ , there are ( k +8 k +28)4 inequivalent irreducible modules of K σ . More precisely, if k = 2 n + 1 , n ≥ , the set { W ( k, i ) j for ≤ i ≤ k − , j = 1 , , ( M i,j ) s for ( i, j ) = ( i, i , i = 2 , , , · · · , n, and ( i, j ) = (2 n + 1 , , s = 0 , ,M i, for ≤ i ≤ k − , M i,j for ≤ i ≤ k, if i = 2 m, ≤ j ≤ m − , if i = 2 m + 1 , ≤ j ≤ m } gives all inequivalent irreducible K σ -modules. If k = 2 n , n ≥ , the set { W ( k, i ) j for ≤ i ≤ k , j = 1 , , ^ W ( k, k j for j = 1 , , ( M i,j ) s for ( i, j ) = ( i, i , i = 2 , , , · · · , n, ( i, j ) = ( n, and ( i, j ) = (2 n, , s = 0 , ,M i, for ≤ i ≤ k − , M i,j for ≤ i ≤ k, if i = 2 m, ≤ j ≤ m − , if i = 2 m + 1 , ≤ j ≤ m } gives all inequivalent irreducible K σ -modules. Remark 2.5.
With the notations in Theorem 2.4, we call W ( k, i ) j and ^ W ( k, k ) j twistedtype modules and ( M i,j ) s , M i,j untwisted modules of type I and type II respectively. Fusion rules for the Z -orbifold of the affine vertexoperator algebra L ( k, In this section, we first recall the definition of weak g -twisted modules, g -twisted modulesand admissible g -twisted modules following [15, 16]. Let L ( k, σ be the Z -orbifold vertexoperator subalgebra of the affine vertex operator algebra L ( k, L ( k,
0) under σ . We then classify and construct the irreducible modules for L ( k, σ . Furthermore, we determine the contragredient modules of irreducible L ( k, σ -modules and the fusion rules for the vertex operator algebra L ( k, σ .Let ( V, Y, , ω ) be a vertex operator algebra (see [26], [34]) and g an automorphism of V with finite order T . Let W { z } denote the space of W -valued formal series in arbitrarycomplex powers of z for a vector space W . Denote the decomposition of V into eigenspaceswith respect to the action of g by V = M r ∈ Z V r , where V r = { v ∈ V | gv = e − πirT v } , i = √− Definition 3.1. A weak g -twisted V -module M is a vector space with a linear map Y M : V → ( End M ) { z } v Y M ( v, z ) = X n ∈ Q v n z − n − ( v n ∈ End M ) which satisfies the following conditions for ≤ r ≤ T − , u ∈ V r , v ∈ V, w ∈ M : Y M ( u, z ) = X n ∈ rT + Z u n z − n − u n w = 0 for n ≫ ,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 ) , where δ ( z ) = P n ∈ Z z n . u m + rT , v n + sT ] = ∞ X i =0 (cid:18) m + rT i (cid:19) ( u i v ) m + n + r + sT − i , (3.2) X i ≥ (cid:18) rT i (cid:19) ( u m + i v ) n + r + sT − i = X i ≥ ( − i (cid:18) mi (cid:19) ( u m + rT − i v n + sT + i − ( − m v m + n + sT − i u rT + i ) , (3.3)where u ∈ V r , v ∈ V s , m, n ∈ Z . Definition 3.2. A g -twisted V -module is a weak g -twisted V -module M which carries a C -grading M = L λ ∈ C M λ , where M λ = { w ∈ M | L (0) w = λw } and L (0) is one of thecoefficient operators of Y ( ω, z ) = P n ∈ Z L ( n ) z − n − . Moreover we require that dim M λ isfinite and for fixed λ, M λ + nT = 0 for all small enough integers n. Definition 3.3. An admissible g -twisted V -module M = ⊕ n ∈ T Z + M ( n ) is a T Z + -gradedweak g -twisted module such that u m M ( n ) ⊂ M ( wt u − m − n ) for homogeneous u ∈ V and m, n ∈ T Z . If g = Id V , we have the notions of weak, ordinary and admissible V -modules [15]. Definition 3.4.
A vertex operator algebra V is called g -rational if the admissible g -twistedmodule category is semisimple. Remark 3.5.
Since K is a rational vertex operator algebra, K σ is C -cofinite and ratio-nal [35], [7], [8], and K is σ -rational [10]. The following lemma about g -rational vertex operator algebras is well known [15]. Lemma 3.6. If V is g -rational, then(1) Any irreducible admissible g -twisted V -module M is a g -twisted V -module, andthere exists a λ ∈ C such that M = ⊕ n ∈ T Z + M λ + n where M λ = 0 . And λ is called theconformal weight of M ; (2) There are only finitely many irreducible admissible g -twisted V -modules up to iso-morphism. Let M = L n ∈ T Z + M ( n ) be an admissible g -twisted V -module, the contragredientmodule M ′ is defined as follows: M ′ = L n ∈ T Z + M ( n ) ∗ , where M ( n ) ∗ = Hom C ( M ( n ) , C ) . The vertex operator Y M ′ ( v, z ) is defined for v ∈ V via h Y M ′ ( v, z ) f, u i = h f, Y M ( e zL (1) ( − z − ) L (0) v, z − ) u i , (3.4)where h f, w i = f ( w ) is the natural paring M ′ × M → C . Remark 3.7. ( M ′ , Y M ′ ) is an admissible g − -twisted V -module [25]. One can also definethe contragredient module M ′ for a g -twisted V -module M . In this case, M ′ is a g − -twisted V -module. Moreover, M is irreducible if and only if M ′ is irreducible. Definition 3.8.
Let ( V, Y ) be a vertex operator algebra and let ( W , Y ) , ( W , Y ) and ( W , Y ) be V -modules. An intertwining operator of type (cid:18) W W W (cid:19) is a linearmap I ( · , z ) : W → Hom ( W , W ) { z } u → I ( u, z ) = X n ∈ Q u n z − n − satisfying:(1) for any u ∈ W and v ∈ W , u n v = 0 for n sufficiently large;(2) I ( L ( − v, z ) = ddz I ( v, z ) ;(3) (Jacobi identity) for any u ∈ V, v ∈ W z − δ (cid:18) z − z z (cid:19) Y ( u, z ) I ( v, z ) − z − δ (cid:18) − z + z z (cid:19) I ( v, z ) Y ( u, z )= z − (cid:18) z − z z (cid:19) I ( Y ( u, z ) v, z ) . The space of all intertwining operators of type (cid:18) W W W (cid:19) is denoted by I V (cid:18) W W W (cid:19) . Let N W W , W = dim I V (cid:18) W W W (cid:19) . These integers N W W , W are usually called the fusionrules . Definition 3.9.
Let V be a vertex operator algebra, and W , W be two V -modules. Amodule ( W, I ) , where I ∈ I V (cid:18) WW W (cid:19) , is called a tensor product (or fusion product)of W and W if for any V -module M and Y ∈ I V (cid:18) MW W (cid:19) , there is a unique V -module homomorphism f : W → M, such that Y = f ◦ I. As usual, we denote ( W, I ) by W ⊠ V W . Remark 3.10.
It is well known that if V is rational, then for any two irreducible V -modules W and W , the fusion product W ⊠ V W exists and W ⊠ V W = X W N WW , W W, where W runs over the set of equivalence classes of irreducible V -modules. Fusion rules have the following symmetric property [25].8 roposition 3.11.
Let W i ( i = 1 , , be V -modules. Then N W W ,W = N W W ,W , N W W ,W = N ( W ) ′ W , ( W ) ′ . We will use the following lemma from [14] later.
Lemma 3.12.
Let V be a vertex operator algebra, and let W and W be irreducible V -modules and W a V -module. If I is a nonzero intertwining operator of type (cid:18) W W W (cid:19) ,then I ( u, z ) v = 0 for any nonzero vectors u ∈ W and v ∈ W . We fix some notations. Let W , W , W be irreducible L ( k, σ -modules. In thissection, we use I (cid:18) W W W (cid:19) to denote the space I L ( k, σ (cid:18) W W W (cid:19) of all intertwiningoperators of type (cid:18) W W W (cid:19) , and use W ⊠ W to denote the fusion product W ⊠ L ( k, σ W for simplicity. We recall the fusion rules for the affine vertex operator algebra of type A (1)1 [36] for later use. Lemma 3.13. L ( k, i ) ⊠ L ( k, L ( k, j ) = X l L ( k, l ) , where | i − j | ≤ l ≤ i + j, i + j + l ∈ Z , i + j + l ≤ k. We notice that since L ( k,
0) is rational, L ( k, σ is rational, and thus L ( k,
0) is σ -rational. Then from [16], we have the following result. Proposition 3.14.
There are precisely k + 1 inequivalent irreducible σ -twisted modulesof L ( k, .Proof. Since L ( k,
0) is σ -rational, from [16], we know that the number of inequivalentirreducible σ -twisted modules of L ( k,
0) is precisely the number of σ -stable irreducibleuntwisted modules of L ( k, L ( k, i ) for 0 ≤ i ≤ k exhaust all the irreduciblemodules for L ( k,
0) with the top level U i = L ij =0 C v i,j . By direct calculation, we have o ( ω aff ) v i,j = ω aff (1) v i,j = 12( k + 2) (cid:16) h (0) + 12 h (0) + 2 f (0) e (0) (cid:17) v i,j = i ( i + 2)4( k + 2) v i,j . (3.5)We see that these lowest weights i ( i +2)4( k +2) are pairwise different for 0 ≤ i ≤ k , which showsthat L ( k, i ) for 0 ≤ i ≤ k are σ -stable irreducible modules. Thus there are totally k + 1inequivalent irreducible σ -twisted modules of L ( k, { h, e, f } is a standard Chevalley basis of sl with brackets [ h, e ] =2 e , [ h, f ] = − f , [ e, f ] = h . Set h ′ = e + f, e ′ = 12 ( h − e + f ) , f ′ = 12 ( h + e − f ) . { h ′ , e ′ , f ′ } is a sl -triple. Let h ′′ = h ′ = ( e + f ), and∆( h ′′ , z ) = z h ′′ (0) exp( ∞ X k =1 h ′′ ( k ) − k ( − z ) − k ) . Note that L ( k, i ) for 0 ≤ i ≤ k are all the irreducible modules for the rational vertexoperator algebra L ( k, Lemma 3.15.
For ≤ i ≤ k , ( L ( k, i ) , Y σ ( · , z )) = ( L ( k, i ) , Y (∆( h ′′ , z ) · , z )) are irreducible σ -twisted L ( k, -modules. As in [28], for u ∈ L ( k,
0) such that σ ( u ) = e − πri u , i = √− r ∈ Z , we use thenotation u n and u ( n ) respectively to distinguish the action of the elements in L ( k,
0) on σ -twisted modules and untwisted modules as follows Y σ ( u, z ) = X n ∈ Z + r u n z − n − , Y ( u, z ) = X n ∈ Z u ( n ) z − n − . Recall that the top level U i = L ij =0 C v i,j of L ( k, i ) for 0 ≤ i ≤ k is an ( i + 1)-dimensionalirreducible module for C h (0) ⊕ C e (0) ⊕ C f (0) ∼ = sl . Let η i = i X j =0 ( − j v i,j , then η i is the lowest weight vector with weight − i in ( i +1)-dimensional irreducible modulefor C h ′ (0) ⊕ C e ′ (0) ⊕ C f ′ (0) ∼ = sl , that is, f ′ (0) η i = 0 and h ′ (0) η i = − iη i , and we have: Lemma 3.16. [28] For the positive integer k ≥ , and ≤ i ≤ k , L (0) η i = (cid:16) i ( i − k )4( k + 2) + k − (cid:17) η i . By Lemma 3.16, we have
Lemma 3.17. L aff (0) η i = (cid:16) i ( i − k )4( k + 2) + k (cid:17) η i . (3.6)We can now construct the k + 1 inequivalent irreducible σ -twisted modules of L ( k, Theorem 3.18. L ( k, i ) for ≤ i ≤ k are k + 1 inequivalent irreducible σ -twisted modulesof L ( k, generated by η i .Proof. We just need to notice that η i is the lowest weight vector of the σ -twisted module L ( k, i ), and h ′ η i = ( h ′ (0) + k ) η i = ( − i + k ) η i , this implies that L ( k, i ) for 0 ≤ i ≤ k are k + 1 inequivalent irreducible σ -twisted modules of L ( k,
0) generated by η i .10e now classify all the irreducible modules of the orbifold vertex operator algebra L ( k, σ . Set u k,i, = η i ∈ L ( k, i )(0) , u k,i, = ( e − f ) − η i ∈ L ( k, i )( 12 ) . (3.7)By applying the results in [15], we have: Proposition 3.19.
For ≤ i ≤ k , let L ( k, i ) + and L ( k, i ) − be the L ( k, σ -modulesgenerated by u k,i, and u k,i, respectively. Then L ( k, i ) + and L ( k, i ) − for ≤ i ≤ k areirreducible modules of L ( k, σ with the lowest weights L aff (0) u k,i, = (cid:16) i ( i − k )4( k + 2) + k (cid:17) u k,i, , L aff (0) u k,i, = (cid:16) i ( i − k )4( k + 2) + k + 816 (cid:17) u k,i, . Combining Proposition 3.14 and the results in [18], we have:
Proposition 3.20.
For ≤ i ≤ k , we have L ( k, i ) = L ( k, i ) + M L ( k, i ) − , where L ( k, i ) + for i = 0 is an irreducible module of L ( k, σ generated by η i with weight i ( i +2)4( k +2) , and L ( k, i ) − for i = 0 is an irreducible module of L ( k, σ generated by e ′ (0) η i withthe same weight i ( i +2)4( k +2) . And L ( k, + is an irreducible module of L ( k, σ generated by with weight , and L ( k, − is an irreducible module of L ( k, σ generated by e ( − withweight . Remark 3.21.
When we consider the basis { e, f, h } of sl with the automorphism τ ( e ) = − e, τ ( f ) = − f, τ ( h ) = h , L ( k, i ) + for i = 0 can also be viewed as an irreducible moduleof L ( k, τ generated by the lowest weight vector v i,i with weight i ( i +2)4( k +2) , and L ( k, i ) − for i = 0 can be viewed as an irreducible module of L ( k, σ generated by e (0) v i,i with thesame weight i ( i +2)4( k +2) . From the above discussion, we obtain the classification of the irreducible modules forthe orbifold vertex operator algebra L ( k, σ . Theorem 3.22.
There are k + 1) inequivalent irreducible modules of L ( k, σ and thelowest weights of these irreducible modules are listed in Proposition 3.19 and Proposition3.20. Remark 3.23.
We call irreducible modules L ( k, i ) ± for ≤ i ≤ k untwisted type modules,and L ( k, i ) ± for ≤ i ≤ k twisted type modules. We now determine the fusion rules for irreducible modules of L ( k, σ . We first provethe following lemma. 11 emma 3.24. For ≤ i, j, l ≤ k, i + j + l ∈ Z , i + j + l ≤ k , let Y ( · , z ) be an intertwiningoperator of L ( k, of type (cid:18) L ( k, l ) L ( k, i ) L ( k, j ) (cid:19) . Define e Y ( v, z ) = Y (∆( h ′′ , z ) v, z ) for v ∈ L ( k, i ) . Then e Y ( · , z ) is an intertwining operator of L ( k, σ of type (cid:18) L ( k, l ) L ( k, i ) L ( k, j ) (cid:19) .Proof. The proof is similar to the proof of Proposition 5.4 of [33]. For simplicity of thenotation, we set ∆( z ) = ∆( h ′′ , z ), then we have ∆( z ) = ,[ L aff ( − , ∆( z )] = − ddz ∆( z ) , and Y L ( k,i ) (∆( z + z ) a, z )∆( z ) = ∆( z ) Y L ( k,i ) ( a, z )for a ∈ L ( k, σ . Thus for a ∈ L ( k, σ , v ∈ L ( k, i ), we have z − δ (cid:18) z − z z (cid:19) Y L ( k,l ) ( a, z ) e Y ( v, z ) − z − δ (cid:18) z − z − z (cid:19) e Y ( v, z ) Y L ( k,j ) ( a, z )= z − δ (cid:18) z − z z (cid:19) Y L ( k,l ) (∆( z ) a, z ) Y (∆( z ) v, z ) − z − δ (cid:18) z − z − z (cid:19) Y (∆( z ) v, z ) Y L ( k,j ) (∆( z ) a, z )= z − δ (cid:18) z − z z (cid:19) Y (cid:0) Y L ( k,i ) (∆( z ) a, z )∆( z ) v, z (cid:1) = z − δ (cid:18) z − z z (cid:19) Y (cid:0) ∆( z ) Y L ( k,i ) ( a, z ) v, z (cid:1) = z − δ (cid:18) z − z z (cid:19) e Y (cid:0) Y L ( k,i ) ( a, z ) v, z (cid:1) So e Y ( · , z ) is an intertwining operator of L ( k, σ of type (cid:18) L ( k, l ) L ( k, i ) L ( k, j ) (cid:19) .We now determine the contragredient modules of irreducible L ( k, σ -modules. Firstwe recall from [17] that the irreducible K -modules M i,j for 0 ≤ i ≤ k, ≤ j ≤ i − V L ⊥ , where L = Z α + · · · + Z α k with h α i , α j i = 2 δ ij , and L ⊥ is the dual lattice of L . More concretely, the top level of M i,j is aone dimensional space spanned by v i,j and v i,j has the explicit form in V L ⊥ : v , = , v i, = X I ⊆ { , , · · · , k }| I | = i e α I / , v i,j = X I ⊆ { , , · · · , k } , | I | = i X J ⊆ I | J | = j e α I − J / − α J / , (3.8)12here α J = P i ∈ J α i for a subset J of { , · · · , k } , and the vertex operator associatedwith e α , α ∈ L ⊥ is defined on V L ⊥ by Y ( e α , z ) = exp (cid:16) ∞ X n =1 α ( − n ) n z n (cid:17) exp (cid:16) − ∞ X n =1 α ( n ) n z − n (cid:17) e α z α (0) . (3.9)From [26, Chapter 8] and [14] (see also [1]), the operator Y ( · , z ) produces the intertwiningoperator for V L of type (cid:18) V λ + λ + L V λ + L V λ + L (cid:19) for λ , λ ∈ L ⊥ . Theorem 3.25.
For ≤ i ≤ k . (1) If i ∈ Z , L ( k, i ) ± are self-dual. If i ∈ Z + 1 , then ( L ( k, i ) ± ) ′ ∼ = L ( k, i ) ∓ . (2) ( L ( k, i ) ± ) ′ = L ( k, k − i ) ± .Proof. First we prove (1). We know that if M is a module of a vertex operator algebra V ,and M ′ is the contragredient module of M , then V ⊆ M ⊠ M ′ . Note that ∈ L ( k, + , v i,i ∈ L ( k, i ) + , and from (3.8) we know that v i,i = X J ⊆ { , , · · · , k }| J | = i e − α J / . (3.10)Since ∈ L ( k, + ⊆ L ( k, i ) + ⊠ ( L ( k, i ) + ) ′ , from (3.9), we can deduce that v i, = X I ⊆ { , , · · · , k }| I | = i e α I / ∈ ( L ( k, i ) + ) ′ . (3.11)Note that v i, = 1 i ! e (0) i v i,i for i = 0. From Remark 3.21, we know that v i, ∈ L ( k, i ) + if i ∈ Z and v i, ∈ L ( k, i ) − if i ∈ Z + 1. That is, if i ∈ Z , L ( k, i ) + is self-dual, and if i ∈ Z + 1, ( L ( k, i ) + ) ′ ∼ = L ( k, i ) − .Thus, if i ∈ Z , L ( k, i ) − is self-dual, and if i ∈ Z + 1, ( L ( k, i ) − ) ′ ∼ = L ( k, i ) + .Next we prove (2), i.e., ( L ( k, i ) + ) ′ = L ( k, k − i ) + . Notice that the top level of theirreducible L ( k, σ -module L ( k, i ) + is one-dimensional and spanned by η i . We denotethe top level of the contragredient module ( L ( k, i ) + ) ′ by η ′ i . From the definition of con-tragredient module (3.4), we know that η i and η ′ i have the same weight. Thus from (3.6)and Proposition 3.19, we know that η ′ i = η i or η ′ i = η k − i . Also from the definition of thecontragredient module (3.4) and noting that L aff (0) h ′ = h ′ , L aff (1) h ′ = 0, we have that h o ( h ′ ) η ′ i , η i i = −h η ′ i , o ( h ′ ) η i i , where o ( h ′ ) = h ′ wt ( h ) − = h ′ . Since h ′ .η i = ( − i + k ) η i , it shows that η ′ i = η k − i , which impliesthat ( L ( k, i ) + ) ′ = L ( k, k − i ) + . It follows immediately that ( L ( k, i ) − ) ′ = L ( k, k − i ) − .13or 0 ≤ i ≤ k, ≤ j ≤ k, ≤ l ≤ k such that i + j + l ∈ Z , noticing that i + j − l / ∈ Z is equivalent to i + j − l + 2 ∈ Z , we definesign( i, j, l ) + = ( + , if i + j − l ∈ Z , − , if i + j − l / ∈ Z , and sign( i, j, l ) − = ( − , if i + j − l ∈ Z , + , if i + j − l / ∈ Z . The following theorem together with Proposition 3.11 and Theorem 3.25 give all thefusion rules for the Z -orbifold affine vertex operator algebra L ( k, σ . Theorem 3.26.
The fusion rules for the Z -orbifold affine vertex operator algebra L ( k, σ are as follows: L ( k, i ) + ⊠ L ( k, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k L ( k, l ) sign ( i,j,l ) ± , (3.12) L ( k, i ) − ⊠ L ( k, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k L ( k, l ) sign ( i,j,l ) ∓ , (3.13) L ( k, i ) + ⊠ L ( k, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k L ( k, l ) sign ( i,j,l ) ± , (3.14) L ( k, i ) − ⊠ L ( k, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k L ( k, l ) sign ( i,j,l ) ∓ . (3.15) Proof.
Let Y ( · , z ) be an intertwining operator of L ( k,
0) of type (cid:18) L ( k, l ) L ( k, i ) L ( k, j ) (cid:19) .From Lemma 3.24, we know that e Y ( · , z ) is an intertwining operator of L ( k, σ of type (cid:18) L ( k, l ) L ( k, i ) L ( k, j ) (cid:19) , where e Y ( v, z ) = Y (∆( h ′′ , z ) v, z ) for v ∈ L ( k, i ). Thus we have e Y ( η i , z ) = Y (∆( h ′′ , z ) η i , z ) = z − i Y ( η i , z ) , (3.16)14here η i = P ij =0 ( − j v i,j is the lowest weight vector of the σ -twisted module L ( k, i ).From (3.5), we know that η i has the weight i ( i +2)4( k +2) in L ( k, i ) for 0 ≤ i ≤ k . For simplicity,we denote a i = i ( i +2)4( k +2) . From (3.6), we know that η i has the weight i ( i − k )4( k +2) + k in L ( k, i )for 0 ≤ i ≤ k . We denote ˜ a i = i ( i − k )4( k +2) + k . From Lemma 3.13, we know that the fusionrule of the affine vertex operator algebra L ( k,
0) is L ( k, i ) ⊠ L ( k, L ( k, j ) = X l L ( k, l ) , where | i − j | ≤ l ≤ i + j, i + j + l ∈ Z , i + j + l ≤ k. From (3.16), we have e Y ( η i , z ) η j = Y (∆( h ′′ , z ) η i , z ) η j = z − i Y ( η i , z ) η j , (3.17)which implies that the fact that e Y is the intertwining operator of L ( k, σ of type L ( k, l ) + L ( k, i ) + L ( k, j ) + ! is equivalent to a i + a j − a l − ˜ a i − ˜ a j + ˜ a l + i ∈ Z , (3.18)that is, i + j − l ∈ Z . And the fact that e Y is the intertwining operator of L ( k, σ oftype L ( k, l ) − L ( k, i ) + L ( k, j ) + ! is equivalent to a i + a j − a l − ˜ a i − ˜ a j + ˜ a l + i ∈ Z , (3.19)that is, i + j − l + 2 ∈ Z . Since i + j + l ∈ Z , it follows that i + j − l + 2 ∈ Z is equivalentto i + j − l / ∈ Z . Thus from the definition of the symbol sign( i, j, l ), we obtain (3.14) and(3.15). Note that σ ( h ′′ ) = h ′′ , thus η i and ∆( h ′′ , z ) η i are in the same irreducible untwistedmodule of L ( k, σ , then by (3.17), (3.14) and (3.15), we obtain (3.12) and (3.13). K σ -modules In this section, we first recall some results on the quantum dimensions of irreducible g -twisted modules and irreducible V G -modules for G being a finite automorphism group ofthe vertex operator algebra V following [21]. Then we determine the quantum dimensionsfor irreducible modules of the orbifold vertex operator algebra K σ .We now recall some notions about quantum dimensions. Let V be a vertex operatoralgebra, g an automorphism of V with order T and M = ⊕ n ∈ T Z + M λ + n a g -twisted V -module. Definition 4.1.
For an homogeneous element v ∈ V , a trace function associated to v isdefined as follows: M ( v, q ) = tr M o ( v ) q L (0) − c/ = q λ − c/ X n ∈ T Z + tr M λ + n o ( v ) q n , where o ( v ) = v ( wt v − is the degree zero operator of v , c is the central charge of thevertex operator algebra V and λ is the conformal weight of M . It is proved [37, 16] that Z M ( v, q ) converges to a holomorphic function in the domain | q | < V is C -cofinite. We denote the holomorphic function Z M ( v, q ) by Z M ( v, τ ).Here and below, τ is in the upper half plane H and q = e πiτ . Note that if v = 1 is thevacuum vector, then Z M (1 , q ) is the formal character of M and we denote Z M (1 , q ) and Z M (1 , τ ) by χ M ( q ) and χ M ( τ ) respectively for simplicity. χ M ( q ) is called the character of M . Let V be a rational, C -cofinite, and selfdual vertex operator algebra of CFT type,and G a finite automorphism group of V . Let g ∈ G and M a g -twisted V -module. Thequantum dimension of M over V is defined to beqdim V M = lim y → χ M ( iy ) χ V ( iy ) , where y is real and positive[21].From [35] and [7], we have Theorem 4.2. If V is a regular, selfdual vertex operator algebra of CFT type, and G issolvable, then V G is a regular, selfdual vertex operator algebra of CFT type. From now on, we assume V is a rational, C -cofinite vertex operator algebra of CFTtype with V ∼ = V ′ . Let M ∼ = V, M , · · · , M d denote all inequivalent irreducible V -modules. Moreover, we assume the conformal weights λ i of M i are positive for all i > . From Theorem 4.2, the orbifold parafermion vertex operator algebra K σ satisfies all theassumptions.The following result shows that the quantum dimensions are multiplicative undertensor product [13] . Proposition 4.3.
Let V and M i for ≤ i ≤ d be as above. Thenqdim V (cid:0) M i ⊠ M j (cid:1) = qdim V M i · qdim V M j for i, j = 0 , · · · , d. Recalling from [17], let L = Z α + · · · + Z α k with h α i , α j i = 2 δ ij and let γ = α + · · · + α k ,then h γ, γ i = 2 k . V Z γ is the vertex operator algebra associated with a rank one lattice Z γ and as a V Z γ ⊗ K -module (note that K = M , ), L ( k, i ) has a decomposition: L ( k, i ) = k − M j =0 V Z γ +( i − j ) γ/ k ⊗ M i,j for 0 ≤ i ≤ k, (4.20)16here V Z γ +( i − j ) γ/ k are the irreducible modules of the lattice vertex operator algebra V Z γ .Since every irreducible V Z γ -module is a simple current, we haveqdim V Z γ V Z γ +( i − j ) γ/ k = 1 . (4.21)We get the following result on the quantum dimension of the orbifold parafermionvertex operator algebra K σ . Theorem 4.4.
The quantum dimensions for all irreducible K σ -modules areqdim K σ W ( k, i ) j = √ k sin π ( i +1) k +2 sin πk +2 for ≤ i ≤ k, i = k if k is even , j = 1 , , (4.22) qdim K σ W ( k, k j = qdim K σ ^ W ( k, k j = √ k π ( k +1) k +2 sin πk +2 for j = 1 , , (4.23) qdim K σ ( M i,j ) s = sin π ( i +1) k +2 sin πk +2 , s = 0 , for ( M i,j ) s being the untwisted K σ -module of type I .qdim K σ M i,j = 2 sin π ( i +1) k +2 sin πk +2 (4.25) for M i,j being the untwisted K σ -module of type II .Proof. Since the quantum dimensions of irreducible modules L ( k, i ) of affine vertex oper-ator algebra L ( k,
0) are qdim L ( k, L ( k, i ) = sin π ( i +1) k +2 sin πk +2 for 0 ≤ i ≤ k . From Proposition 4.1 of [12], we know thatqdim L ( k, L ( k, i ) = sin π ( i +1) k +2 sin πk +2 . Since from [24], qdim K M i,j = sin π ( i +1) k +2 sin πk +2 , (4.26)together with (4.21), we haveqdim V Z γ ⊗ K V Z γ +( i − j ) γ/ k ⊗ M i,j = sin π ( i +1) k +2 sin πk +2 . V Z γ ⊗ K L ( k, i ) = k sin π ( i +1) k +2 sin πk +2 . From Proposition 4.1 of [12], we haveqdim V Z γ ⊗ K L ( k, i ) = k sin π ( i +1) k +2 sin πk +2 . (4.27)Recall from [28] that all the irreducible twisted modules W ( k, i ) of K come from L ( k, i )for 0 ≤ i ≤ k , or more precisely, for the fixed i = k , W ( k, i ) is the only irreducibletwisted module of K , and if i = k , there are two irreducible twisted modules W ( k, k )and ^ W ( k, k ) of K . Note that if i = k , as the twisted module of the vertex operatoralgebra V Z γ ⊗ K , L ( k, i ) has a decomposition: L ( k, i ) = V T ai Z γ ⊗ W ( k, i ) , (4.28)where a i = 1 or 2 depending on i , V T ai Z γ ∈ { V T Z γ , V T Z γ } , and V T Z γ , V T Z γ are the irreducibletwisted V Z γ -modules [9]. For i = k , L ( k, k V T a k Z γ ⊗ W ( k, k V T ′ a k Z γ ⊗ ^ W ( k, k , (4.29)as a V Z γ ⊗ K -twisted module, where V T a k Z γ , V T ′ a k Z γ ∈ { V T Z γ , V T Z γ } . From [21], we know thatqdim V Z γ V T i Z γ = √ k for i = 1 ,
2. Together with (4.27), (4.28), (4.29), we haveqdim K W ( k, i ) = √ k sin π ( i +1) k +2 sin πk +2 for i = k . qdim K W ( k, k K ^ W ( k, k √ k π ( i +1) k +2 sin πk +2 . From the Theorem 4.4 of [21], we haveqdim K σ W ( k, i ) j = √ k sin π ( i +1) k +2 sin πk +2 for i = k , j = 1 ,
2, which proves (4.22). Furthermore,qdim K σ W ( k, k j = qdim K σ ^ W ( k, k j = √ k π ( k +1) k +2 sin πk +2 j = 1 ,
2, proving (4.23). Sinceqdim K M i,j = sin π ( i +1) k +2 sin πk +2 , from Corollary 4.5 of [21], we haveqdim K σ M i,j = 2 sin π ( i +1) k +2 sin πk +2 , for M i,j being the untwisted K σ -module of type II , which proves (4.25). Finally we haveqdim K σ ( M i,j ) s = sin π ( i +1) k +2 sin πk +2 , s = 0 , M i,j ) s being the untwisted K σ -module of type I . We obtain (4.24). Z -orbifold of the parafemion ver-tex operator algebra K K σ . To emphasize the action of the automor-phism σ , we denote twisted type modules W ( k, i ) by W ( k, i ) + and W ( k, i ) by W ( k, i ) − ,and we denote ^ W ( k, k ) by ^ W ( k, k ) + and ^ W ( k, k ) by ^ W ( k, k ) − . We denote untwistedmodules ( M i,j ) of type I by ( M i,j ) + and ( M i,j ) by ( M i,j ) − . For the irreducible K σ -modules W and W , we use W ⊠ W to denote the fusion product W ⊠ K σ W forsimplicity in this section.We first give the fusion rules for all the untwisted type modules. Theorem 5.1.
The fusion rules for the irreducible untwisted type modules of the Z -orbifold parafermion vertex operator algebra K σ are as follows:(1) If k ∈ Z + 1 , i.e., k = 2 n + 1 for n ≥ , we have ( M k, ) + ⊠ ( M i,j ) ± = ( M i,j ) ± , (5.1) where ( i, j ) = ( i, i ) , i = 2 , , , · · · , n, or ( i, j ) = (2 n + 1 , . ( M k, ) − ⊠ ( M i,j ) ± = ( M i,j ) ∓ , (5.2) where ( i, j ) = ( i, i ) , i = 2 , , , · · · , n, or ( i, j ) = (2 n + 1 , . ( M i, i ) + ⊠ ( M j, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k ( M l, ( l ) ) sign ( i,j,l ) ± , (5.3)19 M i, i ) − ⊠ ( M j, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k ( M l, ( l ) ) sign ( i,j,l ) ∓ , (5.4) where a means the residue of the integer a modulo k . The following is the same, whichwe will not point out again.(2) If k ∈ Z , i.e., k = 2 n for n ≥ , we have ( M k, ) + ⊠ ( M i,j ) ± = ( M i,j ) ± , (5.5)( M k, ) − ⊠ ( M i,j ) ± = ( M i,j ) ∓ , (5.6) where ( i, j ) = ( i, i ) , i = 2 , , , · · · , n, ( i, j ) = ( n, or ( i, j ) = (2 n, . ( M i, i ) + ⊠ ( M j, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k ( M l, ( l ) ) sign ( i,j,l ) ± , (5.7)( M i, i ) − ⊠ ( M j, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k ( M l, ( l ) ) sign ( i,j,l ) ∓ , (5.8)( M i, i ) + ⊠ ( M k , ) ± = X | k − i | ≤ l < k i + j + l ∈ Z i + j + l ≤ k M l, ( l − k ) + ( M k , ) ± , (5.9)( M i, i ) − ⊠ ( M k , ) ± = X | k − i | ≤ l < k i + j + l ∈ Z i + j + l ≤ k M l, ( l − k ) + ( M k , ) ∓ , (5.10)( M k , ) + ⊠ ( M k , ) ± = X ≤ l ≤ kk + l ∈ Z l ≤ k ( M k − l, ( k − l ) ) ± , (5.11)20 M k , ) − ⊠ ( M k , ) ± = X ≤ l ≤ kk + l ∈ Z l ≤ k ( M k − l, ( k − l ) ) ∓ . (5.12) (3) If k ∈ Z and k ≥ , we have ( M i,i ′ ) + ⊠ M j,j ′ = ( M i,i ′ ) − ⊠ M j,j ′ = X l (cid:16) ( M l, (2 i ′ − i +2 j ′ − j + l ) ) + + ( M l, (2 i ′ − i +2 j ′ − j + l ) ) − (cid:17) + X l ′ M l ′ , (2 i ′ − i +2 j ′ − j + l ′ ) , (5.13) where ( M i,i ′ ) ± are untwisted modules of type I , M j,j ′ are untwisted modules of type II ,and | i − j | ≤ l ≤ i + j, i + j + l ∈ Z , i + j + l ≤ k such that ( M l, (2 i ′ − i +2 j ′ − j + l ) ) ± areirreducible untwisted modules of type I . | i − j | ≤ l ′ ≤ i + j, i + j + l ′ ∈ Z , i + j + l ′ ≤ k such that M l ′ , (2 i ′ − i +2 j ′ − j + l ′ ) are irreducible untwisted modules of type II , Moreover, withfixed i, i ′ , j, j ′ , ( M l, (2 i ′ − i +2 j ′ − j + l ) ) ± for | i − j | ≤ l ≤ i + j, i + j + l ∈ Z , i + j + l ≤ k areinequivalent irreducible modules. M l ′ , (2 i ′ − i +2 j ′ − j + l ′ ) for | i − j | ≤ l ′ ≤ i + j, i + j + l ′ ∈ Z , i + j + l ′ ≤ k are inequivalent irreducible modules.(4) If k ∈ Z and k ≥ , we have M i,i ′ ⊠ M j,j ′ = X l (cid:16) ( M l, (2 i ′ − i +2 j ′ − j + l ) ) + + ( M l, (2 i ′ − i +2 j ′ − j + l ) ) − + ( M l, (2 i ′ − i +2( j − j ′ ) − j + l ) ) + + ( M l, (2 i ′ − i +2( j − j ′ ) − j + l ) ) − (cid:17) + X l ′ (cid:16) M l ′ , (2 i ′ − i +2 j ′ − j + l ′ ) + M l ′ , (2 i ′ − i +2( j − j ′ ) − j + l ′ ) (cid:17) , (5.14) where M i,i ′ , M j,j ′ are untwisted modules of type II , and | i − j | ≤ l ≤ i + j, i + j + l ∈ Z , i + j + l ≤ k such that ( M l, (2 i ′ − i +2 j ′ − j + l ) ) ± , ( M l, (2 i ′ − i +2( j − j ′ ) − j + l ) ) ± are irreducibleuntwisted modules of type I . | i − j | ≤ l ′ ≤ i + j, i + j + l ′ ∈ Z , i + j + l ′ ≤ k suchthat M l ′ , (2 i ′ − i +2 j ′ − j + l ′ ) , M l ′ , (2 i ′ − i +2( j − j ′ ) − j + l ′ ) are irreducible untwisted modules of type II , Moreover, with fixed i, i ′ , j, j ′ , ( M l, (2 i ′ − i +2 j ′ − j + l ) ) ± and ( M l, (2 i ′ − i +2( j − j ′ ) − j + l ) ) ± for | i − j | ≤ l ≤ i + j, i + j + l ∈ Z , i + j + l ≤ k are inequivalent irreducible K σ -modules. M l ′ , (2 i ′ − i +2 j ′ − j + l ′ ) and M l ′ , (2 i ′ − i +2( j − j ′ ) − j + l ′ ) for | i − j | ≤ l ′ ≤ i + j, i + j + l ′ ∈ Z , i + j + l ′ ≤ k are inequivalent irreducible K σ -modules.Proof. Note that ( M k, ) + = K +0 . Let ( M i,j , Y M i,j ) for 1 ≤ i ≤ k, ≤ j ≤ i − K -modules, then the operator Y M i,j gives the nonzero intertwining operatorsfor K of type (cid:18) M i,j K M i,j (cid:19) . Then by Lemma 3.12, Y M i,j ( a, z ) v is nonzero for any nonzerovectors a ∈ K , v ∈ M i,j . Since σY M i,j ( a, z ) σ − = Y M i,j ( σ ( a ) , z ) for a ∈ K , Y M i,j gives the21onzero intertwining operators for K +0 of type (cid:18) ( M i,j ) ± K +0 ( M i,j ) ± (cid:19) and (cid:18) ( M i,j ) ∓ K − ( M i,j ) ± (cid:19) .This implies (5.1), (5.2), (5.5), (5.6).For (5.3), (5.4), (5.7), (5.8), from [24], we know that M i,i ′ ⊠ K M j,j ′ = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k M l, (2 i ′ − i +2 j ′ − j + l ) . (5.15)Thus we have ( M i, i ) + ⊠ ( M j, j ) + ⊆ X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k M l, ( l ) . Since ( M i, i ) + ⊆ L ( k, i ) + , and L ( k, i ) has a decomposition (4.20): L ( k, i ) = k − M j =0 V Z γ +( i − j ) γ/ k ⊗ M i,j for 0 ≤ i ≤ k, we have V + Z γ ⊗ ( M i, i ) + ⊆ L ( k, i ) + , V + Z γ ⊗ ( M j, j ) + ⊆ L ( k, j ) + . Moreover, from Theorem 3.26, we know L ( k, i ) + ⊠ L ( k, j ) + = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k L ( k, l )sign ( i,j,l ) + . (5.16)Together with the facts that V + Z γ ⊠ V + Z γ = V + Z γ , qdim K σ ( M i, i ) + = sin π ( i +1) k +2 sin πk +2 , and qdim K σ (cid:16) ( M i, i ) + ⊠ ( M j, j ) + (cid:17) = qdim K σ ( M i, i ) + · qdim K σ ( M j, j ) + , we can deduce that( M i, i ) + ⊠ ( M j, j ) + = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k ( M l, ( l ) )sign ( i,j,l ) + . Since K − ⊠ ( M i, i ) + ⊠ ( M j, j ) + = ( M i, i ) − ⊠ ( M j, j ) + = ( M i, i ) + ⊠ ( M j, j ) − , and K − ⊠ ( M i, i ) + ⊠ ( M j, j ) − = ( M i, i ) − ⊠ ( M j, j ) − = ( M i, i ) + ⊠ ( M j, j ) + , together with (5.15), we obtain (5.3), (5.4), (5.7), (5.8).22or (5.9) and (5.10), from the fusion rule (5.15) of irreducible K -modules, we have( M i, i ) + ⊠ ( M k , ) + ⊆ X | k − i | ≤ l ≤ k i, l = k i + k l ∈ Z i + k l ≤ k M l, ( l − k ) + M k , . Note that M l, ( l − k ) for | k − i | ≤ l ≤ k + i, l = k are irreducible modules of K σ , i.e., theyare the untwisted modules of type II , and we have M l, ( l − k ) ∼ = M k − l, ( k − l ) . Note that M k , = ( M k , ) + + ( M k , ) − as K σ -module. From Theorem 4.4, we haveqdim K σ ( M i, i ) + = sin π ( i +1) k +2 sin πk +2 , qdim K σ ( M k , ) + = sin π ( k +1) k +2 sin πk +2 , qdim K σ M l, ( l − k ) = 2 sin π ( l +1) k +2 sin πk +2 . By using qdim K σ (cid:16) ( M i, i ) + ⊠ ( M k , ) + (cid:17) = qdim K σ ( M i, i ) + · qdim K σ ( M k , ) + , and noting that if i ≤ k , then l min = k − i, l max = k + i , we have X k − i ≤ l< k sin π ( l +1) k +2 sin πk +2 = X k 23n this case, we notice that | I − J | = | J | and v k , = X I ⊆ { , , · · · , k }| I | = k e α I / ∈ ( M k , ) + . From (3.9), we can deduce that v k , can be obtained from Y ◦ ( v i, i , z ) v k , , where Y ◦ isthe nonzero intertwining operator for V L of type (cid:18) V λ + λ + L V λ + L V λ + L (cid:19) for λ , λ ∈ L ⊥ . Since v k , ∈ ( M k , ) + , this shows that ǫ = +. Similar to the discussion in the end of the proofof (5.3), (5.4), we obtain (5.9) and (5.10).For (5.11) and (5.12), from the fusion rule (5.15) of irreducible K -modules, we have( M k , ) + ⊠ ( M k , ) + ⊆ X ≤ l ≤ kk + l ∈ Z k + l ≤ k M l, ( l − k ) . Note that M l, ( l − k ) ∼ = M k − l, ( k − l ) as K -modules, and M k − l, ( k − l ) = ( M k − l, ( k − l ) ) + ⊕ ( M k − l, ( k − l ) ) − as a K σ -module. From Theorem 4.4, we haveqdim K σ ( M k , ) + = sin π ( k +1) k +2 sin πk +2 , qdim K σ ( M k − l, ( k − l ) ) + = qdim K σ ( M k − l, ( k − l ) ) − = sin π ( k − l +1) k +2 sin πk +2 . By usingqdim K σ (cid:16) ( M k , ) + ⊠ ( M k , ) + (cid:17) = qdim K σ ( M k , ) + · qdim K σ ( M k , ) + , we can deduce that ( M k , ) + ⊠ ( M k , ) + = X ≤ l ≤ kk + l ∈ Z k + l ≤ k ( M l, ( l − k ) ) ǫ l , where ǫ l = + or − . We now prove that ǫ l = +. Since v k , = X I ⊆ { , , · · · , k }| I | = k e α I / ∈ ( M k , ) + , and M k , ∼ = M k , k as K -module, we have v k , k = X J ⊆ { , , · · · , k }| J | = k e − α J / ∈ ( M k , ) + . v k − l, k − l = X I ⊆ { , , · · · , k } , | I | = k − l X J ⊆ I, | J | = k − l e α I − J / − α J / ∈ ( M l, ( l − k ) ) + can be obtained from Y ◦ ( v k , , z ) v k , k , where Y ◦ is the nonzero intertwining operator for V L of type (cid:18) V λ + λ + L V λ + L V λ + L (cid:19) for λ , λ ∈ L ⊥ . This shows that ǫ l = +. Similar to thediscussion in the end of the proof of (5.3), (5.4), we get (5.11) and (5.12).For (5.13), notice that M j,j ′ ⊠ ( M i,i ′ ) + = K +0 ⊠ M j,j ′ ⊠ ( M i,i ′ ) + = K − ⊠ M j,j ′ ⊠ ( M i,i ′ ) + = M j,j ′ ⊠ K − ⊠ ( M i,i ′ ) + = M j,j ′ ⊠ ( M i,i ′ ) − , where M j,j ′ are the untwisted modules of type II , and ( M i,i ′ ) + are the untwisted modulesof type I . From the fusion rule (5.15) of irreducible K -modules, we have I ( M l, (2 i ′ − i +2 j ′ − j + l ) )( M i,i ′ ) + M j,j ′ ! =0 for | i − j | ≤ l ≤ i + j, i + j + l ∈ Z , i + j + l ≤ k . From Theorem 4.4, we haveqdim K σ ( M i,i ′ ) + = sin π ( i +1) k +2 sin πk +2 , qdim K σ M j,j ′ = 2 sin π ( j +1) k +2 sin πk +2 . By using qdim K σ (cid:16) ( M i,i ′ ) + ⊠ M j,j ′ (cid:17) = qdim K σ ( M i,i ′ ) + · qdim K σ M j,j ′ , we can deduce that (5.13) hold. The second assertion follows from Theorem 4.2 of [24]immediately.For (5.14), from the fusion rule (5.15) of irreducible K -modules, we have M i,i ′ ⊠ K M j,j ′ = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k M l, (2 i ′ − i +2 j ′ − j + l ) , where M i,i ′ , M j,j ′ are the untwisted K σ -modules of type II . From [28], we know that M i,i ′ ∼ = M i,i − i ′ , M j,j ′ ∼ = M j,j − j ′ as K σ -module. Thus M i,i ′ ⊠ K σ M j,j ′ = M i,i ′ ⊠ K σ M j,j − j ′ = M i,i − i ′ ⊠ K σ M j,j − j ′ . M i,i ′ ⊠ K M j,j − j ′ = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k M l, (2 i ′ − i +2( j − j ′ ) − j + l ) ,M i,i − i ′ ⊠ K M j,j − j ′ = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k M l, (2( i − i ′ ) − i +2( j − j ′ ) − j + l ) . We claim that M l, (2 i ′ − i +2 j ′ − j + l ) ∼ = M l, (2( i − i ′ ) − i +2( j − j ′ ) − j + l ) for | i − j | ≤ l ≤ i + j, i + j + l ∈ Z , i + j + l ≤ k . If we can prove the claim, then wehave X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k M l, (2 i ′ − i +2 j ′ − j + l ) + X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k M l, (2 i ′ − i +2( j − j ′ ) − j + l ) ⊆ M i,i ′ ⊠ K σ M j,j ′ . Moreover, from Theorem 4.4, we haveqdim K σ ( M i,i ′ ) = 2 sin π ( i +1) k +2 sin πk +2 , qdim K σ M j,j ′ = 2 sin π ( j +1) k +2 sin πk +2 . Then (5.14) follows fromqdim K σ (cid:16) M i,i ′ ⊠ M j,j ′ (cid:17) = qdim K σ M i,i ′ · qdim K σ M j,j ′ . The second assertion follows from Theorem 4.2 of [24] immediately. We now prove theclaim, i.e., M l, (2 i ′ − i +2 j ′ − j + l ) ∼ = M l, (2( i − i ′ ) − i +2( j − j ′ ) − j + l ) for | i − j | ≤ l ≤ i + j, i + j + l ∈ Z , i + j + l ≤ k. If M l, (2 i ′ − i +2 j ′ − j + l ) is the untwisted K σ -modules of type II , then from [28], we have M l, (2 i ′ − i +2 j ′ − j + l ) ∼ = M l,l − (2 i ′ − i +2 j ′ − j + l ) = M l, (2( i − i ′ ) − i +2( j − j ′ ) − j + l ) . If M l, (2 i ′ − i +2 j ′ − j + l ) is the untwisted K σ -modules of type I , we divide the proof of theclaim into three cases:(i) If ( l, (2 i ′ − i + 2 j ′ − j + l )) = ( l, l ), then (2 i ′ − i + 2 j ′ − j + l ) = l , i.e., 2 i ′ − i = j − j ′ , thus M l, (2( i − i ′ ) − i +2( j − j ′ ) − j + l ) = M l, ( l ) = M l, (2 i ′ − i +2 j ′ − j + l ) . l, (2 i ′ − i + 2 j ′ − j + l )) = ( k, ¯0), then (2 i ′ − i + 2 j ′ − j + k ) = ¯0, i.e., 2 i ′ − i = j − j ′ − k , thus M l, (2( i − i ′ ) − i +2( j − j ′ ) − j + l ) = M k, ¯ k = M k, ¯0 = M l, (2 i ′ − i +2 j ′ − j + l ) . (iii) If ( l, (2 i ′ − i + 2 j ′ − j + l )) = ( k , ¯0), then (2 i ′ − i + 2 j ′ − j + k ) = ¯0, i.e.,2 i ′ − i = j − j ′ − k , thus M l, (2( i − i ′ ) − i +2( j − j ′ ) − j + l ) = M k , ( k ) = M k , ¯0 = M l, (2 i ′ − i +2 j ′ − j + l ) . Thus we proved the claim.We now give the fusion products between untwisted type modules and twisted typemodules. Theorem 5.2. The fusion rules for the irreducible untwisted type modules and twistedtype modules of the Z -orbifold parafermion vertex operator algebra K σ are as follows:(1) If k ∈ Z + 1 , ≤ j ≤ k − , we have ( M k, ) + ⊠ W ( k, j ) ± = W ( k, j ) ± , (5.17)( M k, ) − ⊠ W ( k, j ) ± = W ( k, j ) ∓ . (5.18) If k ∈ Z , ≤ j ≤ k , we have ( M k, ) + ⊠ W ( k, j ) ± = W ( k, j ) ± , (5.19)( M k, ) − ⊠ W ( k, j ) ± = W ( k, j ) ∓ , (5.20)( M k, ) + ⊠ ^ W ( k, k ± = ^ W ( k, k ± , (5.21)( M k, ) − ⊠ ^ W ( k, k ± = ^ W ( k, k ∓ . (5.22) (2) For ( M i, i ) + being the untwisted module of type I , we have the following results: f k ∈ Z + 1 , ≤ j ≤ k − , we have ( M i, i ) + ⊠ W ( k, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k W ( k, l ) sign ( i,j,l ) ± , (5.23)( M i, i ) − ⊠ W ( k, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k W ( k, l ) sign ( i,j,l ) ∓ . (5.24) If k ∈ Z + 2 , i + j ∈ Z , or k ∈ Z , i + j ∈ Z + 1 , we have ( M i, i ) + ⊠ W ( k, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k W ( k, l ) sign ( i,j,l ) ± , (5.25) and ( M i, i ) − ⊠ W ( k, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k W ( k, l ) sign ( i,j,l ) ∓ . (5.26) If k ∈ Z + 2 , i + j ∈ Z + 1 , or k ∈ Z , i + j ∈ Z . And i + j < k or | i − j | > k , j = k , we have ( M i, i ) + ⊠ W ( k, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k W ( k, l ) sign ( i,j,l ) ± , (5.27)( M i, i ) − ⊠ W ( k, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k W ( k, l ) sign ( i,j,l ) ∓ . (5.28) If k ∈ Z + 2 , i + j ∈ Z + 1 , or k ∈ Z , i + j ∈ Z . And i + j ≥ k ≥ | i − j | , j = k ,we have ( M i, i ) + ⊠ W ( k, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k W ( k, l ) sign ( i,j,l ) ± + ^ W ( k, k sign ( i,j,l ) ∓ , (5.29)28 M i, i ) − ⊠ W ( k, j ) ± = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k W ( k, l ) sign ( i,j,l ) ∓ + ^ W ( k, k sign ( i,j,l ) ± . (5.30) If k ∈ Z , i ∈ Z + 2 , we have ( M i, i ) + ⊠ W ( k, k ± = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k W ( k, l ) sign ( i, k ,l ) ± + ^ W ( k, k ± , (5.31)( M i, i ) − ⊠ W ( k, k ± = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k W ( k, l ) sign ( i, k ,l ) ∓ + ^ W ( k, k ∓ , (5.32)( M i, i ) + ⊠ ^ W ( k, k ± = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k W ( k, l ) sign ( i, k ,l ) ± + W ( k, k ± , (5.33)( M i, i ) − ⊠ ^ W ( k, k ± = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k W ( k, l ) sign ( i, k ,l ) ∓ + W ( k, k ∓ . (5.34) If k ∈ Z , i ∈ Z , we have ( M i, i ) + ⊠ W ( k, k ± = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k W ( k, l ) sign ( i, k ,l ) ± + W ( k, k ± , (5.35)29 M i, i ) − ⊠ W ( k, k ± = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k W ( k, l ) sign ( i, k ,l ) ∓ + W ( k, k ∓ , (5.36)( M i, i ) + ⊠ ^ W ( k, k ± = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k W ( k, l ) sign ( i, k ,l ) ± + ^ W ( k, k ± , (5.37)( M i, i ) − ⊠ ^ W ( k, k ± = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k W ( k, l ) sign ( i, k ,l ) ∓ + ^ W ( k, k ∓ . (5.38) (3) For M i,i ′ being the untwisted modules of type II , we have the following results:If k ∈ Z + 1 , we have M i,i ′ ⊠ W ( k, j ) + = M i,i ′ ⊠ W ( k, j ) − = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k (cid:16) W ( k, l ) + + W ( k, l ) − (cid:17) , (5.39) If k ∈ Z + 2 , i + j ∈ Z + 1 , or k ∈ Z , i + j ∈ Z . And j = k , i + j < k or | i − j | > k , we have M i,i ′ ⊠ W ( k, j ) + = M i,i ′ ⊠ W ( k, j ) − = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k (cid:16) W ( k, l ) + + W ( k, l ) − (cid:17) . (5.40)30 f k ∈ Z + 2 , i + j ∈ Z + 1 , or k ∈ Z , i + j ∈ Z . And j = k , i + j ≥ k ≥ | i − j | ,we have M i,i ′ ⊠ W ( k, j ) + = M i,i ′ ⊠ W ( k, j ) − = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k (cid:16) W ( k, l ) + + W ( k, l ) − (cid:17) + (cid:16) ^ W ( k, k + + ^ W ( k, k − (cid:17) . (5.41) If k ∈ Z + 2 , i + j ∈ Z , or k ∈ Z , i + j ∈ Z + 1 . And j = k , we have M i,i ′ ⊠ W ( k, j ) + = M i,i ′ ⊠ W ( k, j ) − = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k (cid:16) W ( k, l ) + + W ( k, l ) − (cid:17) . (5.42) If k ∈ Z , i ∈ Z + 1 , we have M i,i ′ ⊠ W ( k, k + = M i,i ′ ⊠ W ( k, k − = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k (cid:16) W ( k, l ) + + W ( k, l ) − (cid:17) , (5.43) M i,i ′ ⊠ ^ W ( k, k + = M i,i ′ ⊠ ^ W ( k, k − = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k (cid:16) W ( k, l ) + + W ( k, l ) − (cid:17) . (5.44) If k ∈ Z , i ∈ Z , i ′ ∈ Z + 1 , we have M i,i ′ ⊠ W ( k, k + = M i,i ′ ⊠ W ( k, k − = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k (cid:16) W ( k, l ) + + W ( k, l ) − (cid:17) + (cid:16) ^ W ( k, k + + ^ W ( k, k − (cid:17) . (5.45)31 f k ∈ Z , i ∈ Z , i ′ ∈ Z , we have M i,i ′ ⊠ W ( k, k + = M i,i ′ ⊠ W ( k, k − = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k (cid:16) W ( k, l ) + + W ( k, l ) − (cid:17) + (cid:16) W ( k, k + + W ( k, k − (cid:17) . (5.46) If k ∈ Z , i ∈ Z , i ′ ∈ Z + 1 , we have M i,i ′ ⊠ ^ W ( k, k + = M i,i ′ ⊠ ^ W ( k, k − = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k (cid:16) W ( k, l ) + + W ( k, l ) − (cid:17) + (cid:16) W ( k, k + + W ( k, k − (cid:17) . (5.47) If k ∈ Z , i ∈ Z , i ′ ∈ Z , we have M i,i ′ ⊠ ^ W ( k, k + = M i,i ′ ⊠ ^ W ( k, k − = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k (cid:16) W ( k, l ) + + W ( k, l ) − (cid:17) + (cid:16) ^ W ( k, k + + ^ W ( k, k − (cid:17) . (5.48) (4) If k ∈ Z , j ∈ Z + 1 , j = k , we have ( M k , ) + ⊠ W ( k, j ) ± = ( M k , ) − ⊠ W ( k, j ) ± = X | k − j | ≤ l < k k j + l ∈ Z j + l ≤ k (cid:16) W ( k, l ) + + W ( k, l ) − (cid:17) . (5.49) If k ∈ Z , j ∈ Z , j = k , we have ( M k , ) + ⊠ W ( k, j ) ± = X | k − j | ≤ l < k k j + l ∈ Z j + l ≤ k (cid:16) W ( k, l ) + + W ( k, l ) − (cid:17) + (cid:16) W ( k, k ± + ^ W ( k, k ± (cid:17) , (5.50)32 M k , ) − ⊠ W ( k, j ) ± = X | k − j | ≤ l < k k j + l ∈ Z j + l ≤ k (cid:16) W ( k, l ) + + W ( k, l ) − (cid:17) + (cid:16) W ( k, k ∓ + ^ W ( k, k ∓ (cid:17) . (5.51) If k ∈ Z + 2 , we have ( M k , ) + ⊠ W ( k, k ± = X ≤ l ≤ k − k + l ∈ Z l ≤ k W ( k, l ) sign ( k , k ,l ) ± . (5.52)( M k , ) − ⊠ W ( k, k ± = X ≤ l ≤ k − k + l ∈ Z l ≤ k W ( k, l ) sign ( k , k ,l ) ∓ . (5.53)( M k , ) + ⊠ ^ W ( k, k ± = X ≤ l ≤ k − k + l ∈ Z l ≤ k W ( k, l ) sign ( k , k ,l ) ± . (5.54)( M k , ) − ⊠ ^ W ( k, k ± = X ≤ l ≤ k − k + l ∈ Z l ≤ k W ( k, l ) sign ( k , k ,l ) ∓ . (5.55) If k ∈ Z , we have ( M k , ) + ⊠ W ( k, k ± = X ≤ l ≤ k − k + l ∈ Z l ≤ k W ( k, l ) sign ( k , k ,l ) ± + W ( k, k ± . ( M k , ) − ⊠ W ( k, k ± = X ≤ l ≤ k − k + l ∈ Z l ≤ k W ( k, l ) sign ( k , k ,l ) ∓ + W ( k, k ∓ . M k , ) + ⊠ ^ W ( k, k ± = X ≤ l ≤ k − k + l ∈ Z l ≤ k W ( k, l ) sign ( k , k ,l ) ± + ^ W ( k, k ± . ( M k , ) − ⊠ ^ W ( k, k ± = X ≤ l ≤ k − k + l ∈ Z l ≤ k W ( k, l ) sign ( k , k ,l ) ∓ + ^ W ( k, k ∓ . Proof. We will prove the case for k ∈ Z + 1 and k ∈ Z + 2, the proof of the case k ∈ Z is similar to the proof of the case k ∈ Z + 2. Note that ( M k, ) + = K +0 , and we have theintertwining operator in Lemma 3.24. Similar to the proof of (5.1) and (5.2) in Theorem5.1, we can obtain (5.17), (5.18), (5.19)-(5.22).For (5.23) and (5.24), from Theorem 3.26, we have L ( k, i ) + ⊠ L ( k, j ) + = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k L ( k, l )sign ( i,j,l ) + . (5.56)From the decomposition (4.20): L ( k, i ) = k − M j =0 V Z γ +( i − j ) γ/ k ⊗ M i,j for 0 ≤ i ≤ k, we have V + Z γ ⊗ ( M i, i ) + ⊆ L ( k, i ) + . From the decomposition (4.28): L ( k, i ) = V T ai Z γ ⊗ W ( k, i ) for i = k , where a i = 1 or 2 depending on i , we have( V T ai Z γ ) + ⊗ W ( k, j ) + ⊆ L ( k, i ) + . Since ( M i, i ) + ⊆ L ( k, i ) + , W ( k, j ) + ⊆ L ( k, i ) + , and V + Z γ ⊠ V + Z γ ( V T ai Z γ ) + = ( V T ai Z γ ) + , by usingthe quantum dimension obtained in Theorem 4.4:qdim K σ ( M i, i ) + = sin π ( i +1) k +2 sin πk +2 , qdim K σ W ( k, j ) + = √ k sin π ( j +1) k +2 sin πk +2 for j = k , K σ (cid:16) ( M i, i ) + ⊠ W ( k, j ) + (cid:17) = qdim K σ ( M i, i ) + · qdim K σ W ( k, j ) + , together with (5.56), we can deduce( M i, i ) + ⊠ W ( k, j ) + = X | i − j | ≤ l ≤ i + ji + j + l ∈ Z i + j + l ≤ k W ( k, l )sign ( i,j,l ) + , where we notice that l = k in this case, then (5.23), (5.24) follows immediately. By thesimilar proof to (5.23) and (5.24), just noticing the definition of ^ W ( k, k ) ± , we can get(5.25)-(5.30).For (5.31), similar to the arguments in the proof of (5.23), but noticing that in thiscase l can take k , and qdim K σ W ( k, k + = √ k π ( k +1) k +2 sin πk +2 , we can obtain that( M i, i ) + ⊠ W ( k, k + = X | i − k | ≤ l < k i + k l ∈ Z i + l ≤ k W ( k, l )sign ( i, k ,l ) + + M. (5.57)Since sign( i, k , k ) + = − , from the definition of ^ W ( k, k ) + , we can deduce that M = ^ W ( k, k ) + or M = W ( k, k ) − . We now prove that M = ^ W ( k, k ) + . From the latticerealization of the irreducible K -modules M i,j , i.e., (3.8) and (3.9), we know that thereexists m ∈ Z such that v i, i − ( m ) v k , k = a i e (0) v k , k , v i, i +1 ( m − v k , k = b i f ( − v k , k for some nonzero complex numbers a i and b i . This implies that( e ′ (0) i +1 η i )( m ) η k = a i e ′ (0) η k , ( e ′ (0) i − η i )( m − η k = b i f ′ ( − η k . Here we use an identification of basis { h, e, f } and { h ′ , e ′ , f ′ } . We can also deducefrom the lowest weight that ( e ′ (0) j η i )( n ) η k = 0 (5.58)35or n > m . Note that in this case i ∈ Z + 1, v i, i is a linear combination of thevectors η i , h (0) η i , · · · , h i − (0) η i , h i +1 (0) η i , · · · , h i (0) η i , by straightforward calcula-tions. It shows that v i, i ∈ L ( k, i ) + , and from the discussion above, we know that L ( k, k ) − L ( k, i ) + L ( k, k ) + ! = 0. From Lemma 3.24, we have e Y ( e ′ (0) j η i , z ) = Y (∆( h ′′ , z ) e ′ (0) j η i , z ) = z j − i Y ( e ′ (0) j η i , z ) . Thus we have ( e ′ (0) i +1 η i ) m − = ( e ′ (0) i +1 η i )( m ). Together with (5.58) and by consideringthe weights of the lattice realization, we obtain that for j > , j ∈ Z + 1,( e ′ (0) i + j η i ) m − η k = ( e ′ (0) i + j η i )( m + j − 12 ) η k = 0 , (5.59)( e ′ (0) i − j η i ) m − η k = ( e ′ (0) i − j η i )( m − j − 12 ) η k = 0 . (5.60)Let v i, i = X ≤ j ≤ ij ∈ Z c j e ′ (0) j η i , then from (5.59) and (5.60), we have v i, i m − η k = c i +1 a i e ′ (0) η k + c i − b i f ′ ( − η k . Note that L ( k, k − = V T a k , + Z γ ⊗ W ( k, k − ⊕ V T ′ a k , + Z γ ⊗ ^ W ( k, k + ⊕ V T a k , − Z γ ⊗ W ( k, k + ⊕ V T ′ a k , − Z γ ⊗ ^ W ( k, k − ,v i, i ∈ ( M i, i ) + , L ( k, k ) − L ( k, i ) + L ( k, k ) + ! = 0, and( e − f ) − η k = ( f ′ − e ′ ) − η k = ( f ′ ( − − e ′ (0)) η k ∈ L ( k, k − , we deduce that v i, i m − η k = c ( f ′ ( − − e ′ (0)) η k for some nonzero complex number c , which means that ^ W ( k, k ) + ( M i, i ) + W ( k, k ) + ! = 0, thatis, M = ^ W ( k, k ) + as required. Thus we have (5.31). Then (5.32) follows immediately.Similarly, we can prove (5.33) and (5.34). 36or (5.35), similar to the analysis of (5.31), in this case, we need to prove that M is W ( k, k ) + in (5.57). By applying the lattice realization of K -module M i,j , we can obtainthat there exists m ∈ Z such that( e ′ (0) i η i )( m ) η k = a i η k , (5.61)for some nonzero complex number a i . By analyzing the weights in L ( k, k ) + , we can get( e ′ (0) j η i )( m ) η k = 0 (5.62)for j ∈ Z , j = i . Similar to the proof of (5.31), and noticing that in this case v i, i is alinear combination of vectors η i , , e ′ (0) η i , · · · , e ′ (0) i η i , · · · , e ′ (0) i η i , i.e., we may write v i, i = X ≤ j ≤ ij ∈ Z c j e ′ (0) j η i with c j = 0 for j ∈ Z , ≤ j ≤ i . Thus from (5.61) and (5.62), we have( v i, i )( m ) η k = c i a i η k = 0 . which means that W ( k, k ) + ( M i, i ) + W ( k, k ) + ! = 0, that is, M = W ( k, k ) + as required. Thuswe have (5.35), and (5.36) follows immediately. Similarly, we can prove (5.37)-(5.38).For (5.39), since M i,i ′ are untwisted modules of type II , they are irreducible as K σ -modules. This shows that M i,i ′ ⊠ W ( k, j ) + = K +0 ⊠ M i,i ′ ⊠ W ( k, j ) + = K − ⊠ M i,i ′ ⊠ W ( k, j ) + = M i,i ′ ⊠ W ( k, j ) − , since from Theorem 4.4, we haveqdim K σ M i,i ′ = 2 sin π ( i +1) k +2 sin πk +2 , qdim K σ W ( k, j ) + = √ k sin π ( j +1) k +2 sin πk +2 for j = k . qdim K σ W ( k, k + = qdim K σ ^ W ( k, k + = √ k π ( k +1) k +2 sin πk +2 . By usingqdim K σ (cid:16) ( M i,i ′ ) + ⊠ W ( k, j ) + (cid:17) = qdim K σ ( M i,i ′ ) + · qdim K σ W ( k, j ) + , K σ are constructed from the twistedtype modules of the affine vertex operator algebra[28], together with Lemma 3.24, we canget that in this case l = k , (5.39) holds. By the same reason as in the proof of (5.39), wecan obtain (5.40)-(5.42), just noticing that in (5.40) and (5.42), l = k .For (5.43), from Theorem 4.4 we notice that on the left side of the equation (5.43),the quantum dimension isqdim K σ (cid:16) M i,i ′ ⊠ W ( k, k + (cid:17) = qdim K σ M i,i ′ · qdim K σ W ( k, k + = √ k sin π ( i +1) k +2 sin πk +2 sin π ( k +1) k +2 sin πk +2 , and note that if i ≤ k , then l min = k − i, l max = k + i . Thus X k − i ≤ l< k sin π ( l +1) k +2 sin πk +2 = X k Theorem 5.3. All the irreducible modules of the Z -orbifold parafermion vertex operatoralgebra K σ are self-dual.Proof. From Theorem 2.4 and Remark 2.5, we know that the irreducible modules of K σ are twisted type modules and untwisted modules of type I and type II , and the lowestweights of each irreducible K σ -modules are listed in Proposition 3.13, Proposition 3.14and Proposition 3.6 in [28]. Let W be an irreducible K σ -module. Since the top level ofan irreducible K σ -module W is one-dimensional, set the top level W = C v and the toplevel of its contragredient modules W ′ = C v ′ . Then o ( ω ) = ω acts on the top level asscalar multiples. From the definition of the contragredient module (3.4), we have h o ( ω ) v ′ , v i = h v ′ , o ( ω ) v i . It follows that v and v ′ have the same weight. From Proposition 3.13 in [28], we knowthat the lowest weights of irreducible twisted type modules of K σ are pairwise different, sothe irreducible twisted type modules of K σ are self-dual. From Proposition 3.6 in [28], weknow that the lowest weights of irreducible untwisted K σ -modules of type II are pairwisedifferent, thus the irreducible untwisted K σ -modules of type II are also self-dual. For thecase of the irreducible untwisted K σ -module of type I , since ( M k, ) + = K σ , it is self-dual.If k ∈ Z , we know that ∈ K σ ⊆ ( M k, ) + ⊠ (( M k, ) + ) ′ , and v k , = X I ⊆ { , , · · · , k }| I | = k e α I / ∈ ( M k , ) + , from (3.9). Then we can deduce that v k , k = X J ⊆ { , , · · · , k }| J | = k e − α J / ∈ (( M k , ) + ) ′ . M k , ) + ) ′ = ( M k , k ) + ∼ = ( M k , ) + , and so ( M k , ) + is self-dual. It follows that( M k , ) − is self-dual.If i ∈ Z , we know that ∈ K σ ⊆ ( M i, i ) + ⊠ (( M i, i ) + ) ′ , and note that from (3.9), v i, i = X I ⊆ { , , · · · , k }| I | = i X J ⊆ I | J | = i e α I − J / − α J / ∈ ( M i, i ) + . Then we can deduce that v i, i ∈ (( M i, i ) + ) ′ . Thus (( M i, i ) + ) ′ = ( M i, i ) + , so ( M i, i ) + isself-dual. It follows that ( M i, i ) − is self-dual. Remark 5.4. From Proposition 3.11, we know that Theorem 5.1, Theorem 5.2 and The-orem 5.3 give the fusion rules of all the irreducible K σ -modules. Remark 5.5. 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