aa r X i v : . [ m a t h . QA ] S e p S -matrix in orbifold theory Chongying Dong Department of Mathematics, University of California, Santa Cruz, CA 95064 USALi Ren School of Mathematics, Sichuan University, Chengdu 610064 ChinaFeng Xu Department of Mathematics, University of California, Riverside, CA 92521, USA
Abstract
The restricted S -matrix of V G is determined for any regular vertex operatoralgebra V and finite automorphism group G of V. As an application, the S -matricesfor cyclic permutation orbifolds of prime orders are computed. The orbifold theory studies a vertex operator algebra V under the action of a finiteautomorphism group G. It is proved recently in [M, CM] that if V is regular and G is solvable then V G is also regular. Under the assumption that V G is regular, everyirreducible V G -module occurs in an irreducible g -twisted V -module [DRX]. In particular,this result holds for G being solvable. If V is rational and V G C -cofinite, then V G is alsorational [Mc]. In this paper, we study the S -matrix for V G under the assumption that V is regular.The modularity of trace functions in the theory of rational vertex operator algebraplays a power role in studying vertex operator algebra. The conformal block of a regularvertex operator algebra spanned by the trace functions on the irreducible V -modules af-fords to a representation of the modular group Γ = SL ( Z ) [Z]. Moreover, the kernel ofthis representation is a congruence subgroup [DLN]. This implies that the trace functionsin rational vertex operator algebras are modular forms on congruence subgroup. In par-ticular, the irreducible characters are modular functions. The results in [Z] are extendedto include an action of finite groups in [DLM5] to get a modular invariance of the tracefunctions in orbifold theory. In the case that V is holomorphic this result provides aframework for the generalized moonshine [N]. Also, the trace functions in orbifold theoryare modular forms on a congruence subgroup without the assumption that V G is rational[DR].It is well known that Γ is generated by S = (cid:18) −
11 0 (cid:19) and T = (cid:18) (cid:19) . Theaction of T on conformal block with respect to a distinguished basis is a diagonal matrixdetermined by the central charge and the conformal weights of irreducible modules. The Partially supported by NSFC 11871351 Supported by NSFC 11671277 Partially supported by NSFC 11871150 S is called the S -matrix which is the key to understand the action of Γ . Thecelebrated Verlinde formula [V, H] exhibits the beauty and importance of the S -matrix. S -matrix has also been used to determine the number of irreducible g -twisted modules[DLM5] and classification of irreducible modules.The so-called restricted S -matrix of V G is the restriction of the S -matrix of V G to theirreducible V G -modules appearing in the twisted modules. There is no doubt that thisshould be the full S -matrix once the C -cofiniteness of V G is established. More explicitly,we give a precise formula for the restricted S -matrix in terms of the S -matrix of tracefunctions of the twisted modules obtained in [DLM5]. The main idea is that the space ofthe conformal block of V G spanned by the trace functions on the irreducible V G -modulesappearing in twisted V -modules in [Z] is equal to the twisted conformal block of V spannedby the trace functions on the irreducible twisted modules in [DLM5]. Some entries of therestricted S -matrix have been computed previously in [DRX] for studying the quantumdimensions and global dimensions for vertex operator algebras V G [DJX]. In the case G isabelian or V is holomorphic, the S -matrices have simpler expressions. The S -matrix forcyclic group G and holomorphic vertex operator algebra has been investigated in [EMS]for the purpose of constructing holomorphic vertex operator algebras with central charge24 . As an application, we determine the S -matrix for the cyclic permutation orbifold ofprime order. Let k be a positive integer. Then V ⊗ k is a vertex operator algebra [FHL] and S k acts on V ⊗ k as automorphisms. For any g ∈ S k , the g -twisted V ⊗ k -module categorywas determined in [BDM] in terms of V -module category. In particular, if g is a cycle,there is an equivalence between V -module category and g -twisted V ⊗ k -module category.For general orbifold theory, one do not know how to construct twisted modules althoughthe number of irreducible twisted modules are known. But for the permutation orbifoldtheory one has an explicit construction of twisted modules in terms of V - modules [BDM].In this paper when k is a prime and g is a k -cycle and G is generated by g we give anexplicit formula of S -matrix of ( V ⊗ k ) G in terms of the action of Γ on the conformal blockof V. The assumption that k is a prime ensures that every power of g is a cycle, sothe S -matrix of ( V ⊗ k ) G has a simple formula. One can consider general case with morecomplicated computation. The S -matrix of the permutation orbifold for any subgroup of S k has been studied in [B] and [KLX] from the point views of conformal field theory andconformal nets.For the general orbifold theory, one can see [DLM3, DLM4] for the study of twistedmodules, [DLM1, DM, MT, HMT, DY, DJX] for Schur-Weyl duality and quantum Galoistheory in orbifold theory, [X] for conformal net approach to the orbifold theory and [DPR]for the connection between the holomorphic orbifold theory and the twisted Drinfelddouble [D]. One can also find the construction of twisted modules in [FLM1, FLM2, L,DL, Li1] in terms of ∆-operators.The paper is organized as follows. In Section 2, we review twisted modules, g -rationality and related results from [DLM3]. The modular invariance result on tracefunctions for the orbifold theory from [DLM5] is given in Section 3. Classification ofthe irreducible V G -modules occurring in irreducible twisted modules from [DLM1, DY,2T, DRX] is present in Section 4. Section 5 is devoted to the study of the restricted S -matrix for any regular vertex operator algebra V and any finite automorphism group G. In particular, we give an explicit formula of the restricted S -matrix of V G in termsof the S -matrix from [DLM5] for the action of Γ on the twisted conformal block of V .Section 6 is an application of Section 5 to the cyclic permutation orbifolds. Various notions of twisted modules for a vertex operator algebra following [DLM3] arereviewed in this section. The concepts such as rationality, regularity, and C -cofinitenessfrom [Z] and [DLM2] are discussed.Let V be a vertex operator algebra and g an automorphism of V of finite order T .Then V is a direct sum of eigenspaces of g : V = M r ∈ Z /T Z V r , where V r = { v ∈ V | gv = e − πir/T v } . We use r to denote both an integer between 0 and T − T in this situation. Definition 2.1. A weak g -twisted V -module M is a vector space equipped with a linearmap Y M : 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: for all ≤ r ≤ T − , u ∈ V r , v ∈ V, w ∈ M , Y M ( u, z ) = X n ∈ rT + Z u n z − n − ,u l w = 0 for l ≫ ,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) − r/T δ (cid:18) z − z z (cid:19) Y M ( Y ( u, z ) v, z ) , where δ ( z ) = P n ∈ Z z n and all binomial expressions (here and below) are to be expandedin nonnegative integral powers of the second variable. Definition 2.2. A g - twisted V -module is a C -graded weak g -twisted V -module M : M = M λ ∈ C M λ uch that dim M λ is finite and for fixed λ, M nT + λ = 0 for all small enough integers n, where M λ = { w ∈ M | L (0) w = λw } and L (0) is the component operator of Y ( ω, z ) = P n ∈ Z L ( n ) z − n − . If w ∈ M λ we refer to λ as the weight of w and write λ = wt w. Let Z + be the set of nonnegative integers. Definition 2.3. An admissible g -twisted V -module is a T Z + -graded weak g -twisted V -module M : M = M n ∈ T Z + M ( n ) such that v m M ( n ) ⊆ M ( n + wt v − m − for homogeneous v ∈ V, m, n ∈ T Z . If g = Id V we have the notions of weak, ordinary and admissible V -modules [DLM2]. Definition 2.4.
A vertex operator algebra V is called g -rational, if the admissible g -twisted module category is semisimple. V is called rational if V is -rational. There is another important concept called C -cofiniteness [Z]. Definition 2.5.
We say that a vertex operator algebra V is C -cofinite if V /C ( V ) isfinite dimensional, where C ( V ) = h v − u | v, u ∈ V i . The following results about g -rational vertex operator algebras are well-known [DLM3,DLM5]. Theorem 2.6. If V is g -rational, then(1) Any irreducible admissible g -twisted V -module M is a g -twisted V -module. More-over, there exists a number λ ∈ C such that M = ⊕ n ∈ T Z + M λ + n where M λ = 0 . The λ iscalled the conformal weight of M ; (2) There are only finitely many irreducible admissible g -twisted V -modules up to iso-morphism.(3) If V is also C -cofinite and g i -rational for all i ≥ then the central charge c andthe conformal weight λ of any irreducible g -twisted V -module M are rational numbers. Definition 2.7.
A vertex operator algebra V is called regular if every weak V -module isa direct sum of irreducible V -modules. A vertex operator algebra V = ⊕ n ∈ Z V n is said to be of CFT type if V n = 0 for negative n and V = C . It is proved in [Li2] and [ABD] that if V is of CFT type, then regularityis equivalent to rationality and C -cofiniteness. Also V is regular if and only if the weakmodule category is semisimple [DYu].In the rest of this paper we assume the following:(V1) V = ⊕ n ≥ V n is a simple, rational , C -cofinite vertex operator algebra of CFT type,4V2) G is a finite automorphism group of V, (V3) The conformal weight of any irreducible g -twisted V -module M is nonnegative andis zero if and only if M = V. We remark that with these assumptions, V G is C -cofinite, rational if G is solvable[M, CM]. Moreover, V is g -rational for any finite order automorphism g [ADJR, DRX]. This section largely follows from [DLM5]. We present the main results on modular in-variance in orbifold theory.For the modular invariance, we need the action of Aut( V ) on twisted modules [DLM5].Let g, h be two automorphisms of V with g of finite order. If ( M, Y M ) is a weak g -twisted V -module, there is a weak h − gh -twisted V -module ( M ◦ h, Y M ◦ h ) where M ◦ h ∼ = M asvector spaces and Y M ◦ h ( v, z ) = Y M ( hv, z ) for v ∈ V. This defines a right action of Aut( V )on weak twisted V -modules and on isomorphism classes of weak twisted V -modules.Symbolically, we write ( M, Y M ) ◦ h = ( M ◦ h, Y M ◦ h ) = M ◦ h.M is called h -stable if M, M ◦ h are isomorphic. It is proved in [DLM5] that if M is anadmissible g -twisted V -module, then M ◦ g and M are isomorphic.Assume that g, h commute. Then h acts on the g -twisted modules. Denote by M ( g )the equivalence classes of irreducible g -twisted V -modules and set M ( g, h ) = { M ∈ M ( g ) | M ◦ h ∼ = M } . Since V is g -rational for all g , both M ( g ) and M ( g, h ) are finite sets.The following lemma is obvious from the definitions. It will be useful later. Lemma 3.1.
Let g, h ∈ G , M an irreducible g -twisted V -module. For any k ∈ G , wehave(1) M ( g, h ) and M ( k − gk, k − hk ) have the same cardinality.(2) M ◦ k ∈ M ( g ) if and only if k ∈ C G ( g ) .(3) M and M ◦ h are isomorphic V G -modules. Let M be an irreducible g -twisted V -module and G M is a subgroup of G consisting of h ∈ G such that M ◦ h and M are isomorphic. Using the Schur’s Lemma gives a projectiverepresentation ϕ of G M on M such that ϕ ( h ) Y ( u, z ) ϕ ( h ) − = Y ( hu, z )for h ∈ G M . If h = 1 we simply take ϕ (1) = 1 . For homogeneous v ∈ V we set o ( v ) = v wt v − which is the degree zero operator of v. Here and below τ is in the complex upper half-plane H and q = e πiτ . For v ∈ V we set Z M ( v, ( g, h ) , τ ) = tr M o ( v ) ϕ ( h ) q L (0) − c/ = q λ − c/ X n ∈ T Z + tr Mλ + n o ( v ) ϕ ( h ) q n (3.1)5hich is a holomorphic function on the H [DLM5, Z]. Note that Z M ( v, ( g, h ) , τ ) is definedup to a nonzero scalar. For short we set Z M ( v, τ ) = Z M ( v, ( g, , τ ) . Then χ M ( τ ) = Z M ( , τ ) is called the character of M. There is another vertex operator algebra (
V, Y [ ] , , ˜ ω ) associated to V in [Z]. Here˜ ω = ω − c/
24 and Y [ v, z ] = Y ( v, e z − e z · wt v = X n ∈ Z v [ n ] z n − for homogeneous v. We also write Y [˜ ω, z ] = X n ∈ Z L [ n ] z − n − . If v ∈ V is homogeneous in the second vertex operator algebra, we denote its weight bywt[ v ] . Let P ( G ) denote the ordered commuting pairs in G. For ( g, h ) ∈ P ( G ) and M ∈ M ( g, h ) , Z M ( v, ( g, h ) , τ ) is a function on V × H . Let W be the vector space spannedby these functions. It is clear that the dimension of W is equal to P ( g,h ) ∈ P ( G ) | M ( g, h ) | [DLM5]. We now define an action of the modular group Γ on W such that Z M | γ ( v, ( g, h ) , τ ) = ( cτ + d ) − wt[ v ] Z M ( v, ( g, h ) , γτ ) , where γ : τ aτ + bcτ + d , γ = (cid:18) a bc d (cid:19) ∈ Γ = SL (2 , Z ) . (3.2)We let γ ∈ Γ act on the right of P ( G ) via( g, h ) γ = ( g a h c , g b h d ) . The following results are established in [DLM5, Z, DLN, DR].
Theorem 3.2.
Let V and G be as before. Then(1) There is a representation ρ : Γ → GL ( W ) such that for ( g, h ) ∈ P ( G ) , γ = (cid:18) a bc d (cid:19) ∈ Γ , and M ∈ M ( g, h ) ,Z M | γ ( v, ( g, h ) , τ ) = X N ∈ M ( g a h c ,g b h d ) γ M,N Z N ( v, ( g, h ) γ, τ ) where ρ ( γ ) = ( γ M,N ) . That is, Z M ( v, ( g, h ) , γτ ) = ( cτ + d ) wt[ v ] X N ∈ M ( g a h c ,g b h d ) γ M,N Z N ( v, ( g a h c , g b h d ) , τ ) . (2) The cardinalities | M ( g, h ) | and | M ( g a h c , g b h d ) | are equal for any ( g, h ) ∈ P ( V ) and γ ∈ Γ . In particular, the number of irreducible g -twisted V -modules is exactly thenumber of irreducible V -modules which are g -stable.(3) The kernal of ρ is a congruence subgroup of SL ( Z ) . That is, each Z M | γ ( v, ( g, h ) , τ ) is a modular form of weight wt[ v ] on the congruence subgroup. In particular, the character χ M ( τ ) is a modular function on the same congruence subgroup. G = { } was obtained in [Z]. (1),(2) were given in [DLM5] for general G. (3) is a result from [DLN] with G = { } and isproved in [DR] for general G. It is well known that the modular group Γ is generated by S = (cid:18) −
11 0 (cid:19) and T = (cid:18) (cid:19) . So the representation ρ is uniquely determined by ρ ( S ) and ρ ( T ) . It isalmost trivial to compute ρ ( T ) once we know the irreducible modules. The matrix ρ ( S ) iscalled the S -matrix of the orbifold theory. Here is a special case of the S -transformation: Z M ( v, − τ ) = τ wt[ v ] X N ∈ M (1 ,g − ) S M,N Z N ( v, (1 , g − ) , τ ) (3.3)for M ∈ M ( g ) and Z N ( v, (1 , g ) , − τ ) = τ wt[ v ] X M ∈ M ( g ) S N,M Z M ( v, τ ) (3.4)for N ∈ M (1) . We will use M V for M (1) . The matrix S = ( S M,N ) M,N ∈ M V is called S -matrix of V . We need the following result from [DLN] later. Proposition 3.3.
The S -matrix is unitary and S V,M = S M,V is positive for any irreducible V -module M. V G In this section we give the irreducible V G -modules appearing in an irreducible g -twisted V -module for some g ∈ G [DRX]. If G is solvable, these are the all irreducible V G -modules.For general G , this is also true if V G is rational and C -cofinite.Let M = ( M, Y M ) be an irreducible g -twisted V -module. We have discussed that G M acts on M projectively. Let α M be the corresponding 2-cocycle in C ( G, C × ) . Then ϕ ( h ) ϕ ( k ) = α M ( h, k ) ϕ ( hk ) for all h, k ∈ G M . We may assume that α M is unitary in thesense that there is a fixed positive integer n such that α M ( h, k ) n = 1 for all h, k ∈ G M . Let C α M [ G M ] = ⊕ h ∈ G M C ¯ h be the twisted group algebra with product ¯ h ¯ k = α M ( h, k ) ¯ hk. It is well known that C α M [ G M ] is a semisimple associative algebra. It follows that M is a module for C α M [ G M ] . Note that G M is a subgroup of C G ( g ) and g lies in G M . Let M = ⊕ n ∈ T Z + M ( n ) and M (0) = 0 . Then ϕ ( g ) acts on M ( n ) as e πin for all n [DRX].Let Λ G M be the set of all irreducible characters λ of C α M [ G M ]. Denote the corre-sponding simple module by W λ . Let M λ be the sum of simple C α M [ G M ]-submodules of M isomorphic to W λ . Then M = ⊕ λ ∈ Λ GM M λ = ⊕ λ ∈ Λ GM W λ ⊗ M λ (4.1)where the multiplicity space M λ of W λ in M is a V G -module.7ecall that the group G acts on set S = ∪ g ∈ G M ( g ) and M ◦ h and M are isomorphic V G -modules for any h ∈ G and M ∈ S . It is clear that the cardinality of the G -orbit | M ◦ G | of M is [ G : G M ] . The following results are obtained in [DRX], and [MT] (alsosee [DLM1], [DY]) .
Theorem 4.1.
Let g, h ∈ G, M an irreducible g -twisted V -module, N an irreducible h -twisted V -module. 1) M λ is nonzero for any λ ∈ Λ G M .
2) Each M λ is an irreducible V G -module.3) M λ and M γ are equivalent V G -module if and only if λ = γ.
4) For any λ ∈ Λ G M and µ ∈ Λ G N , the irreducible V G -modules M λ and N µ areinequivalent if M, N are not in the same orbit of S under the action of G. Decompose S = ∪ j ∈ J O j into a disjoint union of orbits. Let M j for j ∈ J be theorbit representatives of S , O j = { M j ◦ h | h ∈ G } is the orbit of M j under G . For M ∈ S , recall that G M = { h ∈ G | M ◦ h = M } . Let G = ∪ l M i =1 G M g i be a right cosetdecomposition of G . Assume g = 1 is the identity. Then O M = { M ◦ g i | i = 1 , . . . , l M } and G M ◦ g i = g i − G M g i . Then we have Theorem 4.2.
The set { M jλ | j ∈ J, λ ∈ Λ G Mj } gives a complete list of inequivalent irre-ducible V G -modules appearing in the irreducible twisted V -modules. Moreover, if V G isrational and C -cofinite, this set classifies the irreducible V G -modules. That is, any irre-ducible V G -module is isomorphic to an irreducible V G -submodule M λ for some irreducible g -twisted V -module M and some λ ∈ Λ G M . The following Lemma which is straightforward will be useful in the next section.
Lemma 4.3.
Let M be an irreducible g -twisted V -module. Then for any k ∈ G wemay take α M ◦ k such that α M ◦ k ( k − ak, k − bk ) = α M ( a, b ) for a, b ∈ G M . In particular, C α M ◦ k [ G M ◦ k ] and C α M [ G M ] are isomorphic by sending ¯ a to k − ak. For any λ ∈ Λ G M we define λ ◦ k ∈ Λ G M ◦ k such that ( λ ◦ k )( k − ak ) = λ (¯ a ) for a ∈ G M . We denote the irreducible C α M ◦ k [ G M ◦ k ]-module corresponding to λ ◦ k by W λ ◦ k = W λ ◦ k . S -matrix In this section we investigate the modular transformation among the modular forms Z M iλ ( v, τ ) for i ∈ J and λ ∈ Λ M i without the assumption that V G is rational and C -cofinite. Note that the modular group Γ is generated by S and T. The T -matrix is easyto compute: Z M iλ ( v, τ + 1) = e πi ( − c/ h i,λ ) Z M λ ( v, τ )where h i,λ is a rational number determined by the weight space decomposition M iλ = M n ≥ ( M iλ ) h i,λ + n . T is a diagonal unitary matrix with T M iλ ,M iλ = e πi ( − c/ h i,λ ) for i ∈ J and λ ∈ Λ M i . So our main focus is to show that any τ − wt[ v ] Z M iλ ( v, − τ ) is a linear combination of Z M jµ ( v, τ ) for j ∈ J and µ ∈ Λ M j . That is, we will determine the S M iλ ,M jµ in Z M iλ ( v, − τ ) = τ wt[ v ] X j ∈ J,µ ∈ Λ Mj S M iλ ,M jµ Z M jµ ( v, τ ) . ( S M iλ ,M jµ ) ( i,λ ) , ( j,µ ) is called the restricted S -matrix of V G . We need the following lemma which says that the S -matrix is invariant under theconjugation. Lemma 5.1.
Let M ∈ M ( g, h ) for commuting g, h ∈ G . Assume that N ∈ M ( h, g − ) .For any k ∈ G , we have S M,N = S M ◦ k,N ◦ k .Proof. Note that M ◦ k ∈ M ( k − gk, k − hk ) and N ◦ k ∈ M ( k − hk, k − g − k ). UsingTheorem 3.2 and Lemma 4.3 we have the following computation Z M ( v, ( g, h ) , − /τ ) = Z M ◦ k ( v, ( k − gk, k − hk ) , − /τ )= τ wt[ v ] X N ∈ M ( k − hk,k − g − k ) S M ◦ k,N Z N ( v, ( khk − , k − g − k ) , τ )= τ wt[ v ] X N ∈ M ( h,g − ) S M ◦ k,N ◦ k Z N ◦ k ( v, ( khk − , k − g − k ) , τ )= τ wt[ v ] X N ∈ M ( h,g − ) S M ◦ k,N ◦ k Z N ( v, ( h, g − ) , τ ) . Comparing this result with Z M ( v, ( g, h ) , − /τ = τ wt[ v ] X N ∈ M ( h,g − ) S M,N Z N ( v, ( h, g − ) , τ )and using the fact that { Z N ( v, ( h, g − ) , τ ) | N ∈ M ( h, g − ) } are linearly independentfunctions on V × H [DLM5] gives the identity S M,N = S M ◦ k,N ◦ k . (cid:3) The following result essentially gives the S -matrix of V G . Theorem 5.2.
Let M be an irreducible g -twisted V -module. For v ∈ V G , λ ∈ Λ G M , wehave Z M λ ( v, τ ) = 1 | G M | X h ∈ G M Z M ( v, ( g, h ) , τ ) λ ( h ) ,Z M λ ( v, − /τ ) = τ wt[ v ] | G M | X h ∈ G M X N ∈ M ( h,g − ) S M,N X µ ∈ Λ GN µ ( g − ) Z N µ ( v, τ ) λ ( h ) where ¯ h is the element in twisted group algebra C α M [ G M ] corresponding to h ∈ G M and ¯ x for a complex number x is the complex conjugate. roof. Recall M = ⊕ λ ∈ Λ GM W λ ⊗ M λ . We have1 | G M | X h ∈ G M Z M ( v, ( g, h ) , τ ) λ ( h ) = 1 | G M | X h ∈ G M tr M ho ( v ) q L (0) − c λ ( h )= 1 | G M | X h ∈ G M X µ ∈ Λ GM (tr W µ h )(tr M µ o ( v )) q L (0) − c λ ( h )= 1 | G M | X h ∈ G M X µ ∈ Λ GM µ ( h ) λ ( h ) Z M µ ( v, τ )Using the the orthogonality property of the irreducible characters of C α M [ G M ] gives thefirst equality.For the second equality we use Theorem 3.2 and the first equation to see that Z M λ ( v, − /τ ) = τ wt[ v ] | G M | X h ∈ G M X N ∈ M ( h,g − ) S M,N Z N ( v, ( h, g − ) , τ ) λ ( h )= τ wt[ v ] | G M | X h ∈ G M X N ∈ M ( h,g − ) S M,N X µ ∈ Λ GN µ ( g − ) λ ( h ) Z N µ ( v, τ ) , as desired. (cid:3) From [Z] and Theorem 4.2, { Z M jλ ( v, τ ) | j ∈ J, λ ∈ Λ G Mj } are linearly independentvectors in the conformal block of V G . According to the orbifold theory conjecture, thesevectors are expected to form a basis of the conformal block. Note that each Z M ( v, ( g, h ) , τ )for M ∈ M ( g, h ) with g, h ∈ G commuting is also a vector in the conformal block of V G . The first equation in Theorem 5.2 in fact implies the subspace of the conformal blockspanned Z M jλ ( v, τ ) is equal to the the subspace spanned by Z M j ( v, ( g j , h ) , τ ) for j ∈ J where M j is a g j -twisted V -module.Taking M = M i in Theorem 5.2 we can find the S M iλ ,M jµ after the identification of N appearing in Z M λ ( v, − /τ ) with M j . To give an explicit formula for S M iλ ,M jµ we introducesome notations. Let C i,j be an least subset of G such that { M j ◦ k | k ∈ C i,j } = O j ∩ ( ∪ h ∈ G Mi M ( h, g − i )) . Clearly, the choice of C i,j is not canonical and C i,j could be an empty set. Note that V ◦ h ∼ = V for all h ∈ G. So V itself is an orbit with G V = G. We assume that 0 ∈ J and M = V. In this case C ,j consists of k ∈ G such that M j ◦ k gives all the modules inorbit O j . Here is an explicit expression of the S -matrix of V G . Theorem 5.3.
Let i, j ∈ J and λ ∈ Λ G Mi and µ ∈ Λ G Mj . Then S M iλ ,M jµ = 1 | G M i | X k ∈ C i,j S M i ,M j ◦ k λ ( k − g j k ) µ ( kg − i k − )10 f C i,j is not empty, and S M iλ ,M jµ = 0 otherwise. Moreover, if i = 0 we have a simpleformula: S M λ ,M jµ = 1 | G M j | S M ,M j λ ( g − j ) dim W µ for all j. Proof.
We first note that ∪ h ∈ G Mi M ( h, g − i ) = { M j ◦ k | k ∈ C i,j , j ∈ J } . Now in Theorem5.2, replacing
N, h by M j ◦ k, k − g j k, respectively gives Z M iλ ( v, − /τ ) = τ wt[ v ] | G M i | X j ∈ J X k ∈ C i,j X µ ∈ Λ GMj ◦ k S M i ,M j ◦ k λ ( k − g j k ) µ ( g − i ) Z ( M j ◦ k ) µ ( v, τ ) . Let µ ∈ Λ G Mj and k ∈ C i,j . Recall from Section 3 that µ ◦ k ∈ Λ G Mj ◦ k such that ( µ ◦ k )( k − ak ) = µ ( a ) for a ∈ G M j . In particular, ( µ ◦ k )( g − i ) = µ ( kg − i k − ) . Since Λ G Mj ◦ k =Λ G Mj ◦ k and Z ( M j ◦ k ) µ ◦ k ( v, τ ) = Z M jµ ( v, τ ) for µ ∈ Λ G Mj we have Z M iλ ( v, − /τ ) = τ wt[ v ] | G M i | X j ∈ J X k ∈ C i,j X µ ∈ Λ GMj S M i ,M j ◦ k λ ( k − g j k ) µ ( kg − i k − ) Z M jµ ( v, τ )or equivalently, S M iλ ,M jµ = 1 | G M i | X k ∈ C i,j S M i ,M j ◦ k λ ( k − g j k ) µ ( kg − i k − ) . If i = 0 then g = 1 , | C ,j | = | G || G Mj | ,λ ( k − g j k ) = λ ( k − g j k ) = λ ( g j ) = λ ( g − j ) ,µ ( kg − k − ) = dim W µ . Using Lemma 5.1 yields S M ,M j ◦ k = S M ◦ k,M j ◦ k = S M ,M j and S M λ ,M jµ = 1 | G M j | S M ,M j λ ( g − j ) dim W µ . The proof is complete. (cid:3)
Corollary 5.4.
Assume that G is abelian. Let i, j, λ, µ be as before. Then S M iλ ,M jµ = 1 | G M i | X k ∈ C i,j S M i ,M j ◦ k λ ( g j ) µ ( g − i ) if C i,j is not empty, and S M iλ ,M jµ = 0 otherwise. roof. As G is abelian, k − g j k = g j and kg − i k − = g − i . The result follows from Theorem5.3 immediately. (cid:3)
Next we assume that G is holomorphic. Then for each g ∈ G there is a uniqueirreducible g -twisted V -module V ( g ) [DLM5] and G V ( g ) = C G ( g ) . In this case { g i | i ∈ J } is a set of conjugacy class representatives, M i = V ( g i ) , V ( g ) ◦ k = V ( k − gk ) and | C i,j | isexactly the cardinality of the intersection of the conjugate class of g j in G with C G ( g i ) . Ifwe also assume that G is abelian, then J = G, then C g,h = { h } and C G ( g ) = G. Corollary 5.5.
Let V be holomorphic and i, j, λ, µ be as before. We have S V ( g i ) λ ,V ( g j ) µ = 1 | C G ( g i ) | X k ∈ C i,j S V ( g i ) ,V ( k − g j k ) λ ( k − g j k ) µ ( kg − i k − ) . If G is abelian the formula simplifies to S V ( g ) λ ,V ( h ) µ = 1 | G | S V ( g ) ,V ( h ) λ ( h ) µ ( g − ) . Our next example comes from the cyclic permutation orbifolds from [BDM]. Let V bea vertex operator algebra satisfying conditions (V1)-(V3) with G being automorphismgroup of V . For a fixed prime positive integer k , consider the tensor product vertexoperator algebra U = V ⊗ k [FHL]. Any element g of the permutation group S k acts on U in the obvious way. Take g = (1 , , . . . , k ) to be be a k -cycle permutation of the vertexoperator algebra . Our goal is to determine the S matrix of ( V ⊗ k ) G where G is the cyclicgroup generated by g. According to Corollary 5.4 we only need to know the S -matrix inTheorem 3.2. In the permutation orbifolds, we can use the the action of the modulargroup on the conformal block of V to give an explicit formula of S -matrix of ( V ⊗ k ) G dueto the connection between V -modules and g -twisted ( V ⊗ k ) G -modules [BDM].Suppose that M , M , . . . , M p are the irreducible V -modules up to isomorphism. Then V ⊗ k is ratioanl and all irreducible modules are { M i ⊗ M i · · · ⊗ M i k | i , . . . , i k ∈ { , . . . , p }} . It is known from [DLM5] that the number of irreducible g -stable V ⊗ k -modules is equalto the number of of irreducible V -modules and the number of irreducible g -twisted V ⊗ k -modules up to isomorphism is equal to the number of irreducible V -modules. It is provedin [BDM] that there is equivalent functor T g from the category of V -module categoryto the g -twisted V ⊗ k -module category. In particular, T kg ( M ) , T kg ( M ) , . . . , T kg ( M p ) arethe irreducible g -twisted V ⊗ k -modules. There is k Z + -gradation on T kg ( M i ) such that T kg ( M i ) = ⊕ n ≥ T kg ( M i )( nk ) and T kg ( M i )( nk ) ∼ = M i ( n ) as vector space, and Y g ( v, z ) = P m ∈ k Z v m z − m − for v ∈ V ⊗ k . v ∈ V denote by v j ∈ V ⊗ k the vector whose j -th tensor factor is v and whoseother tensor factors are 1. Then gv j = v j +1 for j = 1 , . . . , k , where k + 1 is understood tobe 1. Let W be a g -twisted V ⊗ k -module, and let η = e − πi/k . Then Y g ( v j +1 , z ) = lim z /k → η − j z /k Y g ( v , z ) . Since V ⊗ k is generated by v j for v ∈ V and j = 1 , , . . . , k , the vertex operaotrs Y g ( v , z )for v ∈ V determined all the vertex operators Y g ( u, z ) on W for any u ∈ V ⊗ k .In (End V )[[ z /k , z − /k ]] , define ∆ k ( z ) = exp (cid:0) P j ∈ Z + a j z − j/k L ( j ) (cid:1) k L (0) z (1 /k − L (0) (see[BDM] for detials). Let M be a V -module. The action of V ⊗ k on T g ( M ) is uniquelydetermined by Y g ( v , z ) = Y (∆ k ( z ) v, z /k ) for v ∈ V. If v ∈ V n is a highest weight vectorthen ∆ k ( z ) v = k − n z (1 /k − n v and Y g ( v , z ) = k − n z (1 /k − n Y ( v, z /k ).For the Virasoro vector ω of V , ∆ k ( z ) ω = z /k − k ( ω + ( k − c z − /k ). We write Y g (¯ ω, z ) = P n ∈ Z L g ( n ) z − n − , where ¯ v = P kj =1 v j for any v ∈ V. We have Y g (¯ ω, z ) = P k − i =0 lim z /k η − i z /k Y g ( ω , z ) . It follows that L g (0) = k L (0) + ( k − ) c k . Lemma 6.1.
Let v ∈ V , then o (¯ v ) = ko ( v ) . Moreover, if v ∈ V n is a highest weightvector for the Virasoro algebra, then o (¯ v ) = k − n +1 o ( v ) . Proof.
It is good enough to show the result for homogeneous v. Note that Y g ( gv , z ) = lim z /k → e πik z /k Y g ( v , z ) = X m ∈ Z v mk ( e πik z k ) − m − k = X m ∈ Z v mk e − mπik z − m − kk . Comparing the coefficients of z − wt v , we have o ( v ) = o ( gv ) = o ( v ) and o (¯ v ) = ko ( v ).If v ∈ V n is a highest weight vector, we have Y g ( v , z ) = k − n z (1 /k − n Y ( v, z /k ) and o ( v ) = k − n o ( v ). The result follows. (cid:3) Here is a general result which will be used later.
Lemma 6.2.
Let σ be an automorphism of V of order T and M = P n ≥ M nT + λ a σ -twisted V -module where λ is the weight of M. Then M ◦ σ r and M are isomorphic for anyinteger r and Z M ( v, ( σ, σ r ) , τ ) = e πir ( c − λ ) Z M ( v, τ + r ) . Proof.
We have mentioned already that for any σ -twisted module is σ -stable. Thus any σ -twisted module is also σ r -stable. By the definition of trace function, we have Z M ( v, ( σ, σ r ) , τ ) = tr M o ( v ) ϕ ( σ r ) q L (0) − c = X n ≥ tr M nT + λ o ( v ) e πirnT q nT + λ − c = e πir ( c − λ ) X n ≥ tr M nT + λ o ( v )( e πir q ) L (0) − c = e πir ( c − λ ) Z M ( v, τ + r ) , as desired. (cid:3)
13e first assume that g = (1 , , . . . , k ) with k being a prime. This assumption willmake the computations much easier as σ r is also a k -cycle for r = 1 , ..., k − . Recall that M , M , . . . , M p are the irreducible V -modules up to isomorphism. Let M j = ⊕ n ≥ M jλ j + n ,where j = 0 , . . . , p . Then the irreducible g r -twisted V ⊗ k -module are T kg r ( M j ) and T kg r ( M j ) = ⊕ n ≥ T kg r ( M j ) λjk + ( k − c k + nk where T kg r ( M j ) λjk + ( k − c k + nk = M jλ j + n as vector spaces. Lemma 6.3.
Suppose that r, s ∈ N with r positive such that s ≡ ra ( mod k ) for someinteger a, and v ∈ V is a highest weight vector for the Virasoro algebra, then Z T kgr ( M j ) (¯ v, ( g r , g s ) , τ ) = ǫ πia ( − λjk + c k ) k − wt v +1 Z M j ( v, τ + ak ) . Proof.
A straightforward calculation using Lemmas 6.1 and 6.2 gives Z T kgr ( M j ) (¯ v, ( g r , g s ) , τ ) = e πia ( − ( λjk + ( k − c k )+ kc ) Z T kgr ( M j ) (¯ v, τ + a )= ǫ πia ( − λjk + c k ) X n ≥ tr T kgr ( M j ) λjk + ( k − c k + nk o (¯ v )( e πi ( τ + a ) ) ( L g (0) − kc ) = ǫ πia ( − λjk + c k ) X n ≥ tr M jn + λj k − wt v +1 o ( v ) e πi ( τ + a )( L (0) k + ( k − c k − c ) = ǫ πia ( − λjk + c k ) k − wt v +1 Z M j ( v, τ + ak ) , as expected. (cid:3) Lemma 6.4.
Let < r, s, a, b < k be positive integers such that s ≡ ra , − r ≡ sb modulo k. Then S T kgr ( M j ) ,T kgs ( M i ) = e πia ( − λjk + c k ) − πib ( − λik + c k ) p X l =0 S M j ,M i A r,sl,i , where A r,sl,i is the entry of ρ ( A r,s ) defined in Theorem 3.2 with A r,s = (cid:18) k − b − a abk (cid:19) . Proof.
We first prove that A r,s ∈ SL ( Z ) . Clearly, the determinant of A is 1 . So we onlyneed to show that ab + 1 is divizable by k. Since k is a prime, a, b are invertible modulo k and ab ∼ = − k which is equivalent to the fact that abk is an integer.From Lemma 6.3 we see that Z T kgr ( M j ) (¯ v, ( g r , g s ) , − /τ ) = ǫ πia ( − λjk + c k ) k − wt v +1 Z M j ( v, − τ + ak )= ǫ πia ( − λjk + c k ) k − wt v +1 Z M j ( v, − τk − aτ )= ǫ πia ( − λjk + c k ) k − wt v +1 (¯ τ wt v ) p X i =0 S M j ,M i Z M i ( v, ¯ τ )14here ¯ τ = τk − aτ . Note that ¯ τ = A r,s τ + bk and Z M i ( v, ¯ τ ) = ( 1 − aτk ) wt v p X l =0 A r.si,l Z M l ( v, τ + bk ) . Thus, Z T kgr ( M j ) (¯ v, ( g r , g s ) , − /τ )= ǫ πia ( − λjk + c k ) k − wt v +1 ( τ k − aτ ) wt v p X i =0 S M j ,M i ( 1 − aτk ) wt v p X l =0 A r,si,l Z M l ( v, τ + bk )= ǫ πia ( − λjk + c k ) k − wt v +1 τ wt v p X i,l =0 S M j ,M i A r,si,l Z M l ( v, τ + bk )= ǫ πia ( − λjk + c k ) k − wt v +1 τ wt v p X i,l =0 S M j ,M l A r,sl,i Z M i ( v, τ + bk )On the other hand, Z T kgr ( M j ) (¯ v, ( g r , g s ) , − /τ )= τ wt v p X i =0 S T kgr ( M j ) ,T kgs ( M i ) Z T kgs ( M i ) (¯ v, ( g s , g − r ) , τ )= τ wt v p X i =0 S T kgr ( M j ) ,T kgs ( M i ) ǫ πib ( − λik + c k ) k − wt v +1 Z M i ( v, τ + bk ) . Comparing the right sides of these equations, we have S T kgr ( M j ) ,T kgs ( M i ) = e πia ( − λjk + c k ) − πib ( − λik + c k ) p X l =0 S M j ,M i A r,sl,i . We should point out that the matrix A r,s depends on r, s only. (cid:3) The following result is easy:
Lemma 6.5.
Let M be an irreducible V -module. For g ∈ G , v ∈ V and < r < k then(1) Z M ⊗ k ( , (1 , g r ) , τ ) = χ M ( kτ ) (2) Z M ⊗ k (¯ v, (1 , g r ) , τ ) = kZ M ( v, kτ ) . Proof. (1) follows from (2) by noting that ¯ = k ( ⊗ · · · ⊗ ) . So we only need to prove(2). Let { w α | α ∈ A } be a basis of M such that each w α is homogeneous and a generalizedeigenvector of o ( v ) . Then { w α ⊗ · · · ⊗ w α k | α i ∈ A } is a basis of M ⊗ k and the action of G on M ⊗ k preserves this basis. Note that each w α ⊗ · · · ⊗ w α k is a generalized eigenvector of o (¯ v ) with eigenvalue P ki =1 λ i where λ i is the eigenvalue of w α i . Moreover, for any h ∈ G, ( w α ⊗· · ·⊗ w α k ) is also a generalized eigenvector of o (¯ v ) with the same eigenvalue P ki =1 λ i . If there are at least two different α i in w α ⊗ · · · ⊗ w α k , then P k − s =0 C g s w α ⊗ · · · ⊗ w α k isa k -dimensional subspace of M ⊗ k and has a basis consisting of generalized eigenvectorsof o (¯ v ) g r with eigenvalues ( P ki =1 λ i ) e πisk for s = 0 , ..., k − . Thus the trace contributionfrom this subspace is 0 . So we only need to compute the trace contribution from vectors w α ⊗ · · · w α for α ∈ A. Let λ α be the eigenvalue of w α . Note that ω V ⊗ k = ¯ ω. Denote the corresponding L (0) by L V ⊗ k (0) . Then w α ⊗· · · w α is a generalized eigenvector of o (¯ v ) g r q L V ⊗ k (0) − kc with eigenvalue kλ α q k (wt w α − c ) . Thus we have Z ( M ) ⊗ k (¯ v, (1 , g r ) , τ ) = tr M ⊗ k o (¯ v ) g r q L V ⊗ k (0) − kc/ = X α ∈ A kλ α q k (wt w α − c ) = kZ M ( v, kτ ) . The proof is finished. (cid:3)
Lemma 6.6.
Let g ∈ G , M i , r be as before, i = 0 , . . . , p . Then S ( M i ) ⊗ k ,T kgr ( M j ) = S M i ,M j = S T kgr ( M i ) , ( M j ) ⊗ k . Proof.
Using Lemma 6.5 yields Z ( M i ) ⊗ K (¯ v, (1 , g r ) , − τ ) = kZ M i ( v, − τk ) = k ( τk ) wt v p X j =0 S M i ,M j Z M j ( v, τk ) . By Lemma 6.3 with s = a = 0 ,Z T kgr ( M j ) (¯ v, ( g r , , τ ) = k − wt v +1 Z M j ( v, τk ) . So Z ( M i ) ⊗ K (¯ v, (1 , g r ) , − τ ) = τ wt v p X j =0 S M i ,M j Z T kgr ( M j ) (¯ v, ( g r , , τ )On the other hand, Z ( M i ) ⊗ K ( v, (1 , g r ) , − τ ) = τ wt v p X j =0 S ( M i ) ⊗ k ,T kgr ( M j ) Z T kgr ( M j ) ( v, ( g r , , τ ) . Comparing the right sides of these equations gives S ( M i ) ⊗ k ,T kgr ( M j ) = S M i ,M j for all i, j. Similarly, Z T kgr ( M i ) (¯ v, ( g r , , − τ ) = k − wt v +1 Z M i ( v, − kτ )= k − wt v +1 ( kτ ) wt v p X j =0 S M i ,M j Z M j ( v, kτ )= τ wt v p X j =0 S M i ,M j Z ( M j ) ⊗ k ( v, τ )16nd S M i ,M j = S T kgr ( M i ) , ( M j ) ⊗ k . (cid:3) We are now ready to compute the S -matrix for ( V ⊗ k ) G . First we give a completelist of irreducible ( V ⊗ k ) G -modules following [DRX]. Let i , ..., i k ∈ [0 , p ] where [0 , p ] = { , ..., p } . Set M i ,...,i k = M i ⊗ · · · ⊗ M i k . Then M i ,...,i k is an irreducible ( V ⊗ k ) G -moduleif the cardinality { i , ..., i k } is greater than 1 . Moreover, M i ,...,i k ◦ g r = M i r , ··· ,i k + r for r = 0 , ..., k − j + r is understood to be modulo k. The M i,...,i is adirect sum of irreducible ( V ⊗ k ) G -modules ( M i,...,i ) j for j = 0 , ..., k − M i,...,i ) j = { w ∈ M i,...,i | gw = e − πijk w } = { k − X s =0 e πisjk g s w | w ∈ M i,...,i } and g acts on M i,...,i in an obvious way. We have already mentioned that G acts on each T kg r ( M i ) . So T kg r ( M i ) = ⊕ k − s =0 ( T kg r ( M i )) s is a direct sum of irreducible ( V ⊗ k ) G -modulessuch that g acts on ( T kg r ( M i )) s as e − πisk . Note that Irr( G ) = { λ s | s = 0 , ..., k − } where λ s ( g ) = e − πisk . Note that the symmetric group S k acts on [0 , p ] k \ { ( i, · · · , i ) | i ∈ [0 , p ] } in an obviousway. Let I be a subset of [0 , p ] k \{ ( i, · · · , i ) | i ∈ [0 , p ] } consisting of the orbit representativesunder the action of G. Proposition 6.7.
The irreducible ( V ⊗ k ) G -modules consist of { M i ,...,i k | ( i , ..., i k ) ∈ I } and { ( M i, ··· ,i ) a , ( T kg r ( M i )) a | ≤ i ≤ p, ≤ a ≤ k − , < r < k } . The following theorem gives explicit expressions of the entries of the S -matrix of( V ⊗ k ) G by noting that S -matrix is symmetric [H]. Theorem 6.8. (1) Let i, j = 0 , ..., p, < r, s < k, ≤ a, b < k. Then S ( T kgr ( M i )) a ,N = 1 k S T kgr ( M i ) ,T kgs ( M j ) e πi ( sa + rb ) k if N = ( T kg s ( M j )) b S M i ,M j e πirbk if N = ( M j,...,j ) b where S T kgr ( M i ) ,T kgs ( M j ) is given in Lemma 6.4 and S M i ,M j is the entry of S -matrix of V .(2) Let ( i , ..., i k ) , ( t , ..., t k ) ∈ I. and i ∈ [0 , p ] , ≤ b < k. Then S M i ,...,ik ,N = ( P k − r =0 Q kj =1 S M ij ,M tj + r if N = M t ,...,t k Q kj =1 S M ij ,M i if N = ( M j,...,j ) b (3) Let i, j ∈ [0 , p ] and a, b ∈ [0 , k − . Then S ( M i,...,i ) a , ( M j,...,j ) b = 1 k S kM i ,M j roof. (1) In Corollary 5.4, let g i = g r , g j = g s ′ with s ′ = s or 1 and λ = λ a , µ = λ b , then λ a ( g s ) λ b ( g − r ) = e πi ( s ′ a + rb ) k . The result follows.(2) Let i , ..., i k ∈ [0 , p ] , v , ..., v k ∈ V. Then Z M i ,...,ik ( v ⊗ · · · ⊗ v k , τ ) = Z M i ( v , τ ) · · · Z M ik ( v k , τ )So we have Z M i ,...,ik ( v ⊗ · · · ⊗ v k − τ ) = τ wt v k Y j =1 p X t j =0 S M ij ,M tj Z M tj ( v j , τ ) = τ wt v X ( t ,...,t k ) ∈ [0 ,p ] k k Y j =1 S M ij ,M tj Z M t ,...,tk ( v ⊗ · · · ⊗ v k , τ ) . So S M i ,...,ik ,M t ,...,tk = Q kj =1 S M ij ,M tj for irreducible V ⊗ k -modules.Let ( i , ..., i k ) , ( t , ..., t k ) ∈ I. Using Corollary 5.4 with M i = M i ,...,i k , M j = M t ,...,t k , g i = g j = 1 , C i,j = G and λ = µ = 1 gives S M i ,...,ik ,M t ,...,tk = k − X r =0 k Y j =1 S M ij ,M tj + r . Now let i ∈ [0 , p ] and b = 0 , ..., k − . Using Corollary 5.4 again with M i = M i ,...,i k , M j = M i,...,i , g i = g j = 1 , λ = 1 , µ = λ b gives S M i ,...,ik , ( M i,...,i ) b = k Y j =1 S M ij ,M i . (3) can be proved similarly. (cid:3) In principle one can compute the S -matrix of V ⊗ k for any k. The idea is clear but thecomputation will be more complicated as g r could be a product of several disjoint cycles. References [ABD] T. Abe, G. Buhl and C. Dong, Rationality, Regularity, and C -cofiniteness, Trans. Amer. Math. Soc. (2004), 3391-3402.[ADJR] C. Ai, C. Dong, X. Jiao and L. Ren, The irreducible modules and fusion rules forthe parafermion vertex operator algebras,
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