Some applications and constructions of intertwining operators in Logarithmic Conformal Field Theory

Transcription

1 Some applications and constructions of intertwining operators in Logarithmic Conformal Field Theory Dražen Adamović and Antun Milas To Jim and Robert with admiration ABSTRACT We discuss some applications of fusion rules and intertwining operators in the representation theory of cyclic orbifolds of the triplet vertex operator algebra We prove that the classification of irreducible modules for the orbifold vertex algebra Wp Am follows from a conjectural fusion rules formula for the singlet vertex algebra modules In the p = 2 case, we compute fusion rules for the irreducible singlet vertex algebra modules by using intertwining operators This result implies the classification of irreducible modules for W2 Am, conjectured previously in [4] The main technical tool is a new deformed realization of the triplet and singlet vertex algebras, which is used to construct certain intertwining operators that can not be detected by using standard free field realizations 1 Introduction The orbifold theory with respect to a finite group of automorphisms is an important part of vertex algebra theory There is a vast literature on this subject in the case of familiar rational vertex algebras, including lattice vertex algebras It is widely believed that all irreducible representations of the orbifold algebra V G G is finite arise from the ordinary and g-twisted V -modules via restriction to the subalgebra This is known to hold if V G is regular [14] In fact, one reason for a great success of orbifold theory in the regular case comes from the underlying modular tensor category structure eg, quantum dimensions, Verlinde formula, etc For irrational C 2 -cofinite vertex algebra this theory is much less developed and only a few known examples have been studied, such as the Z 2 -orbifold of the symplectic fermion vertex operator superalgebra [1] and ADE-type orbifolds for the triplet vertex algebra Wp As we had demonstrated in [4]-[6] jointly with X Lin, for the triplet vertex algebra, we have very strong evidence that all irreducible modules for Wp G arise in this way Let us briefly summarize the main results obtained in those papers For the A-type cyclic orbifolds, based on the Key words and phrases W -algebras, logarithmic conformal field theory, orbifolds DA is partially supported by the Croatian Science Foundation under the project 264 AM was partially supported by a Simons Foundation grant # 1790 and support from the Max Planck Institute for Mathematics, Bonn 1

2 2 DRAŽEN ADAMOVIĆ AND ANTUN MILAS analysis of the Zhu algebra, we obtained classification of irreducible modules for Wp A2 for small p [4] After that, in [5], we studied the D-series dihedral orbifolds of Wp Among many results, we classified irreducible Wp D2 -modules again for small p Finally, in [6] we reduced the classification for all A and D-type orbifolds to a combinatorial problem related to certain constant term identities However, these constant term identities are very difficult to prove even for moderately small values of p In this paper we revisit the classification of modules for the cyclic orbifolds by using a different circle of ideas Instead of working with the Zhu algebra we employ the singlet vertex subalgebra to classify irreducible Wp Am -modules Here is the content of the paper with the main results In Section 2, we review several basic notions pertaining to singlet, doublet and triplet vertex algebras, including A n -orbifolds After that, we prove that the classification problem for Wp An, as conjectured in [4], follows from a certain fusion rules conjecture for the singlet vertex algebra module and simple currents In Section, we study fusion rules for the singlet vertex algebra for p = 2 By using Zhu s algebra we obtain an upper bound on the fusion rules of typical modules In Section 4, which is the most original part of the paper, we obtain a new realization of the doublet and thus of the singlet and triplet vertex algebra for p = 2 and p = We also have a conjectural realization for all p 4 Finally, in Section 5, by using the construction in Section 4, we construct all intertwining operators among singlet modules when p = 2 Combined with the previously obtained upper bound for generic modules, this proves that some special singlet algebra modules are simple currents, in a suitable sense Using these results, we prove the completeness of classification of irreducible modules for the orbifold algebra W2 Am Acknowledgments We dedicate this paper to Jim Lepowsky and Robert Wilson who made substantial contributions to the study of vertex operators and vertex operator algebras It was back in the 90s that both authors started investigating affine Lie algebras highly influenced by papers of Lepowsky and Wilson We thank the referee for providing us with many constructive comments 2 Classification of irreducible modules for orbifold Wp Am : A fusion rules approach Let L = Zα be a rank one lattice with α, α = 2p p 1 Let V L = Uĥ<0 C[L] denote the corresponding lattice vertex algebra [20], where ĥ is the affinization of h = Cα, and C[L] is the group algebra of L Let M1 be the Heisenberg vertex subalgebra of V L generated by α 11 with the conformal vector ω = α p 1 4p 2p α 21 With this choice of ω, V L has a vertex operator algebra structure of central charge c 1,p = 1 6p 12 p

3 APPLICATIONS AND CONSTRUCTIONS OF INTERTWINING OPERATORS Let Q := Res x Y e α, x = e α 0, where we use e β to denote vectors in the group algebra of the dual lattice of L The triplet vertex algebra Wp is strongly generated by ω and three primary vectors F = e α, H = QF, E = Q 2 F of conformal weight 2p 1 [] We use Hn := Res x x 2p 2+n Y H, x The doublet vertex algebra Ap see [16, 11] is a generalized vertex algebra strongly generated by ω and a = e α/2, a + = Qe α/2 We use M1 to denote the singlet vertex algebra It is defined as a vertex subalgebra of Wp generated by ω and H It was proved in [2] that the irreducible N graded M1 modules are parametrized by highest weights with respect to L0, H0 which have the form λ t := 1 tt + 2p 2, 4p t 2p 1 t C The irreducible M1-module M t with highest weight λ t is realized as an irreducible subquotient of the M1 module For n N, we define: M1, t t α := M1 e 2p α 2p π n := M np = M1e nα/2 and π n := M np = M1Q n e nα/2 We know [] that π j, j Z, are irreducible as M1 modules and π j M1, jα/2 = M1 e jα/2 Recall that the vertex algebra Wp Am is a Z m -orbifold of Wp and is generated by ω, F m = e mα, E m = Q 2m F m, H = Qe α ; for details see [4] Therefore Wp Am is an extension of the singlet algebra and we have: Wp Am = n Z π 2nm THEOREM 21 [4] For every 0 i 2pm 2 1, there exists a unique up to equivalence irreducible Wp Am module L i which decomposes as the following direct sum of M1 m modules = M i n Z m 2pmn L i m Moreover, L i m is realized as an irreducible subquotient of V i 2pm α+zmα DEFINITION 22 We say that an irreducible N graded M1 module W 1 is a simplecurrent in the category N graded M1 modules if for every irreducible N graded module W 2, there is a unique irreducible N graded M1 module W such that the vector space of intertwining operators I W W 1 W 2 is 1 dimensional and I N W 1 W 2 = 0 for any other irreducible N graded M1 module which is not isomorphic to W We write W 1 W 2 = W

4 4 DRAŽEN ADAMOVIĆ AND ANTUN MILAS Note that we require the simple current property to hold only on irreducible M1 modules; studying fusion products = tensor products with indecomposable modules is a much harder problem which we do not address here CONJECTURE 2 Let p Z 2 Modules π j, j Z, are simple currents in the category of ordinary, N graded M1 modules, and the following fusion rules holds: π j M t = M t jp This conjecture is in agreement with the fusion rules result from [12] based on the Verlinde formula of characters In Section, we shall prove this conjecture in the case p = 2 LEMMA 24 Assume that Conjecture 2 holds Assume that M is an irreducible Wp Am module generated by a highest weight vector v t for M1 of the L0, H0 weight λ t Then t 1 m Z PROOF By Conjecture 2, we see that π 2m sends M t := M1v t to M t 2mp, where M t 2mp = M1v t 2mp is an irreducible M1 module of highest weight 1 t 2mp t 2mpt 2mp 2p + 2, 4p 2p 1 But the difference between conformal weights of v t and v t 2mp must be an integer This leads to the following condition mt p mp = 1 1 t + 2mpt + 2mp 2p + 2 tt 2p + 2 Z 4p 4p Thus, mt Z THEOREM 25 Assume that Conjecture 2 holds Then the vertex operator algebra Wp Am has precisely 2m 2 p irreducible modules, all constructed in [4] PROOF Clearly, any irreducibe Wp Am -module M is non logarithmic ie, L0 acts semisimple Such module has weights bounded from below, so M must contain a singular vector v t for M1 Then by using Lemma 24 we get that as a M1 module, M = n Z M t 2mp where t 1 m Z and M t 2mp is the irreducible highest weight M1 module with highest weight 1 t 2mp λ t 2mp = t 2mpt 2mp 2p + 2, 4p 2p 1 Now, the assertion follows from Theorem 21 REMARK 26 A different approach to the problem of classification of irreducible Wp Am modules was presented in [6] It is based on certain constant term identities and it does not require any knowledge of fusion rules among M1-modules

5 APPLICATIONS AND CONSTRUCTIONS OF INTERTWINING OPERATORS 5 Fusion rules for M1 and a proof of Conjecture 2 for p = 2 Let V be a vertex operator algebra, W 1, W 2, W be V modules, and Y be an intertwining operator of type W W 1 W 2 For u W 1, we use Yu, z = r C u rz r 1, u r HomW 2, W Let U i be a V submodule of W i, i = 1, 2 Define the inner fusion product 1 U 1 U 2 = span C {u r v u U 1, v U 2, r C} Then U 1 U 2 is a V submodule of W Let M1, λ be as before Then M1, λ is an M1 module and e λ of highest weight 1 t tt 2p + 2,, t = λ, α 4p 2p 1 We specialize now p = 2 For t / Z, the irreducible M1 module M t remains irreducible as a Virasoro module [1] and it can be realized as M1, λ Denote by AM1 the Zhu associative algebra of M1 For a M1-module M, denote by AM the AM1-bimodule with the left and right action given by 1 + x dega a m = Res x Y a, xm, x 1 + x dega 1 m a = Res x Y a, xm, x respectively It is known [17], [21] that there is an injective map from the space of intertwining operators of type M M 1 M 2 to Hom AV AM 1 AV M 2 0, M 0 where M i 0 is the top degree of M i Next, we analyze the space AM 1 AV M 2 0 in the case when V = M1, M 1 = M1, λ and M 2 = M1, µ We denote by L V ir c, 0 the simple Virasoro vertex algebra of central charge c Clearly, M1, λ can be viewed as a Virasoro vertex algebra module As such, we have A V ir M1, λ = C[x, y] as an AL V ir c, 0-bimodule Therefore, AM1, λ = C[x, y]/i, where I is the ideal coming from extra relations in M1 Observe also that inside AM1, λ AM1 C µ, where C µ = Cv µ is the onedimensional AM1-module spanned by the highest weight vector v µ, we have the following relations here v AM1, λ: µ, α v [H] v λ v v µ = 0, µ, α µ, α 2 v [ω] v λ v v µ = 0, 1 + x Res x x 2 Y H, xv = H 2 + H 1 + H 0 + H 1 v OM1, λ Let W λ, µ := AM1, λ AM1 C µ = C[x]/J

6 6 DRAŽEN ADAMOVIĆ AND ANTUN MILAS Then we have THEOREM 1 Suppose t = α, λ / Z, and s = α, µ / Z Then s + ts + t 2 px = x x In particular, W λ, µ is at most 2-dimensional s + t 2s + t 4 J PROOF Irreducibility of M1, λ with respect to Virasoro algebra, implies that H i v λ, i 2 can be written in the Virasoro basis We omit these explicit formulas here obtained by a straightforward tedious computations Together with H 2 + H 1 + H 0 + H 1 v λ = 0, we obtain a new relation fx, y = 0 inside AM1, λ This relation turns into x t + st + s 2 x t + s 2t + s 4 x s ts t 2 x inside W λ, µ Similarly, s 0 = v λ [H] v µ v v µ s = H 1 + 2H 0 + H 1 v λ v µ turns into x t + st + s 2 x s t + 2s t = 0 t + s 2t + s 4 x s2 + t2 st 2 + s 4 + t 4 = 0 inside W λ, µ = 0 Now we have to analyze common roots of these two polynomials Clearly, they have x t+st+s 2 x t+s 2t+s 4 in common It remains to analyse whether s ts t 2 = s2 + t2 st 2 + s 4 + t 4 or s ts t + 2 = s2 + t2 st 2 + s 4 + t 4 The first equation holds if t = 2 for all s and also for s = 0 for all t The second equation holds for s = 2 for all t and also for t = 0 for all s But this means either s or t are integral The initial assumptions on s and t now yield the claim REMARK 2 By using exactly the same method we can prove that for p = the space W λ, µ is at most -dimensional, as predicted in [12] We shall use the following result on intertwining operators for the Virasoro algebra LEMMA Assume that there is a non-trivial intertwining operator I, z of type M1, ν L V ir 2, n 2 + n/2 M1, µ such that Iv n, z = r C vn r z r 1 and there is r 0 C such that 2 v n r 0 v µ = λv ν, λ 0 Here v n is highest weight vector in L V ir 2, n 2 + n/2 Then ν = µ + i n/2α, for 0 i n

7 APPLICATIONS AND CONSTRUCTIONS OF INTERTWINING OPERATORS 7 PROOF This is essentially proven in Theorem 51 in [22] THEOREM 4 Let p = 2 Modules π j, j Z are simple currents in the category of ordinary, N graded M1 modules PROOF Let us here consider the case j > 0 the case j < 0 can be treated similarly Then t = 2j / {0, 2} As above, let M s and M r be irreducible M1 modules with highest weight λ s and λ r, respectively By using the proof of Theorem 1 we see that if Mr I 0, π j M s then exactly one of the following holds: 1 s / {0, 2} and r {s 2j, s 2j 2}, 2 s = 0, r {2j, 2j + 2, 2j + 4, 2j 2, 2j, 2j + 2}, s = 2, r {2j, 2j + 2, 2j + 4, 2 2j, 2j, 2j + 2} Cases 2 and come from analyzing additional solutions By using the following relation in AM1, λ we get that H v λ v λ H = H 0 + 2H 1 + H 2 v λ r = s + t or r = s + t 2 or r = 1/5 s + t ± s + 4t 12st + 4t 2 This implies 1 s / {0, 2} and r {s 2j, s 2j 2}, 2 s = 0, r { 2j 2, 2j, 2j + 2, 4+2j }, s = 2, r { 2j 2, 2j, 2j + 2, 4+2j } On the other hand since L V ir 2, j 2 + j/2 is an irreducible Virasoro submodule of π j, then in the category of L V ir 2, 0 modules we have a non-trivial intertwining operator of the type M r L V ir 2, j 2 + j/2 M s Now, applying Virasoro fusion rules computed in Lemma we get r {s 2j, s 2j + 4,, s + 2j 4, s + 2j} By analyzing all cases above, we see that the only possibility is r = s 2j The proof follows 4 Deformed realization of the triplet and singlet vertex algebra 41 Definition of deformed realization In this section we present a new realization of the triple and doublet vertex operator algebra This is then used to show that the sufficient conditions obtained in Theorem 1 are in fact necessary Let p Z >0 Let V L be the lattice vertex algebra associated to the lattice L = Zα 1 + Zα 2, α 1, α 1 = 2p 1, α 2, α 2 = 1, α 1, α 2 = 0 Let α = α 1 + α 2 Define ω = 1 4p α 12 + p 1 2p α 2 + p 1 p e 2α2

8 DRAŽEN ADAMOVIĆ AND ANTUN MILAS Then ω generates the Virasoro vertex operator algebra Lc 1,p, 0 of central charge c 1,p = 1 6p 1 2 /p More generally, PROPOSITION 41 Let ω V be a conformal vector of central charge c and v V 1 degree one a primary vector for ω such that [v n, v m ] = 0 for all m, n Z Then ω = ω + v is a conformal vector of the same central charge Consider the following operator acting on V L S = e α1+α2 0 + e α1 α2 0 LEMMA 42 The operator S is a screening operator for the Virasoro algebra generated by ω PROOF We have to prove that [e α1+α2 0 + e α1 α2 This follows directly from the relations [e α1+α2 0, L n] = 0, L 0e α1 α2 = 0 [e α1 α2 0, e 2α2 n+1 0, L n + p 1 p e 2α2 n+1 ] = 0 ] = 0, [eα1+α2 0, e 2α2 n+1 ] = α α 2 1e α1 α2 n+1 [Ln, e α1 α2 0 ] = L 1 e α1 α2 n+1 = p 1 p Let Wp be the vertex subalgebra of V L generated by ω, F = e α, H = SF, E = S 2 F Let Ãp be the generalized vertex algebra generated by a = e α/2, α1 1 + α 2 1e α1 α2 n+1 a + = Sa = S p 1 αe α/2 + S p 2 α 1 α 2 e α/2 2α2 Clearly, Wp Ãp Let W 0 p be the vertex subalgebra of Wp generated by ω and H = SF = S 2p 1 α + S 2p α 1 α 2 e 2α2 CONJECTURE 4 We have as vertex algebras Ãp = Ap, Wp = Wp, W0 p = M1, THEOREM 44 Conjecture 4 holds for p = 2 and p = PROOF Let Case p = 2 a = e α/2, a + = Sa = S p 1 αe α/2 + S p 2 α 1 α 2 e α/2 2α2

9 APPLICATIONS AND CONSTRUCTIONS OF INTERTWINING OPERATORS 9 Then a + = α 1e α/2 + e α/2 2α2 Direct calculation shows that a n a + = 2δ n,1 1, a n a + = 0 n 0, and by supercommutator formulae we have {a n, a + m} = 2nδ n+m,0, {a ± n, a ± m} = 0 n, m Z This implies a and a + generate the subalgebra isomorphic to the symplectic fermion vertex superalgebra A2 cf [1], [19], [2] Since the even part of the symplectic fermions is isomorphic to the triplet vertex algebra Wp for p = 2, the assertion holds Case p = Let F p/2 be the generalized lattice vertex algebra associated to the lattice L = Zϕ, ϕ, ϕ = p/2 with generators e ±ϕ cf [15] and V 4/ sl 2 the universal affine vertex algebra at the admissible level 4 As in [], one shows that for p = 4 e = a e ϕ, f = 2 9 a+ e ϕ, h = 4 ϕ defines a non-trivial homomorphism Φ : V 4/ sl 2 à F /2 which maps singular vector in V 4/ sl 2 to zero calculation is completely analogous to that in [] Therefore 4 gives an embedding of the simple affine vertex algebra V 4/ sl 2 into à F /2 By construction, we have that the vacuum space ΩV 4/ sl 2 = {v V 4/ sl 2 hnv = 0 n 1} is generated by a + and a and therefore In [], it was proven that So we have ΩV 4/ sl 2 = à The vacuum space ΩV 4/ sl 2 is isomorphic to the doublet vertex algebra A The parafermionic vertex algebra Ksl 2, 4/ = {v V 4/ sl 2 hnv = 0 n 0} is isomorphic to W 0 = M1 à = ΩV 4/ sl 2 = A, W 0 = Ksl2, 4/ = W 0 Since the triplet W is a Z 2 -orbifold of A, we also have that W = W This proves the conjecture in the case p =

10 10 DRAŽEN ADAMOVIĆ AND ANTUN MILAS 42 On the proof of Conjecture 4 for p 4 At this point, we do not yet have a uniform proof of Conjecture 4 for all p The problem is that Wp and M1 are not members of a family of generic vertex algebras and W algebras In order to solve this difficulty, one can study the embeddings of Wp and M1 into affine vertex algebras and minimal W algebras cf [], [1] The case p = was alredy discussed in the proof of Theorem 44 Similar arguments can be repeated in the p = 4 case For p = 4 one can show that 5 a + e ϕ, a e ϕ generate the simple vertex algebra W 2 with central charge 2/2 whose vacuum space is generated by a + and a By using the realization from [1], we get that a + and a generate the doublet vertex algebra A4 So Conjecture 4 also holds for p = 4 For p 5, one needs to prove that 5 define the Feigin-Semikhatov W algebra W p 2 embedded into Ap F p/2 But this calculation is much more complicated 4 Deformed action for p = 2 Now let p = 2 Then the singlet vertex algebra M1 is isomorphic to a subalgebra of V L generated by ω = 1 α α e 2α2, H = Se α = S α + α 1 1 α 2 2e 2α2 Let r, s C Consider the M1 module We have: where Fr, s := M1v r,s, v r,s = e r α1+sα2 L0v r0 = h r+1,1 v r,0, L0v r, 1 = h r,1 v r, 1 L0v r,1 = h r+2,1 v r, v r, 1, L0v r,1 1 v r, 1 = h r+2,1 v r,1, 2h r,1 r H0v r0 = v r,0, r 1 H0v r, 1 = v r, 1, r + 1 H0v r,1 = v r,1 + r 1v r, 1, h r,1 = PROPOSITION 45 1 Assume that r / Z Then r 12 r 1 4 = r 22 1 P r := M1v r,1 = M1v + r,1 M1v r, 1

11 APPLICATIONS AND CONSTRUCTIONS OF INTERTWINING OPERATORS 11 where v + r,1 = v r,1 + p 1 p 1 v r, 1 = v r,1 + 1 h r+2,1 h r,1 r 1 v r, 1, vr,1 = v r, 1 are highest weight vectors for M1 of L0, H0 weight r + 1 r 1 h r+2,1,, h r,1, respectively In particular, P r = M r+1 Mr 1 2 Assume that r = p 1 = 1 Then P 1 := M1v 1,1 is a Z 0 -graded logarithmic M1 module The lowest component P 1 0 is 2 dimensional, spanned by v 1,1 and v 1, 1 For r Z \ {0}, see also Remark 52 PROOF Parts 1 and 2 follow immediately due to irreducibility of M t and because L0 forms a Jordan block on the top component, respectively REMARK 46 In [10], the authors presented an explicit realization of certain logarithmic modules for Wp Our deformed realization of Wp gives an alternative realization of these logarithmic modules We plan to return to these realizations in our forthcoming publications 5 Intertwining operators between typical M1 and Wp Am modules: the p = 2 case In this section we fix p = 2 THEOREM 51 1 Assume that t 1, t 2, t 1 + t 2 / Z Then in the category of M1 modules, there exists non trivial intertwining operators of types Mt1+t 2 M t1 M t2, In particular, the following fusion rules holds: Mt1+t 2 2 M t1 M t2 M t1 M t2 = M t1+t 2 + M t1+t 2 2 PROOF In our realization, M1 is a vertex subalgebra of the vertex algebra U := M α1 1 V Z2α2, where M α1 1 is the Heisenberg vertex algebra, generated by α 1, and V Z2α2 is the lattice vertex algebra associated to the even lattice Z2α 1 First we notice that in the category of U modules we have a non-trivial intertwining operator Y of type M α1 1, t1+t2 1 α 1 V α2+z2α 2 M α1 1, t1 α 1 V Z2α2 M α1 1, t2 1 α 1 V α2+z2α 2

12 12 DRAŽEN ADAMOVIĆ AND ANTUN MILAS Clearly, M t1 = M1v t1,0 M α1 1, t 1 α 1 V Z2α2 M t2 = M1v t + M 2 1,1 α 1 1, t 2 1 α 1 V α2+z2α 2 M t1+t 2 = M1v t + M 1+t 2 1,1 α 1 1, t 1 + t 2 1 α 1 V α2+z2α 2 M t1+t 2 2 = M1vt M 1+t 2 1,1 α 1 1, t 1 + t 2 1 α 1 V α2+z2α 2 By abusing the notation, let Y denotes the restriction to M t1 as defined above We have to first check that Y satisfies the L 1-property for the new Virasoro generator L 1 = L e 2α2 0 Observe that YL 1v t1,0 = YL 1v t1,0, x = d dx Yv t 1,0, x For other vectors, the L 1 property follows from the fact that M t1 is a cyclic module By using Theorem 1 and the fact that M t1 M t2 is completely reducible, we see that the corresponding fusion product 1, defined via the intertwining operator Y, must satisfy M t1 M t2 M t1+t 2 M t1+t 2 2 By construction, we have that w = v t1+t 2 1,1 + 1 t 2 2 v t 1+t 2 1, 1 = v + t 1+t 2 1,1 + t 1 t 2 2t 1 + t 2 2 v t 1+t 2 1,1 M t 1 M t2, which implies that singular vectors v ± t 1+t 2 1,1 belong to M t 1 M t2 So we get that M t1 M t2 = M t1+t 2 M t1+t 2 2 as desired REMARK 52 One can show that for t 1, t 2 / Z, t 1 + t 2 Z, our free field realizations in Proposition 45 also gives the existence of a logarithmic intertwining operator of type Pt1+t 2 M t1 Moreover, observe that the condition t 1, t 2 / Z is not necessary here and we may assume t 1, t 2 {4n + 1, 4n 1 n Z} This leads to a new family of intertwining operator presumably obtained by the restriction of an intertwining operator coming from a triple of symplectic fermions modules We shall study these intertwining operators in more details in our future publications M t1 We have a similar result in the category of Wp Am modules: THEOREM 5 Assume that i 1, i 2, i {0,, 2pm 2 1} and i 1, i 2, i 1 + i 2 / mz, where m is a positive integer Then in the category of Wp Am modules, there exists non trivial intertwining operators of type if and only if L i/m L i1/m L i2/m i i 1 + i 2 mod2m 2 p or i i 1 + i 2 2m mod2m 2 p REMARK 54 Results in this paper, together with Remark 52, provide a rigorous proof of the main conjecture in [12] pertaining to fusion rules of M1 for p = 2 Almost all results in the paper can be extended for the N = 1 singlet/triplet vertex algebras introduced and studied by the authors [9] work in progress

13 APPLICATIONS AND CONSTRUCTIONS OF INTERTWINING OPERATORS 1 References [1] Abe, T, A Z 2 -orbifold model of the symplectic fermionic vertex operator superalgebra, Mathematische Zeitschrift : [2] Adamović D, Classif ication of irreducible modules of certain subalgebras of free boson vertex algebra, J Algebra , , [] Adamović, D, A construction of admissible A 1 1 -modules of level 4/, J Pure Appl Algebra , [4] Adamović D, Lin X, and Milas A, ADE subalgebras of the triplet vertex algebra Wp: A-series, Commun Contemp Math , 15002, 0 pages, [5] Adamović D, Lin X, and Milas A, ADE subalgebras of the triplet vertex algebra Wp: D-series, , , 4 pages [6] Adamović D, Lin X, and Milas A, Vertex Algebras Wp Am andwp Dm and Constant Term Identities, SIGMA , 019, 16 pages [7] Adamović D and Milas A, Logarithmic intertwining operators and W2, 2p 1 algebras, J Math Phys , 0750, 20 pages [] Adamović D and Milas A, On the triplet vertex algebra Wp, Adv Math , [9] Adamović D and Milas A, The N= 1 triplet vertex operator superalgebras, Comm Math Phys : [10] Adamović D and Milas A, Lattice construction of logarithmic modules for certain vertex algebras, Selecta Math NS [11] Adamović, D and Milas, A The doublet vertex operator superalgebras Ap and A 2,p, Contemporary Mathematics [12] Creutzig T and Milas A, False theta functions and the Verlinde formula, Adv Math , [1] Creutzig, T and Ridout D, and Wood, S, Coset constructions of logarithmic 1, p models, Lett Math Phys , no 5, 55 5 [14] Dong C, Ren, L and Xu, F, On Orbifold Theory, arxiv: [15] Dong C, Lepowsky J, Generalized vertex algebras and relative vertex operators, Progress in Math 112, Birkhauser Boston, 199 [16] Feigin, B and Tipunin, I, Characters of coinvariants in 1, p logarithmic models, New trends in quantum integrable systems, World Sci Publ, Hackensack, NJ, 2011, pp 5 60, [17] Frenkel I, Zhu Y, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math J , no 1, [1] Iohara K, and Koga Y,Representation theory of the Virasoro algebra, Springer Science Business Media, 2010 [19] Kac, V G Vertex algebras for beginners, second ed, University Lecture Series, vol 10, American Mathematical Society, Providence, RI, 199 [20] Lepowsky J and Li H, Introduction to vertex operator algebras and their representations, Progress in Mathematics, Vol 227, Birkhäuser Boston, Inc, Boston, MA, 2004 [21] Li H, Determining fusion rules by AV-modules and bimodules, J Algebra ?556 [22] Lin, X,Fusion rules of Virasoro vertex operator algebras, Proceedings of the American Mathematical Society : [2] Runkel I, A braided monoidal category for free super-bosons, J Math Phys , no 4, , 59 pp DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ZAGREB, CROATIA address: adamovic@mathhr CURRENT ADDRESS: MAX PLANCK INSTITUTE FOR MATHEMATICS, VIVATSGASSE 7, BONN, GER- MANY PERMANENT ADDRESS: DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY AT ALBANY SUNY, ALBANY, NY address: amilas@mathalbanyedu