Generalized parafermions of orthogonal type
aa r X i v : . [ m a t h . QA ] O c t GENERALIZED PARAFERMIONS OF ORTHOGONAL TYPE
THOMAS CREUTZIG, VLADIMIR KOVALCHUK, AND ANDREW R. LINSHAWA
BSTRACT . There is an embedding of affine vertex algebras V k ( gl n ) ֒ → V k ( sl n +1 ) , and thecoset C k ( n ) = Com ( V k ( gl n ) , V k ( sl n +1 )) is a natural generalization of the parafermion alge-bra of sl . It was called the algebra of generalized parafermions by the third author and wasshown to arise as a one-parameter quotient of the universal two-parameter W ∞ -algebra oftype W (2 , , . . . ) . In this paper, we consider an analogous structure of orthogonal type,namely D k ( n ) = Com ( V k ( so n ) , V k ( so n +1 )) Z . We realize this algebra as a one-parameterquotient of the two-parameter even spin W ∞ -algebra of type W (2 , , . . . ) , and we clas-sify all coincidences between its simple quotient D k ( n ) and the algebras W ℓ ( so m +1 ) and W ℓ ( so m ) Z . As a corollary, we show that for the admissible levels k = − (2 n −
2) + (2 n +2 m − for b so n the simple affine algebra L k ( so n ) embeds in L k ( so n +1 ) , and the cosetis strongly rational. As a consequence, the category of ordinary modules of L k ( so n +1 ) atsuch a level is a braided fusion category.
1. I
NTRODUCTION
For n ≥ , the natural embedding of Lie algebras gl n ֒ → sl n +1 defined by a (cid:18) a − tr ( a ) (cid:19) , induces a vertex algebra homomorphism(1.1) V k ( gl n ) ֒ → V k ( sl n +1 ) . The coset vertex algebra(1.2) C k ( n ) = Com ( V k ( gl n ) , V k ( sl n +1 )) was called the algebra of generalized parafermions in [LIV]. The reason for this terminologyis that for n = 1 , C k (1) is isomorphic to the parafermion algebra N k ( sl ) = Com ( H , sl ) ,where H denotes the Heisenberg algebra corresponding to the Cartan subalgebra h ⊆ sl .By Theorem 8.1 of [LIV], C k ( n ) is of type W (2 , , . . . , n + 3 n + 1) , i.e., it has a minimalstrong generating set consisting of one field in each weight , , . . . , n + 3 n + 1 . Thisgeneralizes the case n = 1 , which appears in [DLY]. When k is a positive integer, (1.1)descends to a map of simple affine vertex algebras L k ( gl n ) ֒ → L k ( sl n +1 ) , and the cosetCom ( L k ( gl n ) , L k ( sl n +1 )) coincides with the simple quotient C k ( n ) of C k ( n ) . By Theorem13.1 of [ACL], we have an isomorphism(1.3) C k ( n ) ∼ = W ℓ ( sl k ) , ℓ = − k + k + nk + n + 1 . In particular, C k ( n ) is strongly rational , that is, C -cofinite and rational. This generalizesthe case n = 1 , which was proved earlier in [ALY]. T. C. is supported by NSERC Discovery Grant useful perspective on C k ( n ) is that these algebras all arise in a uniform way as quo-tients of the universal two-parameter W ∞ -algebra W ( c, λ ) of type W (2 , , . . . ) ; see The-orem 8.2 of [LIV]. This realization gives a nice conceptual explanation for the isomor-phisms appearing in (1.3). Each one-parameter quotient of W ( c, λ ) corresponds to anideal in C [ c, λ ] , or equivalently, a curve in the parameter space C called the truncationcurve . The truncation curves for W ℓ ( sl m ) and C k ( n ) are given by Equations 7.8 and 8.4 of[LIV], and the above isomorphisms correspond to intersection points on these curves.The algebras C k ( n ) appear naturally as building blocks for affine vertex algebras oftype A . It is convenient to replace C k ( n ) with ˜ C k ( n ) = H ⊗ C k ( n ) , where H is a rank oneHeisenberg vertex algebra. Then we haveCom ( V k ( gl n − ) , V k ( gl n )) ∼ = ˜ C k ( n − , so V k ( gl n ) can be regarded as an extension of V k ( gl n − ) ⊗ ˜ C k ( n − . Iterating this proce-dure, we see that V k ( gl n ) is an extension of(1.4) H ⊗ ˜ C k (1) ⊗ ˜ C k (2) ⊗ · · · ⊗ ˜ C k ( n − . Note that if k is a positive integer, the simple quotient L k ( gl n ) is then an extension of H ⊗ ˜ C k (1) ⊗ ˜ C k (2) ⊗ · · · ⊗ ˜ C k ( n − ∼ = W ℓ ( gl k ) ⊗ W ℓ ( gl k ) ⊗ · · · ⊗ W ℓ n ( gl k ) , where ℓ i = − k + k + n − ik + n +1 − i . In [ACL], this was regarded as a noncommutative analogue ofthe Gelfand-Tsetin subalgebra of U ( gl n ) . Similarly, we may regard the subalgebra (1.4) asthe universal version of this structure.The algebras C k ( n ) also appear as building blocks for various W -(super)algebras. Forexample, an important conjecture of Ito [I] asserts that the principal W -algebra W ℓ ( sl n +1 | n ) has a coset realization as(1.5) Com ( V k +1 ( gl n ) , V k ( sl n +1 ) ⊗ F (2 n )) , where F (2 n ) denotes the rank n free fermion algebra, and ( ℓ + 1)( k + n + 1) = 1 . Ito’sconjecture was stated in this form in [CLII], and these algebras have the same strong gen-erating type by Lemma 7.12 of [CLII]. In the case n = 1 , the conjecture clearly holdsbecause both sides are isomorphic to the N = 2 superconformal algebra. The first non-trivial case n = 2 was proven in [GL]. It was also shown in [GL] that the coset (1.5) isnaturally an extension of W r ( gl n ) ⊗ C k ( n ) for r = − n + n + kn + k +1 . An important ingredientin the proof of Ito’s conjecture will be to show that W ℓ ( sl n +1 | n ) is indeed an extensionof W r ( gl n ) ⊗ C k ( n ) . Note that C k ( n ) is itself a subalgebra of a W -superalgebra of sl n +1 | n corresponding to a small hook-type nilpotent element [CLIII]. Generalized parafermion algebras of orthogonal type.
There are two different analoguesof C k ( n ) in the orthogonal setting. We have natural embeddings so n ֒ → so n +1 ֒ → so n +2 ,which induce homomorphisms of affine vertex algebras(1.6) V k ( so n ) ֒ → V k ( so n +1 ) ֒ → V k ( so n +2 ) . The cosets Com ( V k ( so n ) , V k ( so n +1 )) and Com ( V k ( so n +1 ) , V k ( so n +2 )) both have actionsof Z , and we define(1.7) D k ( n ) = Com ( V k ( so n ) , V k ( so n +1 )) Z , E k ( n ) = Com ( V k ( so n +1 ) , V k ( so n +2 )) Z . Both these algebras arise as one-parameter quotients of the universal even-spin W ∞ -algebra W ev ( c, λ ) constructed recently by Kanade and the third author in [KL]. Such uotients of W ev ( c, λ ) are in bijection with a family of ideals I in the polynomial ring C [ c, λ ] , or equivalently, the truncation curves V ( I ) ⊆ C . The main result in this paper isthe explicit description of the truncation curve for D k ( n ) for all n ; see Theorem 3.3. Theproof is based on the coset realization of principal W -algebras of type D and a certainlevel-rank duality appearing in [ACL], which implies that(1.8) D m ( n ) ∼ = W ℓ ( so m ) Z , ℓ = − (2 m −
2) + 2 m + 2 n − m + 2 n − . This is analogous to the isomorphisms (1.3) in type A . Since a similar coset realization oftype B principal W -algebras is not available, we are currently unable to obtain an explicitdescription of E k ( n ) , and in this paper we only study D k ( n ) .As in type A , there is a similar description of affine vertex algebras of orthogonal typeas extensions of Gelfand-Tsetlin type subalgebras. Clearly V k ( so n +2 ) is an extension of H ⊗ D k (1) ⊗ E k (1) ⊗ D k (2) ⊗ E k (2) ⊗ · · · ⊗ D k ( n − ⊗ E k ( n − ⊗ D k ( n ) ⊗ E k ( n ) , and similarly, V k ( so n +1 ) is an extension of H ⊗ D k (1) ⊗ E k (1) ⊗ D k (2) ⊗ E k (2) ⊗ · · · ⊗ D k ( n − ⊗ E k ( n − ⊗ D k ( n ) . Additionally, D k ( n ) is a building block for various W -(super)algebras. For example, con-sider the principal W -superalgebra W ℓ ( osp n | n ) where ( ℓ + 1)( k + 2 n −
1) = 1 . Note that and n − are the dual Coxeter numbers of osp n | n and so n +1 , respectively. The freefermion algebra F (2 n ) carries an action of L ( so n ) , and it is expected that(1.9) W ℓ ( osp n | n ) ∼ = Com ( V k +1 ( so n ) , V k ( so n +1 ) ⊗ F (2 n )) . This algebra appears in physics in the duality of N = 1 superconformal field theories andhigher spin supergravities [CHR, CV], and this conjecture appeared in this context. Notethat central charges coincide. It is apparent that the coset appearing in (1.9) is an extensionof W r ( so n ) ⊗ D k ( n ) where r = − (2 n −
2) + k +2 n − k +2 n − . As in the case of Ito’s conjecture, animportant step in the proof of (1.9) will be to show that W ℓ ( osp n | n ) is also an extensionof this structure. Applications.
The first application of our main result is to classify all isomorphisms be-tween the simple quotient D k ( n ) and the simple algebras W ℓ ( so m +1 ) and W ℓ ( so m ) Z .Using results of [KL], this can be achieved by finding the intersection points between thetruncation curve for D k ( n ) , and the truncation curves for W ℓ ( so m +1 ) and W ℓ ( so m ) Z , re-spectively. In the type A case, we find only one family of points where C k ( n ) is isomorphicto a strongly rational W -algebra of type A ; these appear in (1.3). In the orthogonal setting,the situation is more interesting. In addition to the isomorphisms (1.8) when k is a posi-tive integer, we also find that for k = − (2 n −
2) + (2 n + 2 m − , we have an embeddingof simple affine vertex algebras L k ( so n ) → L k ( so n +1 ) , and an isomorphism C k ( n ) = Com ( L k ( so n ) , L k ( so n +1 )) Z ∼ = W ℓ ( so m +1 ) , ℓ = − (2 m −
1) + 2 m + 2 n − m + 2 n + 1 . Since ℓ is a nondegenerate admissible level for so m +1 , W ℓ ( so m +1 ) is strongly rational [ArI,ArII]. These are new examples of cosets of non-rational vertex algebras by admissiblelevel affine vertex algebras, which are strongly rational.This coset is also closely related to level-rank duality. Recall that n (2 m + 1) freefermions carry an action of L n ( so m +1 ) ⊗ L m +1 ( so n ) . The levels shifted by the respective ual Coxeter number are n + 2 m − in both cases. Therefore L k ( so n +1 ) is an extensionof L k ( so n ) ⊗ W ℓ ( so m +1 ) , where ℓ = − (2 m −
1) + m +2 n − m +2 n +1 , i.e., both levels k and ℓ shiftedby the respective dual Coxeter numbers are of the form (2 m + 2 n − /v for v = 2 and v = 2 + 2 m + 2 n − . In particular, the shifted levels have the same numerator as theoriginal level-rank duality and the two denominators only differ by a multiple of the nu-merator. Note that under certain vertex tensor category assumptions the tensor productof two vertex algebras can be extended to a larger vertex algebra with a certain multi-plicity freeness condition if and only if the two vertex algebras have subcategories thatare braid-reversed equivalent, see [CKMII, Main Thm. 3] for the precise statement. Ap-plied to our setting, this means that there are vertex algebra extensions of L k ( so n ) and W ℓ ( so m +1 ) that have subcategories of modules that are braid-reversed equivalent.The theory of vertex algebra extensions, especially [CKMII, Thm. 5.12], then impliesthat the category of ordinary modules of L k ( so n +1 ) at level k = − (2 n −
2) + (2 n + 2 m − is fusion, i.e. a rigid braided semisimple tensor category. This proves special cases ofConjecture 1.1 of [CHY].Finally, our rationality results for C k ( n ) suggest the existence of a new series of principal W -superalgebras of osp n | n which are strongly rational. By Corollary 14.2 of [ACL], thecoset Com ( L k +1 ( so n ) , L k ( so n +1 ) ⊗F (2 n )) is strongly rational when k is a positive integer.In view of the conjectured isomorphism (1.9), this implies that for k a positive integer and ℓ satisfying ( ℓ + 1)( k + 2 n −
1) = 1 , W ℓ ( osp n | n ) is strongly rational. Similarly, it followsfrom Corollary 1.1 of [CKMII] that for k = − (2 n −
2) + (2 n + 2 m − and ℓ satisfying ( ℓ + 1)( k + 2 n −
1) = 1 , the coset Com ( L k +1 ( so n ) , L k ( so n +1 ) ⊗ F (2 n )) is again stronglyrational. This motivates the following Conjecture 1.1.
For k = − (2 n −
2) + (2 n + 2 m − and ℓ satisfying ( ℓ + 1)( k + 2 n −
1) = 1 , W ℓ ( osp n | n ) is strongly rational. The conjecture is true for the N = 2 super Virasoro algebra, i.e. the case n = 1 [Ad].Otherwise strong rationality for principal W -superalgebras of orthosymplectic type iscompletely open. There is, however, a C -cofiniteness results in the case of osp | n [CGN,Cor. 5.19]. 2. V ERTEX ALGEBRAS
We shall assume that the reader is familiar with vertex algebras, and we use the samenotation and terminology as the papers [LIV, KL]. We first recall the universal two-parameter vertex algebra W ev ( c, λ ) of type W (2 , , . . . ) , which was recently constructedin [KL]. It is defined over the polynomial ring C [ c, λ ] and is generated by a Virasoro field L of central charge c , and a weight primary field W , and is strongly generated by fields { L, W i | i ≥ } where W i = W W i − for i ≥ . The idea of the construction is asfollows.(1) All structure constants in the OPEs of L ( z ) W i ( w ) and W j ( z ) W k ( w ) for i ≤ and j + 2 k ≤ , are uniquely determined as elements of C [ c, λ ] by imposing theJacobi identities among these fields.(2) This data uniquely and recursively determines all OPEs L ( z ) W i ( w ) and W j ( z ) W k ( w ) over the ring C [ c, λ ] if a certain subset of Jacobi identities are imposed.
3) By showing that the algebras W k ( sp m ) all arise as one-parameter quotients of W ev ( c, λ ) after a suitable localization, we show that all Jacobi identities hold. Equiv-alently, W ev ( c, λ ) is freely generated by the fields { L, W i | i ≥ } , and is the uni-versal enveloping algebra of the corresponding nonlinear Lie conformal algebra[DSK]. W ev ( c, λ ) is simple as a vertex algebra over C [ c, λ ] , but there is a certain discrete familyof prime ideals I = ( p ( c, λ )) ⊆ C [ c, λ ] for which the quotient W ev ,I ( c, λ ) = W ev ( c, λ ) /I · W ev ( c, λ ) , is not simple as a vertex algebra over the ring C [ c, λ ] /I . We denote by W ev I ( c, λ ) the simplequotient of W ev ,I ( c, λ ) by its maximal proper graded ideal I . After a suitable localization,all one-parameter vertex algebras of type W (2 , , . . . , N ) for some N satisfying somemild hypotheses, can be obtained as quotients of W ev ( c, λ ) in this way. This includes theprincipal W -algebras W k ( so m +1 ) and the orbifolds W k ( so m ) Z . The generators p ( c, λ ) forsuch ideals arise as irreducible factors of Shapovalov determinants, and are in bijectionwith such one-parameter vertex algebras.We also consider W ev ,I ( c, λ ) for maximal ideals I = ( c − c , λ − λ ) , c , λ ∈ C . Then W ev ,I ( c, λ ) and its quotients are vertex algebras over C . Given maximal ideals I = ( c − c , λ − λ ) and I = ( c − c , λ − λ ) , let W and W be the simple quotients of W ev ,I ( c, λ ) and W ev ,I ( c, λ ) . Theorem 8.1 of [KL] gives a simple criterion for W and W to be isomorphic. Aside from a few degenerate cases, we must have c = c and λ = λ .This implies that aside from the degenerate cases, all other coincidences among the sim-ple quotients of one-parameter vertex algebras W ev ,I ( c, λ ) and W ev ,J ( c, λ ) , correspond tointersection points of their truncation curves V ( I ) and V ( J ) .We shall need the following result which is analogous to Theorem 6.2 of [LIV]. Theorem 2.1.
Let W be a vertex algebra of type W (2 , , . . . , N ) which is defined over somelocalization R of C [ c, λ ] /I , for some prime ideal I . Suppose that W is generated by the Virasorofield L and a weight primary field W . If in addition, the graded character of W agrees with thatof W ev ( c, λ ) up to weight , then W is a quotient of W I ( c, λ ) after localization.Proof. First, note that Theorem 3.10 of [KL] holds without the simplicity assumption; seeRemark 5.1 of [LIV] for a similar statement in the case of the algebra W ( c, λ ) of type W (2 , , . . . ) . By Theorem 3.10 of [KL], it suffices to prove that the OPEs L ( z ) W i ( w ) and W j ( z ) W k ( w ) for i ≤ and j + 2 k ≤ in W are the same as the corresponding OPEsin W ev ( c, λ ) if the structure constants are replaced with their images in R . In this notation, W i = W W i − for i ≥ . But this is automatic because the graded character assump-tion implies that there are no null vectors of weight w ≤ in the (possibly degenerate)nonlinear conformal algebra corresponding to { L, W i | ≤ i ≤ N } . (cid:3)
3. G
ENERALIZED PARAFERMIONS OF ORTHOGONAL TYPE
For n ≥ , the natural embedding so n ֒ → so n +1 induces a vertex algebra homomor-phism V k ( so n ) → V k ( so n +1 ) . he action of so n on V k ( so n +1 ) given by the zero modes of the generating fields inte-grates to an action of the orthogonal group O n . Therefore the cosetCom ( V k ( so n ) , V k ( so n +1 )) = V k ( so n +1 ) so n [ t ] has a nontrivial action of Z , we define(3.1) D k ( n ) = Com ( V k ( so n ) , V k ( so n +1 )) Z . It has Virasoro element L so n +1 − L so n with central charge(3.2) c = kn (2 k + 2 n − k + 2 n − k + 2 n − . Note that the case n = 1 , D k ( n ) ∼ = N k ( sl ) Z which is of type W (2 , , , , by Theorem10.1 of [KL]. Lemma 3.1.
For all n ≥ , D k ( n ) is of type W (2 , , . . . , N ) for some N satisfying N ≥ n +3 n .We conjecture, but do not prove, that N = 2 n + 3 n . Moreover, for generic values of k , D k ( n ) isgenerated by the weight primary field W .Proof. By Theorem 6.10 of [CLII], we have lim k →∞ D k ( n ) ∼ = H (2 n ) O n , and a strong generating set for H (2 n ) O n corresponds to a strong generating set for D k ( n ) for generic values of k . Here H (2 n ) denotes the rank n Heisenberg vertex algebra. Itwas shown in [LII], Theorem 6.5, that H (2 n ) O n has the above strong generating type.By Lemma 4.2 of [LI], the weights and fields generate H (2 n ) O n . In fact, it is easy tocheck that only the weight field is needed, and that it can be replaced with a primaryfield which also generates the algebra. Finally, the statement that D k ( n ) inherits theseproperties of H (2 n ) O n for generic values of k is also clear; the argument is similar to theproof of Corollary 8.6 of [CLI]. (cid:3) Corollary 3.2.
For all n ≥ , there exists an ideal K n ⊆ C [ c, λ ] and a localization R n of C [ c, λ ] /K n such that D k ( n ) is the simple quotient of W ev ,K n R n ( c, λ ) .Proof. This holds for n = 1 by Theorem 10.1 of [KL]. For n > , the simplicity of D k ( n ) as a vertex algebra over C [ k ] follows from the simplicity of H (2 n ) O n , which follows from[DLM]. In view of Theorems 2.1 and 3.1, it then suffices to show that the graded char-acters of D k ( n ) and W ev ( c, λ ) agree up to weight . This follows from Weyl’s secondfundamental theorem of invariant theory for O n , since there are no relations among thegenerators of weight less than n + 6 n + 2 . (cid:3) Theorem 3.3.
For all n ≥ , D k ( n ) is isomorphic to a localization of the quotient W ev K n ( c, λ ) ,where the ideal K n ⊆ C [ c, λ ] is described explicitly via the parametrization k ( c n ( k ) , λ n ( k )) iven by c n ( k ) = kn (2 k + 2 n − k + 2 n − k + 2 n − , λ n ( k ) = ( k + 2 n − k + 2 n − p n ( k )7( k − k + n − n − q n ( k ) r n ( k ) ,p n ( k ) = −
112 + 188 k − k − k + 12 k + 744 n − kn + 857 k n − k n + 36 k n − n + 2534 kn − k n + 188 k n + 1632 n − kn + 304 k n − n + 152 kn ,q n ( k ) = 20 − k + 6 k − n + 28 kn + 28 n ,r n ( k ) = 44 − k + 22 k − n + 73 kn + 10 k n + 88 n + 10 kn . (3.3) Proof.
Let n be fixed. In view of Corollary 3.2 and the fact that all structure constants in D k ( n ) are rational functions of k , there is some rational function λ n ( k ) of k such that D k ( n ) is obtained from W ev ( c, λ ) by setting c = c n ( k ) and λ = λ n ( k ) , and then taking the simplequotient. It is not obvious yet that λ n ( k ) is a rational function of n as well.For k a positive integer, it is well known [KW] that the map V k ( so n ) → V k ( so n +1 ) descends to a homomorphism of simple algebras L k ( so n ) → L k ( so n +1 ) . Letting D k ( n ) denote the simple quotient of D k ( n ) , it is apparent from Lemma 2.1 of [ACKL] and The-orem 8.1 of [CLII] that Com ( L k ( so n ) , L k ( so n +1 )) is simple and coincides with the simplequotient of Com ( V k ( so n ) , V k ( so n +1 )) . Moreover, taking Z -invariants preserves simplic-ity, hence D k ( n ) ∼ = Com ( L k ( so n ) , L k ( so n +1 )) Z . Next, by Corollary 1.3 of [ACL], for all n ≥ and m ≥ , we have an isomorphism(3.4) (cid:0)(cid:0) L m ( so n +1 ) ⊕ L m (2 mω ) (cid:1) so n [ t ] (cid:1) Z × Z ∼ = W ℓ ( so m ) , ℓ = − (2 m − n + 2 m − n + 2 m − . In this notation, ω denotes the first fundamental weight of so n +1 and L m (2 mω ) denotesthe simple quotient of the corresponding Weyl module.Note that (cid:0) L m ( so n +1 ) so n [ t ] (cid:1) Z = D m ( n ) is manifestly a subalgebra of the left handside of (3.4). Also, the lowest-weight component of L m (2 mω ) has conformal weight m .If m > , the left-hand side then has a unique primary weight field which lies in D m ( n ) .Similarly, since W ℓ ( so m ) has strong generators in weights , , . . . , m and m , for m > the right hand side has a unique primary weight field, which lies in the Z -orbifold W ℓ ( so m ) Z .Since D k ( n ) is generated by the weight field as a one-parameter vertex algebra, theweight field must generate D m ( n ) for all m sufficiently large. By Corollary 6.1 of [KL], W ℓ ( so m ) Z is generated by the weight field as a one-parameter vertex algebra; equiv-alently, this holds for generic values of ℓ . By the same argument as Proposition A.4 of[ALY], the vertex Poisson structure on the associated graded algebra gr W ℓ ( so m ) withrespect to Li’s canonical filtration, is independent of ℓ for all noncritical values of ℓ . Inparticular this holds for the subalgebra ( gr W ℓ ( so m )) Z = gr ( W ℓ ( so m ) Z ) . It followsfrom the same argument as Proposition A.3 of [ALY] that W ℓ ( so m ) Z is generated by theweights and fields for all noncritical values of ℓ , and the same therefore holds for thesimple quotient W ℓ ( so m ) Z . Finally, for ℓ = − (2 m −
2) + n +2 m − n +2 m − , it is straightforwardto verify that the Virasoro field can be generated from the weight field, so the weight field generates the whole algebra. herefore if m is sufficiently large, we obtain(3.5) D m ( n ) ∼ = W ℓ ( so m ) Z , ℓ = − (2 m −
2) + 2 m + 2 n − m + 2 n − . In fact, we will see later (Theorem 4.1) that this holds for all m ≥ .Finally, the truncation curve that realizes W ℓ ( so m ) Z as a quotient of W ev ( c, λ ) is givenby Theorem 6.3 of [KL], and in parametric form by Equation (B.1) of [KL]. In view of(3.5), we must have λ n (2 m ) = λ m ( ℓ ) for ℓ = − (2 m −
2) + n +2 m − n +2 m − for m sufficiently large,where λ m ( ℓ ) is given by Equation (B.1) of [KL]. If follows that for infintely many valuesof k , λ n ( k ) is given by the above formula (3.3). Since λ n ( k ) is a rational function of k , thisequality holds for all k where it is defined. This completes the proof. (cid:3)
4. C
OINCIDENCES
In this section, we shall use Theorem 3.3 to classify all coincidences between the simplequotient D k ( n ) and the Z -orbifold W ℓ ( so m ) Z , as well as W ℓ ( so m +1 ) . We also classify allcoincidences between D k ( n ) and D ℓ ( m ) for m = n . Theorem 4.1.
For n ≥ and m ≥ , aside from the critical levels k = − n + 2 and k = − n + 1 , and the degenerate cases c = , − , all isomorphisms D k ( n ) ∼ = W ℓ ( so m ) Z appear onthe following list:(1) k = 2 m, ℓ = − (2 m −
2) + 2 n + 2 m − n + 2 m − ,(2) k = − (2 n − − n − m − , ℓ = − (2 m −
2) + 2 m − n − m − ,(3) k = − (2 n −
2) + n − mm , ℓ = − (2 m −
2) + m − nm .Proof. Recall first that W ℓ ( so m ) Z is realized as the simple quotient of W ev ,J m ( c, λ ) , wherethe ideal J m ⊆ C [ c, λ ] is given in parametrized form by Equation (B.1) of [KL]. First, weexclude the values of k and ℓ which are poles of the functions λ n ( k ) given by (3.3), and λ m ( ℓ ) given by Equation (B.1) of [KL], since at these values, D k ( n ) and W ℓ ( so m ) Z are notquotients of W ev ( c, λ ) . For all other noncritical values of k and ℓ , D k ( n ) and W ℓ ( so m ) Z are obtained as quotients of W ev ,I n ( c, λ ) and W ev ,J m ( c, λ ) , respectively. By Corollary 8.2 of[KL], aside from the degenerate cases given by Theorem 8.1 of [KL], all other coincidences D k ( n ) ∼ = W ℓ ( so m ) Z correspond to intersection points on the truncation curves V ( K n ) and V ( J m ) . A calculation shows that V ( K n ) ∩ V ( J m ) consists of exactly five points ( c, λ ) ,namely, (cid:18) − , − (cid:19) , (cid:18) , − (cid:19) , (cid:18) mn (4 m + 2 n − m + n − m + 2 n − , λ (cid:19) , (cid:18) − mn (3 − m − n + 4 mn )2 m − n − , λ (cid:19) , (cid:18) − (2 mn + m − n )(2 mn − m − n ) m − n , λ (cid:19) . (4.1) ere λ = ( m + n − m + 2 n − g m − m + n − n − gh ,f = −
28 + 94 m − m − m + 48 m + 186 n − mn + 857 m n − m n + 144 m n − n + 1267 mn − m n + 376 m n + 408 n − mn + 304 m n − n + 76 mn ,g = 10 − m + 12 m − n + 28 mn + 14 n ,h = 22 − m + 44 m − n + 73 mn + 20 m n + 44 n + 10 mn . (4.2) λ = (1 − m + 2 n ) f − m + 2 mn )( − − n + 4 mn ) gh ,f = 14 − m − m + 24 m + 74 n − mn + 873 m n − m n + 144 m n + 80 n − mn − m n + 452 m n − m n − n + 264 mn − m n + 256 m n − m n + 72 mn − m n − m n + 32 m n ,g = −
10 + 19 m − m − n + 22 mn − m n − n − mn + 8 m n ,h = 11 − m + 22 n + 15 mn − m n − mn + 20 m n . (4.3) λ = ( n − m ) f m − n − m − n + 2 mn ) gh ,f = − m + 19 m + 68 m n − m n − mn − m n + 302 m n − m n − n + 204 mn − m n + 80 m n − n + 100 mn − m n − m n + 16 m n ,g = − m + 7 mn − n − mn + 4 m n ,h = − m − m + 22 n + 5 mn + 10 n − mn + 20 m n . (4.4)By Theorem 8.1 of [KL], the first two intersection points occur at degenerate values of c .By replacing the parameter c with the levels k and ℓ , we see that the remaining intersectionpoints yield the nontrivial isomorphisms in Theorem 4.1. Moreover, by Corollary 8.2 of[KL], these are the only such isomorphisms except possibly at the values of k, ℓ excludedabove.Finally, suppose that k is a pole of the function λ n ( k ) given by (3.3). It is not difficult tocheck that the corresponding values of ℓ for which c n ( k ) = c m ( ℓ ) , are not poles of λ m ( ℓ ) . Asabove, c n ( k ) and λ n ( k ) are given by (3.3), and c m ( ℓ ) and λ m ( ℓ ) are given by Equation (B.1)of [KL]. It follows that there are no additional coincidences at the excluded points. (cid:3) Next, we classify the coincidences between D k ( n ) and W ℓ ( so m +1 ) . Theorem 4.2.
For n ≥ and m ≥ , aside from the critical levels k = − n + 2 and k = − n + 1 ,and the degenerate cases c = , − , all isomorphisms D k ( n ) ∼ = W ℓ ( so m +1 ) appear on thefollowing list:(1) k = − (2 n −
2) + 12 (2 n + 2 m − , ℓ = − (2 m −
1) + 2 m + 2 n − m + 2 n + 1 , k = − (2 n −
2) + 2 n − m − m + 2 , ℓ = − (2 m −
1) + 2 m − n + 12 m + 2 ,(3) k = − (2 n − − nm , ℓ = − (2 m −
1) + m − nm ,(4) k = − (2 n − − n − m − , ℓ = − (2 m −
1) + 2 m − m − n + 1 .(5) k = − (2 n −
2) + 2( n − m − m + 1 , ℓ = − (2 m −
1) + 2 m + 12( m − n + 1) .Proof. The argument is the same as the proof of Theorem 4.1. First, W ℓ ( so m +1 ) is realizedas the simple quotient of W ev ,I m ( c, λ ) where the ideal I m ⊆ C [ c, λ ] is parametrized explic-itly by Equation (A.3) of [KL]. The above isomorphisms all arise from the intersectionpoints between the truncation curves V ( K n ) for D k ( n ) and V ( I m ) for W ℓ ( so m +1 ) . A cal-culation shows that there are exactly intersection points: the degenerate points ( , − ) and ( − , − ) , and the five nontrivial ones appearing above. One then has to rule outadditional coincidences at the points where D k ( n ) does not arise as a quotient of W ev ( c, λ ) ,namely, the poles of λ n ( k ) . The details are straightforward and are left to the reader. (cid:3) Finally, we classify all isomorphisms D k ( m ) ∼ = D ℓ ( n ) for n = m . Theorem 4.3.
For m, n ≥ and n = m , aside from the degenerate cases c = , − and poles of c n ( k ) , λ n ( k ) and c m ( k ) , λ m ( k ) the complete list of isomorphisms D k ( m ) ∼ = D ℓ ( n ) is the following:(1) k = − (2 m −
2) + 2( m − n , ℓ = − (2 n − − m + 2 n − m − ,(2) k = − (2 m − − m + 2 n − n − , ℓ = − (2 n −
2) + 2( n − m . The proof is similar to the proof of Theorem 4.1 and is omitted.5. S
OME RATIONAL COSETS
By composing the map V k ( so n ) → V k ( so n +1 ) with the quotient map V k ( so n +1 ) → L k ( so n +1 ) , we obtain an embedding ˜ V k ( so n ) ֒ → L k ( so n +1 ) , where ˜ V k ( so n ) denotes the quotient of V k ( so n ) by the kernel J k of the above compo-sition. In general, it is a difficult and important problem to determine when J k is themaximal proper graded ideal, or equivalently, when ˜ V k ( so n ) = L k ( so n ) . In the casewhere k is an admissible level for d so n , Lemma 2.1 of [ACKL] would then imply thatCom ( L k ( so n ) , L k ( so n +1 )) is simple, and hence its orbifold Com ( L k ( so n ) , L k ( so n +1 )) Z would be simple as well [DLM]. Additionally, Theorem 8.1 of [CLII] would imply thatCom ( L k ( so n ) , L k ( so n +1 )) Z coincides with the simple quotient D k ( n ) of D k ( n ) . This isparticularly interesting in the cases where D k ( n ) is strongly rational.We conclude by proving this for first family in Theorem 4.2. These are new examplesof cosets of non-rational vertex algebras by admissible level affine vertex algebras, whichare strongly rational. emma 5.1. For n ≥ and m ≥ , we have an embedding of simple affine vertex algebras L k ( so n ) ֒ → L k ( so n +1 ) , k = − (2 n −
2) + 12 (2 n + 2 m − . Proof.
We proceed by induction on m . In the case m = 0 , we have k = − n + , and itis well known that there exists a conformal embedding L k ( so n ) ֒ → L k ( so n +1 ) , see e.g.section 3 of [AKMPP]. Next, we assume the result for m − , so that k = − n + + m − .Recall that the rank n + 1 free fermion algebra F (2 n + 1) admits an action of L ( so n +1 ) ,as well as an action of L ( so n ) via the embedding L ( so n ) ֒ → L ( so n +1 ) . The image of L ( so n ) lies in the subalgebra F (2 n ) ⊆ F (2 n + 1) .Since k is admissible for so n +1 , it is known [KW] that we have a diagonal embeddingof simple affine vertex algebras(5.1) L k +1 ( so n +1 ) ֒ → L k ( so n +1 ) ⊗ F (2 n + 1) . By induction, we have the map L k ( so n ) ֒ → L k ( so n +1 ) . Then we have an embedding(5.2) L k +1 ( so n ) ֒ → L k ( so n ) ⊗ F ( n ) ֒ → L k ( so n +1 ) ⊗ F (2 n + 1) , where F (2 n ) ֒ → F (2 n + 1) is the isomoprhism onto the first n copies. Since the image of(5.2) lies in the image of (5.1), it follows that L k +1 ( so n ) embeds in L k +1 ( so n +1 ) . (cid:3) This has the following immediate consequence.
Corollary 5.2.
For n ≥ , m ≥ , and k = − (2 n −
2) + (2 n + 2 m − , we have an isomorphismCom ( L k ( so n ) , L k ( so n +1 )) Z ∼ = W ℓ ( so m +1 ) , ℓ = − (2 m −
1) + 2 m + 2 n − m + 2 n + 1 . In particular, Com ( L k ( so n ) , L k ( so n +1 )) Z is strongly rational.Proof. This follows from Theorem 4.2 together with the fact that Com ( L k ( so n ) , L k ( so n +1 )) Z is simple, and the map D k ( n ) → Com ( L k ( so n ) , L k ( so n +1 )) Z is surjective. (cid:3) Recall that the category of ordinary modules of an affine vertex algebra at admissiblelevel is semisimple [ArIII] and a vertex tensor category [CHY]. Conjecturally, this cat-egory is fusion [CHY] and this has been proven for simply-laced Lie algebras [C]. Fortype so n +1 and level k = − (2 n −
2) + (2 n + 2 m − this conjecture is also true. First,Com ( L k ( so n ) , L k ( so n +1 )) is a simple current extension, call it V ℓ ( so m +1 ) , of W ℓ ( so m +1 ) and thus rational as well [Li]. It follows that L k ( so n +1 ) is a simple Z -graded exten-sion of L k ( so n ) ⊗ V ℓ ( so m +1 ) in a rigid vertex tensor category C of L k ( so n ) ⊗ V ℓ ( so m +1 ) -modules, namely the Deligne product of the categories of ordinary L k ( so n ) -modules and V ℓ ( so m +1 ) -modules. Every ordinary module for L k ( so n +1 ) must be an object in this cate-gory C . This means that as a braided tensor category the category of ordinary modules of L k ( so n +1 ) is equivalent to the category of local modules for L k ( so n +1 ) viewed as an al-gebra object in C [HKL, CKMI]. All assumptions of Theorem 5.12 of [CKMII] are satisfied(with U = V ℓ ( so m +1 ) and V = L k ( so n ) ) and so Corollary 5.3.
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