Trialities of orthosymplectic \mathcal{W}-algebras
aa r X i v : . [ m a t h . R T ] F e b TRIALITIES OF ORTHOSYMPLECTIC W -ALGEBRAS THOMAS CREUTZIG AND ANDREW R. LINSHAWA
BSTRACT . Trialities of W -algebras are certain nontrivial isomorphisms between the affinecosets of three different W -(super)algebras. We recently proved a family of trialities in type A that were conjectured by Gaiotto and Rapˇc´ak, and in this paper we prove an analogousresult in types B , C , and D . The key idea is to identify the affine cosets of these algebras withone-parameter quotients of the universal two-parameter even spin W ∞ -algebra which wasrecently constructed by Kanade and the second author. Special cases of our result includeFeigin-Frenkel duality in types B , C , and D , as well as the coset realization of principal W -algebras of type D . Another special case is a new coset realization of principal W -algebras oftypes B and C , which is quite different from the simply-laced cases. As an application, weprove the rationality of the affine vertex superalgebra L k ( osp | n ) , the minimal W -algebra W k − / ( sp n +2 , f min ) , and the coset Com ( L k ( sp m ) , L k ( sp n )) , for all integers k, n, m ≥ with m < n . We also prove the rationality of some families of principal W -superalgebras of osp | n and osp | n , and subregular W -algebras of so n +1 .
1. I
NTRODUCTION
Trialities of W -algebras are nontrivial isomorphisms between the affine cosets of threedifferent W -(super)algebras, as one-parameter vertex algebras. In recent work, we proveda family of trialities among W -(super)algebras of type A which was conjectured by Gaiottoand Rapˇc´ak in [GR]; see Theorem 1.1 of [CL3]. This theorem is a common generalizationof both Feigin-Frenkel duality and the coset realization of principal W -algebras of sl n [FF2,ACL], as well as Feigin-Semikhatiov duality between subregular W -algebras of sl n , andprincipal W -superalgebras of sl n | [FS, CGN]. The key idea of the proof was to identifyall these affine cosets with one-parameter quotients of the universal two-parameter vertexalgebra W ( c, λ ) of type W (2 , , . . . ) . The existence and uniqueness of this structure wasconjectured for many years in the physics literature [YW, GG, Pro1, Pro2], and was recentlyproven by the second author in [L2]. One-parameter quotients of W ( c, λ ) are in bijectionwith a family of curves in the parameter space C called truncation curves, and Theorem1.1 of [CL3] follows from the explicit ideals that define these curves.In this paper, we will prove an analogous triality theorem which involves eight familiesof W -(super)algebras of types B , C , and D . First, g will be either so n +1 , sp n , so n , or osp n | r , and will decompose as g = a ⊕ b ⊕ ρ a ⊗ ρ b . Here a and b are Lie sub(super)algebras of g , and ρ a , ρ b transform as the standard represen-tations of a , b , respectively, and can have either even or odd parity. Key words and phrases. vertex algebra; W -algebra; nonlinear Lie conformal algebra; coset construction.T. C. is supported by NSERC Discovery Grant et f b ∈ g be the nilpotent element which is principal in b and trivial in a , and let W k ( g , f b ) be the corresponding W -(super)algebra. In all cases, W k ( g , f b ) is of type W (cid:18) dim a , , , . . . , m, (cid:18) d b + 12 (cid:19) d a (cid:19) . In particular, there are dim a fields in weight which generate an affine vertex (super)algebraof a . The fields in weights , , . . . , m are even and are invariant under a . The d a fields inweight d b +12 can be even or odd, and transform as the standard a -module.For n, m ≥ we have the following cases where b = so m +1 .(1) Case 1B : g = so n +2 m +2 , a = so n +1 .(2) Case 1C : g = osp m +1 | n , a = sp n .(3) Case 1D : g = so n +2 m +1 , a = so n .(4) Case 1O : g = osp m +2 | n , a = osp | n .For n ≥ and m ≥ we have the following cases where b = sp m .(1) Case 2B : g = osp n +1 | m , a = so n +1 .(2) Case 2C : g = sp n +2 m , a = sp n .(3) Case 2D : g = osp n | m , a = so n .(4) Case 2O : g = osp | n +2 m , a = osp | n .It is convenient to replace the level k with the critically shifted level ψ = k + h ∨ , where h ∨ is the dual Coxeter number of g . For i = 1 , and X = B, C, D, O , we denote thecorresponding W -algebras by W ψiX ( n, m ) . In the cases i = 1 , and X = C , we denote thecorresponding affine cosets by C ψiC ( n, m ) . In the cases X = B, D, O , there is an additionalaction of Z , and C ψiX ( n, m ) denotes the Z -orbifold of the affine coset. We will also definethe algebras W ψ X ( n, in a different way so that our results hold uniformly for all n, m ≥ .Our main result is that there are four families of trialities among the algebras C ψiX ( n, m ) . Main Theorem 1. (Theorem 4.1) For all integers m ≥ n ≥ , we have the following isomorphismsof one-parameter vertex algebras. (1.1) C ψ B ( n, m ) ∼ = C ψ ′ O ( n, m − n ) ∼ = C ψ ′′ B ( m, n ) , ψ ′ = 14 ψ , ψ + 1 ψ ′′ = 2 , (1.2) C ψ C ( n, m ) ∼ = C ψ ′ C ( n, m − n ) ∼ = C ψ ′′ C ( m, n ) , ψ ′ = 12 ψ , ψ + 1 ψ ′′ = 1 , (1.3) C ψ D ( n, m ) ∼ = C ψ ′ D ( n, m − n ) ∼ = C ψ ′′ O ( m, n − , ψ ′ = 12 ψ , ψ + 1 ψ ′′ = 1 , (1.4) C ψ O ( n, m ) ∼ = C ψ ′ B ( n, m − n ) ∼ = C ψ ′′ D ( m + 1 , n ) , ψ ′ = 1 ψ , ψ + 12 ψ ′′ = 1 . Special cases of this result include Feigin-Frenkel duality in types B , C , and D [FF2], aswell as a version for principal W -superalgebras of osp | n which was recently proven in[CGe]. The special case C ψ D ( n, ∼ = C ψ ′′ O (0 , n − , ψ + 1 ψ ′′ = 1 , n ≥ f (1.3) provides a new proof the coset realization of principal W -algebras of type D [ACL].The special case C ψ B ( n, ∼ = C ψ ′′ B (0 , n ) , ψ + 1 ψ ′′ = 2 of (1.1) recovers the coset realization of the principal W -superalgebra of osp | n [CGe]. Moreimportantly, the special case C ψ C ( n, ∼ = C ψ ′′ C (0 , n ) , ψ + 1 ψ ′′ = 1 of (1.2) provides a new coset realization of principal W -algebras of type B , since C ψ C ( n,
0) =
Com ( V k ( sp n ) , V k ( osp | n )) , k = −
12 ( ψ + 2 n + 1) . C ψ ′′ C (0 , n ) = W ψ ′′ − n +1 ( so n +1 ) . This is quite different from the coset realizations of W k ( g ) for g simply-laced given in [ACL]since it involves affine vertex superalgebras. Finally, the special case C ψ D (1 , m ) ∼ = C ψ ′ D (1 , m − , ψ ′ = 12 ψ of (1.3) provides an alternative proof of the duality between the Heisenberg cosets of W ψ ′ − m +1 ( so m +1 , f subreg ) and W ψ − m ( osp | m ) appearing in [CGN].The key idea in the proof of Main Theorem 1 is to identity all the algebras C ψiX ( n, m ) asone-parameter quotients of the universal even spin two-parameter W ∞ -algebra W ev ( c, λ ) constructed by Kanade and the second author in [KL]. Such quotients of W ev ( c, λ ) are inbijection with a family of plane curves called truncation curves. By explicitly computingthe truncation curves for these algebras, Main Theorem 1 follows from symmetries of ourformulas for the defining ideals. We thus identify four distinct N × N families of truncationsof W ev ( c, λ ) , which we conjecture to account for all of its truncations. A further observationis that we are able to give a uniform description of all these truncations by replacing theinteger parameters n and m in one of the formulas by half-integers. We mention that thealgebras C ψiX ( n, m ) were called orthosymplectic Y -algebras by Gaiotto and Rapˇc´ak in [GR],and some of the trialities we prove were conjectured in [GR], as well as the paper [Pro3] ofProch´azka.1.1. Rationality results.
As an application of Main Theorem 1, we prove many new ratio-nality results in Section 7.
Main Theorem 2.
Denote by F ( n ) the vertex superalgebra of n free fermions, by L k ( g ) the simpleaffine vertex superalgebra of the simple Lie superalgebra g at level k , and by W ℓ ( g ) the simpleprincipal W -superalgebra of g at level ℓ .(1) For all n, k ∈ Z ≥ , L k ( osp | n ) is lisse and rational (Theorem 7.1).(2) For all m ≥ , W ψ − m − / ( osp | m ) is lisse and rational at the following levels:(a) ψ = m − m + r ) , where m + r and r are coprime (Theorem 7.2),(b) ψ = m m +2 r ) , where r and m are coprime (Theorem 7.2),(c) ψ = m m +2 r − , where r − and m are coprime (Theorem 7.3).(3) For all m ≥ , W ψ − m − ( so m +3 , f subreg ) is lisse and rational at the following levels:(a) ψ = m +2 r m +2 , where m + 1 and r + 1 are coprime (Theorem 7.5), b) ψ = m +2 r +12 m +1 where r and m + 1 are coprime (Theorem 7.6),(c) ψ = m )1+2 m (Corollary 7.3),(d) ψ = m m − (Corollary 7.4).(4) For all m ≥ , W ψ − m ( osp | m +2 ) is lisse and rational at the following levels:(a) ψ = m m +2 r , where m + 1 and r + 1 are coprime (Corollary 7.1),(b) ψ = m +12(2 m +2 r +1) , where r and m + 1 are coprime (Corollary 7.2),(c) ψ = m +14(2+ m ) (Corollary 7.3),(d) ψ = m − m (Corollary 7.4).(5) For all n ≥ and r ≥ , W r − / ( sp n +2 , f min ) is lisse and rational.(6) For all k, n, m ∈ Z ≥ with n > m , the following cosets are lisse and rational:(a) Com ( L k − ( sp n ) , L k ( sp n ) ⊗ L − ( sp n )) (Theorem 7.8),(b) Com ( L k ( sp n − m ) , L k ( sp n )) (Corollary 7.6),(c) Com ( L n ( sp k ) , L n − m ( sp k ) ⊗ F (4 mk )) (Corollary 7.7). The importance of Main Theorem 2 (1) is that it completes the classification of lisse andrational affine vertex superalgebras L k ( g ) , for g a simple Lie superalgebra. When g is aLie algebra, it is a celebrated result of Frenkel and Zhu [FZ] that L k ( g ) is lisse and rationalif and only if k ∈ N . When g is not a Lie algebra, Gorelik and Kac [GK] showed that L k ( g ) is lisse only when g = osp | n and k ∈ N ; see [AL] for an alternative perspective.However, the rationality was previously known only for osp | [CFK]. The proof of MainTheorem 2 (1) involves exhibiting L k ( osp | n ) as an extension of the rational vertex algebra L k ( sp n ) ⊗ W ℓ ( sp n ) for ℓ = − ( n + 1) + k + n k +2 n .A celebrated result of Arakawa [Ar1, Ar2] is that for a simple Lie algebra g , W k ( g ) is lisseand rational when k is a nondegenerate admissible level for b g . Again, a similar statementis expected to be true for W k ( osp | m ) . Based on the coset realization of W k ( osp | m ) , wemake the following conjecture. Conjecture 1.1. (Conjecture 7.1) For all m ≥ , W ψ − m − / ( osp | m ) where ψ = p p + q ) , is lisseand rational if(1) p, q ∈ N are coprime,(2) p ≥ m − if q is odd,(3) p ≥ m if q is even.We expect that W k ( osp | m ) is lisse and rational if and only if k = ψ − m − or k = ψ − m − ,which is the Feigin-Frenkel dual level.Main Theorem 2 (2) proves several cases of Conjecture 1.1 by identifying W ψ − m − / ( osp | m ) in these cases with a simple current extension of a known rational vertex algebra of theform W s ( sp r ) or W s ( so r ) Z . In the case n = 1 , where W k ( osp | ) is just the N = 1 super-conformal algebra, Conjecture 1.1 is already known ([KWan, Ad, M, BMRW]), and we givean alternative proof; see Theorem 7.4.Main Theorem 2 (3) proves several cases of the Kac-Wakimoto rationality conjecture[KW4], which was later refined by Arakawa and van Ekeren [AvE]. Let g be a simpleLie algebra and k = − h ∨ + pq an admissible level for b g . The associated variety of L k ( g ) is then the closure of a nilpotent orbit O q which depends only on the denominator q . If f ∈ g is a nilpotent lying in O q , the simple W -algebra W k ( g , f ) is known to be non-zero nd lisse [Ar1]. Such pairs ( f, q ) are called exceptional pairs in [AvE], and they generalizethe notion of exceptional pair due to Kac and Wakimoto [KW4] and Elashvili, Kac, andVinberg [EKV]. The corresponding W -algebras are called exceptional and were conjecturedby Arakawa to be rational in [Ar1], generalizing the original conjecture of [KW4]. Veryrecently, Arakawa and van Ekeren proved rationality of all exceptional W -algebras in type A , and all exceptional subregular W -algebras of simply-laced types [AvE].The type B subregular W -algebra W k ( so m +3 , f subreg ) for m ≥ is exceptional when k = − (2 m + 1) + pq is admissible and q = 2 m + 2 or m + 1 ; see Table 1 of [AvE].Main Theorem 2 (3) proves rationality in all cases where q = 2 m + 2 and all cases where q = 2 m + 1 and p is odd, generalizaing the result for m = 1 of Fasquel [F]. In thesecases, we identify Com ( H (1) , W k ( so m +3 , f subreg )) Z with a known rational vertex algebraof the form W s ( sp r ) or W s ( so r ) Z . In the missing cases where q = 2 m + 1 and p is even,Com ( H (1) , W k ( so m +3 , f subreg )) Z is identified with a (conjecturally) rational algebra of theform W r ( osp | r ) Z , so these cases would follow from Conjecture 1.1. Note that Main Theo-rem 2 (4) follows immediately from part (3) together with the duality between subregular W -algebras of type B and principal W -superalgebras of osp | m appearing in [CGN].It turns out that the cases q = 2 m + 1 and q = 2 m + 2 do not account for all rationalalgebras of the form W k ( so m +3 , f subreg ) . We will show thatCom ( H (1) , W k ( so m +3 , f subreg )) Z ∼ = W s ( osp | r ) Z , for k = − (2 m +1)+ m − r +1)1+2 m − r and s = − ( r + )+ m +1 − r m +1 − r . In the case m ≥ r − , Conjecture 1.1would then imply the rationality of W k ( so m +3 , f subreg ) ; see Remark 7.1. In the case r = 1 ,these examples are rational by Main Theorem 2 (4d).The rationality of W r − / ( sp n +2 , f min ) for r ∈ Z ≥ given by Main Theorem 2 (5), is anothercase of the Kac-Wakimoto rationality conjecture. We will show that W r − / ( sp n +2 , f min ) isan extension of the rational vertex algebra L r ( sp n ) ⊗ W s ( sp r ) for s = − ( r + 1) + n + r n +2 r .This was conjectured in [ACKL] and proven in [KL] in the case n = 1 .Main Theorem 2 (6a) is proven by showing that Com ( L k − ( sp n ) , L k ( sp n ) ⊗ L − ( sp n )) is isomorphic to W ℓ ( sp k ) for ℓ = − ( k + 1) + n + k n +2 k , which is a new level-rank duality. Sim-ilarly, Main Theorem 2 (6b) and (6c) are proven by showing that both cosets are extensionsof the rational vertex algebra m O i =1 ( W ℓ i ( sp k ) ⊗ W s i ( sp k )) with ℓ i = − ( k + 1) + n − i + k n − i +2 k and s i = − ( k + 1) + n − i + k n − i +2 k . Note that (6b) impliesthat L k ( sp n ) is an extension of N ni =1 ( W ℓ i ( sp k ) ⊗ W s i ( sp k )) with ℓ i , s i as above. This isanalogous to the statement in [ACL] that L k ( gl n ) is an extension of N ni =1 W ℓ i ( gl k ) with ℓ i = − k = k + n − ik + n − i +1 , which is an analogue of the Gelfand-Tsetlin subalgebra of U ( gl n ) .1.2. Triality from kernel vertex algebras.
Motivated from four-dimensional GL-twisted N = 4 supersymmetric Yang-Mills theories, certain kernel vertex algebras were conjec-turally introduced in [CGa]. The attention was restricted to the case of type A . Here, wewill introduce analogues for orthosymplectic type and explain how this conjectures an-other perspective on triality. The kernel vertex algebras also play an important role in he context of the quantum geometric Langlands program and our conjectures are closelyrelated to the ideas sketched in Section 10.4 of [FG].Let g be either a simple Lie algebra of type B, C, D or osp | n . Let P + denote the set ofdominant weights of g , and R ⊆ P + the subset corresponding to the tensor ring generatedby the standard representation of g ; that is, λ ∈ R if and only if the irreducible highest-weight representation ρ λ is a submodule of some iterated tensor product of the standardrepresentation of g . Let g ′ = so n +1 if g = osp | n and vice versa, and let g ′ = g otherwise.Note that there is a one-to-one correspondence of irreducible finite dimensional non-spinrepresentations of so n +1 of osp | n , such that characters agree. A similar statement alsoholds for the quantum (super)groups [CFLW]. Let τ denote the induced map on dominantweights. If g is neither of type so n +1 nor of type osp | n then let τ be the identity on P + .Motivated from Theorem 4.1 we let φ be generic and φ ′ related to φ by the formula φ + 1 φ ′ = 2 , if g is of type C, φ + 1 φ ′ = 1 , if g is of type D, φ + 1 φ ′ = 1 , if g is of type osp | n , φ + 12 φ ′ = 1 , if g is of type B. Let V φ − h ∨ g ( g ) and V φ ′ − h ∨ g ′ ( g ′ ) be the universal affine vertex (super)algebras of g and g ′ atlevels φ − h ∨ g and φ ′ − h ∨ g ′ . Let M φ ( λ ) = V φ − h ∨ g ( λ ) and M φ ′ ( λ ) = V φ ′ − h ∨ g ′ ( τ ( λ )) be the Weylmodules at these levels whose top level is the irreducible highest-weight representation of g of highest-weight λ , respectively of g ′ of highest-weight τ ( λ ) . Then set A [ g , φ ] := M λ ∈ R M φ ( λ ) ⊗ M φ ′ ( λ ) . More generally, let f, f ′ be nilpotent elements in g , g ′ and let M φf ( λ ) and M φ ′ f ′ ( λ ) be theimages under quantum Hamiltonian reduction of M φ ( λ ) and M φ ′ ( λ ) corresponding to f and f ′ , respectively. Then set A [ g , φ, f, f ′ ] := M λ ∈ R M φf ( λ ) ⊗ M φ ′ f ′ ( λ ) , so that A [ g , φ ] = A [ g , φ, , . The conjecture is that these objects can be given the structureof a simple vertex super algebra for generic φ . In the case that g is of type D and f = 0 and f ′ principal nilpotent, this is the coset Theorem of type D of [ACL]. Moreover, the caseof arbitrary f and f ′ the principal nilpotent is the main Theorem of [ACF] applied to thecoset Theorem. In that paper also many similar algebras are studied. Note that in the caseof g = so we take the convention that h ∨ = 1 and that V k ( so ) = V k ( sl ) . For f = f ′ = 0 ,the case of g = so and g = osp | is proven in [CGa] and the case of g = sp in [CGaL] andthese are the affine vertex superalgebra of d (2 ,
1; 1 − φ ) at level one respectively the minimalquantum Hamiltonian reduction of d (2 ,
1; (1 − φ ) / at level / which is the one-parameterfamily of large N = 4 superconformal algebras at central charge − . The case g = so can also be included and we refer to [CGNS] for discussion of the kernel vertex algebraand its relative semi-infinite cohomology. osets can often also be characterized as relative semi-infinite Lie algebra cohomologies.It seems that the cohomology approach is suitable to put our trialities into a more generalperspective. Let g be a simple Lie algebra, B a basis for g and B ′ a dual basis. Let F ( g ) betwo copies of free fermions in the adjoint representation of g with generators { b x , c x ′ | x ∈ B, x ′ ∈ B ′ } and operator products b x ( z ) c y ′ ( w ) ∼ δ x,y ( z − w ) − . Consider the affine vertexalgebra of g at level − h ∨ , V − h ∨ ( g ) , and let x ( z ) be the field corresponding to x ∈ g . Let d := d be the zero-mode of the field d ( z ) := X x ∈B : x ( z ) c x ′ ( z ) : − X x,y ∈B : (: b [ x,y ] ( z ) c x ′ ( z ) :) c y ′ ( z ) : . It squares to zero, d = 0 . Let e F ( g ) denote the subalgebra of F ( g ) generated by the b x and ∂c x ′ (these are just dim g pairs of symplectic fermions). For a module M for V − h ∨ ( g ) therelative complex is C rel ( g , d ) = (cid:16) M ⊗ e F ( g ) (cid:17) g and this relative complex is preserved by d [FGZ, Prop. 1.4.]. The cohomology is denotedby H rel , •∞ ( g , M ) . As shown in [FGZ] and explained in Section 2.5 of [CFL], it satisfies(1.5) H rel , ∞ ( g , V k ( λ ) ⊗ V − h ∨ − k ( µ )) = ( C if µ = − ω ( λ )0 otherwise . Here ω is the unique Weyl group element that interchanges the fundamental Weyl cham-ber with its negative. It is reasonable to expect that one can construct similar complexeswith similar properties for affine vertex superalgebras and it would be very interesting todo so for at least g = osp | n . In order to include the case n = 0 , we define master chi-ral algebra and cohomology to be trivial, that is so := so := sp := osp | = { } and A [ { } , φ ] := C as well as H rel , ∞ ( { } , C ) = C . Conjecture 1.2.
With the above set-up and f, f ′ nilpotent in g , g ′ .(1) The object A [ g , φ, f, f ′ ] can be given the structure of a one-parameter vertex superalge-bra.(2) For generic φ , A [ g , φ, f, f ′ ] is a simple vertex operator superalgebra extending W φ − h ∨ g ( g , f ) ⊗W φ ′ − h ∨ g ′ ( g ′ , f ′ ) .(3) There exists a generalization of relative semi-infinite Lie superalgebra cohomology for g = osp | n satisfying (1.5).(4) For all integers m ≥ n ≥ , we have the following isomorphisms of one-parametervertex algebras.(a) For ψ ′ = ψ and ψ + ψ ′′ = 2 , W ψ B ( n, m ) ∼ = H rel , ∞ (cid:18) osp | n , W ψ ′ O ( n, m − n ) ⊗ A [ osp | n , − ψ ′ ] (cid:19) , W ψ ′′ B ( m, n ) ∼ = Com (cid:16) V ψ ′ − n − ( osp | n ) , A [ osp | m , ψ ′ , f sp m − n , (cid:17) . (b) For ψ ′ = ψ and ψ + ψ ′′ = 1 , W ψ C ( n, m ) ∼ = H rel , ∞ (cid:18) sp n , W ψ ′ C ( n, m − n ) ⊗ A [ sp n , − ψ ′ ] (cid:19) , W ψ ′′ C ( m, n ) ∼ = Com (cid:16) V ψ ′ − n − ( sp n ) , A [ sp m , ψ ′ , f sp m − n , (cid:17) . c) For ψ ′ = ψ and ψ + ψ ′′ = 1 , W ψ D ( n, m ) ∼ = H rel , ∞ (cid:16) so n , W ψ ′ D ( n, m − n ) ⊗ A [ so n , − ψ ′ ] (cid:17) , W ψ ′′ O ( m, n ) ∼ = Com (cid:16) V ψ ′ − n +1 ( so n ) , A [ so m +1 , ψ ′ , f so m − n +1 , (cid:17) . (d) For ψ ′ = ψ and ψ + ψ ′′ = 1 and m > n in the second case, W ψ O ( n, m ) ∼ = H rel , ∞ (cid:16) so n +1 , W ψ ′ B ( n, m − n ) ⊗ A [ so n +1 , − ψ ′ ] (cid:17) , W ψ ′′ D ( m, n ) ∼ = Com (cid:16) V ψ ′ − n ( so n +1 ) , A [ so m , ψ ′ , f so m − n − , (cid:17) . Both sides of the conjectured isomorphisms have the same affine vertex superalgebraand the same quotient of W ev ( c, λ ) as commuting pair of subalgebras.The relative semi-infinite cohomology part of (4) of the conjecture for n = 0 is Feigin-Frenkel duality [FF2]. The coset part of (4) should be viewed as a generalization of cosetrealization of principal W -algebras, e.g. the case n = 0 of (4)(a) corresponds to the cosetrealization of prinicpal W -superalgebras of osp | n of [CGe] and the case (4)(d) to the cosetrealization of principal W -algebras of type D of [ACL]. Our conjecture is a natural exten-sion to orthosymplectic type of our conjectures for type A made in Section 10 of [CL3].The relative semi-infinite cohomology part of this conjecture can be proven for subregular W -algebras of type A and B and that will be presented in [CGNS]. We sketched a proofstrategy in Section 10 of [CL3] for type A , and we expect that all our conjectures for type A as well as orthosymplectic type can be proven uniformly.Besides its importance in physics and quantum geometric Langlands, our conjecturesprovide a way to relate representation categories in a nice way. For example, if Conjecture4 (a) is correct, then the functor H rel , ∞ (cid:0) osp | n , ? ⊗ A [ osp | n , − ψ ′ ] (cid:1) maps W ψ ′ O ( n, m − n ) -modules to W ψ B ( n, m ) -modules. It is reasonable that such a functor has nice monoidalproperties. Indeed in the type A and subregular W -algebra case, this is true and will bepresented in [CGNS] as well. Also the coset part of the conjecture should be useful to con-nect representation categories. For example, the coset realization of principal W -algebrasof type ADE has been used to prove a braided monoidal equivalence between a simplecurrent twist of ordinary modules of the affine vertex algebra at admissible level, and asubcategory of modules of the principal W -algebra appearing as the coset; see [C, Thm.7.1] for the precise statement. In the case g = sl the admissible level result appearedin [CHY, Thm. 7.4], and a theorem at generic level was also established in [CJORY, Prop.5.5.2]. Note that these results prove variants of a conjecture made in the context of quantumgeometric Langlands, see [AFO, Conj. 6.3]. It is work in progress to study the categoriesof ordinary modules of L k ( osp | n ) at admissible level, and especially to show that they arebraided equivalent to subcategories of the principal W -algebras of type B that appear asthe cosets.1.3. Geometry and conformal field theory.
The Alday-Gaiotto-Tachikawa (AGT) corre-spondence is a relation between four-dimensional gauge theories and two-dimensionalconformal field theories [AGT]. It yields interesting connections to geometry, for examplea celebrated result of Schiffmann and Vasserot [SV] asserts that the principal W -algebraof type A acts on the equivariant cohomology of the moduli space of instantons on C .There is a generalization to equivariant cohomology of Uhlenbeck spaces where principal -algebras of simply-laced type act [BFN] and the authors expect that their constructioncan be generalized to the non simply-laced case. This is interesting as conjecturally the Y M,N,L -algebras of Gaiotto and Rapˇc´ak, that is the cosets of W -superalgebras of the trialityof type A , act on moduli spaces of spiked instantons [RSYZ] and we hope that a nice geo-metric interpretation also exists in the orthosymplectic case. Note that at least a nice four-dimensional physics interpretation exists for them [GR]. In all the above mentioned works,it has been shown that a subalgebra of a Heisenberg vertex algebra, characterized as thekernel of certain screening operators, acts on an equivariant cohomology. Another crucialproblem is thus to develop explicit screening realizations. Naoki Genra has already pro-vided nice screening realizations of W -superalgebras [Ge] and the main obstacle of makingthem more explicit are screening realizations of affine vertex superalgebras. Screening re-alizations should also provide a different proof of the trialities with the advantage that itshould give further insights on the connection of representation theories. From the physicsperspective the screening charges provide the interaction term in the action of the confor-mal field theory. A duality of conformal field theories is then an isomorphism of symmetryalgebras, that is underlying vertex algebras, toghether with a matching of correlation func-tions. Our trialities in type A at low rank have already led to new dualities of conformalfield theories [CH] and likely there are more to be discovered.1.4. Outline.
This paper is organized as follows. In Section 2 we review the basic termi-nology and examples of vertex algebras that we need. We also prove that extensions ofa rational vertex algebra are rational under hypotheses that hold in all our examples. InSection 3, we introduce the W -(super)algebras W ψiX ( n, m ) for i = 1 , and X = B, C, D, O that we need. In Section 4, we state our main result and also discuss the special cases whichrecover Feigin-Frenkel duality and various coset realizations of principal W -algebras. InSection 5 we discuss the free field limits of W ψiX ( n, m ) and the strong generating types ofthe algebras C ψiX ( n, m ) . In Section 6 we prove Main Theorem 1 by explicitly computingthe truncation curves realizing C ψiX ( n, m ) as quotients of W ev ( c, λ ) . In Section 7 we proveMain Theorem 2. In Appendix A, we give the explicit truncation curve for C ψ B ( n, m ) , fromwhich all other truncation curves can be derived. Finally, in Appendices B, C, and D, weclassify the pointwise coincidences between the simple quotients C ψ,iX ( n, m ) , and the alge-bras W s ( sp r ) , W s ( so r ) Z , and W s ( osp | r ) Z . These coincidences are needed in the proof ofMain Theorem 2. 2. V ERTEX ALGEBRAS
We will assume that the reader is familiar with vertex algebras, and we use the samenotation as our previous paper [CL3]. In this section, we briefly recall the definition andbasic properties of free field algebras, W -algebras, and the two-parameter even spin alge-bra W ev ( c, λ ) . We then prove some general results on extensions of rational vertex algebraswhich are needed in the proof of Main Theorem 2.2.1. Free field algebras.
Recall that a free field algebra is a vertex superalgebra V withweight grading V = M d ∈ Z ≥ V [ d ] , V [0] ∼ = C , ith strong generators { X i | i ∈ I } satisfying OPE relations X i ( z ) X j ( w ) ∼ a i,j ( z − w ) − wt ( X i ) − wt ( X j ) , a i,j ∈ C , a i,j = 0 if wt ( X i ) + wt ( X j ) / ∈ Z . Note V is not assumed to have a conformal structure. We now recall the four families ofstandard free field algebras that were introduced in [CL3]. Even algebras of orthogonal type . For each n ≥ and even k ≥ , O ev ( n, k ) is the vertexalgebra with even generators a , . . . , a n of weight k , which satisfy a i ( z ) a j ( w ) ∼ δ i,j ( z − w ) − k . In the case k = 2 , O ev ( n, k ) just the rank n Heisenberg algebra H ( n ) . Note that O ev ( n, k ) has no conformal vector for k > , but for all k it is a simple vertex algebra and has fullautomorphism group the orthogonal group O n . Even algebras of symplectic type . For each n ≥ and odd k ≥ , S ev ( n, k ) is the vertex algebrawith even generators a i , b i for i = 1 , . . . , n of weight k , which satisfy a i ( z ) b j ( w ) ∼ δ i,j ( z − w ) − k , b i ( z ) a j ( w ) ∼ − δ i,j ( z − w ) − k ,a i ( z ) a j ( w ) ∼ , b i ( z ) b j ( w ) ∼ . (2.1)In the case k = 1 , S ev ( n, k ) is just the rank n βγ -system S ( n ) . For k > , S ev ( n, k ) has noconformal vector, but for all k it is simple and has full automorphism group the symplecticgroup Sp n . Odd algebras of symplectic type . For each n ≥ and even k ≥ , S odd ( n, k ) is the vertexsuperalgebra with odd generators a i , b i for i = 1 , . . . , n of weight k , which satisfy a i ( z ) b j ( w ) ∼ δ i,j ( z − w ) − k , b j ( z ) a i ( w ) ∼ − δ i,j ( z − w ) − k ,a i ( z ) a j ( w ) ∼ , b i ( z ) b j ( w ) ∼ . (2.2)In the case k = 2 , S odd ( n, k ) is just the rank n symplectic fermion algebra A ( n ) . Notethat S odd ( n, k ) has no conformal vector for k > , but for all k it is simple and has fullautomorphism group Sp n . Odd algebras of orthogonal type . For each n ≥ and odd k ≥ , we define O odd ( n, k ) to be thevertex superalgebra with odd generators a i for i = 1 , . . . , n of weight k , satisfying(2.3) a i ( z ) a j ( w ) ∼ δ i,j ( z − w ) − k . For k = 1 , O odd ( n, k ) is just the free fermion algebra F ( n ) . As above, O odd ( n, k ) has noconformal vector for k > , but it is simple and has full automorphism group O n .2.2. W -algebras. Let g be a simple, finite-dimensional Lie (super)algebra equipped witha nondegenerate, invariant (super)symmetric bilinear form ( | ) , and let f be a nilpotentelement in the even part of g . Associated to g and f and any complex number k , is the W -(super)algebra W k ( g , f ) . The definition is due to Kac, Roan, and Wakimoto [KRW], andit generalizes the definition for f a principal nilpotent and g a Lie algebra given by Feiginand Frenkel [FF1]. irst, let { q α } α ∈ S be a basis of g which is homogeneous with respect to parity. We definethe corresponding structure constants and parity by [ q α , q β ] = X γ ∈ S f αβγ q γ , | α | = ( q α even , q α odd . The affine vertex algebra of g associated to the bilinear form ( | ) at level k is stronglygenerated by { X α } α ∈ S with OPEs X α ( z ) X β ( w ) ∼ k ( q α | q β )( z − w ) − + X γ ∈ S f αβγ X γ ( w )( z − w ) − . We define X α to be the field corresponding to q α where { q α } α ∈ S is the dual basis of g withrespect to ( | ) .Let f be a nilpotent element in the even part of g , which we complete to an sl -triple { f, x, e } ⊆ g satisfying [ x, e ] = e, [ x, f ] = − f, [ e, f ] = 2 x . Then g decomposes as an sl -module as follows. g = M k ∈ Z g k , g k = { a ∈ g | [ x, a ] = ka } . Write S = S k S k and S + = S k> S k , where S k corresponds to a basis of g k .As in [KW3], one defines a complex C ( g , f, k ) = V k ( g ) ⊗ F ( g + ) ⊗ F ( g ) , where F ( g + ) isa free field superalgebra associated to the vector superspace g + = L k ∈ Z > g k , and F ( g ) isthe neutral vertex superalgebra associated to g with bilinear form h a, b i = ( f | [ a, b ]) . F ( g + ) is strongly generated by fields { ϕ α , ϕ α } α ∈ S + , where ϕ α and ϕ α have opposite parity then q α . The operator products are ϕ α ( z ) ϕ β ( w ) ∼ δ α,β ( z − w ) − , ϕ α ( z ) ϕ β ( w ) ∼ ∼ ϕ α ( z ) ϕ β ( w ) .F ( g ) is strongly generated by fields { Φ α } α ∈ S and Φ α and q α have the same parity. Theiroperator products are Φ α ( z )Φ β ( w ) ∼ h q α , q β i ( z − w ) − ∼ ( f | [ q α , q β ])( z − w ) − . There is a Z -grading on C ( g , f, k ) by charge, and a weight one odd field d ( z ) of chargeminus one, d ( z ) = X α ∈ S + ( − | α | : X α ϕ α : − X α,β,γ ∈ S + ( − | α || γ | f αβγ : ϕ γ ϕ α ϕ β : + X α ∈ S + ( f | q α ) ϕ α + X α ∈ S : ϕ α Φ α : , (2.4)whose zero-mode d is a square-zero differential on C ( g , f, k ) . The W -algebra W k ( g , f ) isdefined to be the homology H ( C ( g , f, k ) , d ) . It has Virasoro element L = L sug + ∂x + L ch + ne , where L sug = 12( k + h ∨ ) X α ∈ S ( − | α | : X α X α : ,L ch = X α ∈ S + ( − m α : ϕ α ∂ϕ α : +(1 − m α ) : ( ∂ϕ α ) ϕ α :) ,L ne = 12 X α ∈ S : ( ∂ Φ α )Φ α : . (2.5)Here m α = j if α ∈ S j . The central charge of L is computed to be(2.6) c ( g , f, k ) = k sdim g k + h ∨ − k ( x | x ) − X α ∈ S + ( − | α | (12 m α − m α + 2) − sdim g . Denote by g f the centralizer of f in g , and let a = g f ∩ g , which is a Lie subsuperalgebraof g . By [KW3, Thm. 2.1], W k ( g , f ) contains an affine vertex superalgebra of type a . Inparticular, W k ( g , f ) contains elements I α for α ∈ g ,(2.7) I α := X α + X β,γ ∈ S + ( − | γ | f αβγ : ϕ γ ϕ β : + ( − | α | X β ∈ S f βαγ Φ β Φ γ and satisfying(2.8) [ I αλ I β ] = f αβ γ I γ + λ (cid:18) k ( q α | q β ) + 12 (cid:16) κ g ( q α , q β ) − κ g ( q α , q β ) − κ ( q α , q β ) (cid:17)(cid:19) , with κ the supertrace of g on g . The key structural theorem is the following. Theorem 2.1. [KW3, Thm 4.1]
Let g be a simple finite-dimensional Lie superalgebra with aninvariant bilinear form ( | ) , and let x, f be a pair of even elements of g such that ad x is diago-nalizable with eigenvalues in Z and [ x, f ] = − f . Suppose that all eigenvalues of ad x on g f arenon-positive, so that g f = L j ≤ g fj . Then(1) For each q α ∈ g f − j , ( j ≥ ) there exists a d -closed field K α of conformal weight j , withrespect to L .(2) The homology classes of the fields K α , where { q α } is a basis of g f , strongly and freely generatethe vertex algebra W k ( g , f ) .(3) H ( C ( g , f, k ) , d ) = W k ( g , f ) and H j ( C ( g , f, k ) , d ) = 0 if j = 0 . One can also consider the reduction of a module, i.e for a V k ( g ) -module M , the homologyof the complex H ( M ⊗ F ( g + ) ⊗ F ( g ) , d ) is a W k ( g , f ) -module that we denote by H k,f ( M ) ,that is(2.9) H k,f ( M ) := H ( M ⊗ F ( g + ) ⊗ F ( g ) , d ) . Translation of W -algebras. We summarize a very useful result of Tomoyuki Arakawa,Boris Feigin and one of us [ACF]. For this we consider two affine vertex algebras of type g , the first one, V , of level k and the second one, L , of level ℓ . Denote the generators ofthe first one by X α and the generators of the second one by Y α . As complexes we take the revious one C ( g , f, k ) = V ⊗ F ( g + ) ⊗ F ( g ) and also C ( g , f, k, ℓ ) = V ⊗ L ⊗ F ( g + ) ⊗ F ( g ) .In addition to the differential d of last subsection, we also define the differential d ′ as thezero-mode of the field d ′ ( z ) = d ( z ) + X α ∈ S + ( − | α | : Y α ϕ α : . Then the homology with respect to d is just the quantum Hamiltonian reduction on the V subalgebra of V ⊗ L , while the homology with differential d ′ is the reduction with respect tothe diagonal action at level k + ℓ . We now restrict to ℓ being a positive integer and consider L = L ℓ ( g ) the simple affine vertex algebra of g at level ℓ . The main result of [ACF] is Theorem 2.2. [ACF]
As vertex algebras H ( C ( g , f, k ) , d ) ⊗ L ∼ = H ( C ( g , f, k, ℓ ) , d ′ ) . The conformal vector of the right-hand side is not the sum of the standard conformalvectors of the left-hand side.
Remark . Let a = g f ∩ g . Then the affine subalgebras of W k ( g , f ) and W k + ℓ ( g , f ) are V s ( a ) and V t ( a ) for some levels s, t depending on k, k + ℓ . Moreover, L has an action of L t − s ( a ) . Re-call that I α is non-trivial in H ( C ( g , f, k ) , d ) and I α + Y α is non-trivial in H ( C ( g , f, k, ℓ ) , d ′ ) .The homology classes [ I α + Y α ] ′ , [ I α ] of these fields generate homomorphic images N s ( a ) of V s ( a ) and N t ( a ) of V t ( a ) inside H ( C ( g , f, k, d ) ⊗ L and H ( C ( g , f, k, ℓ ) , d ′ ) . Considerthe cases: • If V = V k ( g ) then both homologies are subalgebras of C ( g , f, k, ℓ ) (see the discussionbefore [ACF, Lem. 4.1]) and so in this instance the subalgebra V t ( a ) of W k + ℓ ( g , f ) ⊆ H ( C ( g , f, k, ℓ ) , d ′ ) is the diagonal subalgebra of V s ( a ) ⊗ L t − s ( a ) ⊆ H ( C ( g , f, k ) , d ) ⊗ L generated by I α + Y α . • If H ( C ( g , f, k ) , d ) is a homomorphic image of W k ( g , f ) . Then the embedding V t ( a ) ⊆ V s ( a ) ⊗ L t − s ( a ) ⊆ W k ( g , f ) ⊗ L induces the embedding N t ( a ) ⊆ N s ( a ) ⊗ L t − s ( a ) ⊆ H ( C ( g , f, k ) , d ) ⊗ L , mapping [ I α + Y α ] ′ to [ I α ] + Y α .2.4. Singular Vectors.
In this subsection, we compute some conformal weights of singularvectors in principal W -algebras at nondegenerate admissible levels. We use Lemma 2.1. [CL3, Lem. 3.3]
Let g be a simple Lie algebra, ¯ ρ, ¯ ρ ∨ its Weyl vector and Weyl covectorand set ¯ α = − θ if ( v, r ∨ ) = 1 and ¯ α = − θ s if ( v, r ∨ ) = r ∨ . Here r ∨ is the lacity of g and θ, θ s arethe highest root and highest short root. Set ¯ λ = n uv ¯ α ∨ − ( ¯ ρ, ¯ α ∨ ) ¯ α Denote by Sing ( V ) the weight of the singular vector of V of lowest conformal weight. Then for k = − h ∨ + uv of (co)principal admissible weight, singular vectors of affine and principal W -algebrahave weight(1) Sing ( V k ( g )) = v u ¯ λ (¯ λ + 2 ¯ ρ ) ,(2) Sing ( W k ( g )) = v u ¯ λ (¯ λ + 2 ¯ ρ ) − ¯ λ ¯ ρ ∨ for k a nondegenerate admissible level. Corollary 2.1.
Denote by Sing ( V ) the weight of the singular vector of V of lowest conformalweight. Then for k = − h ∨ + uv of (co)principal admissible weight, we have(1) Sing ( V k ( sp n )) = v ( u − n ) for v odd, and Sing ( V k ( sp n )) = v ( u − n + 1) for v even,(2) Sing ( V k ( so n +1 )) = v ( u − n + 2) for v odd, and Sing ( V k ( so n +1 )) = v ( u − n + 1) for v even,
3) Sing ( W k ( sp n )) = ( v − n +1)( u − n ) for v odd, and Sing ( W k ( sp n )) = ( v − n +2)( u − n +1) for v even and for k a nondegenerate admissible level,(4) Sing ( W k ( so n +1 )) = ( v − n + 1)( u − n + 2) for v odd, and Sing ( W k ( so n +1 )) = ( v − n )( u − n + 1) for v even and for k a nondegenerate admissible level.Proof. Consider the lattice Z n with orthonormal basis ǫ , . . . , ǫ n . We embed root and corootsin rescalings of this lattice in the standard way, e.g. the simple positive roots of so n +1 are ǫ − ǫ , . . . , ǫ n − − ǫ n , ǫ n , and for sp n they are ǫ − ǫ √ , . . . , ǫ n − − ǫ n √ , √ ǫ n . Especially,(1) g = so n +1 and v odd. Then θ = θ ∨ = ǫ + ǫ , ρ ∨ = nǫ + ( n − ǫ + · · · + ǫ n , ρ = ((2 n − ǫ + (2 n − ǫ + · · · + ǫ n ) and so ρθ ∨ = 2 n − and ρ ∨ θ = 2 n − . It followsthat ¯ λ = ( u − n + 2) θ .(2) g = so n +1 and v even. Then θ s = ǫ and θ ∨ s = 2 ǫ . Thus ρθ ∨ s = 2 n − and θ s ρ ∨ = n . Itfollows that ¯ λ = ( u − n + 1) θ s .(3) g = sp n and v odd. Then θ = √ ǫ and ρ ∨ = √ ((2 n − ǫ + (2 n − ǫ + · · · + ǫ n ) , ρ = √ ( nǫ + ( n − ǫ + · · · + ǫ n ) . Thus ρθ ∨ = n and θρ ∨ = 2 n − and ¯ λ = ( u − n ) θ .(4) g = sp n and v even. Then θ s = √ ( ǫ + ǫ ) and θ ∨ s = √ ǫ + ǫ ) . Thus ρθ ∨ s = 2 n − and ρ ∨ θ s = 2 n − and ¯ λ = ( u − n + 1) θ s .Inserting in the formula of Lemma 2.1 gives the claim. (cid:3) Remark . Let ≤ m < n , then there is an embedding of ι : sp n − m → sp n of sp n − m in sp n sending the simple roots of sp n − m to the simple roots ǫ m +1 − ǫ m +2 √ , . . . , ǫ n − − ǫ n √ , √ ǫ n of sp n − m . Denote by X x , Y y the fields of V k ( sp n − m ) and V k ( sp n ) corresponding to theelements x ∈ sp n − m , y ∈ sp n . Then ι induces an embedding of V k ( sp n − m ) in V k ( sp n ) sending X x to Y ι ( x ) for x ∈ sp n − m . This embedding can be characterized via nilpotentelements as follows: Let f i = e −√ ǫ and g i be the subspace of g f i i − of √ ǫ i weight zero for i = 1 , . . . , m and g := sp n so that g i ∼ = sp n − i and especially g m = ι ( sp n − m ) .2.5. Universal two-parameter even spin W ∞ -algebra. We briefly recall the universal two-parameter vertex algebra W ev ( c, λ ) of type W (2 , , . . . ) , which was conjectured to existin the physics literature [CGKV], and was constructed in [KL]. It is defined over thepolynomial ring C [ c, λ ] and is generated by a Virasoro field L of central charge c , and aweight primary field W . In addition, it is strongly and freely generated by the fields { L, W i | i ≥ } where W i = W W i − for i ≥ . W ev ( c, λ ) is simple as a vertex algebra over C [ c, λ ] , but there is a discrete family of primeideals 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 some mildhypotheses, can be obtained as quotients of W ev ( c, λ ) in this way; see [CKoL, Thm. 2.1].The distinct generators of such ideals arise as irreducible factors of Shapovalov determi-nants, and are in bijection with such one-parameter vertex algebras.We also consider W ev ,I ( c, λ ) for maximal ideals I = ( c − c , λ − λ ) , c , λ ∈ C . hen 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 ) . A criterion for W and W to be isomorphic is given by [KL, Thm. 8.1];aside from a few degenerate cases, we must have c = c and λ = λ . This implies thataside from the degenerate cases, all other coincidences among the simple quotients of one-parameter vertex algebras W ev ,I ( c, λ ) and W ev ,J ( c, λ ) , correspond to intersection points oftheir truncation curves V ( I ) and V ( J ) .We record a slight improvement to the results of [KL] for later use. By Theorem 8.1, case(4), for c = 1 , − , the Virasoro algebra Vir c is realized as a quotient of W ev ( c, λ ) by setting(2.10) λ = ± p ( c − c − , and taking the simple quotient. This occurs because the weight field becomes singular,and hence all higher descendants W k for k ≥ also vanish in the simple quotient. It isstraightforward to check that the truncation curve for W k ( sp n ) in the case n = 1 coincideswith (2.10). It follows that all results such as [KL, Thm. 9.3], which classify isomorphismsbetween certain algebras and W k ( sp n ) for n ≥ , actually hold for n = 1 as well.Similarly, the truncation curve for W k ( so n ) Z given in [KL, Thm. 6.3] for n ≥ , in factholds for n = 2 as well. In this case, W k ( so ) Z ∼ = (cid:0) Vir c ⊗ Vir c (cid:1) Z , c = − (1 + 2 k )(4 + 3 k )2 + k , where Z acts by permutation. This truncation curve was computed in [MPS] and is easilyseen to coincide with the specialization of the curve for W k ( so n ) Z to the case n = 2 .Therefore [KL, Thm. 9.4] holds for all n ≥ and m ≥ , and other results of this kind canbe similarly improved.2.6. Extensions of rational vertex operator algebras.
An extension of a lisse ( C -cofinite)vertex algebra is also lisse [ABD, Prop. 5.2]. We also need a general result that says thatextensions of a rational vertex algebra are rational as well. One such statement is [HKL,Thm. 3.5]. One assumption, however, is a positivity assumption on conformal weightsof modules, and this assumption is not satisfied in most of our cases of interest. Anothersuch statement is [CKM2, Cor. 1.1], which however only applies to Z -graded vertex algebraextensions. We need to consider Z -graded vertex superalgebra extensions, and so we nowensure that the rationality result generalizes to this setting.We first recall the main basic theorems of [KO, HKL, CKL, CKM1, CKM2] using [CY, Sec-tion 2]. Let V be a vertex operator algebra and C = ( C , ⊠ , , A • , • , • , l • , r • , R • , • ) be a categoryof V -modules with a natural vertex and braided tensor category structure in the sense of[HLZ1]-[HLZ9]. The tensor bifunctor is denoted by ⊠ , the tensor identity is just he vertexoperator algebra V itself and wil be denoted by . The associativity constraint, the left andright unit constraint and the braiding are denoted by A , l, r, R . Definition 2.1. [KO] An algebra is a triple ( A, µ A , ι A ) with A an object in C an multiplication µ A : A ⊠ A → A and an embedding of tensor unit ι A : C → A that satisfy(1) Multiplication is associative: µ A ◦ (id A ⊠ µ A ) = µ A ◦ ( µ A ⊠ id A ) ◦ A A,A,A : A ⊠ ( A ⊠ A ) → A (2) Multiplication is unital: µ A ◦ ( ι A ⊠ id A ) = l A : C ⊠ A → A and µ A ◦ (id A ⊠ ι A ) = r A : A ⊠ C → A A, µ A , ι A ) is a commutative algebra if additionally(3) Multiplication is commutative: µ A ◦ R A,A = µ A : A ⊠ A → A .We will use the short-hand notation A for an algebra ( A, µ A , ι A ) . Definition 2.2. [KO] Let A be an algebra, define C A to be the category of pairs ( X, µ X ) ,where X is an object in C and µ X : A ⊠ X → X is a morphism in C subject to(1) Unit property: l X = µ X ◦ ( ι A ⊠ id X ) : C ⊠ X → X (2) Associativity: µ X ◦ (id A ⊠ µ X ) = µ X ◦ ( µ A ⊠ id X ) ◦ A A,A,X : A ⊠ ( A ⊠ X ) → X .A morphism f ∈ Hom C A (( X , µ X ) , ( X , µ X )) is a morphism f ∈ Hom C ( X , X ) such that µ X ◦ (id A ⊠ f ) = f ◦ µ X .When A is commutative, define C locA to be the full subcategory of C A containing localobjects: those ( X, µ X ) such that µ X ◦ R X,A ◦ R
A,X = µ X .Super commutative algebras are defined similarly in [CKL]. The structural categoricalresults are summarized in the following theorem. Theorem 2.3.
Let C be a braided tensor category and let A be a commutative algebra in C . Thenthe following results hold: (1) The category C A is a tensor category ( [KO, Thm. 1.5] , [CKM1, Thm. 2.53] ). (2) The subcategory C locA is a braided tensor category ( [KO, Thm. 1.10] , [CKM1, Thm. 2.55] ]). (3) The induction functor F A : C → C A is monoidal ( [KO, Thm. 1.6] , [CKM1, Thm. 2.59] ). (4) The induction functor satisfies Frobenius reciprocity, that is, it is left adjoint to the forgetfulfunctor G A from C A to C : (2.11) Hom C A ( F A ( W ) , X ) = Hom C ( W, G A ( X )) for objects W in C and X in C A (see for example [KO, Thm. 1.6(2)] , [CKM1, Lem. 2.61] ). (5) Let W be an object in C . Then F A ( W ) is in C locA if and only if the monodromy is trivial, that is M A,W := R W,A ◦ R
A,W = id A ⊠ W ( [CKM1, Prop. 2.65] ). (6) Let C be the full subcategory of objects in C that induce to C locA . Then the restriction of theinduction functor F A : C → C locA is a braided tensor functor ( [CKM1, Thm. 2.67] ). The next theorem tells us that we can use the categorical results to study extensions ofvertex algebras.
Theorem 2.4.
Let V be a vertex operator algebra, and let C be a category of V -modules with anatural vertex tensor category structure. Then the following results hold: (1) A vertex operator algebra extension V ⊆ A in C is equivalent to a commutative associative al-gebra in the braided tensor category C with trivial twist and injective unit ( [HKL, Thm. 3.2] ). (2) The category of modules in C for the extended vertex operator algebra A is isomorphic to thecategory of local C -algebra modules C locA ( [HKL, Thm. 3.4] ). (3) The results in (1) and (2) hold for a vertex operator superalgebra extension: The vertex operatorsuperalgebra extension V ⊆ A in C such that V is in the even subalgebra A is equivalent toa commutative associative superalgebra in C whose twist θ satisfies θ = id A . The categoryof generalized modules for the vertex operator superalgebra A is isomorphic to the category oflocal C -superalgebra modules C locA ( [CKL, Thm. 3.13, 3.14] ). The isomorphism given in [HKL, Thm. 3.4] and [CKL, Thm. 3.14] between the categoryof modules in C for the extended vertex operator (super)algebra A and the category of lo-cal C -algebra modules C locA is an isomorphism of vertex tensor (super)categories ( [CKM1,Thm. 3.65] ). A tensor category is called a fusion category if it is semi-simple with finitely many in-equivalent simple objects and every object is rigid. In particular in that case there is a traceand thus a notion of dimension of objects. The following theorem gives conditions underwhich a vertex algebra extension has a semi-simple representation category.
Theorem 2.5. [CKM2, Theorem 5.12]
Suppose U and W are braided fusion categories of modulesfor simple self-contragredient vertex operator algebras U and W , respectively, and A = M i ∈ I U i ⊗ W i is a simple Z -graded vertex operator algebra extension of U ⊗ W in C = U ⊠ W where the U i aredistinct simple modules in U including U = U and the W i are modules in W such that dim Hom W ( W, W i ) = δ i, . Then dim C A > and the category of (grading-restricted, generalized) A -modules in C is a braidedfusion category. We need to generalize this theorem to the case where A is possibly a vertex operatorsuperalgebra and possibly Z -graded. Let V be a vertex operator algebra and C a categoryof V -modules. We assume this category to be braided fusion. Let A ⊇ V be an object in C that itself carries the structure of a vertex algebra or vertex superalgebra. We assume V to be Z -graded, but A does not need to be. However, we assume A to be Z -graded, A = A ¯0 ⊕ A ¯1 , such that A ¯0 is a Z -graded vertex algebra containing V , V ⊆ A ¯0 . In otherwords, A ¯0 is a Z -orbifold of A and so A ¯1 is a self-dual simple current [Mc] . Let C loc A and C loc A ¯0 be the categories of local A and A ¯0 -modules that lie in C . Note that C A = ( C A ¯0 ) A by[DMNO, Section 3.6] and since a local A -module is necessarily local as a module for thesubalgebra A ¯0 , also C loc A = ( C loc A ¯0 ) loc A .Let M be a simple object in C loc A ¯0 and let F : C loc A ¯0 → C A be the reduction functor and G therestriction functor. We have G ( F ( M )) ∼ = M ⊕ N with N = A ¯1 ⊠ M . The modules M and N are graded by conformal weight, i.e. M = M n ∈ Z + h M M n , N = M n ∈ Z + h N N n , for some complex numbers h M , h N . The monodromy R A ¯1 ,M ◦ R M,A ¯1 is either the identityon N or minus the identity on N , see [CKL] for details. In the latter case, we call anysubmodule of F ( M ) a twisted module and the subcategory of C A whose objects are directsums of twisted modules is denoted by C tw A . The module F ( M ) is either simple or a directsum of two simple modules [CKM1, Prop. 4.18 and Cor. 4.22]. In the latter case one has M ∼ = N and by Frobenius reciprocity Hom C A ¯0 ( M, M ⊕ M ) = Hom C A ( F ( M ) , F ( M )) , i.e.these two simple summands of F ( M ) need to be inequivalent. We are interested in threecases and their properties are given in [CKM1, Section 4.2]. A simpler proof of this statement is given in Appendix A of [CKLR], observing that the argument thereis the same for vertex superalgebras. A is a Z -graded vertex algebra, especially conformal weights of A ¯1 are in Z + . In thiscase F ( M ) is always simple. Moreover it is local if and only if h M = h N + mod 1 .Otherwise it is twisted. See Section 4.2.2 and especially Lemma 4.29 of [CKM1].(2) A is a Z -graded vertex superalgebra, especially conformal weights of A ¯1 are in Z + . F ( M ) is local if and only if h M = h N + mod 1 and otherwise it is twisted. F ( M ) issimple if it is local. if F ( M ) is twisted then either it is simple or M ∼ = N and F ( M ) is adirect sum of two simple objects. This type of extension is Section 4.2.3 of [CKM1].(3) A is a Z -graded vertex superalgebra. In this case F ( M ) is always simple. Moreover itis local if and only if h M = h N mod 1 . Otherwise it is twisted. See Section 4.2.1 andespecially Lemma 4.26 of [CKM1].If we assume that A is a simple vertex (super)algebra, then the action of A on a non-zero A -module cannot have a kernel and so especially M and N cannot be zero. Assume that A is simple.Assume that V = U ⊗ W is the tensor product of two vertex operator algebras andassume that A ¯0 ∼ = M i ∈ I U i ⊗ W i for some index set I . Here the U i are distinct simple U -modules and we set U = U and W = W . Also, assume that the U i and W i are objects of vertex tensor categories C U of U -modules and C W of W -modules, that are braided fusion categories Then the Deligneproduct C = C U ⊠ C W is a vertex tensor category as well [CKM2, Thm. 5.5]. The W i arenot necessarily distinct, but one requires that Hom ( W, W i ) = 0 for i = 0 . Under theseassumptions C loc A ¯0 is a braided fusion category as well by Theorem 2.5. We now prove that C loc A and C tw A are also semi-simple.The following is similar to the proof of [CGN, Thm. 5.13]. Let D be either C loc A or C tw A . Let X, Y be two simple modules in D and consider an exact sequence E : 0 → X → Z → Y → . We show that it splits. Let M be a direct summand of G ( Y ) . There are two cases(1) Y ∼ = F ( M ) and G ( Y ) = M ⊕ N with N = A ¯1 ⊠ M and M = N or(2) F ( M ) ∼ = Y ⊕ W with W = Y and G ( F ( M )) ∼ = M ⊕ M .We use Frobenius reciprocity. In the first caseHom C A ( Y, Z ) =
Hom C A ( F ( M ) , Z ) = Hom C loc A ¯0 ( M, G ( Z )) = Hom C loc A ¯0 ( M, G ( X ) ⊕ G ( Y ))= Hom C loc A ¯0 ( M, G ( X ⊕ Y )) = Hom C A ( F ( M ) , X ⊕ Y ) = Hom C A ( Y, X ⊕ Y ) and hence E splits. In the second caseHom C A ( Y ⊕ W, Z ) =
Hom C A ( F ( M ) , Z ) = Hom C loc A ¯0 ( M, G ( Z )) = Hom C loc A ¯0 ( M, G ( X ) ⊕ G ( Y ))= Hom C loc A ¯0 ( M, G ( X ⊕ Y )) = Hom C A ( F ( M ) , X ⊕ Y ) = Hom C A ( Y ⊕ W, X ⊕ Y ) and hence E splits as well. Summarizing: Proposition 2.1. ( Z vertex superalgebra generalization of [CKM2, Thm. 5.12]) Let A = A ¯0 ⊕ A ¯1 be a simple vertex (super)algebra of one of the three types listed above extendingthe Z -graded self-contragredient simple vertex algebra U ⊗ W . Assume that A ¯0 ∼ = M i ∈ I U i ⊗ W i or some index set I . Here the U i are distinct simple U -modules and we set U = U and W = W . Also assume that the U i and W i are objects of vertex tensor categories that are braided fusioncategories C U , C W of U respectively W -modules. Assume that Hom ( W, W i ) = 0 for i = 0 . Let C = C U ⊠ C W . Then both C loc A and C tw A are semisimple. Proposition 2.2.
Let V be a lisse vertex superalgebra of CFT-type and U be the affine subalgebragenerated by the weight one subspace of V . Let W = Com ( U, V ) and assume that W is self-contragredient. Then W is simple and if W is rational, then so is V .Proof. By [DM] applied to the even subalgebra of V , U is necessarily the tensor product ofa Heisenberg vertex algebra and an integrable affine vertex algebra L . Especially the actionof U on V is completely reducible and so W is simple by [CGN, Prop. 5.4]. The commutant ˜ U = Com ( W, V ) of W in V is an extension of L ⊗ V L for some lattice vertex algebra V L (see[CGN, Lem. 5.8] or [DM]). L needs to be positive definite for V being of CFT-type. Sincethe (categorical) dimension of any simple lattice vertex algebra module is one and the oneof any integrable representation is positive, it follows that ˜ U is rational by [KO, Thm. 3.3].Hence Proposition 2.1 (respectively already [CKM2, Thm. 5.12] if V is an integer gradedvertex algebra) applies to V as an extension of ˜ U ⊗ W , and so V is rational as well. (cid:3) By [Li, Cor. 3.2], a simple integer graded vertex operator algebra is self-dual if its con-formal weight one space is in the kernel of the Virasoro mode L . This holds especially ifthe conformal weight one space vanishes. Corollary 2.2. (Corollary of [Li, Cor. 3.2])
Let W be a simple integer graded vertex operatoralgebra of CFT-type with no fields of conformal weight one, then W is its own contragredient dual. Especially Proposition 2.2 applies if W is a simple rational principal W -algebra associ-ated to a simple Lie algebra or an order two orbifold of a simple rational principal W -algebra.We need another corollary of Proposition 2.1. For this let V be a simple vertex (su-per)algebra and W , W be simple vertex (super)subalgebras. Let L, W be simple ver-tex (super)subalgebras of W , such that Com ( W , V ) = W , Com ( L, W ) = W and suchthat W , W are actually integer graded self-contragredient vertex operator algebras. As-sume that there are braided fusion categorues C L , C , C of modules for the vertex algebras L, W , W , such that V is an object in C := C L ⊠ C ⊠ C . Then W corresponds to a commu-tative (super)algebra object in D := C L ⊠ C , that we also denote by W . We thus have aninduction functor F : D → D W with right adjoint denoted by G . For an object M in D W one has by Frobenius reciprocityHom D W ( W , M ) ∼ = Hom D ( W ⊗ L, G ( M )) . Hence a simple object M in D W with the property that Hom D ( W ⊗ L, G ( M )) is non-zeronecessarily is isomorphic to W . This impliesCom ( W , V ) = Com ( W ⊗ L, V ) and thus W = Com ( W , V ) = Com ( W ⊗ L, V ) =
Com ( W , Com ( L, V )) . The setting of Proposition 2.1 thus holds for W = U, W = W, Com ( L, V ) = A , i.e. Corollary 2.3.
With the above setting, the categories of local and twisted Com ( L, V ) -modules thatlie in C ⊠ C are semisimple. Especially if W and W are rational and lisse, then so is Com ( L, V ) . . H OOK - TYPE W - ALGEBRAS IN TYPES B , C , AND D In this section, we define the eight families of W -(super)algebras that we need in a uni-fied framework. First, let g be a simple Lie (super)algebra of type B , C , or D ; in particular, g is either so n +1 , sp n , so n , or osp n | r . We further assume that g admits a decomposition(3.1) g = a ⊕ b ⊕ ρ a ⊗ ρ b , with the following properties.(1) a and b are Lie sub(super)algebras of g . Here b is either so m +1 or sp m , and a can be so n +1 , sp n , so n , or osp | n .(2) ρ a and ρ b transform as the standard representations of a and b , respectively.(3) ρ a and ρ b have the same parity, which can be even or odd.Note that if a = osp | n , ρ a even means that ρ a ∼ = C n | as a vector superspace, whereas ρ a odd means that ρ a ∼ = C | n . If g = osp m | n , we use the following convention for its dualCoxeter number h ∨ .(3.2) h ∨ = ( m − n − type B n +2 − m type C , sdim ( osp m | n ) = ( m − n )( m − n − . In this notation, type B (respectively C ) means that the subalgebra b ⊆ g is of type B (respectively C ), and the bilinear form on osp m | n is normalized so that it coincides withthe usual bilinear form on b . The cases we need are the following. • Case 1B: g = so n +2 m +2 , b = so m +1 , a = so n +1 , ρ a ⊗ ρ b even. • Case 1C: g = osp m +1 | n , b = so m +1 , a = sp n , ρ a ⊗ ρ b odd. • Case 1D: g = so n +2 m +1 , b = so m +1 , a = so n , ρ a ⊗ ρ b even. • Case 1O: g = osp m +2 | n , b = so m +1 , a = osp | n , ρ a ⊗ ρ b odd. • Case 2B: g = osp n +1 | m , b = sp m , a = so n +1 , ρ a ⊗ ρ b odd. • Case 2C: g = sp n +2 m , b = sp m , a = sp n , ρ a ⊗ ρ b even. • Case 2D: g = osp n | m , b = sp m , a = so n , ρ a ⊗ ρ b odd. • Case 2O: g = osp | n +2 m , b = sp m , a = osp | n , ρ a ⊗ ρ b even.Corresponding to (3.1), we have an embedding V k ( b ) ⊗ V ℓ ( a ) ֒ → V k ( g ) , where the level ℓ is given as follows.(1) In cases 1B, 1D, 2C, and 2O, ℓ = k ,(2) In cases 1C and 1O, ℓ = − k ,(3) In cases 2B and 2D, ℓ = − k .Let f b ∈ g be the nilpotent element which is principal in b and trivial in a . The cor-responding W -algebras W k ( g , f b ) will be called hook-type W -algebras since they are anal-ogous to the hook-type W -algebras of type A introduced in [CL3]. Let d a = dim ρ a and d b = dim ρ b . In particular, d b = 2 m + 1 if b = so m +1 , and d b = 2 m if b = sp m .It follows from the decomposition (3.1) that in all cases, W k ( g , f b ) is of type W (cid:18) dim a , , , . . . , m, (cid:18) d b + 12 (cid:19) d a (cid:19) . The affine subalgebra is V t ( a ) for some level t , which we describe below. The fields inweights , , . . . , m are even and are invariant under a . The d a fields in weight d b +12 can e even or odd, and they transform as the standard a -module. By [CL3, Cor. 3.5], we mayassume without loss of generality that the fields in weights , , . . . , m lie in the affinecoset Com ( V t ( a ) , W k ( g , f b )) , and that the d a fields in weight d b +12 are primary for the actionof V t ( a ) .Write g = L d ρ d , where ρ d denotes the d -dimensional representation of the sl -triple { f, x, e } . Then each ρ d gives rise to a field of conformal weight d +12 in W k ( g , f b ) , and thecorresponding ghosts give rise to a central charge contribution(3.3) c d = − ( d − d − d − . The central charge c of W k ( g , f b ) is then computed to be c = c g + c dilaton + c ghost ,c g = k sdim g k + h ∨ g ,c dilaton = − k × ( m ( m + 1)(2 m + 1) b = so m +1 m (4 m − b = sp m ,c prin = 6 m − m ,c ghost = c prin + sd a c d b . (3.4)Here the formula for c d b is obtained by specializing (3.3). Finally, the level t of the affinesubalgebra V t ( a ) ⊆ W k ( g , f b ) is given by(3.5) t = ℓ ± ( d b − × ( a = so n , so n +1 , a = sp n , osp | n . where we have + if ρ a ⊗ ρ b is even and − if it is odd. For a = osp | n , we must replace d a by d a − sd a , where sd a means superdimension. Recall that in this case, ρ a is called even(respectively odd) if the n -dimensional standard module for sp n is even (respectivelyodd). We will always replace k with the critically shifted level ψ = k + h ∨ , where h ∨ denotesthe dual Coxeter number of g . We now describe the examples we need in more detail.3.1. Case 1B.
For g = so n +2 m +2 , we have ψ = k + 2 n + 2 m . We define W ψ B ( n, m ) := W k ( so n +2 m +2 , f so m +1 ) , which has affine subalgebra V ψ − n ( so n +1 ) . We consider the following extreme cases.(1) If m ≥ and n = 0 , f so m +1 is also the principal nilpotent in so n +2 , so W ψ B (0 , m ) = W ψ − m ( so m +2 ) .(2) For m = 0 and n ≥ , f so ∈ so n +2 is the zero nilpotent, so W ψ B ( n,
0) = V ψ − n ( so n +2 ) .(3) If m = n = 0 , W ψ B (0 ,
0) = V ψ ( so ) , which is just the rank one Heisenberg algebra H (1) .3.2. Case 1C.
For g = osp m +1 | n , we have ψ = k + 2 m − n − . We define W ψ C ( n, m ) := W k ( osp m +1 | n , f so m +1 ) , which has affine subalgebra V − ψ/ − n − / ( sp n ) . Here we are using the convention (3.2) that osp m +1 | n has dual Coxeter number m − n − .
1) If m ≥ and n = 0 , g = so m +1 and f so m +1 is the principal nilpotent, so W ψ C (0 , m ) = W ψ − m +1 ( so m +1 ) .(2) If m = 0 and n ≥ , g = osp | n and f so = 0 , so W ψ C ( n,
0) = V ψ +2 n +1 ( osp | n ) . Notethat even for m = 0 , we use the convention (3.2) that osp | n has dual Coxeter number − n − . With this choice, we have the embedding V − ψ − n − / ( sp n ) → V ψ +2 n +1 ( osp | n ) .(3) If m = n = 0 , W ψ C (0 ,
0) = C .3.3. Case 1D.
For g = so n +2 m +1 , we have ψ = k + 2 n + 2 m − . We define W ψ D ( n, m ) := W k ( so n +2 m +1 , f so m +1 ) , which has affine subalgebra V ψ − n +1 ( so n ) .(1) If m ≥ and n = 0 , g = so m +1 and f so m +1 is principal, so W ψ D (0 , m ) = W ψ − m +1 ( so m +1 ) .(2) If m = 0 and n ≥ , g = so n +1 and f so = 0 , so W ψ D ( n,
0) = V ψ − n +1 ( so n +1 ) .(3) If m ≥ and n = 1 , g = so m +3 and f so m +1 ∈ so m +3 is the subregular nilpotent, so W ψ D (1 , m ) = W ψ − m − ( so m +3 , f subreg ) . In this case, the affine subalgebra V ψ − ( so ) isjust H (1) .(4) If m = n = 0 , W ψ D (0 ,
0) = C .3.4. Case 1O.
For g = osp m +2 | n , we have ψ = k + 2 m − n . We define W ψ O ( n, m ) = W k ( osp m +2 | n , f so m +1 ) , which has affine subalgebra V − ψ/ − n ( osp | n ) . We are using the convention (3.2) that osp m +2 | n has dual Coxeter number m − n , whereas the dual Coxeter number of osp | n is taken tobe n +12 .(1) If m ≥ and n = 0 , g = so m +2 and f so m +1 is the principal nilpotent, so W ψ O (0 , m ) = W ψ − m ( so m +2 ) .(2) If m = 0 and n ≥ , we have g = osp | n and f so = 0 , so W ψ O ( n,
0) = V ψ +2 n ( osp | n ) .As above, even for m = 0 , we use the convention (3.2) that osp | n has dual Coxeternumber − n . We then have an embedding V − ψ/ − n ( osp | n ) → V ψ +2 n ( osp | n ) , where the dual Coxeter number of osp | n is chosen to be n +12 .(3) If m = n = 0 , W ψ O (0 ,
0) = V ψ ( osp | ) = H (1) .3.5. Case 2B.
For g = osp n +1 | m , we have ψ = k + m − n + 1 / . For m ≥ and n ≥ , wedefine W ψ B ( n, m ) := W k ( osp n +1 | m , f sp m ) , which has affine subalgebra V − ψ − n +2 ( so n +1 ) . We are using the convention (3.2) that osp n +1 | m has dual Coxeter number m − n +12 .(1) If m ≥ and n = 0 , g = osp | m and f sp m is the principal nilpotent, so W ψ B (0 , m ) is theprincipal W -superalgebra W ψ − m − / ( osp | m ) .(2) If m = 1 and n ≥ , g = osp n +1 | and f sp is the minimal nilpotent, so W ψ B ( n,
1) = W ψ + n − / ( osp n +1 | , f min ) .
3) If m = 0 and n ≥ , we need a different definition. We set W ψ B ( n,
0) := V − ψ − n +1 ( so n +1 ) ⊗ F (2 n + 1) . Here F (2 n +1) is the rank n +1 free fermion algebra, which has an action of L ( so n +1 ) .Therefore W ψ B ( n, has a diagonal action of V − ψ − n +2 ( so n +1 ) .(4) If m = n = 0 , we define W ψ B (0 ,
0) = F (1) .3.6. Case 2C.
For g = sp n +2 m , we have ψ = k + n + m + 1 . For m ≥ and n ≥ , we define W ψ C ( n, m ) := W k ( sp n +2 m , f sp m ) , which has affine subalgebra V ψ − n − / ( sp n ) .(1) If m ≥ and n = 0 , g = sp m and f sp m is the principal nilpotent, so W ψ C (0 , m ) = W ψ − m − ( sp m ) .(2) If m = 1 and n ≥ , g = sp n +2 and f sp m is the minimal nilpotent. Then W ψ C ( n,
1) = W ψ − n − ( sp n +2 , f min ) . (3) If m = 0 and n ≥ , we define W ψ C ( n,
0) := V ψ − n − ( sp n ) ⊗ S ( n ) . Here S ( n ) is the rank n βγ system, which has an action of L − / ( sp n ) . Therefore W ψ C ( n, has a diagonal action of V ψ − n − / ( sp n ) .(4) If m = n = 0 , we define W ψ C (0 ,
0) = C .3.7. Case 2D.
For g = osp n | m , we have ψ = k + m − n + 1 . For m ≥ and n ≥ , we define W ψ D ( n, m ) := W k ( osp n | m , f sp m ) , which has affine subalgebra V − ψ − n +3 ( so n ) . We are using the convention (3.2) that osp n | m has dual Coxeter number m − n + 1 .(1) If m ≥ and n = 0 , g = sp m and f sp m is the principal nilpotent. Then W ψ D (0 , m ) = W ψ − m − ( sp m ) .(2) If m ≥ and n = 1 , g = osp | m and f sp m is the principal nilpotent, so W ψ D (1 , m ) is theprincipal W -superalgebra W ψ − m ( osp | m ) . Note that in this case, the affine subalgebra V − ψ +1 ( so ) is just the Heisenberg algebra H (1) .(3) If m = 1 and n ≥ , g = osp n | , and f sp is the minimal nilpotent so W ψ D ( n,
1) = W ψ + n − ( osp n | , f min ) .(4) If m = 0 and n ≥ , we define W ψ D ( n,
0) := V − ψ − n +2 ( so n ) ⊗ F (2 n ) . Here F (2 n ) is the rank n free fermion algebra, which admits an action of L ( so n ) .Then W ψ D ( n, admits a diagonal action of V − ψ − n +3 ( so n ) .(5) If m = n = 0 , we define W ψ D (0 ,
0) = C . .8. Case 2O.
For g = osp | n +2 m , we have ψ = k + m + n + 1 / . For m ≥ and n ≥ , wedefine W ψ O ( n, m ) := W k ( osp | n +2 m , f sp m ) , which has affine subalgebra V ψ − n − ( osp | n ) . We are using the convention (3.2) that osp | n +2 m has dual Coxeter number m +2 n +12 , and the dual Coxeter number of osp | n is taken to be n +12 .(1) If m ≥ and n = 0 , g = osp | m and f sp m is principal. Then W ψ O (0 , m ) = W ψ − m − / ( osp | m ) .(2) If m = 1 and n ≥ , g = osp | n +2 , and f sp is the minimal nilpotent, so W ψ O ( n,
1) = W ψ − n − / ( osp | n +2 , f min ) .(3) If m = 0 and n ≥ , we define W ψ O ( n,
0) := V ψ − n − / ( osp | n ) ⊗ S ( n ) ⊗ F (1) . Recall that S ( n ) ⊗ F (1) admits an action of L − / ( osp | n ) , so W ψ O ( n, admits a diag-onal action of V ψ − n − ( osp | n ) .(4) If m = n = 0 , we define W ψ O (0 ,
0) = F (1) .3.9. Affine cosets.
Our main objects of study are the affine cosets of these W -algebras. Infact, in the case where a is either so n , so n +1 , or osp | n , the action of a integrates to anaction of the corresponding connected (super)group SO n , SO n +1 , or SOsp | n , and thisaction further extends to the double cover O n , O n +1 , or Osp | n . In these cases, we needto take the Z -orbifold of the corresponding affine coset. Here is the list of these algebras.3.10. Case 1B. (3.6) C ψ B ( n, m ) = Com ( V ψ − n ( so n +1 ) , W ψ B ( n, m )) Z m ≥ , n ≥ , Com ( V ψ − n ( so n +1 ) , V ψ − n ( so n +2 )) Z m = 0 , n ≥ , W ψ − m ( so m +2 ) Z m ≥ , n = 0 , H (1) Z m = n = 0 . In all cases, C ψ B ( n, m ) has central charge c = − ( ψ + mψ − m − n − mψ − m − n − ψ + 2 mψ − m − n )( ψ − ψ . Case 1C. (3.7) C ψ C ( n, m ) = Com ( V − ψ/ − n − / ( sp n ) , W ψ C ( n, m )) m ≥ , n ≥ , Com ( V − ψ/ − n − / ( sp n ) , V ψ +2 n +1 ( osp | n )) m = 0 , n ≥ , W ψ − m +1 ( so m +1 ) m ≥ , n = 0 , C m = n = 0 . If we define the central charge of C to be zero, then in all cases, C ψ C ( n, m ) has central charge c = − ( − m + n + mψ )(1 − m + 2 n + ψ + 2 mψ )( − − m + 2 n + 2 ψ + 2 mψ )( ψ − ψ . .12. Case 1D. (3.8) C ψ D ( n, m ) = Com ( V ψ − n +1 ( so n ) , W ψ D ( n, m )) Z m ≥ , n > , Com ( H (1) , W ψ − m − ( so m +3 , f subreg )) Z m ≥ , n = 1 , Com ( V ψ − n +1 ( so n ) , V ψ − n +1 ( so n +1 ) Z m = 0 , n ≥ , W ψ − m +1 ( so m +1 ) m ≥ , n = 0 , C m = n = 0 . In all cases, C ψ D ( n, m ) has central charge c = − ( − m − n + mψ )(1 − m − n + ψ + 2 mψ )( − − m − n + 2 ψ + 2 mψ )( ψ − ψ . Case 1O. (3.9) C ψ O ( n, m ) = Com ( V − ψ/ − n ( osp | n ) , W ψ O ( n, m )) Z m ≥ , n ≥ , Com ( V − ψ/ − n ( osp | n ) , V ψ +2 n ( osp | n )) m = 0 , n ≥ , W ψ − m ( so m +2 ) Z m ≥ , n = 0 , H (1) Z m = n = 0 . In all cases, C ψ O ( n, m ) has central charge c = − ( − − m + n + ψ + mψ )( − − m + 2 n + 2 mψ )( − m + 2 n + ψ + 2 mψ )( ψ − ψ . Case 2B. (3.10) C ψ B ( n, m ) = Com ( V − ψ − n +2 ( so n +1 ) , W ψ B ( n, m )) Z m ≥ , n ≥ , Com ( V − ψ − n +2 ( so n +1 ) , V − ψ − n +1 ( so n +1 ) ⊗ F (2 n + 1)) Z m = 0 , n ≥ , W ψ − m − / ( osp | m ) Z m ≥ , n = 0 , F (1) Z m = n = 0 . In all cases, C ψ B ( n, m ) has central charge c = − ( − m + n − ψ + 2 mψ )(1 − m + 2 n + 4 mψ )( − − m + 2 n + 2 ψ + 4 mψ )2 ψ (2 ψ − . Case 2C. (3.11) C ψ C ( n, m ) = Com ( V ψ − n − / ( sp n ) , W ψ C ( n, m )) m ≥ , n ≥ , Com ( V ψ − n − / ( sp n ) , V ψ − n − ( sp n ) ⊗ S ( n )) m = 0 , n ≥ , W ψ − m − ( sp m ) m ≥ , n = 0 , C m = n = 0 . In all cases, C ψ C ( n, m ) has central charge c = − ( − m − n + 2 mψ )( − − m − n + ψ + 2 mψ )( − − m − n − ψ + 4 mψ ) ψ (2 ψ − . .16. Case 2D. (3.12) C ψ D ( n, m ) = Com ( V − ψ − n +3 ( so n ) , W ψ D ( n, m )) Z m ≥ , n > , Com ( H (1) , W ψ − m ( osp | m )) Z m ≥ , n = 1 , Com ( V − ψ − n +3 ( so n ) , V − ψ − n +2 ( so n ) ⊗ F (2 n )) Z m = 0 , n ≥ , W ψ − m − ( sp m ) m ≥ , n = 0 , C m = n = 0 . In all cases, C ψ D ( n, m ) has central charge c = − ( − m + n + 2 mψ )( − − m + n + ψ + 2 mψ )( − − m + 2 n − ψ + 4 mψ ) ψ (2 ψ − . Case 2O. (3.13) C ψ O ( n, m ) = Com ( V ψ − n − ( osp | n ) , W ψ O ( n, m )) Z m ≥ , n ≥ , Com ( V ψ − n − ( osp | n ) , V ψ − n − / ( osp | n ) ⊗ S ( n ) ⊗ F (1)) Z m = 0 , n ≥ , W ψ − m − / ( osp | m ) Z m ≥ , n = 0 , F (1) Z m = n = 0 . In all cases, C ψ O ( n, m ) has central charge c = − ( − m − n − ψ + 2 mψ )(1 − m − n + 4 mψ )( − − m − n + 2 ψ + 4 mψ )2 ψ (2 ψ − . We shall regard ψ as a formal variable and the algebras C ψiX ( n, m ) for i = 1 , and X = B, C, D, O , as one-parameter vertex algebras with parameter ψ . If ψ ∈ C is a complexnumber, C ψ iX ( n, m ) will always denote the specialization of C ψiX ( n, m ) to the value ψ = ψ .For generic values of ψ , this coincides with the actual coset, although it can be a propersubalgebra of the coset if ψ is a negative rational number. Theorem 3.1.
For i = 1 , and X = B, C, D, O , C ψiX ( n, m ) is simple as a one-parameter vertexalgebra; equivalently this holds for generic values of ψ .Proof. In all cases where W ψiX ( n, m ) is a quantum Hamiltonian reduction, namely the caseswhere n + m ≥ , and m ≥ when i = 2 , the simplicity of W ψiX ( n, m ) and of its affine cosetfollows from parts (1) and (2) of [CL3, Thm. 3.6]. In the cases X = B, D, O where C ψiX ( n, m ) is the Z -orbifold of the affine coset, the simplicity follows from [DLM].In the cases W ψ X ( n, , the simplicity of the affine coset follows from [CGN, Prop. 5.4],and in the cases X = B, D, O , the simplicity of C ψ X ( n, again follows from [DLM]. Finally,the claim is obvious in the all cases when n = m = 0 . (cid:3)
4. M
AIN RESULT
The main result in this paper is analogous to [CL3, Thm. 1.1].
Theorem 4.1.
For all integers m ≥ n ≥ , we have the following isomorphisms of one-parametervertex algebras. (4.1) C ψ B ( n, m ) ∼ = C ψ ′ O ( n, m − n ) ∼ = C ψ ′′ B ( m, n ) , ψ ′ = 14 ψ , ψ + 1 ψ ′′ = 2 , C ψ C ( n, m ) ∼ = C ψ ′ C ( n, m − n ) ∼ = C ψ ′′ C ( m, n ) , ψ ′ = 12 ψ , ψ + 1 ψ ′′ = 1 , (4.3) C ψ D ( n, m ) ∼ = C ψ ′ D ( n, m − n ) ∼ = C ψ ′′ O ( m, n − , ψ ′ = 12 ψ , ψ + 1 ψ ′′ = 1 , (4.4) C ψ O ( n, m ) ∼ = C ψ ′ B ( n, m − n ) ∼ = C ψ ′′ D ( m + 1 , n ) , ψ ′ = 1 ψ , ψ + 12 ψ ′′ = 1 . Note that C ψ D ( n, m ) for m ≥ n and C ψ D ( n, m ) for m < n belong in different families, andsimilarly for C ψ O ( n, m ) . For the rest of this section we discuss some special cases of thisresult.4.1. Special cases of (4.1) . In the case n = 0 of (4.1), we have C ψ B (0 , m ) = W ψ − m − / ( osp | m ) Z , C ψ ′ O (0 , m ) = W ψ ′ − m − / ( osp | m ) Z . The isomorphism C ψ B (0 , m ) ∼ = C ψ ′ O (0 , m ) is the Z -invariant part of Feigin-Frenkel dualityfor principal W -algebras of osp | m , which was proven in a different way in [CGe]. Remark . A special case of Theorem 6.3 is that the OPE algebra of W ψ ′ − m − / ( osp | m ) ,which is a simple current extension of W ψ ′ − m − / ( osp | m ) Z by an odd field of weight m +12 , is uniquely determined by W ψ ′ − m − / ( osp | m ) Z . Therefore our result implies thefull Feigin-Frenkel duality W ψ − m − / ( osp | m ) ∼ = W ψ ′ − m − / ( osp | m ) .In the case m = 0 of (4.1), we have C ψ B ( n,
0) =
Com ( V − ψ − n +2 ( so n +1 ) , V − ψ − n +1 ( so n +1 ) ⊗ F (2 n + 1)) Z , C ψ ′′ B (0 , n ) = W ψ ′′ − n − / ( osp | n ) Z . Therefore the isomorphism C ψ B ( n, ∼ = C ψ ′′ B (0 , n ) implies that both W ψ ′′ − n − / ( osp | n ) and Com ( V − ψ − n +2 ( so n +1 ) , V − ψ − n +1 ( so n +1 ) ⊗ F (2 n + 1)) are simple current extensionsof W ψ ′′ − n − / ( osp | n ) Z by an odd field in weight n +12 . As above, Theorem 6.3 then implies W ψ ′′ − n − / ( osp | n ) ∼ = Com ( V − ψ − n +2 ( so n +1 ) , V − ψ − n +1 ( so n +1 ) ⊗ F (2 n + 1)) ∼ = Com ( V − ψ − n +2 ( so n +1 ) , V − ψ − n +1 ( so n +1 ) ⊗ L ( so n +1 )) . (4.5)We recover the coset realization of principal W -superalgebras of osp | n , which was provenin a different way in [CGe].4.2. Special cases of (4.2) . In the case n = 0 , the isomorphism C ψ C (0 , m ) ∼ = C ψ ′ C (0 , m ) for ψ ′ = ψ , is just Feigin-Frenkel duality in types B and C , since C ψ C (0 , m ) = W ψ − m +1 ( so m +1 ) and C ψ ′ C (0 , m ) ∼ = W ψ ′ − m − ( sp m ) .In the case m = 0 , we have C ψ C ( n,
0) =
Com ( V − ψ/ − n − / ( sp n ) , V ψ +2 n +1 ( osp | n )) , C ψ ′′ C (0 , n ) = W ψ ′′ − n +1 ( so n +1 ) , o the isomorphism C ψ C ( n, ∼ = C ψ ′′ C (0 , n ) yields a new coset realization of type B and C principal W -algebras. Recall that we are using the convention (3.2) that osp | n has dualCoxeter number − n − . If we instead use the dual Coxeter number n +12 , so that V k ( sp n ) embeds in V k ( osp | n ) , then we have C ψ C ( n,
0) =
Com ( V k ( sp n ) , V k ( osp | n )) for k = − ( ψ +2 n + 1) . We obtain Corollary 4.1.
For all n ≥ , we have the following isomorphism of one-parameter vertex algebras (4.6) Com ( V k ( sp n ) , V k ( osp | n )) ∼ = W r ( so n +1 ) , r = − (2 n −
1) + 1 + 2 k + 2 n k + n ) . This realization of W r ( so n +1 ) is very different from the coset realization of W ℓ ( g ) for g simply-laced given in [ACL] since it involves affine vertex superalgebras. Although we arenot aware of this statement being previously conjectured in the literature, both algebraswere known to have the same strong generating type and graded character; see [CL1, Cor.7.4] and [CL2, Cor 5.7].4.3. Special cases of (4.3) . In the case n = 0 , the isomorphism C ψ D (0 , m ) ∼ = C ψ ′ D (0 , m ) isagain Feigin-Frenkel duality in types B and C , since C ψ D (0 , m ) = W ψ − m − ( sp m ) , C ψ ′ D (0 , m ) = W ψ ′ − m +1 ( so m +1 ) . In the case n = 1 and m ≥ , we have C ψ D (1 , m ) = Com ( H (1) , W ψ − m ( osp | m )) Z , C ψ ′ D (1 , m −
1) =
Com ( H (1) , W ψ ′ − m +1 ( so m +1 , f subreg )) Z . Therefore the isomorphism C ψ D (1 , m ) ∼ = C ψ ′ D (1 , m − recovers the Z -invariant part of theduality(4.7) Com ( H (1) , W ψ ′ − m +1 ( so m +1 , f subreg )) ∼ = Com ( H (1) , W ψ − m ( osp | m )) , of Genra, Nakatsuka and one of us [CGN]. Remark . The isomorphism C ψ D (1 , m ) ∼ = C ψ ′ D (1 , m − can be used to give a new proof of(4.7) as follows. Both Com ( H (1) , W ψ ′ − m +1 ( so m +1 , f subreg )) and Com ( H (1) , W ψ − m ( osp | m )) are simple current extensions of C ψ D (1 , m ) , where the extension is generated by an evenfield ω in weight m + 1 . The generators of C ψ D (1 , m ) and ω do not close under OPE, andnew strong generators are needed in weights m + 3 , m + 5 , . . . . It can be shown thatthere is a unique simple current extension of C ψ D (1 , m ) with these properties, such that ω (4 m +1) ω = 0 . The proof is similar to, but more involved than the proof of Theorem 6.3, andis omitted.In the case n = 1 and m = 0 of (4.3), we have W ψ D (1 ,
0) := H (1) ⊗ F (2) and C ψ D (1 ,
0) =
Com ( H (1) , H (1) ⊗F (2)) Z ∼ = F (2) O . Also, C ψ O (0 , ∼ = H (1) Z . We recover the isomorphism H (1) Z ∼ = F (2) O . .4. Special cases of (4.4) . In the case n = 0 , the isomorphism C ψ O (0 , m ) ∼ = C ψ ′ B (0 , m ) is justthe Z -invariant part of Feigin-Frenkel duality in type D , since C ψ O (0 , m ) = W ψ − m ( so m +2 ) Z , C ψ ′ B (0 , m ) = W ψ ′ − m ( so m +2 ) Z . As in Remark 4.1, this statement together with Theorem 6.3, gives a new proof of the fullduality.In the case m = 0 and n ≥ , we have C ψ ′′ D ( n, ∼ = Com ( V − ψ ′′ − n +3 ( so n ) , V − ψ ′′ − n +2 ( so n ) ⊗ F (2 n )) Z ∼ = Com ( V − ψ ′′ − n +3 ( so n ) , V − ψ ′′ − n +2 ( so n ) ⊗ L ( so n )) Z . (4.8)Since C ψ O (0 , n − ∼ = W ψ − n +2 ( so n ) Z , the isomorphism C ψ O (0 , n − ∼ = C ψ ′′ D ( n, recoversthe Z -invariant part of the coset realization of principal W -algebras of type D proven in[ACL]. Finally, this statement together with Theorem 6.3, gives a new proof of the cosetrealization of W ψ ′ − n +2 ( so n ) .4.5. Sketch of proof.
The proof of Theorem 4.1 involves the following steps.(1) Using the free field limits of W ψiX ( n, m ) , together with some classical invariant theory,we find strong generating sets for C ψiX ( n, m ) for i = 1 , and X = B, C, D, O . If a = sp n ,and if a = so n or a = so n +1 and ρ a ⊗ ρ b is odd, we will find minimal strong generatingsets. In the remaining cases, namely, a = osp | n , and a = so n or a = so n +1 and ρ a ⊗ ρ b is even, we will not find minimal strong generating sets at this stage, but wewill deduce them later as a consequence of Theorem 4.1.(2) We show that in all cases, C ψiX ( n, m ) has a subalgebra ˜ C ψiX ( n, m ) generated by the weights and field, which is isomorphic to a quotient W of W ev ,I iX,n,m ( c, λ ) , for some ideal I iX,n,m ⊆ C [ c, λ ] . In particular, W ψiX ( n, m ) is an extension of V t ( a ) ⊗ W by d a fields ofappropriate parity in weight d b +12 which transform as the standard a -module.(3) We show that the existence of such an extension of V t ( a ) ⊗ W uniquely and explicitlyspecifies the ideal I iX,n,m .(4) We compute coincidences between the simple quotient ˜ C ψ,iX ( n, m ) and principal W -algebras of type C , by finding intersection points on their truncation curves. UsingCorollary 2.1, we prove that ˜ C ψiX ( n, m ) = C ψiX ( n, m ) as one-parameter vertex algebras.In particular, C ψiX ( n, m ) is isomorphic to the simple quotient W ev I iX,n,m ( c, λ ) in all cases.(5) The isomorphisms in Theorem 4.1 all follow from the explicit formulas for I iX,n,m .5. L ARGE LEVEL LIMITS
In this section, we describe the large level limits of W ψiX ( n, m ) and the strong generatingtypes of C ψiX ( n, m ) .5.1. Case 1C.
Recall that W ψ C ( n, m ) has affine subalgebra V − ψ/ − n − / ( sp n ) , even fields inweights , , . . . , m which are invariant under sp n , and n odd fields of weight m + 1 ,which transform as the standard representation of sp n . By [CL3, Cor. 3.5], we mayassume without loss of generality that the fields in weights , , . . . , m commute with − ψ/ − n − / ( sp n ) , and the weight m +1 fields are primary with respect to V − ψ/ − n − / ( sp n ) .By [CL3, Cor. 3.4], the free field limit of W ψ C ( n, m ) is O ev (2 n + n, ⊗ O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ S odd ( n, m + 2) . Theorem 5.1.
For n + m ≥ , C ψ C ( n, m ) is of type W (2 , , . . . , m )(1 + n ) − as a one-parameter vertex algebra. Equivalently, this holds for generic values of ψ .Proof. First, it follows from [CL3, Lem. 4.2] that C ψ C ( n, m ) has limit O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ (cid:0) S odd ( n, m + 2) (cid:1) Sp n . By [CL3, Thm. 4.3], (cid:0) S odd ( n, m + 2) (cid:1) Sp n is of type W (2 m + 2 , m + 4 , . . . , m )(1 + n ) − . Since O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) is of type W (2 , , . . . , m ) , it follows from [CL3,Lem. 4.2] that C ψ C ( n, m ) is of type W (2 , , . . . , m )(1 + n ) − . (cid:3) Case 2B.
Recall that for m ≥ , W ψ B ( n, m ) has affine subalgebra V − ψ − n +2 ( so n +1 ) ,even fields in weights , , . . . , m which commute with V − ψ − n +2 ( so n +1 ) , and n + 1 oddfields of weight m +12 , which are primary with respect to V − ψ − n +2 ( so n +1 ) and transformas the standard representation of so n +1 . The free field limit of W ψ B ( n, m ) is therefore O ev (2 n + n, ⊗ O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ O odd (2 n + 1 , m + 1) . Theorem 5.2.
For n + m ≥ , C ψ B ( n, m ) is of type W (2 , , . . . , m + 1)( n + 1) − as a one-parameter vertex algebra. Equivalently, this holds for generic values of ψ .Proof. By [CL3, Lem. 4.2] as above, C ψ B ( n, m ) has limit O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ (cid:0) O odd (2 n + 1 , m + 1) (cid:1) O n +1 . By [CL3, Thm. 4.4], (cid:0) O odd (2 n + 1 , m + 1) (cid:1) O n +1 is of type W (2 m + 2 , m + 4 , . . . , m + 1)( n + 1) − , so the claim follows as above. (cid:3) Case 2C.
Recall that for m ≥ , W ψ C ( n, m ) has affine subalgebra V ψ − n +3 / ( sp n ) , evenfields in weights , , . . . , m which commute with V ψ − n +3 / ( sp n ) , and n even fields ofweight m +12 , which are primary with respect to V ψ − n +3 / ( sp n ) and transform as the stan-dard representation of sp n . The free field limit of W ψ C ( n, m ) is therefore O ev (2 n + n, ⊗ O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ S ev ( n, m + 1) . Theorem 5.3.
For n + m ≥ , C ψ C ( n, m ) is of type W (2 , , . . . , n )(1 + m + n ) − as a one-parameter vertex algebra. Equivalently, this holds for generic values of ψ . roof. By [CL3, Lem. 4.2], C ψ C ( n, m ) has limit O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ (cid:0) S ev ( n, m + 1) (cid:1) Sp n . By [CL3, Thm. 4.2], (cid:0) S ev ( n, m + 1) (cid:1) Sp n is of type W (2 m + 2 , m + 4 , . . . , n )(1 + m + n ) − , so the claim follows. (cid:3) Case 2D.
Recall that for m ≥ , W ψ D ( n, m ) has affine subalgebra V − ψ − n +3 ( so n ) ,even fields in weights , , . . . , m which commute with V − ψ − n +3 ( so n ) , and n odd fieldsof weight m +12 , which are primary with respect to V − ψ − n +3 ( so n ) and transform as thestandard representation of so n . The free field limit of W ψ D ( n, m ) is therefore O ev (2 n − n, ⊗ O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ O odd (2 n, m + 1) . Theorem 5.4.
For n + m ≥ , C ψ D ( n, m ) is of type W (2 , , . . . , m + 1)(2 n + 1) − as a one-parameter vertex algebra. Equivalently, this holds for generic values of ψ .Proof. By [CL3, Lem. 4.2], C ψ D ( n, m ) has limit O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ (cid:0) O odd (2 n, m + 1) (cid:1) O n . By [CL3, Thm. 4.4], (cid:0) O odd (2 n, m + 1) (cid:1) O n is of type W (2 m + 2 , m + 4 , . . . , m + 1)(2 n + 1) − , so the claim follows. (cid:3) Next, we consider the cases C ψ B ( n, m ) and C ψ D ( n, m ) where we are not able to find aminimal strong generating set at this stage.5.5. Case 1B.
Recall that W ψ B ( n, m ) has affine subalgebra V ψ − n ( so n +1 ) , even fields inweights , , . . . , m which commute with V ψ − n ( so n +1 ) , and n + 1 even fields in weight m + 1 which are primary with respect to V ψ − n ( so n +1 ) and transform as the standard rep-resentation of so n +1 . The free field limit of W ψ B ( n, m ) is then O ev (2 n + n, ⊗ O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ O ev (2 n + 1 , m + 2) . Lemma 5.1.
As one-parameter vertex algebras, C ψ B ( n, m ) is of type W (2 , , . . . , N ) for some N satisfying N ≥ n )(3 + 2 m + 2 n ) − .Proof. By [CL3, Lem. 4.2], C ψ B ( n, m ) has limit O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ (cid:0) O ev (2 n + 1 , m + 2) (cid:1) O n +1 . By [CL3, Thm. 4.5], (cid:0) O ev (2 n + 1 , m + 2) (cid:1) O n +1 is of type W (2 m + 2 , m + 4 , . . . , N ) for some N ≥ n )(3 + 2 m + 2 n ) − , so the claim follows. (cid:3) We will see later in Corollary 6.1 that N = 2(1 + n )(3 + 2 m + 2 n ) − . .6. Case 1D.
Recall that W ψ D ( n, m ) has affine subalgebra V ψ − n +1 ( so n ) , even fields inweights , , . . . , m which commute with V ψ − n +1 ( so n ) , and n additional even fields ofweight m + 1 which are primary with respect to V ψ − n +1 ( so n ) and transform as the stan-dard representation of so n . The free field limit of W ψ D ( n, m ) is therefore O ev (2 n − n, ⊗ O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ O ev (2 n, m + 2) . Lemma 5.2.
As one-parameter vertex algebras, C ψ D ( n, m ) is of type W (2 , , . . . , N ) for some N satisfying N ≥ m + n )(1 + 2 n ) − .Proof. First, [CL3, Lem. 4.2] shows that C ψ D ( n, m ) has limit O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ (cid:0) O ev (2 n, m + 2) (cid:1) O n . Again by [CL3, Thm. 4.5], (cid:0) O ev (2 n, m + 2) (cid:1) O n is of type W (2 m + 2 , m + 4 , . . . , N ) forsome N ≥ m + n )(1 + 2 n ) − , so the claim follows. (cid:3) As above, Corollary 6.1 shows that N = 2(1 + m + n )(1 + 2 n ) − .Finally, we consider the cases C ψ O ( n, m ) and C ψ O ( n, m ) . We need two new ingredients: thedescription of orbifolds of certain free field algebras under Osp | n , and the adaptation ofthe method of studying affine cosets by passing to their orbifold limits developed in [CL1],to cosets of V k ( osp | n ) . We begin by restating the versions of Sergeev’s first and secondfundamental theorems of invariant theory for Osp | n that we need; these are specializa-tions of [S1, Thm. 1.3] and [S2, Thm. 4.5]. First, we consider the invariants in the ring offunctions on a sum of copies of the standard module with odd parity. Theorem 5.5.
For k ≥ , let U k be copy of the standard Osp | n -module C | n , which has oddsubspace spanned by { x k,i , y k,i | i = 1 , . . . , n } , and even subspace spanned by z k . Then the ring ofinvariant polynomial functions R = C [ M k ≥ U k ] Osp | n is generated by the quadratics q a,b = 12 n X i =1 ( x i,a y i,b + x i,b y i,a ) − z a z b a, b ≥ . Let Q a,b be commuting indeterminates satisfying Q a,b = Q b,a . The kernel of the map C [ Q a,b ] → R, Q a,b q a,b is generated by polynomials p I of degree n + 2 in the variables Q a,b corresponding to a rectangularYoung tableau of size × (2 n + 2) , filled by entries from a standard sequence I of length n + 4 fromthe set of indices { , , , . . . } . The entries must weakly increase along rows and strictly increasealong columns. For the invariants in the ring of functions on a sum of copies of the standard modulewith even parity, a few modifications are needed. heorem 5.6. For k ≥ , let U k be copy of the standard Osp | n -module C n | , with even subspacespanned by { x k,i , y k,i | i = 1 , . . . , n } and odd subspace spanned by z k . Then the ring of invariantpolynomial functions R = C [ M k ≥ U k ] Osp | n is generated by the quadratics q a,b = 12 n X i =1 ( x i,a y i,b − x i,b y i,a ) − z a z b a, b ≥ . Let Q a,b be commuting indeterminates satisfying Q a,b = − Q b,a . The kernel of the map C [ Q a,b ] → R, Q a,b q a,b is generated by polynomials p I of degree n + 2 in the variables Q a,b corresponding to a rectangularYoung tableau of size × (2 n + 2) , filled by entries from a standard sequence I of length n + 4 fromthe set of indices { , , , . . . } . The entries must strictly increase along rows and weakly increasealong columns. In both cases, the precise form of the relations can be found in [S2], but is not needed forour purposes. We only need the conformal weight of the relations which can be read offfrom the entries in the corresponding Young tableau.Next, we recall that V k ( osp | n ) comes from a deformable family in the sense of [CL1]as follows. Let κ be a formal variable satisfying κ = k , and let F be the ring of complex-valued rational functions in κ of degree at most zero, with possible poles only at κ = 0 .There is vertex algebra V over F such that V / ( κ − √ k ) V ∼ = V k ( osp | n ) for all k = 0 , Here ( κ − √ k ) V denotes the ideal generated by κ − √ k . In the notation of [CL1], V ∞ = lim κ →∞ V ∼ = H (2 n + n ) ⊗ A ( n ) , where H (2 n + n ) denotes the Heisenberg algebra of rank n + n = dim sp n and A ( n ) denotes the rank n symplectic fermion algebra.We now consider vertex algebras W k which admit a homomorphism V k ( osp | n ) → W k with the following properties:(1) There exists a deformable family W defined over ring F K of rational functions of de-gree at most zero in κ , with poles in some at most countable set K , such that W / ( κ − √ k ) W ∼ = W k , for all √ k / ∈ K. (2) The map V k ( osp | n ) → W k is induced by a map of deformable families V → W .(3) W ∞ = lim κ →∞ W decomposes as W ∞ ∼ = V ∞ ⊗ ˜ W ∼ = H (2 n + n ) ⊗ A ( n ) ⊗ ˜ W , for some vertex subalgebra ˜ W ⊆ W ∞ .(4) The action of osp | n on W infinitesimally generates an action of the Lie supergroupSOsp | n , and W decomposes into finite-dimensional SOsp | n -modules.Under these circumstances, we obtain heorem 5.7. (1) C = Com ( V , W ) is a deformable family, and C / ( κ − √ k ) C ∼ = C k = Com ( V k ( osp | n ) , W k ) , for generic k .(2) SOsp | n acts on ˜ W , and we have an isomorphism C ∞ ∼ = Com ( V ∞ , V ∞ ⊗ ˜ W ) SOsp | n ∼ = Com ( H (2 n + n ) ⊗ A ( n ) , H (2 n + n ) ⊗ A ( n ) ⊗ ˜ W ) SOsp | n ∼ = ˜ W SOsp | n (5.1) (3) For generic k , W k admits a decomposition (5.2) W k ∼ = M λ ∈ P + V k ( λ ) ⊗ C k ( λ ) , where P + denotes the set of dominant weights of osp | n , V k ( λ ) are the corresponding Weylmodules, and the multiplicity spaces C k ( λ ) are irreducible C k -modules. The proof of the first two statements is the same as the proof of [CL1, Thm. 6.10], andonly uses the fact that finite-dimensional SOsp | n -modules are completely reducible. Sim-ilarly, the proof of the third statement is the same as the proof of [CL3, Thm. 4.12]. It isapparent than in our main examples, namely W k = W ψ O ( n, m ) and W ψ O ( n, m ) these hy-potheses are satisfied. Moreover, these algebras are in fact modules over the double coverOsp | n of SOsp | n , and it is the Z -orbifold of the coset that we actually need to study.5.7. Case 1O.
Recall that W ψ O ( n, m ) has affine subalgebra V − ψ/ − n ( osp | n ) , even fields inweights , , . . . , m which commute with V − ψ/ − n ( osp | n ) , and n odd fields and one evenfield of weight m + 1 , which are primary with respect to V − ψ/ − n ( osp | n ) and transform asthe standard representation of osp | n . The free field limit of W ψ O ( n, m ) is therefore O ev (2 n + n, ⊗S odd ( n, ⊗O ev (1 , ⊗O ev (1 , ⊗· · ·⊗O ev (1 , m ) ⊗S odd ( n, m +2) ⊗O ev (1 , m +2) . Lemma 5.3.
As one-parameter vertex algebras, C ψ O ( n, m ) has a strong generating set of type W (2 , , . . . ) . If this truncates to W (2 , , . . . , N ) for some N , we must have N ≥ m )(1 + n ) − .Proof. First, it follows from Theorem 5.7 that C ψ O ( n, m ) has limit O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ (cid:0) S odd ( n, m + 2) ⊗ O ev (1 , m + 2) (cid:1) Osp | n . We assign S odd ( n, m + 2) ⊗ O ev (1 , m + 2) the good increasing filtration where the weight m + 1 generators { a i , b i | i = 1 , . . . , n } of S odd ( n, m + 2) , and the weight m + 1 generator a of O ev (1 , m + 2) all have degree . Thengr (cid:18)(cid:0) S odd ( n, m +2) ⊗O ev (1 , m +2) (cid:1) Osp | n (cid:19) ∼ = gr (cid:18) S odd ( n, m +2) ⊗O ev (1 , m +2) (cid:19) Osp | n ∼ = R, where R is the ring of invariants in Theorem 5.5. Then (cid:0) S odd ( n, m +2) ⊗O ev (1 , m +2) (cid:1) Osp | n is strongly generated by the corresponding fields ω a,b = 12 n X i =1 : ( ∂ a a i )( ∂ b b i ) : + : ( ∂ b a i )( ∂ a b i ) : −
12 : ( ∂ a a )( ∂ b a ) : , a, b ≥ , hich have weight m + 2 + a + b . As usual, there are linear relations among these fieldsand their derivatives, and the subsets { ∂ k ω a, | a ≥ } , { ω a,b | a ≥ b ≥ } span the same vector space. Therefore the fields { ω a, | a ≥ } , which have weight m +2 + 2 a , are a strong generating set. This shows that (cid:0) S odd ( n, m + 2) ⊗ O ev (1 , m + 2) (cid:1) Osp | n has a strong generating set of type W (2 m + 2 , m + 4 , . . . ) , which proves this first statementsince O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) is of type W (2 , , . . . , m ) .Next, the relation of minimal weight given by Theorem 5.5 corresponds to the × (2 n +2) ,Young tableau with bottom row consisting of ’s and top row consisting of ’s. This relationtherefore has weight m )(1 + n ) . If there exists a decoupling relation(5.3) ω a, = P ( ω , , ω , , . . . , ω a − , ) , where P is a normally ordered polynomial in ω , , ω , , . . . , ω a − , and their derivatives, theweight a + 2 m + 2 of this relation must therefore be at least m )(1 + n ) . By apply-ing the operator ( ω , ) (2 m +1) repeatedly, it is easy to construct similar decoupling relationsexpressing ω b, for all b > a as normally ordered polynomials in ω , , ω , , . . . , ω a − , andtheir derivatives. Therefore if (5.3) is such a relation of minimal weight, then C ψ O ( n, m ) would be of type W (2 , , . . . , N ) for N = 2 a + 2 m . (cid:3) We will see later (Corollary 6.1) that C ψ O ( n, m ) is of type W (2 , , . . . , m )(1 + n ) − ,so the relation of weight m )(1 + n ) must in fact be a decoupling relation.5.8. Case 2O.
Recall that W ψ O ( n, m ) has affine subalgebra V ψ − n − ( osp | n ) , even fields inweights , , . . . , m which commute with V ψ − n − ( osp | n ) , and n even fields and one oddfield of weight m +12 , which are primary with respect to V ψ − n − ( osp | n ) and transform asthe standard representation of osp | n . The free field limit of W ψ O ( n, m ) is therefore O ev (2 n + n, ⊗S odd ( n, ⊗O ev (1 , ⊗O ev (1 , ⊗· · ·⊗O ev (1 , m ) ⊗S ev ( n, m +1) ⊗O odd (1 , m +1) . Lemma 5.4.
As one-parameter vertex algebras, C ψ O ( n, m ) has a strong generating set of type W (2 , , . . . ) . If this truncates to W (2 , , . . . , N ) for some N , we must have N ≥ n )(1 + m + n ) − .Proof. First, it follows from Theorem 5.7 that C ψ O ( n, m ) has limit O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ (cid:0) S ev ( n, m + 1) ⊗ O odd (1 , m + 1) (cid:1) Osp | n . So to prove the first statement, it suffices to show that (cid:0) S ev ( n, m +1) ⊗O odd (1 , m +1) (cid:1) Osp | n is of type W (2 m + 2 , m + 4 , . . . ) . The argument is similar to the proof of Lemma 5.3, andis based on Theorem 5.6. First, in terms of the weight m +12 generators { a i , b i | i = 1 , . . . , n } of S ev ( n, m + 1) , and the weight m +12 generator φ of O odd (1 , m + 1) , we have stronggenerators ω a,b = 12 n X i =1 : ( ∂ a a i )( ∂ b b i ) : − : ( ∂ b a i )( ∂ a b i ) : −
12 : ( ∂ a φ )( ∂ b φ ) : , a, b ≥ , which have weight m + 1 + a + b . Not all of these are necessary, and it is easy to see thatthe subset { ω a +1 , | a ≥ } suffices to strongly generate. Since these fields have weights m + 2 , m + 4 , . . . , this proves the first statement. ext, the relation of minimal weight among these generators corresponds to the × (2 n +2) Young tableau with both rows consisting of , , . . . , n + 1 , so this relation has weight n )(1 + m + n ) . If there exists a decoupling relation for any of the generating fields, thelowest possible weight where this could occur is therefore n )(1 + m + n ) . (cid:3) Again, Corollary 6.1 implies that the relation of weight n )(1 + m + n ) is in fact adecoupling relation.5.9. On subalgebras of C ψiX ( n, m ) . Even though C ψiX ( n, m ) is of type W (2 , , . . . ) , it is notyet obvious that it can be obtained as a quotient of W ev ,I iX,n,m ( c, λ ) because it remains toshow that it is generated by the weights and fields. This will be shown in the nextsection, and the following weaker statement will be needed. Lemma 5.5.
For i = 1 , and X = B, C, D, O , C ψiX ( n, m ) is generated by the fields in weights , , . . . , m + 4 .Proof. It suffices to show that the free field limit has this property. In all cases, this limit hasthe form O ev (1 , ⊗ O ev (1 , ⊗ · · · ⊗ O ev (1 , m ) ⊗ A G , where A is a free field algebra and G is either O n +1 , Sp n , O n , or Osp | n . In all cases, itis straightforward to check that the fields in weights m + 2 and m + 4 are sufficient togenerate all the fields in higher weights m + 6 , m + 8 , . . . which strongly generate A G .The proof is similar to the proof of [L1, Lem. 4.2], and is omitted. (cid:3) Lemma 5.6.
For i = 1 , and X = B, C, D, O , let ˜ C ψiX ( n, m ) ⊆ C ψiX ( n, m ) be the subalgebragenerated by the weights and fields. Then ˜ C ψiX ( n, m ) is a one-parameter quotient of W ev ( c, λ ) for some ideal I iX,n,m .Proof. We prove this only ˜ C ψ D ( n, m ) since the proof in the other cases is similar. First, for n ≥ and m = 0 , C ψ D ( n, ∼ = Com ( V ψ − n +1 ( so n ) , V ψ − n +1 ( so n +1 )) Z , which is generated by the weights and fields and arises as a quotient of W ev ( c, λ ) [CKoL]. In particular, it coincides with ˜ C ψ D ( n, . Similarly, for n = 0 and m ≥ , C ψ D (0 , m ) ∼ = W k ( so m +1 ) so the same holds by [KL, Cor. 5.2].We assume next that n ≥ and m ≥ . Let { ω r | r ≥ } be the generators of C ψ D ( n, m ) given by Lemma 5.2 that correspond to the generators of the limiting algebra. Without lossof generality, we may assume that ω = L and W = ω , that is, ω has chosen to be primarywith respect to L and normalized as in [KL]. Set W r = W W r − , for r ≥ .For ≤ r ≤ m + n + 1)(2 n + 1) , we can write W r = λ r ω r + · · · , λ r ∈ C , where the remaining terms are normally ordered monomials in { L, ω s | ≤ s < r } . If λ r = 0 for all r , then ˜ C ψ ( n, m ) = C ψ ( n, m ) . Otherwise, let M ≥ be the first integersuch that λ M +1 = 0 . It is then apparent that { L, W , . . . , W M } close under OPE, so that ˜ C ψ D ( n, m ) is of type W (2 , , . . . , M ) . t suffices to prove that the generators { L, W r | ≤ r ≤ } satisfy the OPE relations of[KL]; equivalently, all Jacobi identities of type ( W a , W b , W c ) for a + b + c ≤ hold as aconsequence of (2.6)-(2.9) of [KL]. In this notation, W = L , as in [KL].By [CKoL, Thm. 2.1], this condition is automatic if the graded character of ˜ C ψ D ( n, m ) co-incides with that of W ev ( c, λ ) up to weight . Since the first relation among the generators { L, ω r | r ≥ } of C ψ D ( n, m ) and their derivatives occurs in weight m + n + 1)(2 n + 1) and n, m ≥ , there are no normally ordered relations in C ψ D ( n, m ) among these fields in weightbelow . Therefore the character of C ψ D ( n, m ) coincides with that of W ev ( c, λ ) in weightup to . If M ≥ , ˜ C ψ D ( n, m ) and C ψ D ( n, m ) have the same graded character up to weight , so the conclusion holds.Finally, suppose that M ≤ . Since λ M +1 = 0 and λ r = 0 for ≤ r ≤ M , there canbe no nontrivial normally ordered relations among the generators { L, W , . . . , W M } of ˜ C ψ D ( n, m ) , since this property holds for the corresponding fields { L, ω , . . . , ω M } . Equiva-lently, all Jacobi relations among { L, W , . . . , W M } of type ( W a , W b , W c ) , a + 2 b + 2 c ≤ M + 2 , must hold as a consequence of (2.6)-(2.9) of [KL] alone. Therefore the OPEs W i ( z ) W j ( w ) for i + 2 j ≤ M are the same as those of W I, ev ( c, λ ) for some ideal I ⊆ C [ c, λ ] .If we use the same procedure as the construction W ev ( c, λ ) given by [KL, Thm. 3.9],beginning with the fields L, W , . . . , W M and the OPEs W i ( z ) W j ( w ) for i + 2 j ≤ M ,we can formally define new fields W M +2 r = ( W ) r W M for all r ≥ , and then define theOPE algebra of all fields { L, W , . . . , W M , W M +2 r | r ≥ } recursively so that they are thesame as the OPEs in W I, ev ( c, λ ) . In particular, this realizes ˜ C ψ D ( n, m ) as a one-parameterquotient of W I, ev ( c, λ ) by some vertex algebra ideal I containing a field in weight M + 2 of the form W M +2 − P ( L, W , . . . , W M ) , where P is a normally ordered polynomial in L, W , . . . , W M and their derivatives. (cid:3) Corollary 5.1.
For n + m ≥ , W ψiX ( n, m ) is an extension of V t ( a ) ⊗ W , where W is a quotient of W ev ,I iX,n,m ( c, λ ) , for some ideal I iX,n,m ⊆ C [ c, λ ] .
6. P
ROOF OF MAIN RESULT
Step 1: Computation of truncation curves.
In this subsection, we shall compute theideals I iX,n,m ⊆ C [ c, λ ] such that ˜ C ψiX ( n, m ) is realized as a quotient of W ev ,I iX,n,m ( c, λ ) . Moreprecisely, we will parametrize the corresponding variety V ( I iX,n,m ) ⊆ C by giving a ratio-nal map Φ iX,n,m : C \ P → V ( I iX,n,m ) , Φ iX,n,m ( ψ ) = (cid:0) c ( ψ ) , λ ( ψ ) (cid:1) . Here P is the finite set consisting of poles c ( ψ ) and λ ( ψ ) . Note first that in the cases n = 0 , m ≥ , and X = C, D , there is nothing to prove because ˜ C ψ C (0 , m ) = C ψ C (0 , m ) = W ψ − m +1 ( so m +1 ) = ˜ C ψ D (0 , m ) = C ψ D (0 , m ) , ˜ C ψ C (0 , m ) = C ψ C (0 , m ) = W ψ − m − ( sp m ) = ˜ C ψ D (0 , m ) = C ψ D (0 , m ) . (6.1)The truncation curves for type B and C principal W -algebra already appear in [KL], andhave the desired form. The following cases must also be treated separately, and will bediscussed briefly at the end of this section. ˜ C ψ D (1 , m ) and ˜ C ψ D (1 , m ) , where a = so ,(2) ˜ C ψ B (0 , m ) , ˜ C ψ O (0 , m ) , ˜ C ψ B (0 , m ) , and ˜ C ψ O (0 , m ) where a = 0 . In these cases, W ψiX (0 , m ) isa simple current extension of C ψiX (0 , m ) or order two.In all other cases, a is simple and our approach will be uniform, and from now on we as-sume this to be the case. As in Corollary 5.1, we consider extensions of V t ( a ) ⊗ W by d a fields of weight d b +12 and appropriate parity, which transform in the standard representa-tion ρ a of a . We will show that this data uniquely determines the truncation curve for W ,or equivalently, the formula λ ( ψ ) .Let p be a vector in this copy of ρ a which is primary with respect the action of V t ( a ) . With-out loss of generality, we may take p to be a highest-weight vector in this representation of a . This forces the following OPEs:(6.2) L ( z ) p ( w ) ∼ (cid:18) µ − Cas t + h ∨ a (cid:19) p ( w )( z − w ) − + (cid:18) ∂p + · · · (cid:19) ( w )( z − w ) − . Here µ = d b +12 , Cas is the Casimir eigenvalue of the standard representation of a , and h ∨ a isthe dual Coxeter number of a . W ( z ) p ( w ) ∼ k p ( w )( z − w ) − + (cid:18) k ∂p + · · · (cid:19) ( w )( z − w ) − + (cid:18) k ∂ p + k : Lp : + · · · (cid:19) ( w )( z − w ) − + (cid:18) k ∂ p + k : ( ∂L ) p : + k : L∂p : + · · · (cid:19) ( w )( z − w ) − , (6.3)(6.4) W ( z ) p ( w ) ∼ k p ( w )( z − w ) − + (cid:18) k ∂p + · · · (cid:19) ( w )( z − w ) − + · · · , (6.5) W ( z ) p ( w ) ∼ k p ( w )( z − w ) − + · · · . We impose the following Jacobi identities L (2) ( W p ) − W ( L (2) p ) − ( L (0) W ) (3) p − L (1) W ) (2) p − ( L (2) W ) (1) p, (6.6) L (3) ( W p ) − W ( L (3) p ) − ( L (0) W ) (3) p − L (1) W ) (2) p − L (2) W ) (1) p − ( L (3) W ) (0) p, (6.7) L (4) ( W p ) − W ( L (4) p ) − ( L (0) W ) (3) p − L (1) W ) (3) p − L (2) W ) (2) p − L (3) W ) (1) p − ( L (4) W ) (0) p, (6.8) L (3) ( W p ) − W ( L (3) p ) − ( L (0) W ) (4) p − L (1) W ) (3) p − L (2) W ) (2) p − ( L (3) W ) (1) p, (6.9) L (2) ( W p ) − W ( L (2) p ) − ( L (0) W ) (4) p − L (1) W ) (3) p − ( L (2) W ) (2) p, (6.10) L (2) ( W p ) − W ( L (2) p ) − ( L (0) W ) (2) p − L (1) W ) (1) p − ( L (2) W ) (0) p, (6.11) W ( W p ) − W ( W p ) − ( W W ) (6) p, (6.12) ( W p ) − W ( W p ) − ( W W ) (6) p − W W ) (5) p − W W ) (4) p − W W ) (3) p − ( W W ) (2) p, (6.13) W ( W p ) − W ( W p ) − ( W W ) (6) p − ( W W ) (5) p, (6.14)(6.15) W ( W p ) − W ( W p ) − ( W W ) (8) p, (6.16) W ( W p ) − W ( W p ) − ( W W ) (8) p − ( W W ) (7) p. Note that (6.6) has weight µ +1 , and a computation shows that the coefficient of ∂p dependsonly on k , k , k together with the level t of a , and the parameters n, m . Similarly, (6.7) hasweight µ + 1 , and the coefficient of ∂p depends only on k , k , k , k together with t, n, m .Next, (6.8), (6.9), and (6.10) all have weight µ , and hence are scalar multiples of p ; theseequations depend only on k , . . . , k , together with t, n, m . Also, (6.11) has weight µ + 2 ,and the coefficient of ∂ p depends only on k , k , k together with t, n, m .Next, (6.12), (6.13), and (6.14) all have weight µ , and hence are scalar multiples of p ;these equations depend only on k , . . . , k , together with λ, t, n, m . Finally, (6.15) and (6.16),all have weight µ , and hence are scalar multiples of p ; these equations depend only on k , . . . , k , together with λ, t, n, m .Using the Mathematica package of Thielemans [T], we can solve these equations to ob-tain a unique solution for k , . . . , k and λ as functions of t, n, m . We then set t to be thelevel of the affine subalgebra V t ( a ) , which depends on ψ and n . Solving for λ in terms of ψ, n, m , and using the formulas for c = c ( λ ) appearing in Subsection 3.9, gives the explicitrational parametrizations Φ iX,n,m : C \ P → V ( I iX,n,m ) , Φ iX,n,m ( ψ ) = (cid:0) c ( ψ ) , λ ( ψ ) (cid:1) , for i = 1 , and X = B, C, D, O . The explicit formula for Ψ B,n,m ( ψ ) is given in AppendixA, but we do not give the others because as we shall see in the next section, all others canbe obtained from this one together with various symmetries.Finally, we comment on how this argument must be modified in the cases where a = so ,or a = 0 and we take a Z -orbifold. First, in the case ˜ C ψ D (1 , m ) , the affine subalgebra isa Heisenberg algebra H (1) , and we normalize the generator J so that the two fields p ± transforming as ρ a satisfy(6.17) J ( z ) p ± ( w ) ∼ ± ( z − w ) − . Then J satisfies(6.18) J ( z ) J ( w ) ∼ ( ψ − z − w ) − . We replace the level t in the above argument by ψ − , we replace (6.2) with(6.19) L ( z ) p ( w ) ∼ (cid:18) ψ + 2 mψ − m − ψ − (cid:19) p ( w )( z − w ) − + (cid:18) ∂p + · · · (cid:19) ( w )( z − w ) − , and we solve the same system of equations to determine λ .Next, in the case ˜ C ψ D (1 , m ) , if we normalize the Heisenberg field J so that (6.17) holds,we have(6.20) J ( z ) J ( w ) ∼ (1 − ψ )( z − w ) − . gain, we replace the level t by − ψ , we replace (6.2) with(6.21) L ( z ) p ( w ) ∼ (cid:18) ψ + 2 mψ − m ψ − (cid:19) p ( w )( z − w ) − + (cid:18) ∂p + · · · (cid:19) ( w )( z − w ) − , and we apply the same procedure.Finally, in the remaining cases ˜ C ψ B (0 , m ) , ˜ C ψ O (0 , m ) , ˜ C ψ B (0 , m ) , and ˜ C ψ O (0 , m ) , we let p bea generator of the simple current extension of C ψiX (0 , m ) of weight µ , and we replace (6.2)with(6.22) L ( z ) p ( w ) ∼ µp ( w )( z − w ) − + (cid:18) ∂p + · · · (cid:19) ( w )( z − w ) − . The variable t no longer appears, and rest of the argument is the same.6.2. Step 2: Symmetries of truncation curves.Theorem 6.1.
For m ≥ n ≥ and m + n ≥ , we have the following identities Φ B,n,m ( ψ ) = Φ O,n,m − n (cid:0) ψ (cid:1) = Φ B,m,n (cid:0) ψ ψ − (cid:1) , Φ C,n,m ( ψ ) = Φ C,n,m − n (cid:0) ψ (cid:1) = Φ C,m,n (cid:0) ψψ − (cid:1) , Φ D,n,m ( ψ ) = Φ D,n,m − n (cid:0) ψ (cid:1) = Φ O,m,n − (cid:0) ψ ψ − (cid:1) , Φ O,n,m ( ψ ) = Φ B,n,m − n (cid:0) ψ (cid:1) = Φ D,m +1 ,n (cid:0) ψ ψ − (cid:1) . (6.23) Proof.
The explicit formulas for Φ iX,n,m ( ψ ) in all cases can be computed using the approachin the previous subsection. These symmetries follow immediately from our formulas. (cid:3) It turns out that all eight functions Φ iX,n,m ( ψ ) can be expressed uniformly in terms ofone of them. From (6.23), it it clear that within each of the four triality classes, there is auniform expression, so what remains is to find an expression that relates the expressionsfrom different triality classes. The explicit formula for Φ B,n,m ( ψ ) appears in Appendix Aas (A.1). Here n, m are nonnegative integers, but if we are allowed to replace them withhalf-integers, we obtain the following. Theorem 6.2. (6.24) Φ O,n,m ( ψ ) = Φ B,n,m + (cid:0) ψ (cid:1) , (6.25) Φ D,n,m ( ψ ) = Φ B,n − ,m ( ψ ) , (6.26) Φ C,n,m ( ψ ) = Φ B,n + ,m + (cid:0) ψ (cid:1) . By (6.1), we can recover Φ B,n,m ( ψ ) , Φ D,n,m ( ψ ) , and Φ C,n,m ( ψ ) , from Φ O,n,m + n ( ψ ) , Φ D,n,m + n ( ψ ) ,and Φ C,n,n + m ( ψ ) , respectively. Together with (6.24) - (6.26), this shows that all functions Φ iX,n,m ( ψ ) for i = 1 , and X = B, C, D, O , can be recovered from these symmetries togetherwith the explicit formula (A.1) for Φ B,n,m ( ψ ) . .3. Step 3: Exhaustiveness.
The last step in the proof of Theorem 4.1 is to show that ˜ C ψiX ( n, m ) = C ψiX ( n, m ) for i = 1 , and X = B, C, D, O . The isomorphisms in Theorem 4.1then follow immediately from the symmetries in Theorem 6.1. For a particular value of ψ ∈ C , let ˜ C ψ,iX ( n, m ) and C ψ,iX ( n, m ) denote the simple quotients of these algebras.In view of Lemmas 5.5 and 5.6, it suffices to show that ˜ C ψiX ( n, m ) contains the strongfields of C ψiX ( n, m ) in weights , , . . . , m + 4 . We give the proof only for ˜ C ψ B ( n, m ) since theargument in the other cases is the same. The truncation curve (A.1) for ˜ C ψ B ( n, m ) and thetruncation curve for W s ( sp r ) , which appears in Appendix A of [KL], intersect at the point ( c, λ ) given in (A.2). This intersection gives rise to the following isomorphism:(6.27) ˜ C ψ, B ( n, m ) ∼ = W s ( sp r ) , ψ = 1 + 2 m − n m + 2 r ) , s = − ( r + 1) + 1 + 2 m + 2 r n + r ) . Note that s is a nondegenerate admissible level for b sp r whenever m + 2 r and n + r arecoprime. By Corollary 2.1, for ψ and r are sufficiently large, the universal algebra W s ( sp r ) has a singular vector in weight m + 1)( n + 1) , and no singular vector in lower weight.Also, by [KL, Rem. 5.3], W s ( sp r ) is generated by the weights and fields for all non-critical values of s , hence this holds for the simple quotient W s ( sp r ) as well. It follows that ˜ C ψ, B ( n, m ) contains all fields in weights , , . . . , m +1)( n +1) − , so it must coincide with C ψ, B ( n, m ) . Since this holds at infinitely many values of ψ and r , it holds for the universalobjects as well. This shows that ˜ C ψ B ( n, m ) = C ψ B ( n, m ) as one-parameter vertex algebras.Repeating this argument in the other cases completes the proof of Theorem 4.1.As a consequence of Theorem 4.1 and the minimal strong generating types for C ψ B ( n, m ) and C ψ D ( n, m ) given earlier, we immediately obtain Corollary 6.1.
For n + m ≥ , we have the following minimal strong generating types as one-parameter vertex algebras.(1) C ψ B ( n, m ) is of type W (2 , , . . . , n )(3 + 2 m + 2 n ) − ,(2) C ψ D ( n, m ) is of type W (2 , , . . . , m + n )(1 + 2 n ) − ,(3) C ψ O ( n, m ) is of type W (2 , , . . . , m )(1 + n ) − ,(4) C ψ O ( n, m ) is of type W (2 , , . . . , n )(1 + m + n ) − . A remarkable feature of the truncation curves is that their pairwise intersection pointsare all rational points. We expect, but do not prove, that these four families of curvesaccount for all nontrivial truncations of W ev ( c, λ ) ; an equivalent conjecture is also due toProch´azka [Pro3]. In Appendices B, C, and D, we will give the explicit classification of co-incidences between the simple quotients C ψ,iX ( n, m ) and the algebras W s ( sp r ) , W s ( so r ) Z ,and W s ( osp | r ) Z ; certain isomorphisms of this kind will be needed for our rationality re-sults in Section 7.6.4. Uniqueness and reconstruction.
The algebras W ψiX ( n, m ) satisfy a uniqueness theo-rem which is analogous to [CL3, Thm. 9.1 and Thm. 9.8]. Theorem 6.3.
For all n, m with n + m ≥ , i = 1 , , and X = B, C, D, O , the full OPE algebra of W ψiX ( n, m ) is determined completely from the structure of C ψiX ( n, m ) , the action of the Lie algebra a on the fields which transform as the standard representation ρ a , and the nondegeneracy conditionon these fields given by [CL3, Thm. 3.5] . In particular,
1) If A ψiX ( n, m ) is a one-parameter vertex algebra which extends V t ( a ) ⊗ C ψiX ( n, m ) by d a fields inconformal weight d b +12 of correct parity, which are primary with respect to V t ( a ) as well as thetotal Virasoro field, and satisfy the nondegeneracy condition, then A ψiX ( n, m ) ∼ = W ψiX ( n, m ) ,as one-parameter vertex algebras.(2) The same result holds if we specialize to a particular value of ψ , and replace A ψiX ( n, m ) and W ψiX ( n, m ) by their simple quotients A ψ,iX ( n, m ) and W ψ,iX ( n, m ) . In the cases where a is simple, the proof is the same as the proof of [CL3, Thm. 9.1] in thecase m > , and is omitted. In the cases W ψiD (1 , m ) where a = so , the affine subalgebra is aHeisenberg algebra H (1) , and we normalize the generator J such that (6.17) holds. By thesame argument as the proof of [CL3, Thm. 9.1] in the case m = 1 , all OPEs in W ψiD (1 , m ) areuniquely determined by the structure of C ψiD (1 , m ) and (6.17), (6.18), and (6.20).Finally, in the cases C ψ B (0 , m ) , C ψ O (0 , m ) , C ψ B (0 , m ) , and C ψ O (0 , m ) , a is zero and C ψiX (0 , m ) is just the Z -orbifold of C ψiX (0 , m ) . In these cases, the argument showing the uniquenessof order two simple current extensions of C ψiX (0 , m ) by one field in the appropriate weightand parity, is even easier and is left to the reader.7. R ATIONALITY RESULTS
By combining Theorem 4.1 with the theory of extensions of rational vertex superalgebras,we prove many new rationality results in this section.7.1.
Affine vertex superalgebras of osp | n . Among the most fundamental examples ofrational vertex algebras are the simple affine vertex algebras L k ( g ) at positive integer level k [FZ]. For Lie superalgebras, it is known that the only examples of lisse affine vertexsuperalgebras are L k ( osp | n ) for k ≥ [GK], but the rationality is only known for n = 1 [CFK]. In this case, it is a consequence of the fact that L k ( osp | ) is an extension of L k ( sp ) times a rational Virasoro algebra. This perspective generalizes naturally to the case n > ,where the Virasoro algebra is replaced by a principal W -algebra of type C . Theorem 7.1.
For all n, k ∈ Z ≥ , the vertex superalgebra L k ( osp | n ) is lisse and rational, and isan extension of L k ( sp n ) ⊗ W ℓ ( sp n ) , for ℓ = − ( n + 1) + k + n k +2 n .Proof. [KW5, Prop. 8.1 and 8.2] tells us that L k ( sp n ) embeds into L k ( osp | n ) if k is a posi-tive integer. Since L k ( sp n ) is rational, L k ( osp | n ) is completely reducible as a module for L k ( sp n ) . It follows that Com ( L k ( sp n ) , L k ( osp | n )) is simple by [CGN, Prop. 5.4]. ThereforeCom ( L k ( sp n ) , L k ( osp | n )) coincides with the simple quotient C ψ, C ( n, by [CL1, Thm.8.1], where k = − ( ψ + 2 n + 1) . By Corollary 4.1 together with Feigin-Frenkel duality,Com ( L k ( sp n ) , L k ( osp | n )) ∼ = W ℓ ( sp n ) , ℓ = − ( n + 1) + 1 + k + n k + 2 n , which is lisse and rational [Ar2]. We thus have that both L k ( osp | n ) and its even subalgebra L k ( osp | n ) Z are extensions of a lisse vertex algebra. This extension must be of finite index;otherwise, at least one of the finitely many irreducible modules of the lisse vertex algebramust appear with infinite multiplicity. This is impossible since conformal weight spacesof L k ( osp | n ) , and its even subalgebra L k ( osp | n ) Z , are finite dimensional. It follows thatboth these extensions are lisse. Rationality of L k ( osp | n ) Z follows from Proposition 2.2,and rationality of L k ( osp | n ) then follows from [CGN, Thm. 5.13]. (cid:3) .2. Rationality of W k ( osp | n ) . A celebrated result of Arakawa [Ar2] says that for a simpleLie algebra g , W ℓ ( g ) is lisse and rational when ℓ is a nondegenerate admissible level for b g .When g is simply-laced, recall from [ACL] that(7.1) Com ( L k +1 ( g ) , L k ( g ) ⊗ L ( g )) ∼ = W ℓ ( g ) , where ℓ = − h ∨ + k + h ∨ k + h ∨ + 1 . In particular, this realizes W ℓ ( g ) for all nondegenerate admissible levels ℓ .We consider the analogous diagonal coset for type B . First, if k is an admissible level for b so n +1 we have an embedding L k +1 ( so n +1 ) ֒ → L k ( so n +1 ) ⊗ L ( so n ) [KW2]. Additionally, L ( so n +1 ) acts on the free fermion algebra F (2 n + 1) , andCom ( L k +1 ( so n +1 ) , L k ( so n +1 ) ⊗ L ( so n +1 )) ∼ = Com ( L k +1 ( so n +1 ) , L k ( so n +1 ) ⊗ F (2 n + 1)) . In the notation of Theorem 4.1, recall the isomorphism C ψ B ( n, ∼ = Com ( V − ψ − n +2 ( so n +1 ) , V − ψ − n +1 ( so n +1 ) ⊗ F (2 n + 1)) Z ∼ = C ψ ′ B (0 , n ) ∼ = W ψ ′ − n − / ( osp | n ) Z , ψ + 1 ψ = 2 . (7.2)Suppose that the level − ψ − n + 1 is admissible for b so n +1 , that is, − ψ − n + 1 = − (2 n −
1) + pq , where p, q ∈ N are coprime and p ≥ n − if q is odd, and p ≥ n is q is even. In thiscase, by [CL1, Thm. 8.1 and Rem. 8.3] the simple quotient C ψ ′ , B (0 , n ) coincides withCom ( L k +1 ( so n +1 ) , L k ( so n +1 ) ⊗ F (2 n + 1)) Z , which we expect to be lisse and rational byanalogy with the simply-laced case. This motivates the following conjecture. Conjecture 7.1.
The principal W -superalgebra W ψ ′ − n − / ( osp | n ) where ψ ′ = p p + q ) , is lisseand rational if(1) p, q ∈ N are coprime,(2) p ≥ n − if q is odd,(3) p ≥ n if q is even.Moreover, we expect that W k ( osp | n ) is rational and lisse if and only if k = ψ ′ − n − or k = ψ ′ − n − , which is the Feigin-Frenkel dual level by (4.1).As in the case of W k ( g ) for a Lie algebra g , we expect that rational vertex superalgebras W k ( osp | n ) will serve as building blocks for many non-principal rational W -superalgebras.In the next subsection, we will give examples of subregular W -algebras of so m +3 and prin-cipal W -superalgebras of osp | n +2 with this property.Using the realization C ψ B (0 , m ) ∼ = W ψ − m − / ( osp | m ) Z , we are able to prove some casesof Conjecture 7.1 using the coincidences appearing in Appendices B and C. Theorem 7.2. (1) For k = − ( m + ) + m − m + r ) , W k ( osp | m ) is lisse and rational when m + r and r arecoprime.(2) For k = − ( m + ) + m m +2 r ) , W k ( osp | m ) is lisse and rational when r and m arecoprime. roof. By Theorem B.3, for k = − ( m + ) + m − m + r ) , we have C ψ, B (0 , m ) = W ψ − m − / ( osp | m ) Z ∼ = W s ( sp r ) , s = − ( r + 1) + 1 + 2 r m + r ) . Since s is a nondegenerate admissible level for b sp r , the first statement follows.Similarly, Theorem B.3 also shows that for k = − ( m + ) + m m +2 r ) , we have C ψ, B (0 , m ) = W ψ − m − / ( osp | m ) Z ∼ = W s ( sp r ) , s = − ( r + 1) + 1 + 2 m + 2 r r . Again, s is a nondegenerate admissible level for b sp r , so the second statement follows. (cid:3) We have a similar result coming from coincidences with algebras of the form W r ( so n ) Z . Theorem 7.3.
For k = − ( m + ) + m m +2 r − , W k ( osp | m ) is lisse and rational when r − and m are coprime.Proof. By Theorem C.3, for k = − ( m + ) + m m +2 r − we have C ψ, B (0 , m ) = W ψ − m − / ( osp | m ) Z ∼ = W s ( so r ) Z , s = − (2 r −
2) + 2 r − m + 2 r − . Since s is a nondegenerate admissible level for b so r , the claim follows. (cid:3) In the case of W k ( osp | ) , it was shown in [CFL] that the diagonal coset C r = Com ( V r +2 ( sl ) , V r ( sl ) ⊗ L ( sl )) is a quotient of W ev ( c, λ ) with parametrization c = 3 r (6 + r )2(2 + r )(4 + r ) , λ = − r + 2)( r + 4)( − − r − r + 132 r + 11 r )7( r − r + 8)(68 + 42 r + 7 r )(352 + 354 r + 59 r ) . A calculation shows that C r ∼ = C ψ B (0 ,
1) = W ψ − / ( osp | ) Z , where ψ and r are related by ψ = r r ) or ψ = r r ) . Since the simple quotient C r, is lisse and rational whenever r isadmissible for b sl , we obtain Theorem 7.4. W ψ − / ( osp | ) is lisse and rational when ψ = r r ) or ψ = r r ) , and r isadmissible for b sl . Subregular W -algebras of type B . Recall that W k ( so m +3 , f subreg ) for m ≥ is excep-tional in the sense of [AvE] when k = − (2 m + 1) + pq is admissible and q = 2 m + 2 or m + 1 ;see Table 1 of [AvE]. It is therefore lisse [Ar1], but the rationality is only known in the case m = 1 [F]. We will prove the rationality in all cases where q = 2 m + 2 , and in all caseswhere q = 2 m + 1 and p is odd. In the missing cases where q = 2 m + 1 and p is even, wewill see that rationality would follow from Conjecture 7.1.Recall that Com ( H (1) , W ψ − m − ( so m +3 , f subreg )) Z can be identified with C ψ, D (1 , m ) . Weset ψ = pq as above, and we begin with the case q = 2 m + 2 . Theorem 7.5.
For all ψ = m +2 r m +2 such that m +1 and r +1 are coprime, W ψ − m − ( so m +3 , f subreg ) is lisse and rational. roof. By Theorem B.2, we have C ψ, D (1 , m ) ∼ = Com ( H (1) , W ψ − m − ( so m +3 , f subreg )) Z ∼ = W s ( sp r ) , s = − ( r + 1) + 2 m + 2 r + 32(2 r + 1) . Note that the first isomorphism holds by [CL1, Thm. 8.1 and Rem. 8.3]. Under the abovearithmetic condition, s is a nondegenerate admissible level for b sp r , so C ψ, D (1 , m ) is lisseand rational. Therefore Com ( H (1) , W ψ − m − ( so m +3 , f subreg )) , being a simple current ex-tension of C ψ, D (1 , m ) is also lisse and rational. It then follows from Proposition 2.2 that W ψ − m − ( so m +3 , f subreg ) is lisse and rational as well. (cid:3) Recall from (4.3) in the case n = 1 thatCom ( H (1) , W ψ − m − ( so m +3 , f subreg )) Z ∼ = Com ( H (1) , W ψ ′ − m ( osp | m +2 )) Z , ψ ′ = 12 ψ . This is the Z -invariant part of the duality (4.7) proved in [CGN]. We obtain Corollary 7.1.
For ψ ′ = m m +2 r such that m + 1 and r + 1 are coprime, W ψ ′ − m ( osp | m +2 ) islisse and rational. The fact that W ψ ′ − m ( osp | m +2 ) is lisse was also pointed out in [CGN] as a consequenceof (4.7) together with the lisseness of W ψ − m − ( so m +3 , f subreg ) .Next, we consider the case where q = 2 m + 1 and p is odd. Theorem 7.6.
For ψ = m +2 r +12 m +1 such that r and m + 1 are coprime, W ψ − m − ( so m +3 , f subreg ) islisse and rational.Proof. By Theorem C.2 and [CL1, Thm. 8.1 and Rem. 8.3], we have C ψ, D (1 , m ) ∼ = Com ( H (1) , W ψ − m − ( so m +3 , f subreg )) Z ∼ = W s ( so r ) Z , s = − (2 r −
2) + 2 r m + 2 r + 1 . As above, s is a nondegenerate admissible level for b so r , so C ψ, D (1 , m ) is lisse and ra-tional. Therefore Com ( H (1) , W ψ − m − ( so m +3 , f subreg )) is also lisse and rational, and so is W ψ − m − ( so m +3 , f subreg ) . (cid:3) Corollary 7.2.
For ψ ′ = m +12(2 m +2 r +1) such that r and m + 1 are coprime, W ψ ′ − m ( osp | m +2 ) is lisseand rational. The missing cases where q = 2 m + 1 and p is even correspond to coincidences between C ψ, D (1 , m ) and algebras W s ( osp | r ) Z . By Theorem D.2, we have C ψ, D (1 , m ) ∼ = W s ( osp | r ) Z , ψ = 2( m + r + 1)2 m + 1 , s = − ( r + 12 ) + m + r + 11 + 2 r , which is expected to be rational by Conjecture 7.1. Therefore Conjecture 7.1 together withProposition 2.2 would imply the rationality of W ψ − m − ( so m +3 , f subreg ) in these cases.For r = 1 , we have C ψ, D (1 , m ) ∼ = W s ( osp | ) Z ∼ = C a, , where ψ = 2(2 + m )1 + 2 m , s = −
32 + 2 + m , a = − m . Since a is admissible for b sl , it follows from Theorem 7.4 that orollary 7.3. For ψ = m )1+2 m , W ψ − m − ( so m +3 , f subreg ) is lisse and rational. Similarly, for ψ ′ = m +14(2+ m ) , W ψ ′ − m ( osp | m +2 ) is lisse and rational. It is natural to ask whether the examples where k = − (2 m + 1) + pq is admissible and q = 2 m + 2 or m + 1 , account for all cases where W k ( so m +3 , f subreg ) is lisse and rational. Itturns out that this is not the complete list. For example, we have isomorphisms C ψ, D (1 , m ) ∼ = W s ( osp | ) Z ∼ = C a, ,ψ = 2 m m − , s = −
32 + m m − , a = 4 m − . (7.3)Since a is a positive integer for m > , we obtain Corollary 7.4.
For ψ = m m − , W ψ − m − ( so m +3 , f subreg ) is lisse and rational. Similarly, for ψ ′ = m − m , W ψ ′ − m ( osp | m +2 ) is lisse and rational.Remark . The examples in Corollary 7.4 fit into the third family of coincidences in Theo-rem D.2, namely, C ψ, D (1 , m ) ∼ = W s ( osp | r ) Z , ψ = 2( m − r + 1)1 + 2 m − r , s = − ( r + 12 ) + m + 1 − r m + 1 − r . Since m +1 − r m +1 − r = p + q p for p = 2 m − r + 1 and q = 1 , Conjecture 7.1 would imply that W s ( osp | r ) Z is lisse and rational whenever m ≥ r − , and hence that the followingalgebras are lisse and rational: W ψ − m − ( so m +3 , f subreg ) , ψ = 2( m − r + 1)1 + 2 m − r , m ≥ r − , W ψ ′ − m ( osp | m +2 ) , m − r m − r + 1) , m ≥ r − . (7.4)7.4. Minimal W -algebras of type C . Here we prove another case of the Kac-Wakimotorationality conjecture, which involves the minimal W -algebras W r − / ( sp n +2 , f min ) for allintegers r, n ≥ . Recall that in the case m = 1 , W ψ C ( n,
1) = W ψ − n − ( sp n +2 , f min ) , and hasaffine subalgebra V ψ − n − / ( sp n ) . If we specialize to the case ψ = n +2 r for r a positiveinteger, it was shown in [ACKL] that we have an induced embedding of simple vertexalgebras L r ( sp n ) → W r − / ( sp n +2 , f min ) . By [CL1, Thm. 8.1], the cosetCom ( L r ( sp n ) , W r − / ( sp n +2 , f min )) is simple and coincides with the simple quotient C ψ, C ( n, of C ψ C ( n, . Theorem 7.7.
For all n ≥ and r ≥ , W r − / ( sp n +2 , f min ) is lisse and rational, and is anextension of L r ( sp n ) ⊗ W s ( sp r ) for s = − ( r + 1) + n + r n +2 r .Proof. By Theorem B.4 and [CL1, Thm. 8.1], for ψ = n +2 r and r a positive integer wehave(7.5) C ψ, C ( n, ∼ = Com ( L r ( sp n ) , W r − / ( sp n +2 , f min ) ∼ = W s ( sp r ) , s = − ( r +1)+ 1 + n + r n + 2 r . Since s is a nondegenerate admissible level for W s ( sp r ) , C ψ, C ( n, is lisse and rational.Therefore W r − / ( sp n +2 , f min ) is an extension of L r ( sp n ) ⊗ W s ( sp r ) , and hence is also lisseand rational by Proposition 2.2. (cid:3) he isomorphism (7.5) was first conjectured in [ACKL], and was shown in [KL] to beequivalent to the explicit truncation curve; see Conjecture 7.4 of [KL]. This curve is alsogiven by Φ C,n,m ( ψ ) in the case m = 1 , which can be obtained from the formula (A.1) for Φ B,n,m ( ψ ) , together with (6.23) and (6.26). Remark . In the case r = 0 and ψ = n , C ψ, C ( n, is the simple quotient of W I, ev ( c, λ ) ,where I is the maximal ideal generated by c and ( λ + n +12 n n +12 n ) ) . Using (3.6) and (3.8) of[KL] and the recursive structure of the OPE algebra of W ev ( c, λ ) , it is not difficult to checkthat the generators L and W of W I, ev ( c, λ ) lie in the maximal proper ideal, so C ψ, C ( n, ∼ = C . Therefore Theorem 7.7 holds for r = 0 as well. This provides an alternative proof of thefact that L ( sp n ) ֒ → W − / ( sp n +2 , f min ) is a conformal embedding for all n ≥ [AKMPP].7.5. Cosets of type C . It is a longstanding conjecture that if
A ⊆ V are both lisse and ratio-nal vertex algebras, the coset C = Com ( A , V ) is also lisse and rational. This is a theorem if A is a lattice vertex algebra [CKLR], but otherwise is it known only in isolated examples. Infact, there are even more general situations where coset vertex algebras can be lisse and ra-tional. For example, (7.1) implies that when g is simply-laced, Com ( L k +1 ( g ) , L k ( g ) ⊗ L ( g )) is lisse and rational for all admissible levels k . We expect the following generalization ofthis statement to hold. Conjecture 7.2.
Let g be a simple, finite-dimensional Lie algebra, r a positive integer, and k an admissible level for b g . Then the coset Com ( L k + r ( g ) , L k ( g ) ⊗ L r ( g )) is lisse and rational.This is known for all admissible levels k in the special case g = sl and r = 2 [Ad, CFL],and also when k is a positive integer and r = 2 in the case of E [Lin] . The next resultgives another special case and will be useful for proving the rationality of other interestingcosets later. Theorem 7.8.
For k ∈ Z ≥ , the cosetCom ( L k − ( sp n ) , L k ( sp n ) ⊗ L − ( sp n )) ∼ = W ℓ ( sp k ) , with ℓ = − ( k + 1) + n + k n +2 k . In particular, this coset is lisse and rational.Proof. Note that [KW1, Cor. 4.1] tells us that L k − ( sp n ) embeds into L k ( sp n ) ⊗ L − ( sp n ) if k is a positive integer. Hence we get that Com ( L k − ( sp n ) , L k ( sp n ) ⊗ L − ( sp n )) is simpleas well, again by [CGN, Prop. 5.4], which applies since L k ( sp n ) ⊗ L − ( sp n )) is an ordinarymodule for L k − ( sp n ) , and that category is completely reducible [Ar3]. Thus by TheoremB.4 and [CL1, Thm. 8.1]Com ( L k − ( sp n ) , L k ( sp n ) ⊗ S ( n )) ∼ = W ℓ ( sp k ) , ℓ = − ( k + 1) + 1 + n + k n + 2 k . The vertex algebra S ( n ) decomposes as S ( n ) ∼ = L − ( sp n ) ⊕ L − ( ω ) with ω the firstfundamental weight of sp n , i.e., the top level of L − ( ω ) is the standard representationof sp n . Since ω is not in the root lattice of sp n , L k − ( sp n ) cannot be a submodule of L k ( sp n ) ⊗ L − ( ω ) . It follows thatCom ( L k − ( sp n ) , L k ( sp n ) ⊗ L − ( sp n )) ∼ = Com ( L k − ( sp n ) , L k ( sp n ) ⊗ S ( n )) , which completes the proof. (cid:3) Note that the argument of [Lin] for admissible k also applies if one uses [CKM2, Thm. 5.5] he category of ordinary modules of L k − ( sp n ) is semisimple [Ar3] and we denote by P k the set of weights such that L k − ( λ ) is an ordinary module for L k − ( sp n ) . We have(7.6) L k ( sp n ) ⊗ L − ( sp n ) ∼ = M λ ∈ P k ∩ Q L k − ( λ ) ⊗ M ( λ ) . Here each multiplicity space M ( λ ) is either a direct sum of W ℓ ( sp k ) -modules or zero. Infact M ( λ ) can only be non-zero if λ is in the root lattice Q of sp n and so we restrict the sumto P k ∩ Q . Finally, M (0) ∼ = W ℓ ( sp k ) .Let f min be a minimal nilpotent element. The minimal reduction functor H k,f min at level k , see (2.9), has the property that for an irreducible highest weight module L k ( λ ) of theaffine vertex algebra of g , the reduction H k,f min ( L k ( λ )) is an irreducible ordinary module ofthe minimal W -algebra W k ( g , f min ) as long as k is not a positive integer [Ar4]. We aim todetermine the λ , such that H k,f min ( L k − ( λ )) ∼ = W k − ( sp n , f min ) . Lemma 7.1.
For g = sp n and k ∈ Z ≥ , H k,f min ( L k − ( λ )) ∼ = W k − ( sp n , f min ) implies λ = mω with m ∈ { , k + 1 } .Proof. The minimal W -algebra has an affine subalgebra of type sp n − . The top level of H k,f min ( L k − ( λ )) is described in [Ar4, (66)] (see also [KW3, (6.14)]), and has highest-weight λ restricted to the Cartan subalgebra of sp n − . The conformal weight of the top level isthe conformal weight of the top level of L k − ( λ ) minus λ ( x ) , where x is in the Cartansubalgebra of the sl -triple for the quantum Hamiltonian reduction; see Section 2.2.In the case of g = sp n , we embed the root system as usual in Z n with orthonormal basis { ǫ , . . . , ǫ n } . Then simple positive roots are α = √ ( ǫ − ǫ ) , . . . , α n − = √ ( ǫ n − − ǫ n ) , α n = √ ǫ n . The longest short co-root is θ ∨ s = √ ǫ + ǫ ) and the Weyl vector is ρ = √ ( nǫ +( n − ǫ + · · · + ǫ n ) . The sl -triple corresponds to the longest root, that is √ ǫ and so itfollows that the top level of H k,f min ( L k − ( λ )) has sp n − weight zero if and only if λ = mω is a multiple of the first fundamental weight ω = ǫ √ . Moreover λ is an admissible weightif and only if λθ ∨ s ≤ k + 2 n + 1 − h = 2 k + 1 with h = 2 n the Coxeter number of sp n . Theconformal weight of the top level is for λ = mω , λ ( λ + 2 ρ )2 k + 2 n + 1 − λω = m (cid:18) m + 2 n k + 2 n + 1 − (cid:19) , i.e. it vanishes if either m = 0 or m = 2 k + 1 . (cid:3) Recall from Remark 7.2 that H − ,f min ( L − ( sp n )) ∼ = C . Applying Theorem 2.2 to (7.6) with L = L k ( sp n ) and V = L − ( sp n ) yields L k ( sp n ) ∼ = L k ( sp n ) ⊗ C ∼ = L k ( sp n ) ⊗ H − ,f min ( L − ( sp n )) ∼ = H k − ,f min ( L k ( sp n ) ⊗ ( L − ( sp n )) ∼ = M λ ∈ P k ∩ Q H k − ,f min ( L k − ( λ )) ⊗ M ( λ ) . (7.7)Since (2 k + 1) ω is not in the root lattice Q of sp n , we can use Lemma 7.1 to conclude thatCom ( W k − ( sp n , f min ) , L k ( sp n )) ∼ = W ℓ ( sp k ) , ℓ = − ( k + 1) + 1 + n + k n + 2 k , ecall Theorem 7.7 saying thatCom ( L k ( sp n − ) , W k − ( sp n , f min ) ∼ = W s ( sp k ) , s = − ( k + 1) + n + k n + 2 k . We can thus employ Corollary 2.3 with V = L k ( sp n ) , W = W ℓ ( sp k ) , W = W k − ( sp n , f min ) ,W = W s ( sp k ) and L = L k ( sp n − ) to conclude that Corollary 7.5.
For k ∈ Z ≥ and n ∈ Z ≥ the coset Com ( L k ( sp n − ) , L k ( sp n )) is lisse and rationaland is an extension of W ℓ ( sp k ) ⊗W s ( sp k ) with ℓ = − ( k +1)+ n + k n +2 k and s = − ( k +1)+ n + k n +2 k .Remark . By Remark 2.1 the embedding of L k ( sp n − ) in L k ( sp n ) is the standard onedescribed in Remark 2.2 (and m = 1 ).The standard coset conformal vector of the coset Com ( L k ( sp n − ) , L k ( sp n )) is the dif-ference of the Sugawara vectors of L k ( sp n ) and L k ( sp n − ) . Note that the conformal vec-tor of W ℓ ( sp k ) ⊗ W s ( sp k ) is not the standard coset conformal vector. The contragredientdual and being of CFT-type depend on the choice of a conformal vector, e.g. the cosetCom ( L k ( sp n − ) , L k ( sp n )) is self-contragredient and of CFT-type with the standard cosetconformal vector. On the other hand, neither the lisse nor rationality properties depend ona choice of conformal vector.We can iterate, i.e. apply Corollary 2.3 with V = L k ( sp n ) , L = L k ( sp n − m +1) ) , W = Com ( L k ( sp n − m ) , L k ( sp n )) , W = L k ( sp n − m ) , W = Com ( L k ( sp n − m +1) ) , L k ( sp n − m )) .Then the induction hypothesis is that W = Com ( L k ( sp n − m ) , L k ( sp n )) is rational andlisse. The base case has just been proven and the induction step is again Corollary 2.3. Corollary 7.6.
For k ∈ Z ≥ , n, m ∈ Z ≥ and n > m the coset Com ( L k ( sp n − m ) , L k ( sp n )) islisse and rational and is an extension of (7.8) m O i =1 ( W ℓ i ( sp k ) ⊗ W s i ( sp k )) with ℓ i = − ( k + 1) + n − i + k n − i +2 k and s i = − ( k + 1) + n − i + k n − i +2 k . As above, the embedding of L k ( sp n − m ) in L k ( sp n ) is the standard one described inRemark 2.2.This coset is isomorphic to another interesting coset via level-rank duality. For this weuse that L k ( sp n ) and L n ( sp k ) form a commuting pair in F (4 nk ) [KP, Prop. 2]; a detailedproof is given in the appendix of [ORS]. We can thus apply the idea of the proof of [ACL,Thm. 13.1], namelyCom ( L k ( sp n − m ) , L k ( sp n )) ∼ = Com ( L k ( sp n − m ) , Com ( L n ( sp k ) , F (4 nk ))) ∼ = Com ( L k ( sp n − m ) ⊗ L n ( sp k ) , F (4 nk )) ∼ = Com ( L n ( sp k ) , Com ( L k ( sp n − m ) , F (4 nk ))) ∼ = Com ( L n ( sp k ) , Com ( L k ( sp n − m ) , F (4( n − m ) k )) ⊗ F (4 mk )) ∼ = Com ( L n ( sp k ) , L n − m ( sp k ) ⊗ F (4 mk )) . Corollary 7.6 thus gives us orollary 7.7. For k, n, m ∈ Z ≥ and n > m , the coset Com ( L n ( sp k ) , L n − m ( sp k ) ⊗ F (4 mk )) is lisse and rational and is an extension of m O i =1 ( W ℓ i ( sp k ) ⊗ W s i ( sp k )) with ℓ i = − ( k + 1) + n − i + k n − i +2 k and s i = − ( k + 1) + n − i + k n − i +2 k . Gelfand-Tsetlin algebras in types B , C and D . In [ACL], it was shown that for all k, n ∈ Z ≥ , the coset Com ( L k ( gl n − ) , L k ( gl n )) is isomorphic to W ℓ ( gl k ) for ℓ = − k + k + n − k + n .Iterating this construction shows that L k ( gl n ) is an extension of N ni =1 W ℓ i ( gl k ) with ℓ i = − k = k + n − ik + n − i +1 . This was regarded in [ACL] as a noncommutative, affine analogue of theGelfand-Tsetlin subalgebra Γ of U ( gl n ) . Even though N ni =1 W ℓ i ( gl k ) is noncommutative, itsZhu algebra is commutative, and it maps to Γ via the Zhu functor [Z].For types B and D , a similar observation appears in [CKoL]. For k, n ∈ Z , the cosets(7.9) D k ( n ) = Com ( L k ( so n ) , L k ( so n +1 )) Z , E k ( n ) = Com ( L k ( so n +1 ) , L k ( so n +2 )) Z , were called generalized parafermion algebras of orthogonal types. Then L k ( so n +2 ) and L k ( so n +1 ) are extensions 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 ) , H ⊗ D k (1) ⊗ E k (1) ⊗ D k (2) ⊗ E k (2) ⊗ · · · ⊗ D k ( n − ⊗ E k ( n − ⊗ D k ( n ) . (7.10)The algebras (7.10) are also analogues of the Gelfand-Tsetlin algebra, and they also havecommutative Zhu algebras. Since D k ( m ) ∼ = C k +2 m − , D ( m, and E k ( m ) ∼ = C k +2 m, B ( m, , itfollows from Theorems C.1 and C.2 that for r ∈ N and k = 2 r , D r ( m ) ∼ = W s ( so r ) Z , s = − (2 r −
2) + 2 m + 2 r − m + 2 r − , E r ( m ) ∼ = W s ( so r ) Z , s = − (2 r −
2) + 2 m + 2 r − m + 2 r . (7.11)In particular, if k = 2 r is even, the Gelfand-Tsetlin subalgebras of L r ( so n +1 ) and L r ( so n +2 ) are tensor products of rational vertex algebras of the form W s ( so r ) Z .Similarly, for r ∈ N and k = 2 r + 1 , it follows from Theorems D.1 and D.2 that D r +1 ( m ) ∼ = W s ( osp | r ) Z , s = − ( r + 12 ) + m + r m + 2 r − , E r +1 ( m ) ∼ = W s ( osp | r ) Z , s = − ( r + 12 ) + m + r m + 2 r + 1 . (7.12)So if k = 2 r + 1 is odd, the Gelfand-Tsetlin subalgebras of L r +1 ( so n +1 ) and L r +1 ( so n +2 ) are tensor products of algebras of the form W s ( osp | r ) Z , which are expected to be rationalby Conjecture 7.1.In type C , it follows from Corollary 7.5 that for all k, n ∈ Z ≥ , L k ( sp n ) is an extension ofthe rational vertex algebra n O i =1 ( W ℓ i ( sp k ) ⊗ W s i ( sp k )) with ℓ i = − ( k + 1) + n − i + k n − i +2 k and s i = − ( k + 1) + n − i + k n − i +2 k . Again, we regard this as aGelfand-Tsetlin subalgebra of L k ( sp n ) , and its Zhu algebra is commutative. PPENDIX
A. E
XPLICIT TRUNCATION CURVE FOR C ψ B ( n, m ) Here we give the explicit parametrization of the truncation curve for C ψ B ( n, m ) . For all n, m with n + m ≥ , C ψ B ( n, m ) ∼ = W ev I B,n,m ( c, λ ) , where the ideal I B,n,m is described explicitly via the parametrization c B,n,m ( ψ ) = − ( − m + n − ψ + 2 mψ )(1 − m + 2 n + 4 mψ )( − − m + 2 n + 2 ψ + 4 mψ )2 ψ (2 ψ − ,λ B,n,m ( ψ ) = − ψ (2 ψ − f − m + n + ψ + 2 mψ )( − − m + 2 n + 4 mψ )(1 − m + 2 n − ψ + 4 mψ ) gh ,f = − m + 80 m − m + 19 n − m n + 80 m n + 240 mn − m n − n + 160 m n − mn + 16 n + 49 ψ + 114 mψ − m ψ − m ψ + 160 m ψ − nψ + 728 mnψ + 1440 m nψ − m nψ − n ψ − mn ψ + 960 m n ψ + 160 n ψ − m n ψ + 160 mn ψ − ψ − mψ + 2184 m ψ + 2240 m ψ − m ψ + 228 nψ − mnψ − m nψ + 1920 m nψ + 728 n ψ + 1920 mn ψ − m n ψ − n ψ + 640 m n ψ + 392 ψ + 760 mψ − m ψ − m ψ + 1280 m ψ − nψ + 2912 mnψ + 5760 m nψ − m nψ − mn ψ + 1280 m n ψ − ψ − mψ + 2912 m ψ + 5120 m ψ − m ψ + 304 nψ − m nψ + 1280 m nψ + 608 mψ − m ψ + 512 m ψ ,g = − m − mn + 4 n + 14 ψ − m ψ + 16 mnψ − ψ + 16 m ψ ,h = 5 m − m − n + 60 m n − mn + 20 n + 49 ψ − mψ + 120 m ψ + 10 nψ − m nψ + 120 mn ψ − ψ + 40 mψ − m ψ − nψ + 240 m nψ − mψ + 160 m ψ . (A.1)Using this explicit parametrization (A.1) as well as the truncation curve for W s ( sp r ) which appears in Appendix A of [KL], it is easy to verify that these curves intersect at the oint ( c, λ ) given by c = − r ( − − m + 4 n − mr + 4 nr )(1 + 2 m + 2 n + 2 r − mr + 4 nr )2( n + r )(1 + 2 m + 2 r ) ,λ = − n + r )(1 + 2 m + 2 r ) f r )(2 n + r − mr + 2 nr )(1 + 2 m − mr + 4 nr ) gh ,f = − n − mn − m n − m n + 136 n + 544 mn + 544 m n + 96 n + 192 mn − r − mr − m r + 64 m r + 304 m r − nr − mnr − m nr − m nr + 92 n r − mn r − m n r + 1824 n r + 3264 mn r − n r − r − mr − m r + 608 m r − nr − mnr − m nr + 4832 m nr + 640 n r − mn r − m n r + 4176 n r + 6432 mn r − n r − r − mr + 2368 m r + 2272 m r − m r − nr − mnr + 928 m nr + 3840 m nr + 2240 n r − mn r − m n r + 1312 n r + 2560 mn r − n r − r + 608 mr + 2912 m r − m r − nr − mnr + 1600 m nr − m nr − n r + 640 mn r + 1920 m n r − n r − mn r + 640 n r + 608 mr − m r − m r + 256 m r − nr + 2432 mnr + 384 m nr − m nr − n r − mn r + 1536 m n r + 128 n r − mn r + 256 n r g = − − m − m + 14 n + 28 mn − n − r − mr − nr − mnr + 16 n r − r + 16 m r − mnr + 16 n r ,h = − n − mn − r − mr − m r − nr + 20 mnr + 40 n r − r − mr + 40 nr − mnr + 120 n r − mr + 80 m r + 40 nr − mnr + 80 n r . (A.2)In the next three Appendices, we classify coincidences between the simple quotients C ψ, B ( n, m ) , C ψ, D ( n, m ) , C ψ, B ( n, m ) , and C ψ, C ( n, m ) and the algebras W s ( sp r ) , W s ( so r ) Z ,and W s ( osp | r ) Z . There are coincidences at central charges c = 0 , , − , − , , where thealgebra degenerate; see [KL, Thm. 8.1]. Aside from these points, all additional coincidencecorrespond to intersection points on the truncation curves. This follows from [KL, Cor.8.2], together with a case-by-case analysis to rule out possible additional coincidences atpoints where the formula for λ is not defined. The details are omitted since the argument issimilar to the proof of special cases appearing in Section 9 of [KL]. Via our triality results,similar coincidences can be found for C ψ, C ( n, m ) , C ψ, D ( n, m ) , C ψ, O ( n, m ) , and C ψ, O ( n, m ) and these are also omitted.A PPENDIX
B. C
OINCIDENCES WITH TYPE C PRINCIPAL W - ALGEBRAS
Theorem B.1. ( Type 1B ) We have the following coincidences. C ψ, B ( n, m ) ∼ = W s ( sp r ) , for m, n ≥ and r ≥ .(1) ψ = 1 + m + n + r m , s = − ( r + 1) + 1 + m + n + r n + r ) , ψ = 2( m + n )1 + 2 m + 2 r , s = − ( r + 1) + 1 − n + 2 r m + 2 r ) ,(3) ψ = 1 + 2 m + 2 n + 2 r m , s = − ( r + 1) + 1 + 2 n + 2 r m + 2 n + 2 r ) ,(4) ψ = 1 + m + n m + r , s = − ( r + 1) + 1 + m + r r − n ) ,(5) ψ = 2( m + n − r )1 + 2 m − r , s = − ( r + 1) + r − m − n r − m − ,(6) ψ = 1 + 2 m + 2 n − r m − r ) , s = − ( r + 1) + r − m r − m − n − . Theorem B.2. ( Type 1D ) We have the following coincidences C ψ, D ( n, m ) ∼ = W s ( sp r ) , for m, n ≥ and r ≥ .(1) ψ = m + n + rm , s = − ( r + 1) + n + r m + n + r ) .(2) ψ = 2 m + 2 n − m + 2 r + 1 , s = − ( r + 1) + 1 − n + r m + 2 r ,(3) ψ = 1 + 2 m + 2 n + 2 r m ) , s = − ( r + 1) + 1 + 2 m + 2 n + 2 r r + 2 n − .(4) ψ = 1 + 2 m + 2 n m + r ) , s = − ( r + 1) + 1 + m + r − n + 2 r ,(5) ψ = 2 m + 2 n − r − m − r + 1 , s = − ( r + 1) + 1 − m − n + 2 r r − m − ,(6) ψ = m + n − rm − r , s = − ( r + 1) + r − m r − m − n ) . Theorem B.3. ( Type 2B ) We have the following coincidences C ψ, B ( n, m ) ∼ = W s ( sp r ) , for m, n ≥ and r ≥ .(1) ψ = 1 + 2 m − n + 2 r m ) , s = − ( r + 1) + 1 + 2 m − n + 2 r r − n ) ,(2) ψ = 1 + 2 m − n m + 2 r ) , s = − ( r + 1) + 1 + 2 m + 2 r n + r ) ,(3) ψ = m − n + r m − , s = − ( r + 1) + 1 − n + 2 r m − n + r ) ,(4) ψ = m − n − r m − r − , s = − ( r + 1) + 1 − m + 2 r n − m + r ) ,(5) ψ = 2 m − n − r − m − r ) , s = − ( r + 1) + 1 − m + 2 n + 2 r r − m ) ,(6) ψ = 2 m − n − m + r ) , s = − ( r + 1) + 1 + 2 n + 2 r m + r ) . heorem B.4. ( Type 2C ) We have the following coincidences C ψ, C ( n, m ) ∼ = W s ( sp r ) , for m, n ≥ and r ≥ .(1) ψ = 1 + m + n + r m , s = − ( r + 1) + 1 + m + n + r n + 2 r ,(2) ψ = 1 + m + n m + 2 r , s = − ( r + 1) + 1 + 2 m + 2 r r − n − ,(3) ψ = 1 + 2 m + 2 n + 2 r m − , s = − ( r + 1) + 1 + n + r m + 2 n + 2 r ,(4) ψ = m + n m + r ) , s = − ( r + 1) + r − n m + r ) ,(5) ψ = m + n − r m − r ) , s = − ( r + 1) + r − m − n r − m ) ,(6) ψ = 1 + 2 m + 2 n − r m − r − , s = − ( r + 1) + 1 − m + 2 r r − m − n − . A PPENDIX
C. C
OINCIDENCES WITH ORBIFOLDS OF TYPE D PRINCIPAL W - ALGEBRAS
Theorem C.1. ( Type 1B ) We have the following coincidences C ψ, B ( n, m ) ∼ = W s ( so r ) Z , for m, n ≥ and r ≥ .(1) ψ = 2( m + n + r )1 + 2 m , s = − (2 r −
2) + 2 n + 2 r − m + n + r ) .(2) ψ = 1 + 2 m + 2 n m + r ) , s = − (2 r −
2) + 2 r − n − m + r ) ,(3) ψ = 1 + m + n − r m − r , s = − (2 r −
2) + r − m − n − r − m − , Theorem C.2. ( Type 1D ) We have the following coincidences C ψ, D ( n, m ) ∼ = W s ( so r ) Z , for m, n ≥ and r ≥ .(1) ψ = 2 m + 2 n + 2 r −
11 + 2 m , s = − (2 r −
2) + 2( n + r − m + 2 n + 2 r − .(2) ψ = m + nm + r , s = − (2 r −
2) + r − nm + r ,(3) ψ = 1 + 2 m + 2 n − r m − r ) , s = − (2 r −
2) + 2 r − m − n − r − m − . Theorem C.3. ( Type 2B ) We have the following coincidences C ψ, B ( n, m ) ∼ = W s ( so r ) Z , for m, n ≥ and r ≥ .(1) ψ = 2 m − n + 2 r − m , s = − (2 r −
2) + 2 r − n − m − n + 2 r − , ψ = 1 + 2 m − n − r m − r ) , s = − (2 r −
2) + 2 r − m − n + 2 r − m − ,(3) ψ = m − n m + 2 r − , s = − (2 r −
2) + 2 n + 2 r − m + 2 r − . Theorem C.4. ( Type 2C ) We have the following coincidences C ψ, C ( n, m ) ∼ = W s ( so r ) Z , for m, n ≥ and r ≥ .(1) ψ = m + n + r m , s = − (2 r −
2) + n + rm + n + r ,(2) ψ = 1 + 2 m + 2 n m + 2 r − , s = − (2 r −
2) + 2( r − n − m + 2 r − ,(3) ψ = 1 + m + n − r m − r , s = − (2 r −
2) + 2( r − m − n − r − m − , A PPENDIX
D. C
OINCIDENCES WITH ORBIFOLDS OF W s ( osp | r ) Z Theorem D.1. ( Type 1B ) We have the following coincidences C ψ, B ( n, m ) ∼ = W s ( osp | r ) Z , for m, n ≥ and r ≥ .(1) ψ = 1 + 2 m + 2 n + 2 r m , s = − ( r + 12 ) + n + r m + 2 n + 2 r ,(2) ψ = 1 + 2 m + 2 n m + 2 r , s = − ( r + 12 ) + r − n m + 2 r ,(3) ψ = 1 + 2 m + 2 n − r m − r , s = − ( r + 12 ) + 2 r − m − n − r − m − . Theorem D.2. ( Type 1D ) We have the following coincidences C ψ, D ( n, m ) ∼ = W s ( osp | r ) Z , for m, n ≥ and r ≥ .(1) ψ = 2( m + n + r )1 + 2 m , s = − ( r + 12 ) + m + n + r n + 2 r − ,(2) ψ = 2( m + n )1 + 2 m + 2 r , s = − ( r + 12 ) + 1 − n + 2 r m + 2 r ) ,(3) ψ = 2( m + n − r )1 + 2 m − r , s = − ( r + 12 ) + r − m − n r − m − . Theorem D.3. ( Type 2B ) We have the following coincidences C ψ, B ( n, m ) ∼ = W s ( osp | r ) Z , for m, n ≥ and r ≥ .(1) ψ = m − n + r m , s = − ( r + 12 ) + r − n m − n + r ) ,(2) ψ = m − n − r m − r ) , s = − ( r + 12 ) + r − m n − m + r ) , ψ = m − n m + r ) , s = − ( r + 12 ) + n + r m + r ) . Theorem D.4. ( Type 2C ) We have the following coincidences C ψ, C ( n, m ) ∼ = W s ( osp | r ) Z , for m, n ≥ and r ≥ .(1) ψ = 1 + 2 m + 2 n + 2 r m , s = − ( r + 12 ) + 1 + 2 n + 2 r m + 2 n + 2 r ) ,(2) ψ = 1 + 2 m + 2 n m + r ) , s = − ( r + 12 ) + m + r r − n − ,(3) ψ = 1 + 2 m + 2 n − r m − r ) , s = − ( r + 12 ) + r − m r − m − n − . Corollary D.1.
All isomorphisms W k ( osp | m ) Z ∼ = W ℓ ( osp | n ) Z occur in the following list: k = − ( m + 12 ) + m + n m , k = − ( m + 12 ) + m m + n ) ,ℓ = − ( n + 12 ) + m + n n , ℓ = − ( n + 12 ) + n m + n ) . (D.1) This has central charge c = − (1 + 2 m )(1 + 2 n )(2 mn − m − n )2( m + n ) . R EFERENCES [Ad] D. Adamovic,
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NIVERSITY OF A LBERTA
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