Duality of subregular W-algebras and principal W-superalgebras
aa r X i v : . [ m a t h . QA ] M a y IPMU-20-0062
DUALITY OF SUBREGULAR W -ALGEBRAS AND PRINCIPAL W -SUPERALGEBRAS THOMAS CREUTZIG, NAOKI GENRA, AND SHIGENORI NAKATSUKA
Abstract.
We prove Feigin-Frenkel type dualities between subregular W -algebras of type A, B and principal W -superalgebras of type sl (1 | n ) , osp (2 | n ).The type A case proves a conjecture of Feigin and Semikhatov.Let ( g , g ) = ( sl n +1 , sl (1 | n + 1)) or ( so n +1 , osp (2 | n )) and let r be thelacity of g . Let k be a complex number and ℓ defined by r ( k + h ∨ )( ℓ + h ∨ ) = 1with h ∨ i the dual Coxeter numbers of the g i . Our first main result is that theHeisenberg cosets C k ( g ) and C ℓ ( g ) of these W -algebras at these dual levelsare isomorphic, i.e. C k ( g ) ≃ C ℓ ( g ) for generic k . We determine the genericlevels and furthermore establish analogous results for the cosets of the simplequotients of the W -algebras.Our second result is a novel Kazama-Suzuki type coset construction: Weshow that a diagonal Heisenberg coset of the subregular W -algebra at level k times the lattice vertex superalgebra V Z is the principal W -superalgebraat the dual level ℓ . Conversely a diagonal Heisenberg coset of the principal W -superalgebra at level ℓ times the lattice vertex superalgebra V √− Z is thesubregular W -algebra at the dual level k . Again this is proven for the universal W -algebras as well as for the simple quotients.We show that a consequence of the Kazama-Suzuki type construction isthat the simple principal W -superalgebra and its Heisenberg coset at level ℓ are rational and/or C -cofinite if the same is true for the simple subregular W -algebra at dual level ℓ . This gives many new C -cofiniteness and rationalityresults. Introduction
Let g be a simple Lie superalgebra, k a complex number and f a nilpotent ele-ment in g that belongs to an sl -triple. Then one associates to the universal affinevertex operator superalgebra V k ( g ) at level k via quantum Hamiltonian redcutionthe universal W -superalgebra W k ( g , f ) [KRW]. Its unique simple quotient is de-noted by W k ( g , f ). If f is principal nilpotent, then one often omits mentioning f and we do so as well. Affine vertex superalgebras and their W -superalgebrasare most important families of vertex algebras due to their essential role in vari-ous aspects of representation theory, geometry and physics. This ranges from thegeometric and quantum geometric Langlands correspondence [F2, AFO] and topo-logical invariants of 4-manifolds [FeGu] to meaningful invariants of three and fourdimensional supersymmetric quantum field theories [AGT, BMR, CG, FrGa, GR]or symmetries of six dimensional conformal field theories [BRvR]. The famousFeigin-Frenkel duality [FF1] of principal W -algebras asserts that for all non-criticallevels k in C one has (see [ACL2, Thm. 5.6] for a short proof) W k ( g ) ≃ W ℓ ( L g )where L g is the Langlands dual Lie algebra of g and the dual level ℓ is defined by r ( k + h ∨ )( ℓ + L h ∨ ) = 1 (1.1) W -algebras and principal W -superalgebras with r the lacity of g and h ∨ , L h ∨ the dual Coxeter numbers of g and L g respec-tively. W -algebras corresponding to non-principal nilpotent elements usually havea non-trivial weight one subspace, i.e., they have an affine vertex operator algebraas subalgebra. In this instance we do not expect an immediate isomorphism of W -algebras, but different W -algebras might have coset subalgebras that are iso-morphic. For example, the coset by the affine subalgebra of the large N = 4superconformal algebra, i.e., the minimal W -superalgebra of d (2 , α ), coincideswith a diagonal coset of the tensor product of two affine vertex algebras of type sl [CFL]. We are interested in such isomorphisms if g is either of type A or B and inthis case we still expect that the levels of the involved W -algebras satisfy a relationof the form (1.1). However, we do not expect that this is an isomorphism betweencosets of W -algebras of Lie algebras that are Langlands dual to each other, but therole of the dual Lie algebra is replaced by some Lie superalgebra. The aim of thispaper is to show that these expectations are true for very interesing W -algebras.Our work is motivated by an impressive work of Boris Feigin and Alexei Semikhatov[FS].1.1. Feigin-Semikhatov W (2) n -algebras. Let g = sl n and g = sl ( n | W -algebra of g is strongly generated by even fields in conformalweights 1 , , . . . , n − n . Theprincipal W -superalgebra of g is strongly generated by even fields in conformalweight 1 , , . . . , n together with a pair of odd fields of conformal weight n +12 . In [FS],Feigin and Semikhatov constructed W (2) n -algebras as subalgebras of the joint kernelof a set of screening charges acting on some free field algebras. These screeningcharges were associated to simple positive roots of g . The subalgebra of the kernelof g screenings is interpreted as a W -superalgebra of type g while the W (2) n -algebra is generated under operator product by two even fields of conformal weight n . Next, also a vertex algebra generated by two weight n fields is constructedinside the tensor product of the affine vertex superalgebra of g and a lattice vertexoperator algebra. Many impressive operator product computations then suggestthat the algebras appearing in these different constructions are isomorphic. Thesecomputations however do not prove any isomorphism and should be interpreted asthe conjecture: Conjecture 1.1. (Feigin and Semikhatov [FS])
Let g = sl n , g = sl ( n | and let f be subregular in g and f be principal nilpotent in g . Let π H i be the Heisenbergsubalgebra of W k i ( g i , f i ) . Then for generic k (1) Com (cid:0) π H , W k ( g , f ) (cid:1) ≃ Com (cid:0) π H , W k ( g ) (cid:1) , where ( k + n )( k + n −
1) = 1 . (2) Com (cid:0) π H , W k ( g , f ) (cid:1) ≃ Com (cid:0) V k ( gl n ) , V k ( g ) (cid:1) , where k + n + k + n = 1 . Note that for n = 2, W k ( g , f ) = V k ( sl ) and W k ( g ) is the N = 2 superVirasoro algebra. In this case the first part of the Conjecture is essentially thewell-known Kazama-Suzuki coset realization of the N = 2 super Virasoro algebra[DPYZ] proven in [CL3, Cor. 8.8]. The second part of the Conjecture for n = 2follows from a nice relation between V k ( sl (2 | L ( d (2 , − ( k + 1))) [BFST, uality of subregular W -algebras and principal W -superalgebras 3 CG]. The second part of the Conjecture for n = 3 is [ACL1, Thm 6.2]. The proofused that in this case [FS] explicitely computed all the necessary operator productalgebras for the inclusion of Com (cid:0) π H , W k ( g , f ) (cid:1) in Com (cid:0) V k ( gl n ) , V k ( g ) (cid:1) .The surjectivity then followed from the computation of strong generators that canbe done in an orbifold limit [CL3] and that limit is determined in [CL1, Thm 4.3].We remark that the cases n = 2 and n = 4 can be handled similarly, but for larger n not enough is known about operator product algebras and so there is need for adifferent proof strategy.In this work, we prove the first part of this Conjecture. The development ofscreening realizations of affine vertex superalgebras is work in progress. We believethat this will then allow us to also treat the second part of the Conjecture. Wenote that a very different proof of the Conjecture is given in [CL4].We actually not only prove the first part of the Conjecture, but we also noticethat a similar statement holds in the case of type B , namely Theorem 4.3 andCorollaries 5.15 and 5.16.Let ( g , g ) = ( sl n +1 , sl (1 | n + 1)) or ( so n +1 , osp (2 | n )). Denote by r the lacityof g , (which is equal to that of g ), and by h ∨ i the dual Coxeter number of g i ,namely, ( r, h ∨ , h ∨ ) = ( (1 , n + 1 , n ) , if ( g , g ) = ( sl n +1 , sl (1 | n + 1)) , (2 , n − , n ) , if ( g , g ) = ( so n +1 , osp (2 | n )) . For k ∈ C \{− h ∨ } , define k ∈ C by the formula r ( k + h ∨ )( k + h ∨ ) = 1 . (1.2)Set the rational numbers x i , ( i = 1 , x , x ) = ((cid:16) n − n, − n n +1 (cid:17) , if ( g , g ) = ( sl n +1 , sl (1 | n + 1)) , (cid:0) − n, − n (cid:1) , if ( g , g ) = ( so n +1 , osp (2 | n )) , (1.3)and S i = {− h ∨ i , x i } . Theorem 1.2.
Let ( g , g ) = ( sl n +1 , sl (1 | n + 1)) , or ( so n +1 , osp (2 | n )) , and ( k , k ) satisfy (1.2) . Then for k ∈ C \ S and k ∈ C \ S , Com (cid:0) π H , W k ( g , f sub ) (cid:1) ≃ Com (cid:0) π H , W k ( g ) (cid:1) and also for the simple quotients Com ( π H , W k ( g , f sub )) ≃ Com ( π H , W k ( g )) . It can be improved to a new variant of Kazama-Suzuki cosets.1.2.
Kazama-Suzuki cosets.
Kazama-Suzuki cosets appeared in the 1980’s asbuilding blocks of sigma models in string theory [KaSu]. The idea is to consider V k ( sl n +1 ) and tensor it with n pairs of free fermions so that they carry the stan-dard representation of gl n and its conjugate and so especially an action of L ( gl n ).The coset by the diagonal V k +1 ( gl n )-action then automatically has odd fields inconformal weight and actually gives rise to an extension of the N = 2 superVirasoro algebra. They are conjecturally isomorphic to principal W -superalgebrasof sl ( n + 1 | n ) and the case n = 1 is the just mentioned relation between the N = 2super Virasoro algebra and V k ( sl ). The case of n = 2 is proven in [GL] andstrong rationality of the Kazama-Suzuki cosets is [ACL2, Cor 14.1]. The idea of Duality of subregular W -algebras and principal W -superalgebras Kazama-Suzuki can be generalized in the following way. Consider some vertex op-erator algebra A k with non-trivial action of V k ( gl n ), so that the tensor productof A k with n -pairs of free fermions has a diagonal V k +1 ( gl n )-action. Then thecommutant by this diagonal action is our new variant of Kazama-Suzuki coset.For a recent related work, see [S2]. We are interested in the case of n = 1, i.e., V k ( gl ) is nothing but a rank one Heisenberg vertex algebra. In this case, there isa remarkable observation due to Boris Feigin, Alexei Semikhatov and Ilya Tipuninthat one can also somehow invert this coset construction [FST]. This has been putto efficient use in studying the representation theory of the N = 2 super Virasoroalgebra and its relation to the one of the simple affine vertex algebra of sl , L k ( sl )[Ad2, Ad3, S1, KoSa, CLRW]. Also the relation between the βγ -system vertex alge-bra and V k ( gl (1 | K i = {− h ∨ i } and S i = {− h ∨ i , x i } .Then we have the following. Theorem 1.3.
Let ( g , g ) = ( sl n +1 , sl (1 | n + 1)) , or ( so n +1 , osp (2 | n )) , and ( k , k ) satisfy (1.2) , k ∈ C \ {− h ∨ } and k ∈ C \ {− h ∨ } . Then the Kazama-Suzuki type coset isomorphisms W k ( g ) ≃ Com (cid:16) π e H , W k ( g , f sub ) ⊗ V Z (cid:17) W k ( g ) ≃ Com (cid:16) π e H , W k ( g , f sub ) ⊗ V Z (cid:17) (1.4) and their inverses W k ( g , f sub ) ≃ Com (cid:16) π e H , W k ( g ) ⊗ V Z √− (cid:17) , W k ( g , f sub ) ≃ Com (cid:16) π e H , W k ( g ) ⊗ V Z √− (cid:17) (1.5) hold. We expect that these isomorphisms will turn out to be very efficient in exploringthe representation theory of the principal W -superalgebras in terms of the one ofthe subregular W -algebras. As a first important step, we deduce C -cofinitenessand rationality results.1.3. Rationality and C -cofiniteness. A well-known result of Tomoyuki Arakawais that principal W -algebras at non-degenerate admissible levels are C -cofinite[Ar2] and rational [Ar4]. The C -cofininteness of certain non-principal W -algebrasincluding subregular W -algebras of type A and B at certain admissible levels hasalso been proven in [Ar2]. They are also conjectured to be rational and this hasbeen proven for type A in [AvE] (including earlier results [Ar3, CL2] for subregular W -algebras of type sl and sl ).In contrast to this, the rationality and C -cofiniteness of principal W -superalgebrasand W -superalgebras in general is open. We only found [Ad3, Ad1] that used acoset construction to prove rationality of the N = 1 and N = 2 super Virasoro alge-bras. Note that the N = 1 super Virasoro algebra is the principal W -superalgebraof osp (1 | C -cofininiteness resultsin the Kazama-Suzuki coset construction. This amounts to essentially showing that uality of subregular W -algebras and principal W -superalgebras 5 certain lattice vertex operator algebras appear as cosets. The key observation forthis is our Lemma 5.8. C -cofiniteness results follow from [CKLR, Lemma 4.6](which is based on [Mi]) modulo a lattice vertex operator algebra assumption. Butthe latter holds in our cases due to Lemma 5.8. Rationality of Heisenberg cosets is[CKLR, Theorem 4.12], again modulo a lattice vertex operator algebra assumption,but that assumption holds due to some Jacobi form argument [Ma]. Our Corollaries5.18 and 5.19 are thus Corollary 1.4. (1)
Let ℓ = − n + nu with u ∈ Z ≥ n and ( u, n ) = 1 . Then W ℓ ( sl (1 | n + 1)) and Com ( π H , W ℓ ( sl (1 | n + 1))) are rational and C -cofinite. (2) Let ℓ = − n + n − u with u ∈ Z ≥ n and ( u, n −
1) = 1 . Then W ℓ ( osp (2 | n )) and Com ( π H , W ℓ ( osp (2 | n ))) are C -cofinite. (3) Let ℓ = − n + nu with u ∈ Z ≥ n and ( u, n ) = 1 . Then W ℓ ( osp (2 | n )) and Com ( π H , W ℓ ( osp (2 | n ))) are C -cofinite. Note that rationality of Heisenberg cosets for the cases sl and sl has beentreated in [ACL1, CL2].1.4. W r ,r ,r -algebras. A famous result of Schiffmann and Vasserot asserts thatthe principal W -algebra of gl r acts on the equivariant cohomology of the mod-uli space of rank r torsion free coherent sheaves on CP with some framing at ∞ [SV]. This result was recently generalized to the moduli space of spiked in-stantons of Nekrasov [RSYZ] and the resulting W r ,r ,r -algebra is characterizedas the intersection of certain screening operators on a Heisenberg vertex algebra.The W r ,r ,r -algebra is defined in [BFM] and physics [PR] conjectures that it isisomorphic to the Y r ,r ,r -algebra of [GR]. These Y r ,r ,r -algebras are defined tobe cosets of W -algebras and superalgebras (times a Heinsenberg vertex algebra ofrank one) and enjoy a triality of isomorphisms if one of the labels is zero [CL4].If two of the labels are zero, then the Y r ,r ,r -algebra and W r ,r ,r -algebra are aprincipal W -algebra of gl r (where r is the non-zero label). If one of the labels iszero, another one is one and the remaining one is r , then the resulting Y r ,r ,r -algebra is the Heisenberg coset of the subregular W -algebra of sl r (again timesa rank one Heisenberg vertex algebra) and the screening realization of [BFM] ofthe W r ,r ,r -algebra is precisely the one that we derive in this paper. This meansthat as a byproduct of our work, we also prove in this instance the conjecture that Y r ,r ,r -algebras and W r ,r ,r -algebras coincide.1.5. Applications.
There are a few possible applications. Firstly and as men-tioned before, the Kazama-Suzuki coset relation between the simple affine vertexalgebra L k ( sl ) and the simple N = 2 super Virasoro algebra at central charge3 k/ ( k + 2) has been used intensively to study the relation of the representationcategories of the two [Ad2, Ad3, S1, KoSa, CLRW]. Similar studies can and shouldbe accomplished for the better understanding of the representation theory of theprincipal W -superalgebras. Of course the rational cases are the easiest one. Wenote that the B ( p ) -algebras of [CRW] have recently been proven to be isomorphicto W − p − p − ( sl p − , f sub ) [ACGY]. These are at admissible levels and the represen-tation theory has been considered in [ACKR]. One can use these results to alsostudy the representation theory of W − p +( p − − ( sl ( p − | W -algebra at the critical Duality of subregular W -algebras and principal W -superalgebras level [CGL1, CGL2, GK] together with some connection to geometry and physics.At the critical level the subregular W -algebra has a large center. Recall that inthe case of affine vertex algebras at the critical level, the spectrum of the centeris isomorphic to the space of opers on the formal disc of the Langlands dual Liealgebra [F2]. This can be seen as Feigin-Frenkel duality at the critical level wherethe dual level goes to infinity. It is surely interesting to study the behaviour ofour duality in the large level/critical level limit. Another related direction that isworth further investigation is to show these type of dualities for the correspondingfinite W -(super)algebras. In general we aim to investigate dualities between cosetsof seemingly different W -(super)algebras in best possible generality. Our presentwork is a step in this direction. In order to derive further dualities with our meth-ods we need a better understanding of free field realizations and screening chargesof W -superalgebras. A first step are those of affine vertex operator superalgebraswhich is work in progress. There are already works on this direction, see, for ex-ample, [IK, R, IMP1, IMP2] for earlier results. Another aim is to prove that the W r ,r ,r -algebras and Y r ,r ,r -algebras coincide. Given the geometrically meaningof the W r ,r ,r -algebras we consider this to be an important problem. Free fieldrealizations can also be used to study correspondences between conformal fieldtheories with W -algebra symmetries and our new free field realizations are used toderive correspondences between principal and subregular W -algebras of type A andmore [CGHL].1.6. Outline.
The proofs of the main theorems go in a few steps. Firstly, weneed to characterize the W -(super)algebras in terms of joint kernels of certainscreening charges acting on some free fied algebras. For this, we utilize a free fieldrealization of V κ ( gl (1 | W -(super)algebras that we study. In Section 4 we then relate thescreening charges of the principal W -superalgebra to the one of the correspondingsubregular W -algebra. This allows us to prove our main theorems for the univeral W -(super)algebras at generic level. Next, in Section 5 we adapt the theory ofHeisenberg cosets developed in [CKLR] to our setting so that we can also determinethe cosets of simple quotients and very importantly C -cofiniteness and rationality.For the latter two its most crucial to show that certain lattice vertex operatoralgebras appear. These results are then applied to our cosets. Acknowledgements.
T.C. appreciates many frutiful discussions on related topicswith Boris Feigin and Andrew Linshaw. We also thank Tomoyuki Arakawa foruseful comments. T.C. is supported by NSERC
Free field realization of V κ ( gl (1 | V κ ( gl (1 | uality of subregular W -algebras and principal W -superalgebras 7 Heisenberg vertex algebra.
We follow the definition of vertex superalge-bras in [K2] and also use the language of the λ -bracket, cf. [DK]. For a vertexsuperalgebra V , we denote by | i the vacuum vector, by ∂ the translation operator,and by Y ( a, z ) = a ( z ) = P n ∈ Z a ( n ) z − n − the field corresponding to a ∈ V .For a finite dimensional commutative Lie algebra h over C with a symmetricbilinear form κ , we denote by b h κ := h [ t, t − ] ⊕ C K the affine Lie algebra of h , whoseLie bracket is defined by[ h ( m ) , h ′ ( n ) ] = mκ ( h | h ′ ) δ m + n, K, [ K, b h κ ] = 0 , h, h ′ ∈ h , m, n ∈ Z , where h ( m ) = ht m . Define a b h κ -module π κ by π κ h := U (cid:16)b h κ (cid:17) ⊗ U ( h [ t ] ⊕ C K ) C , where C is regarded as a h [ t ] ⊕ C K -module, on which h [ t ] acts trivially and K actsas 1, and U ( a ) denotes the universal enveloping algebra of a Lie superalgebra a .There is a unique vertex algebra structure on π κ h with the vacuum vector | i = 1 ⊗ Y ( h ( − | i , z ) = h ( z ) := X n ∈ Z h ( n ) z − n − , h ∈ h . It is called the Heisenberg vertex algebra associated with h at level κ . More gener-ally, for µ ∈ h ∗ , define a b h κ -module by π κ h ,µ := U (cid:16)b h κ (cid:17) ⊗ U ( h [ t ] ⊕ C K ) C µ , where C µ := C is regarded as a h [ t ] ⊕ C K -module, on which h ( n ) , ( n ≥ µ ( h ) δ n, and K acts as 1. It has a unique π κ h -module structure coming fromthe ˆ h κ -modules structure, called the highest weight π κ h -module with highest weight µ . If κ is non-degenerate, π κ h is simple, called a non-degenerate Heisenberg vertexalgebra. The dimension dim h of h is equal to that of subspace of π κ h with conformaldegree 1, called the rank of π κ h .If we fix a non-degenerate bilinear form ( ·|· ) on h , then h is identified with h ∗ by h ( ν ( h ) : h ′ ( h | h ′ )). For κ = k ( ·|· ), we write π k h , (resp. π k h ,µ ) instead of π κ h ,(resp. π κ h ,µ ), and denote by α ( z ) = P n ∈ Z α ( n ) z − n − := ν − ( α )( z ) for α ∈ h ∗ . Wecall π k h the Heisenberg vertex algebra associated with h at level k .2.2. Wakimoto representations of b gl (1 | κ . Let gl (1 |
1) be the Lie superalgebraEnd( C | ) with Lie superbracket[ x, y ] = xy − ( − ¯ x ¯ y yx, x, y ∈ End( C | ) , where ¯ x ∈ Z = { ¯0 , ¯1 } denotes the parity of x ∈ End( C | ). Let { E i,j } ≤ i,j ≤ denotethe elementary matrices of gl (1 |
1) = End( C | ), h = C E , ⊕ C E , , n + = C E , and n − = C E , . Note that the parity of E i,j is i + j . For an even supersymmetricinvariant bilinear form κ on gl (1 | k , k ∈ C such that κ = k κ + k κ , Duality of subregular W -algebras and principal W -superalgebras where κ ( x | y ) = str C | ( xy ) and κ ( x | y ) = − str gl (1 | (ad( x ) ad( y )), and str V (?)denotes the supertrace over a vector superspace V . The non-zero parings of κ are: κ ( E , | E , ) = k + k , κ ( E , | E , ) = − k + k ,κ ( E , | E , ) = − k , κ ( E , | E , ) = k . Define χ i ∈ h ∗ by χ i ( E j,j ) = ( − i +1 δ i,j , i, j ∈ { , } . We identify h with h ∗ by κ , under which E i,i is identified with χ i , ( i = 1 , κ ( E i,i | E j,j ) = χ i ( E j,j ). Let π κ − κ := π κ − κ h be the Heisenberg vertex algebraassociated with h at level κ − κ , which is freely generated by fields χ i ( z ), ( i = 1 , χ i ( z ) χ j ( w ) ∼ ( κ − κ )( E i,i | E j,j )( z − w ) , i, j = 1 , . (2.1)Let b gl (1 | κ := gl (1 | t, t − ] ⊕ C K be the affine Lie superalgebra,[ x ( m ) , y ( n ) ] = [ x, y ] ( m + n ) + mδ m + n, κ ( x, y ) K, x, y ∈ gl (1 | , m, n ∈ Z [ K, b gl (1 | κ ] = 0 , where x ( m ) = xt m for x ∈ gl (1 | m ∈ Z . Define a b gl (1 | κ -module V κ ( gl (1 | V κ ( gl (1 | U (cid:16) b gl (1 | κ (cid:17) ⊗ U ( b gl (1 | κ, + ) C , where C is regarded as a b gl (1 | κ, + (:= gl (1 | t ] ⊕ C K )-module, on which gl (1 | t ]acts trivially and K acts as 1. Then there is a unique vertex superalgebra structureon V κ ( gl (1 | | i = 1 ⊗ Y ( u ( − | i , z ) = u ( z ) := X n ∈ Z u ( n ) z − n − , u ∈ gl (1 | . It is called the universal affine vertex superalgebra of gl (1 |
1) at level κ . If k = 0,then it admits the Segal-Sugawara conformal field T ( z ) := 12 k (cid:16) − k k : ( E , ( z ) + E , ( z )) : + : E , ( z ) : − : E , ( z ) :) − : E , ( z ) E , ( z ) : + : E , ( z ) E , ( z ) : (cid:17) , (2.2)whose central charge is 0, i.e., the superdimension of gl (1 | M gl (1 | be the bc -system vertex superalgebra, which is generated by oddfields b ( z ) , c ( z ) satisfying the OPEs b ( z ) c ( w ) ∼ z − w , b ( z ) b ( w ) ∼ ∼ c ( z ) c ( w ) . The following proposition follows from direct calculations.
Proposition 2.1.
There exists a homomorphism of vertex superalgebras ρ : V κ ( gl (1 | → M gl (1 | ⊗ π κ − κ , which satisfies E , ( z ) b ( z ) , E , ( z ) : c ( z ) ( χ ( z ) + χ ( z )) : + k ∂c ( z ) ,E , ( z )
7→ − : c ( z ) b ( z ) : + χ ( z ) , E , ( z ) : c ( z ) b ( z ) : + χ ( z ) . (2.3) uality of subregular W -algebras and principal W -superalgebras 9 We denote a V κ ( gl (1 | M gl (1 | ⊗ π κ − κ by W κ . Note that T ( z ) maps to: ∂c ( z ) b ( z ) : + 1 − k k : ( χ + χ )( z ) : + 12 k (cid:0) : χ ( z ) : − : χ ( z ) : + ∂ ( χ + χ )( z ) (cid:1) . Lemma 2.2. ρ is injective for all κ .Proof. Define conformal gradings ∆ on V κ ( gl (1 | W κ by setting∆( | i ) = ∆( c ( − | i ) = 0 , ∆( x ( − | i ) = ∆( b ( − | i ) = ∆( χ i ( − | i ) = 1 , x ∈ gl (1 | , i = 1 , A ( n ) B ) = ∆( A ) + ∆( B ) − n −
1. We choose the set A of homogeneousstrong generators to be { X ( − | i} X ∈ gl (1 | for V κ ( gl (1 | { b ( − | i , c ( − | i , χ − | i , χ − | i} for W κ . Then the associated standard filtrations are F n V := Span a i ( − n ) · · · a i r ( − n r ) | i ; a i j ∈ A , X j ∆( a i j ) ≤ n, r ≥ , n i ≥ for V = V κ ( gl (1 | W κ respectively and their associated graded superspacesgr F V := ∞ M n =0 F n VF n − V admit a structure of Poisson vertex superalgebra [Ar1, Li4]. Since ρ preserves thegradings ∆ by (2.3), ρ induces a homomorphism of Poisson vertex superalgebrasgr F ρ : gr F V κ ( gl (1 | → gr F W κ . We have gr F V κ ( gl (1 | C [ ∂ n E i,j | ≤ i, j ≤ , n ∈ Z ≥ ] , gr F W κ = C [ ∂ n b, ∂ n c, ∂ n χ i | i = 1 , , n ∈ Z ≥ ] , where ∂ n A is the image of n ! A ( − n − | i ∈ F ∆( A )+ n V in F ∆( A )+ n V /F ∆( A )+ n − V .Next, define weight gradings wt on gr F V κ ( gl (1 | F W κ by settingwt( ∂ n E , ) = wt( ∂ n b ) = wt( ∂ n c ) = 0 , wt( ∂ n E i,i ) = wt( ∂ n E , ) = wt( ∂ n χ i ) = 1 , i = 1 , AB ) = wt( A )+wt( B ). They yield filtrations G n V = Span { A ∈ V ; wt( A ) ≤ n } on gr F ¯ V for ¯ V = gr F V κ ( gl (1 | F W κ . Since { G m V λ G n V } ⊂ G m + n V [ λ ],the associated graded superspacegr G V := ∞ M n =0 G n VG n − V also has a structure of Poisson vertex superalgebra. Since gr F ρ preserves the weightgradings by (2.3), gr F ρ induces a homomorphism of Poisson vertex superalgebrasgr G gr F ρ : gr G gr F V κ ( gl (1 | → gr G gr F W κ ,E , b, E , c ( χ + χ ) ,E i,i χ i , i = 1 , , where A denotes the image of A ∈ G wt( A ) V in G wt( A ) V /G wt( A ) − V by abuse ofnotation. Since gr G gr F ρ is injective, so is ρ . (cid:3) W -algebras and principal W -superalgebras By using ρ in Proposition 2.1, the W κ -module W κµ := M gl (1 | ⊗ π κ − κ µ , µ ∈ h ∗ becomes a V κ ( gl (1 | b gl (1 | κ with highest weight µ , (cf. [F1]).2.3. Wakimoto representations at generic level.
Here we study the Wakimotorepresentations W κµ for generic level κ . Let µ i := µ ( E i,i ) , i = 1 , . Consider the highest, (resp. lowest), Verma module of gl (1 | V ± n,e := U ( gl (1 | ⊗ U ( b ± ) C v n,e , n, e ∈ C , where b + := Span { E , , N, E } , (resp. b − := Span { E , , N, E } ), with N := 12 ( E , − E , ) , E := E , + E , and C v n,e is the one dimensional b ± -module such that E , , (resp. E , ), acts by0, N by n , and E by e . If e = 0, then they are irreducible and we have anisomorphism V + n,e ≃ V − n − ,e . If e = 0, then they are only indecomposable and wehave the following short exact sequences0 → A n ± → V ± n, → A n → A q = C w q , ( q ∈ C ), is the one dimensional gl (1 | E , , E , , E acts by 0, and N acts by q .Define the V κ ( gl (1 | V ± ,κn,e := U ( b gl (1 | κ ) ⊗ U ( gl (1 | κ, + ) V ± n,e , ˆ A κn := U ( b gl (1 | κ ) ⊗ U ( gl (1 | κ, + ) A n . Lemma 2.3 ([CR2]) . (1) If e = 0 , then ˆ V ± ,κn,e is irreducible for ek / ∈ Q , and thereexists an isomorpshim ˆ V + ,κn,e ≃ ˆ V − ,κn − ,e . (2) If e = 0 , then ˆ V ± ,κn,e admits the following short exact sequence for k = 00 → ˆ A κn ± → ˆ V ± ,κn, → ˆ A κn → . (2.6) (3) ˆ A κn is irreducible for k = 0 .Proof. We include the proof for the completeness of the paper, following the ar-gument in [CR2, Sec. 3.2]. We may assume k = 0. For (1), the isomorphism V + n,e ≃ V − n − ,e induces an isomorphism ˆ V + ,κn,e ≃ ˆ V − ,κn − ,e . The conformal dimension of1 ⊗ v n,e ∈ ˆ V ± ,κn,e with respect to (2.2) is∆ n,e := 12 k (cid:18) − k k e + 2 en ∓ (cid:19) . Suppose that ˆ V − ,κn,e is reducible. Then we have a nontrivial V κ ( gl (1 | V − ,κn ′ ,e → ˆ V − ,κn,e , ∃ n ′ ∈ n + Z , and thus ∆ n ′ ,e = ∆ n,e + j for some j ∈ Z > , that is, ek ( n ′ − n ) = j . Since n ′ − n ∈ Z ,it is impossible when ek / ∈ Q , in which case ˆ V − ,κn,e is irreducible. This completesthe proof of (1). For (3), note that the conformal dimension of 1 ⊗ w n ∈ ˆ A κn with uality of subregular W -algebras and principal W -superalgebras 11 respect to (2.2) is always 0. Suppose that ˆ A κn is reducible. Then we have a nontrivial V κ ( gl (1 | V + ,κn ′ , → ˆ A κn or ˆ V − ,κn ′ , → ˆ A κn , ∃ n ′ ∈ Z such that the image of 1 ⊗ v n, is not contained in C w n . It is impossible since theconformal dimension of 1 ⊗ v n, ∈ ˆ V ± ,κn, is 0. Thus ˆ A κn is irreducible for k = 0. For(2), (2.5) induces an exact sequence of V κ ( gl (1 | A κn ± → ˆ V ± ,κn, → ˆ A κn → U ( b gl (1 | κ ) ⊗ U ( gl (1 | κ, + ) (?). The mapˆ A κn ± → ˆ V ± ,κn, is also injective by (3) for k = 0. This completes the proof of(2). (cid:3) Proposition 2.4.
We have an isomorphism W κµ ≃ ˆ V − ,κn ( µ ) ,e ( µ ) of V κ ( gl (1 | -modules where n ( µ ) = µ + µ − , e ( µ ) = µ − µ for e ( µ ) k Q if e ( µ ) = 0 and for k = 0 if e ( µ ) = 0 .Proof. Define a conformal grading ∆ on W κµ by ∆( | µ i ) = 0 and (2.4): W κµ = M d ≥ W κµ,d . (2.7)Then the subspace W κµ, = C | µ i ⊕ C c ( − | µ i is a ˆ gl (1 | κ, + -module, which is iso-morphic to V − n ( µ ) ,e ( µ ) by V − n ( µ ) ,e ( µ ) → W κµ, ,v n ( µ ) ,e ( µ ) c ( − | µ i , E , v n ( µ ) ,e ( µ )
7→ | µ i . Thus, by the universality of the induced modules, we have a V κ ( gl (1 | V − ,κn ( µ ) ,e ( µ ) → W κ − nα . (2.8)If µ = µ , then the map (2.8) is injective by Lemma 2.3 (1). If µ = µ , then( n ( µ ) , e ( µ )) = ( − n − , V − ,κ − n − , for k = 0 belong to the subspace A − n ⊂ ˆ A κ − n , which is clearly embedded by (2.8).Thus the map (2.8) is injective by Lemma 2.3 (2) also in this case. Therefore, themap (2.8) is injective for generic κ for any highest weight µ ∈ h ∗ .Define a conformal degree ∆ of ˆ V − ,κn ( µ ) ,e ( µ ) by ∆( V − n ( µ ) ,e ( µ ) ) = 0, ∆( X ( − ) = 1,( X ∈ gl (1 | X ( − n − Y ) = ∆( X ) + ∆( Y ) + n + 1. Then the map (2.8)preserves the conformal gradings by Proposition 2.1. Now its surjectivity followsfrom the equality of the characters:ch h b V − n ( µ ) ,e ( µ ) i = 2 ∞ Y n =1 (1 + q n ) (1 + q n ) − = ch[ W κµ ] . Thus (2.8) is an isomorphism. (cid:3) W -algebras and principal W -superalgebras Resolution.
For α = χ + χ ∈ h ∗ , define an intertwining operator S ( z ) : W κ − nα S −→ W κ − ( n +1) α (( z )), ( k = 0), by S ( z ) = : b ( z )e − k R α ( z ) : , where: e − k R α ( z ) : = T − α exp k X n< α ( n ) n z − n ! exp k X n> α ( n ) n z − n ! , (2.9)and T − α is the translation operator π κ − nα → π κ − ( n +1) α sending the highest weightvector to the highest weight vector and commuting with all χ i ( n ) , i = 1 , n = 0.By direct calculation, one can show that the residue S := Z S ( z ) dz satisfies S ( u ( − | i ) = 0, u ∈ gl (1 | S is a V κ ( gl (1 | W κ − nα to W κ − ( n +1) α (cf. [F1]).Using Wakimoto representations, we can extend the injective morphism ρ to along exact sequence as in the following proposition. Proposition 2.5.
The sequence → V κ ( gl (1 | ρ −→ W κ S −→ W κ − α → · · · → W κ − nα S −→ W κ − ( n +1) α → · · · . (2.10) is a complex of V κ ( gl (1 | -modules and exact for k = 0 .Proof. To show that (2.10) is a complex, we have to show (1) Im( ρ ) ⊂ Ker( S ) and(2) S ◦ S = 0. (1) follows from S ( x ( − | i ) = 0, x ∈ gl (1 | W κ to the direct sum of W κ -modules L n ≥ W κ − nα by setting Y ( | − nα i , z ) =: e − nk R α ( z ) : , where | − nα i is a fixed non-zero highest weight vector of π κnα (see (2.9)). This isdue to κ ( nα, mα ) = 0 for n, m ≥ S is the 0-th mode Q (0) of the field corresponding to Q = b ( − | − α i . By the Jacobi identity of the λ -bracket, we have S ◦ S ( a ) = [ Q λ [ Q µ a ]] λ = µ =0 = 12 [[ Q λ Q ] λ + µ a ]] λ = µ =0 = 0 , a ∈ W κ − nα . Thus S ◦ S = 0 follows.It is clear that the complex (2.10) preserves the conformal gradings (2.7). Inparticular, the subcomplex of conformal grading 0 is isomorphic to0 → A → V −− , → V −− , → · · · → V −− n, → V −− n − , → · · · , which is a complex of gl (1 | S ( | − nα i ) = 0 and S ( c ( − | − nα i ) = | − ( n + 1) α i , it is a successive composition of (2.5), and thus exact. Now it followsthat the complex (2.10) for k = 0 is a successive composition of (2.6) since ˆ A κ − n ,( n ≥ k = 0. Thus (2.10) is exact. (cid:3) uality of subregular W -algebras and principal W -superalgebras 13 Free field realizations of W -algebras W -algebras. Let g be a finite dimensional simple Lie superalgebra with a non-degenerate even supersymmetric invariant bilinear form ( ·|· ) such that ( θ | θ ) = 2 fora long root θ of g . Let f a nilpotent element of the even part of g and a Z -gradingΓ : g = L j ∈ Z g j on g , called a good grading, which satisfies [ g i , g j ] ⊂ g i + j , f ∈ g − , and that ad f : g j → g j − is injective for j ≥ and surjective for j ≤ . Thenone can define the W -algebra W k ( g , f ; Γ) associated with g , f, Γ and a complexnumber k via the (generalized) Drinfeld-Sokolov reduction [FF1, KRW].Following [KW, G], we will embed W k ( g , f ; Γ) in an affine vertex superalgebraunder the assumption that Γ is a Z -grading, i.e.,Γ : g = M j ∈ Z g j . Let τ k be the invariant bilinear form on g defined by τ k ( u | v ) = k ( u | v ) + 12 κ g ( u | v ) − κ g ( u | v ) , u, v ∈ g , (3.1)where κ g , κ g are the Killing forms on g , g respectively. Let b g = g [ t, t − ] ⊕ C K denote the affine Lie superalgebra of g at level τ k , which satisfies[ u ( m ) , v ( n ) ] = [ u, v ] ( m + n ) + mτ k ( u | v ) δ m + n, K, u, v ∈ g , m, n ∈ Z , [ K, b g ] = 0 , where u ( m ) = ut m . Define a b g -module V τ k ( g ) by V τ k ( g ) := U ( b g ) ⊗ U ( g [ t ] ⊕ C K ) C , where C is a one dimensional ( g [ t ] ⊕ C K )-module, on which K acts by 1 and g [ t ] acts trivially. There is a unique vertex algebra structure on V τ k ( g ) such that | i = 1 ⊗ Y ( u ( − ⊗ v, z ) = J u ( z ) := P n ∈ Z u ( n ) z − n − ,( u ∈ g ), with OPEs J u ( z ) J v ( w ) ∼ J [ u,v ] ( w ) z − w + τ k ( u | v )( z − w ) , u, v ∈ g . It is called the affine vertex superalgebra of g at level τ k .Let ∆ denote the root system of g and Π the set of its simple roots. The gradingΓ gives the decomposition ∆ = F j ∈ Z ∆ j where ∆ j = { α ∈ ∆ | g α ⊂ g j } . Thenwe have Π = Π ⊔ Π where Π j = Π ∩ ∆ j . For α ∈ Π , let g [ α ] ⊂ g denote theadjoint g -submodule generated by g α . It is simple, and so is its dual module,which we denote by C [ α ] . More explicitly, take a basis { e α } α ∈ I ⊔ ∆ of g consisting ofa basis { e α } α ∈ I of the Cartan subalgebra h and root vectors e α of g α . Let c γα,β bethe structure constants, i.e., [ e α , e β ] = P γ c γα,β e γ , extended to [ u, v ] = P γ c γu,v e γ , u, v, ∈ g . By setting v β ∈ g ∗ to be the dual vector of e β , we have C [ α ] = M β ∈ [ α ] C v β , [ α ] := { β ∈ ∆ | β − α ∈ Z ∆ } , α ∈ Π , and u · v β = X γ ∈ [ α ] c βγ,u v γ , u ∈ g , β ∈ [ α ] . W -algebras and principal W -superalgebras Define a ˆ g -module by M α := U ( b g ) ⊗ U ( g [ t ] ⊕ C K ) C [ α ] , α ∈ Π , where g [ t ] ⊕ C K acts on C [ α ] by u ( n ) = δ n, u , ( u ∈ g , n ≥ K acts by1. It has a unique V τ k ( g )-module structure induced by the ˆ g -module structure.Now, by [G, Sec. 4.3], there exists a cochain complex ( C ′ k , d st(0) ) consisting of a Z ≥ -graded vertex superalgebra C ′ k and an odd vertex operator d st(0) , whose cohomologyis a vertex superalgebra such that H (cid:0) C ′ k , d st(0) (cid:1) ≃ V τ k ( g ) , H (cid:0) C ′ k , d st(0) (cid:1) ≃ M α ∈ Π M α as vertex algebras and as V τ k ( g )-modules respectively. Define an intertwiningoperator S β ( z ) of type (cid:0) M α M α ,V τk ( g ) (cid:1) , ( α ∈ Π ), by S β ( z ) := Y ( v β , z ) , β ∈ [ α ] , (3.2)and a screening operator by Z Q α ( z ) dz := X β ∈ [ α ] ( f | e β ) Z S β ( z ) dz : V τ k ( g ) → M α . (3.3)Note that S β ( z ) is uniquely determined by the property S β ( z ) J u ( w ) ∼ Y ( u · v β , w ) z − w = X γ ∈ [ α ] c βu,γ S γ ( w ) z − w , u ∈ g . (3.4)In [G], the formula ∂S β ( z ) = − ( k + h ∨ ) − ( e β | e − β ) (cid:16) : J h β ( z ) S β ( z ) : + X γ ∈ [ α ] ,δ ∈ ∆ γ = β ( − ¯ γ ¯ δ c δγ, − β : J e δ ( z ) S γ ( z ) : (cid:17) , where h β = [ e β , e − β ] is also used to characterize S β ( z ). We note that this is nothingbut ∂S β ( z ) = Y ( L v β , z ) , where L ( z ) = P n ∈ Z L n z − n − is the Segal-Sugawara conformal field of g . Thus(3.4) is enough to characterize S β ( z ). We have a vertex superalgebra homomor-phism W k ( g , f ; Γ) ֒ → V τ k ( g ) , (3.5)called the Miura map [KW]. The injectivity of this map is proven in [Ar5, F2] if g is a simple Lie algebra and f is a principal nilpotent element, but the proof thereapplies for arbitrary W -algebras W k ( g , f ; Γ), see for example [N]. By [FBZ, G] theimage of the Miura map for generic k coincides with W k ( g , f ; Γ) ≃ \ α ∈ Π Ker Z Q α ( z ) dz. (3.6) uality of subregular W -algebras and principal W -superalgebras 15 Subregular W -algebras. We describe the isomorphism (3.6) more explicitlyin the case g = sl n +1 or so n +1 with a subregular nilpotent element f sub . We havethe natural representation sl n +1 ֒ → gl ( V ), so n +1 ֒ → gl ( V ), where V = C n +1 , V = C n +1 , see e.g., [Hum]. Let { e i } i ∈ I s denote the standard bases of V s withindex sets I = { , . . . , n + 1 } , I = { , , . . . , n, − , . . . , − n } , and e i,j ∈ gl ( V s ) the elementary matrix e i,j · e m = δ j,m e i , ( i, j, m ∈ I s ). Then theCartan subalgebra h of g is spanned by h i = e i,i − e i +1 ,i +1 , ( i = 1 , . . . , n ) , if g = sl n +1 ,h j = ( t j − t j +1 , ( j = 1 , . . . , n − ,t n , ( j = n ) , if g = so n +1 , where t i = e i,i − e − i, − i . The normalized invariant bilinear form ( ·|· ) on g is given by( u | v ) = tr( u ◦ v ) if g = sl n +1 , and by ( u | v ) = tr( u ◦ v ) if g = so n +1 , which givesan isomorphism ν : h ∼ −→ h ∗ . Then the set of simple roots of g is Π = { α i } ni =1 where α i = ν ( h i ) ∈ h ∗ . Choose a subregular nilpotent element f sub of g and a semisimpleelement x of g such that ad x defines a good Z -grading Γ = Γ sub on g as follows: f sub = n X i =2 e i +1 ,i , x = 12( n + 1) n X i =1 ( n − i + 1)( in + i − h i , if g = sl n +1 ,f sub = n − X i =2 ( e i +1 ,i − e − i, − i − ) + e ,n − e − n, , x = n X i =1 ( n − i + 1) t i − t , if g = so n +1 . The weighted Dynkin diagrams corresponding to Γ sub are as follows. g = sl n : ❡ α ❡ α · · · ❡ α n − ❡ α n − ❡ α n . g = so n +1 : ❡ α ❡ α · · · ❡ α n − ❡ α n − > ❡ α n .The Lie subalgebra g = { u ∈ g | [ x, u ] = 0 } decomposes as g = g red0 ⊕ z , g red0 := Span { e , h , f } , z := Span { e h , h , . . . , h n } , where e h = h + h and e = e , , f = e , , if g = sl n +1 ,e = e , − e − , − , f = e , − e − , − , if g = so n +1 . The Lie subalgebra z is commutative and g red0 is isomorphic to sl by sl ∼ −→ g red0 e, h, f e , h , f . W -algebras and principal W -superalgebras The restriction of τ k (3.1) on g is τ k ( u | v ) = ( k + h ∨ )( u | v ) , u, v ∈ z , ( k + h ∨ − u | v ) , u, v ∈ g red0 , , u ∈ z , v ∈ g red0 , where h ∨ is the dual Coxeter number of g , which is n +1, (resp. 2 n − g = sl n +1 ,(resp. so n +1 ). Therefore, the affine vertex algebra V τ k ( g ) decomposes as V τ k ( g ) ≃ V k + h ∨ − ( sl ) ⊗ π k + h ∨ z , (3.7)where π k + h ∨ z is the Heisenberg vertex algebra associated with z at level k + h ∨ (Section 2.1).Since [ α ] = { α , α + α } , [ α i ] = { α i } , i = 3 , . . . , n, (3.8)the screening operators R Q i ( z ) dz := R Q α i ( z ) dz in (3.3) are Z Q i ( z ) = Z S α i ( z ) dz : V τ k ( g ) → M α i , i = 2 , . . . , n, (3.9)First, consider (3.9) in the case i = 3 , . . . , n . The orthogonal decomposition z = z ⊥ i ⊕ z i , z i = C h i , z ⊥ i = { h ∈ z | α i ( h ) = 0 } . induces the decomposition of the Heisenberg vertex algebra π k + h ∨ z ≃ π k + h ∨ z ⊥ i ⊗ π k + h ∨ z i . It follows from (3.7) and (3.8) that we have an isomorphism of V τ κ ( g )-modules M α i ≃ f M α i := V k + h ∨ − ( sl ) ⊗ π k + h ∨ z ⊥ i ⊗ π k + h ∨ z i , − α i v α i
7→ | i ⊗ | i ⊗ | − α i i . Thus S α i ( z ) is identified with Z S α i ( z ) dz = Z : e − k + h ∨ R α i ( z ) : dz : V k + h ∨ − ( sl ) ⊗ π k + h ∨ z ⊥ i ⊗ π k + h ∨ z i → f M α i , where α i ( z ) = J h i ( z ), and, therefore, Z Q i ( z ) dz = Z : e − k + h ∨ R α i ( z ) : dz. (3.10)Next, consider (3.9) in the case α = α . To this end, we use the Wakimoto realiza-tion of V k + h ∨ − ( sl ), (cf. [F1]). ρ sl : V k + h ∨ − ( sl ) ֒ → M sl ⊗ π k + h ∨ a ,J e ( z ) β ( z ) , J h ( z )
7→ − γ ( z ) β ( z ) : + a ( z ) ,J f ( z )
7→ − : γ ( z ) β ( z ) : +( k + h ∨ − ∂γ ( z ) + γ ( z ) a ( z ) , which gives an isomorphism for generic kV k + h ∨ − ( sl ) ≃ Ker (cid:18)Z : β ( z )e − k + h ∨ R a ( z ) : dz : M sl ⊗ π k + h ∨ a → M sl ⊗ π k + h ∨ a, − a (cid:19) . uality of subregular W -algebras and principal W -superalgebras 17 Here M sl is the βγ -system vertex algebra generated by the even fields β ( z ) , γ ( z )with OPEs β ( z ) γ ( w ) ∼ z − w , β ( z ) β ( w ) ∼ ∼ γ ( z ) γ ( w ) , and π k + h ∨ a is the Heisenberg vertex algebra generated by an even field a ( z ) withan OPE a ( z ) a ( w ) ∼ k + h ∨ )( z − w ) . Then it follows from (3.7) and the isomorphism of vertex algebras π k + h ∨ h ∼ −→ π k + h ∨ a ⊗ π k + h ∨ z α ( z ) a ( z ) , α ( z ) J e h ( z ) − a ( z ) , α i ( z ) J h i ( z ) , ( i = 3 , . . . , n ) , that we have a vertex algebra embedding ρ g : V τ k ( g ) ֒ → M sl ⊗ π k + h ∨ h , (3.11)which gives an isomorphism for generic kV τ k ( g ) ≃ Ker (cid:18)Z : β ( z )e − k + h ∨ R a ( z ) : dz : M sl ⊗ π k + h ∨ h → M sl ⊗ π k + h ∨ h , − α (cid:19) . (3.12)It gives a V τ k ( g )-module structure on M sl ⊗ π k + h ∨ h , − α . Let f M α be a V τ k ( g )-submodule generated by the subspace e C [ α ] = C | − α i ⊕ C γ ( − | − α i . Lemma 3.1.
For generic k , M α ≃ f M α as V τ k ( g ) -modules.Proof. The linear map C [ α ] ∼ −→ e C [ α ] , v α
7→ | − α i , v α + α
7→ − γ ( − | − α i gives an isomorphism as ( g [ t ] ⊕ C K )-modules. By the universality of the inducedmodules, it induces a surjective V τ k ( g )-module homomorphism M α ։ f M α . (3.13)Since C [ α ] is simple as a g -module, M α is simple as a V τ k ( g )-module for generic k . Thus (3.13) is an isomorphism for such k . (cid:3) Under the realization (3.12), Lemma 3.1 implies that S α ( z ) is identified with S α ( z ) = : e − k + h ∨ R α ( z ) : : Im( ρ g ) → f M α (( z )) , and thus Z Q ( z ) dz = Z : e − k + h ∨ R α ( z ) : dz. (3.14) W -algebras and principal W -superalgebras We also note that S α + α ( z ) = − : γ ( z )e − k + h ∨ R α ( z ) :. By (3.6), (3.10) and(3.14), we conclude W k ( g , f sub ) ≃ n \ i =2 Ker (cid:16)Z Q i ( z ) dz : V τ k ( g ) → M α i (cid:17) ≃ n \ i =2 Ker (cid:16)Z : e − k + h ∨ R α i ( z ) : dz : Im( ρ g ) → f M α i (cid:17) . The composition Υ of (3.5) and (3.11) gives a vertex algebra embedding of thesubregular W -algebra W k ( g , f sub ) := W k ( g , f sub ; Γ sub )Υ : W k ( g , f sub ) ֒ → M sl ⊗ π k + h ∨ h . (3.15)Now (3.12) implies the following realization of the image Im(Υ ). Theorem 3.2. If k is generic, then we have Im(Υ ) = n \ i =1 Ker Z Q i ( z ) dz for g = sl n +1 , so n +1 , where Q ( z ) =: β ( z )e − k + h ∨ R α ( z ) : , Q i ( z ) =: e − k + h ∨ R α i ( z ) : , ( i = 2 , . . . , n ) . Principal W -superalgebras. We describe the isomorphism (3.6) more ex-plicitly in the case g = sl (1 | n + 1) or osp (2 | n ) with a principal nilpotent element.We have the natural representation sl (1 | n +1) ֒ → gl ( U ), osp (2 | n ) ֒ → gl ( U ), where U = C | n +1 , U = C | n , see e.g., [K1, Sec. 2]. Let { e i } i ∈ J s denote the standardbases of U s with index sets J = J , ¯0 ⊔ J , ¯1 , J , ¯0 = { } , J , ¯1 = { , . . . , n + 1 } ,J = J , ¯0 ⊔ J , ¯1 , J , ¯0 = { + , −} , J , ¯1 = { , . . . , n, − , . . . , − n } , where e i is even, (resp. odd), if i ∈ J s, ¯0 , (resp. i ∈ J s, ¯1 ), and e i,j ∈ gl ( U s ) theelementary matrix e i,j · e m = δ j,m e i , ( i, j, m ∈ J s ). Then the Cartan subalgebra h of g is spanned by h i = ( − e , − e , , ( i = 0) ,e i,i − e i +1 ,i +1 , ( i = 1 , . . . , n ) , if g = sl (1 | n + 1) ,h j = − ( t + t ) , ( j = 0) , ( t j − t j +1 ) , ( j = 1 , . . . , n − ,t n , ( j = n ) , if g = osp (2 | n ) , where t = e + , + − e − , − and t j = e j,j − e − j, − j , (1 ≤ j ≤ n ). The normalizedinvariant bilinear form ( ·|· ) on g is given by ( u | v ) = − str( u ◦ v ), where str denotesthe super trace. It induces an isomorphism ν : h ∼ −→ h ∗ . Then the set of the simpleroots is Π = { α i } ni =0 where α i = ν ( h i ) ∈ h ∗ . Choose a principal nilpotent element f prin of the even part of g and a semisimple element x of g such that ad x defines a uality of subregular W -algebras and principal W -superalgebras 19 good Z -grading Γ = Γ prin on g as follows: f prin = n X i =1 e i +1 ,i , x = n +1 X i =0 (cid:18) n + 12 − i + 1 (cid:19) e i,i − e , , if g = sl (1 | n + 1) ,f prin = n − X i =1 ( e i +1 ,i − e − i, − i − ) + e − n,n , x = n X i =0 (cid:18) n − i + 12 (cid:19) t i − t , if g = osp (2 | n ) . The weighted Dynkin diagrams corresponding to Γ prin are as follows. g = sl (1 | n + 1): ❡ × α ❡ α · · · ❡ α n − ❡ α n − ❡ α n . g = osp (2 | n ): ❡ × α ❡ α · · · ❡ α n − ❡ α n − < ❡ α n .The Lie subalgebra g = { u ∈ g | [ x, u ] = 0 } decomposes as g = g red0 ⊕ z , g red0 := Span { e , h , h , f } , z := Span { e h , h , . . . , h n } , where e = e , , f = e , , ˜ h = h − h , if g = sl (1 | n + 1) ,e = e + , + e − , − , f = e , + − e − , − , ˜ h = h − (1 + δ n, ) h , if g = osp (2 | n ) . The Lie subalgebra z is commutative and g red0 is isomorphic to gl (1 |
1) by ι : gl (1 | ∼ −→ g red0 E , , E , , E , , E ,
7→ − r ( h + h ) , rh , e , f , where r = 1, (resp. 2), is the lacity of g = sl (1 | n + 1), (resp. osp (2 | n )). (See alsoSection 2.2.) The restriction of τ k on g is τ k ( u | v ) = ( k + h ∨ )( u | v ) , u, v ∈ z , ( k κ + k κ )( ι − ( u ) | ι − ( v )) , u, v ∈ g red0 , , u ∈ z , v ∈ g red0 , where k = − r ( k + h ∨ ) , k = r ( k + h ∨ ) + 1 , and h ∨ = n is the dual Coxeter number of g = sl (1 | n + 1), osp (2 | n ). (See alsoSection 2.2 for κ i .) Then the affine vertex superalgebra V τ k ( g ) decomposes as V τ k ( g ) ≃ V κ ( gl (1 | ⊗ π k + h ∨ z , (3.16)where κ = k κ + k κ and π k + h ∨ z is the Heisenberg vertex algebra associated with z at level k + h ∨ , (Section 2.1). Since[ α ] = { α , α + α } , [ α i ] = { α i } , i = 2 , . . . , n, (3.17)the screening operators R Q i ( z ) dz := R Q α i ( z ) dz , ( i = 1 , . . . , n ), are Z Q i ( z ) = Z S α i ( z ) dz : V τ k ( g ) → M α i . (3.18) W -algebras and principal W -superalgebras First, consider (3.18) in the case i = 2 , . . . , n . The orthogonal decomposition g = h ⊥ i ⊕ h i , h i = C h i , h ⊥ i = Span { e , f , h | h ∈ h , α i ( h ) = 0 } induces the decomposition of the affine vertex superalgebra V τ k ( g ) V τ k ( g ) ≃ V τ k ( h ⊥ i ) ⊗ π k + h ∨ h i . It follows from (3.16) and (3.17) that we have an isomorphism of V τ κ ( g )-modules M α i ≃ f M α i := V τ k ( h ⊥ i ) ⊗ π k + h ∨ h i , − α i v α i
7→ | i ⊗ | i ⊗ | − α i i . Thus S α i ( z ) is identified with Z S α i ( z ) dz = Z : e − k + h ∨ R α i ( z ) : dz : V τ k ( h ⊥ i ) ⊗ π k + h ∨ h i → f M α i , where α i ( z ) = J h i ( z ), and, therefore, Z Q i ( z ) z = Z : e − k + h ∨ R α i ( z ) : dz. (3.19)Next, consider (3.18) in the case n = 1. By Proposition 2.1, we have a vertexsuperalgebra embedding ρ gl (1 | : V κ ( gl (1 | ֒ → M gl (1 | ⊗ π κ − κ χ , which is injective by Lemma 2.2 and gives an isomorphism V κ ( gl (1 | ≃ Ker (cid:18)Z : b ( z )e r ( k + h ∨ ) R ( χ + χ )( z ) : dz : M gl (1 | ⊗ π κ − κ χ → M gl (1 | ⊗ π κ − κ χ, − ( χ + χ ) (cid:19) . (3.20)for k = − h ∨ by Proposition 2.5. Here M gl (1 | is the bc -system vertex superalgebraand π κ − κ χ is the Heisenberg vertex algebra generated by even fields χ ( z ), χ ( z )with OPEs (2.1). Then it follows from (3.16) and the isomorphism of vertex algebras π k + h ∨ h ∼ −→ π κ − κ χ ⊗ π k + h ∨ z α ( z )
7→ − r ( χ + χ )( z ) , α ( z ) r χ ( z ) , α i ( z ) J h i ( z ) , ( i = 3 , . . . , n ) ,α ( z ) ( J e h ( z ) + ( χ + χ )( z ) , if g = sl (1 | n + 1) ,J e h ( z ) + δ n, ( χ + χ )( z ) , if g = osp (2 | n ) , that we have a vertex superalgebra embedding ρ g : V τ k ( g ) ֒ → M gl (1 | ⊗ π k + h ∨ h , (3.21)which gives an isomorphism for k = − h ∨ V τ k ( g ) ≃ Ker (cid:18)Z : b ( z )e − k + h ∨ R α ( z ) : dz : M gl (1 | ⊗ π k + h ∨ h → M gl (1 | ⊗ π k + h ∨ h , − α (cid:19) . (3.22) uality of subregular W -algebras and principal W -superalgebras 21 Then it gives a V τ k ( g )-module structure on f M α := M gl (1 | ⊗ π k + h ∨ h , − α . We havean isomorphism f M α ≃ b V + ,κ , − ⊗ π k + h ∨ z (3.23)as V τ k ( g ) ≃ V κ ( gl (1 | ⊗ π k + h ∨ z -module for k / ∈ Q by Lemma 2.3 (1) and Propo-sition 2.4. Note that as a V τ k ( g )-module, f M α is generated by a ( g [ t ] ⊕ C K )-submodule e C [ α ] = C | − α i ⊕ C c ( − | − α i . Lemma 3.3.
For k / ∈ Q , we have M α ≃ f M α as V τ k ( g ) -modules.Proof. It follows from (3.23) and the ( g [ t ] ⊕ C K )-module isomorphism C [ α ] ∼ −→ e C [ α ] , v α
7→ | − α i , v α + α
7→ − c ( − | − α i . (cid:3) Under the realization (3.22), Lemma 3.3 implies that the intertwining operator S α ( z ) is identified with S α ( z ) = : e − k + h ∨ R α ( z ) : : Im( ρ g ) → f M α (( z )) , and thus Z Q ( z ) dz = Z : e − k + h ∨ R α ( z ) : dz. (3.24)We also note that S α + α ( z ) = − : c ( z )e − k + h ∨ R α ( z ) :.By (3.6), (3.19) and (3.24), we conclude W k ( g ) ≃ n \ i =1 Ker (cid:18)Z Q i ( z ) dz : V τ k ( g ) → M α i (cid:19) ≃ n \ i =1 Ker (cid:18)Z : e − k + h ∨ R α i ( z ) : dz : Im( ρ g ) → f M α i (cid:19) . The composition Ψ of (3.5) and (3.21) gives an vertex algebra embedding of theprincipal super W -algebra W k ( g ) = W k ( g , f prin ; Γ prin )Ψ : W k ( g ) ֒ → M gl (1 | ⊗ π k + h ∨ h . (3.25)Now (3.22) implies the following realization of the image Im(Ψ ). Theorem 3.4. If k is generic, then we have Im(Ψ ) = n \ i =0 Ker Z Q i ( z ) dz for g = sl (1 | n + 1) , osp (2 | n ) , where Q ( z ) =: b ( z )e − k + h ∨ R α ( z ) : , Q i ( z ) =: e − k + h ∨ R α i ( z ) : , ( i = 1 , . . . , n ) . W -algebras and principal W -superalgebras Dualities in coset vertex algebras
Coset vertex algebras.
Given a vertex superalgebra V and a subalgebra W ⊂ V , the subspaceCom( W, V ) := { a ∈ V | ∀ b ∈ W, Y ( b, z ) Y ( a, w ) ∼ } , is a vertex subalgebra, called the coset vertex algebra of the pair ( V, W ).For g = sl n +1 , so n +1 , define a filed H ( z ) on M sl ⊗ π k + h ∨ h by H ( z ) = ω ∨ ( z ) − : β ( z ) γ ( z ) : ,ω ∨ = n + 1 n X i =1 ( n − i + 1) α i , if g = sl n +1 , n X i =1 α i , if g = so n +1 . (4.1)Note that ω ∨ represents the first fundamental coweight of g . By direct compu-tations, one can show that H ( z ) lies in the kernels of the screening operators R Q i ( z ) dz , (1 ≤ i ≤ n ), and thus it defines a field on W k ( g , f sub ) by Theorem 3.2.Let π H be the Heisenberg vertex subalgebra of W k ( g , f sub ) generated by H ( z )and consider the coset vertex algebra Com (cid:0) π H , W k ( g , f sub ) (cid:1) . We will describe itin terms of screening operators for generic k .Let L = Z x ⊕ Z y a Z -lattice equipped with a bilinear form ( ·|· ) given by( ax + by | cx + dy ) = ac − bd , π L the Heisenberg vertex algebra associate withthe commutative Lie algebra L ⊗ Z C , (Section 2.1), and V L := M ( m,n ) ∈ Z π L ,mx + ny the lattice vertex superalgebra associated with L and the vertex subalgera V x + y := M n ∈ Z π L ,n ( x + y ) . The Friedan-Martinec-Shenker bosonization [F2, Chap. 7] gives a vertex algebraembedding Υ : M sl ֒ → V x + y β ( z ) : e R ( x + y )( z ) : ,γ ( z )
7→ − : x ( z )e − R ( x + y )( z ) : , whose image is equal to the kernel of the screening operator R : e R x ( z ) : dz Im(Υ ) = Ker Z : e R x ( z ) : dz : V x + y → M n ∈ Z π L , ( n +1) x + ny ! . By composing it with (3.15), we obtain a vertex algebra embeddingΥ := Υ ◦ Υ : W k ( g , f sub ) ֒ → V x + y ⊗ π k + h ∨ h , (4.2) uality of subregular W -algebras and principal W -superalgebras 23 whose image for generic k coincides withIm(Υ) = Ker Z : e R x ( z ) : dz ∩ Ker Z : e − k + h ∨ R ( α − ( k + h ∨ )( x + y ))( z ) : dz ∩ n \ i =2 Ker Z : e − k + h ∨ R α i ( z ) : dz (4.3)Let π k + h ∨ e α ⊂ π L ⊗ π k + h ∨ h be the Heisenberg vertex subalgebra generated by e α ( z ) = x ( z ) , e α ( z ) = ( α − ( k + h ∨ )( x + y ))( z ) , e α i ( z ) = α i ( z ) , ( i = 2 , . . . , n − , e α n ( z ) = rα n ( z ) , where r = 1, (resp. 2), is the lacity of g = sl n +1 , (resp. so n +1 ). Since Υ inducesan isomorphism π H ⊗ π k + h ∨ e α ∼ −→ π L ⊗ π k + h ∨ h ⊗ e α i ( z ) ⊗ e α i ( z ) , ( i = 0 , . . . , n ) H ( z ) ⊗ ω ∨ ( z ) − y ( z ) , we have Com (cid:16) π H , V x + y ⊗ π k + h ∨ h (cid:17) = π k + h ∨ e α . Therefore, Υ restricts toΥ : Com (cid:0) π H , W k ( g , f sub ) (cid:1) ֒ → π k + h ∨ e α , (4.4)and thus we obtain the following proposition. Proposition 4.1.
For generic k , Com (cid:0) π H , W k ( g , f sub ) (cid:1) ≃ Ker π k + h ∨ e α Z : e R e α ( z ) : dz ∩ n − \ i =1 Ker π k + h ∨ e α Z : e − k + h ∨ R e α i ( z ) : dz ∩ Ker π k + h ∨ e α Z : e − r ( k + h ∨ ) R e α n ( z ) : dz. Similarly, for g = sl (1 | n + 1) , osp (2 | n ), consider the field H ( z ) on M gl (1 | ⊗ π k + h ∨ h defined by H ( z ) = ω ∨ ( z )+ : b ( z ) c ( z ) : ,ω ∨ = − n n X i =0 ( n − i + 1) α i , if g = sl (1 | n + 1) , − n − X i =0 α i − α n , if g = osp (2 | n ) . (4.5)Note that ω ∨ represents the 0-th fundamental coweight of g . By direct compu-tations, one can show that H ( z ) lies in the kernels of the screening operators R Q i ( z ) dz , (0 ≤ i ≤ n ), and thus H ( z ) defines a field on W k ( g ) by Theorem 3.4.Let π H be the Heisenberg vertex subalgebra of W k ( g ) generated by H ( z ) andconsider the coset vertex algebra Com (cid:0) π H , W k ( g ) (cid:1) . We will describe it in termsof screening operators for generic k .Let Z = Z φ a Z -lattice equipped with a bilinear form ( mφ | nφ ) = mn , π Z theHeisenberg vertex algebra associate with the commutative Lie algebra Z ⊗ Z C , W -algebras and principal W -superalgebras (Section 2.1), and V Z := M m ∈ Z π Z ,mφ the lattice vertex superalgebra associated with Z . By the boson-fermion correspon-dence, e.g., [FBZ, Chap. 5], we have an isomorphismΨ : M gl (1 | ∼ −→ V Z b ( z ) : e R φ ( z ) : ,c ( z ) : e R − φ ( z ) : . Composing it with (3.25), we obtain a vertex algebra embeddingΨ := Ψ ◦ Ψ : W k ( g ) ֒ → V Z ⊗ π k + h ∨ h , (4.6)whose image for generic k coincides withIm(Ψ) = Ker Z : e − k + h ∨ R ( α − ( k + h ∨ ) φ )( z ) : dz ∩ n \ i =1 Ker Z : e − k + h ∨ R α i ( z ) : dz. (4.7)Let π k + h ∨ e β ⊂ V Z ⊗ π k + h ∨ h be the Heisenberg vertex subalgebra generated by e β ( z ) = − k + h ∨ α ( z ) + φ ( z ) , e β i ( z ) = − k + h ∨ α i ( z ) , ( i = 1 , . . . , n ) . Since Ψ induces an isomorphism π H ⊗ π k + h ∨ e β ∼ −→ π Z ⊗ π k + h ∨ h ⊗ e β i ( z ) e β i ( z ) , ( i = 0 , . . . , n ) H ( z ) ⊗ ω ∨ ( z ) + φ ( z ) , we have Com (cid:16) π H , V Z ⊗ π k + h ∨ h (cid:17) = π k + h ∨ e β . Therefore, Ψ restricts toΨ : Com (cid:0) π H , W k ( g ) (cid:1) ֒ → π k + h ∨ e β , (4.8)and thus we obtain the following proposition. Proposition 4.2.
For generic k , we have an isomorphism Com (cid:0) π H , W k ( g ) (cid:1) ≃ n \ i =0 Ker π k + h ∨ e β Z : e R e β i ( z ) : dz where g = sl (1 | n + 1) or osp (2 | n ) . Feigin-Frenkel type duality.
Let ( g , g ) = ( sl n +1 , sl (1 | n +1)) or ( so n +1 , osp (2 | n )).Denote by r the lacity of g , (which is equal to that of g ), and by h ∨ i the dualCoxeter number of g i . Then( r, h ∨ , h ∨ ) = ( (1 , n + 1 , n ) , if ( g , g ) = ( sl n +1 , sl (1 | n + 1)) , (2 , n − , n ) , if ( g , g ) = ( so n +1 , osp (2 | n )) . For k ∈ C \{− h ∨ } , define k ∈ C by the formula r ( k + h ∨ )( k + h ∨ ) = 1 . (4.9) uality of subregular W -algebras and principal W -superalgebras 25 Theorem 4.3.
Let k , k ∈ C be generic satisfying (4.9) . Then we have an iso-morphism Com (cid:0) π H , W k ( g , f sub ) (cid:1) ≃ Com (cid:0) π H , W k ( g ) (cid:1) , for ( g , g ) = ( sl n +1 , sl (1 | n + 1)) , or ( so n +1 , osp (2 | n )) .Proof. For k , k ∈ C satisfying (4.9), we have an isomorphism π k + h ∨ e α → π k + h ∨ e β e α i ( z ) e β i ( z ) , ( i = 0 , . . . , n ) , since both of the Gram matrices for { e α i } ni =0 and { e β i } ni =0 are · · · n − n − n −K · · · −K K −K · · · −K K · · · n − · · · K −K n − · · · −K K − r K n · · · − r K r K , where K = k + h ∨ . By applying the Feigin-Frenkel duality for the Virasoro vertexalgebras (cf. [FBZ, Chap. 15]), we haveKer π k h ∨ e α Z : e − k h ∨ R e α i ( z ) : dz = Ker π k h ∨ e α Z : e R e α i ( z ) : dz, ( i = 1 , . . . , n − , Ker π k h ∨ e α Z : e − r ( k h ∨ R e α n ( z ) : dz = Ker π k h ∨ e α Z : e R e α n ( z ) : dz, for generic k . Hence, for generic k , k ∈ C satisfying (4.9),Com (cid:0) π H , W k ( g , f sub ) (cid:1) ≃ n \ i =0 Ker π k h ∨ e α Z : e R e α i ( z ) : dz ≃ n \ i =0 Ker π k h ∨ e β Z : e R e β i ( z ) : dz ≃ Com (cid:0) π H , W k ( g ) (cid:1) by Proposition 4.1 and Proposition 4.2. (cid:3) Kazama-Suzuki Coset.
Let ( g , g ) be as in Section 4.2. Let π e H ⊂ W k ( g , f sub ) ⊗ V Z be the Heisenberg vertex subalgebra generated by the field e H ( z ) := φ ( z ) − H ( z ) = − ω ∨ ( z ) + y ( z ) + φ ( z ) (4.10)(see (4.1)). Next, consider the lattice Z √− ⊂ C , i.e., the lattice Z ψ , spanned by ψ ,equipped with a bilinear form ( ·|· ), which satisfies ( mψ | nψ ) = − mn . Let π √− Z bethe Heisenberg vertex algebra associated with the abelian Lie algebra C ⊗ Z Z √− V Z √− := M n ∈ Z π Z √− ,nψ . W -algebras and principal W -superalgebras the lattice vertex superalgebra associated with Z √−
1. Let π e H ⊂ W k ( g ) ⊗ V Z √− ,be the Heisenberg vertex subalgebra generated by the field e H ( z ) := ψ ( z ) + H ( z ) = ω ∨ ( z ) + φ ( z ) + ψ ( z ) (4.11)(see (4.5)). Theorem 4.4.
Let k , k ∈ C be generic satisfying (4.9) . Then we have isomor-phisms (1) W k ( g , f sub ) ≃ Com (cid:16) π e H , W k ( g ) ⊗ V Z √− (cid:17) , (2) W k ( g ) ≃ Com (cid:16) π e H , W k ( g , f sub ) ⊗ V Z (cid:17) ,for ( g , g ) = ( sl n +1 , sl (1 | n + 1)) , or ( so n +1 , osp (2 | n )) .Proof. For g i , let h i denote its Cartan subalgebra, r i its lacity, and h ∨ i its dualCoxeter number. Let { α i } ni =1 , (resp. { β i } ni =0 ), be the set of simple roots of g ,(resp. g ), and α i ( z ) = Y ( α i ( − | i , z ) the corresponding fields on π k + h ∨ h , (resp. β i ( z ) = Y ( β i ( − | i , z ) on π k + h ∨ h ).First, we will show (1). By (4.6), we have a vertex superalgebra embeddingΨ ⊗ id : W k ( g ) ⊗ V Z √− ֒ → V Z ⊗ π k + h ∨ h ⊗ V Z √− . (4.12)Let V Z ( φ + ψ ) ⊂ V Z ⊗ V Z √− be the lattice vertex subalgebra corresponding to thesublattice Z ( φ + ψ ) ⊂ Z + Z √− V X + Y ⊂ V Z ⊗ π k + h ∨ h ⊗ V Z √− the vertex subalgebra generated by V Z ( φ + ψ ) and the Heisenberg vertex subalgebra π X,Y generated by the fields X ( z ) = − k + h ∨ β ( z ) + φ ( z ) , Y ( z ) = 1 k + h ∨ β ( z ) + ψ ( z ) . Let π A ⊂ V Z ⊗ π k + h ∨ h ⊗ V Z √− be the Heisenberg vertex subalgebra generated bythe fields A i ( z ) = r β ( z ) − φ ( z ) − ψ ( z ) , i = 1 ,r β i ( z ) , i = 2 , . . . , n − ,β n ( z ) , i = n. It follows from X ( z ) A i ( w ) ∼ ∼ Y ( z ) A i ( w ) , i = 1 , . . . , n, that V X + Y ⊗ π A ⊂ V Z ⊗ π k + h ∨ h ⊗ V Z √− . By (4.11), we haveCom (cid:16) π e H , V Z ⊗ π k + h ∨ h ⊗ V Z √− (cid:17) = V X + Y ⊗ π A , and thus (4.12) impliesCom (cid:16) π e H , W k ( g ) ⊗ V Z √− (cid:17) ֒ → V X + Y ⊗ π A , uality of subregular W -algebras and principal W -superalgebras 27 whose image for generic k coincides withΨ ⊗ id (cid:16) Com (cid:16) π e H , W k ( g ) ⊗ V Z √− (cid:17)(cid:17) = Ker Z : e R X ( z ) : dz ∩ Ker Z : e − r k h ∨ R ( A + X + Y )( z ) : dz ∩ n − \ i =2 Ker Z : e − r k h ∨ R A i ( z ) : dz ∩ Ker Z : e − k h ∨ R A n ( z ) : dz by (4.7). Since ( A + X + Y ) (0) | n ( φ + ψ ) i = 0, we haveKer Z : e − r k h ∨ R ( A + X + Y )( z ) : dz = M n ∈ Z (cid:18) Ker π X,Y Z : e − r k h ∨ R ( A + X + Y )( z ) : dz (cid:19) ( − | n ( φ + ψ ) i = M n ∈ Z (cid:18) Ker π X,Y Z : e R ( A + X + Y )( z ) : dz (cid:19) ( − | n ( φ + ψ ) i = Ker Z : e R ( A + X + Y )( z ) : dz. by the Feigin-Frenkel duality for the Virasoro vertex algebras, (cf. [FBZ, Chap.15]). Similarly, we haveKer Z : e − r k h ∨ R A i ( z ) : dz = Ker Z : e R A i ( z ) : dz, i = 1 , . . . , n − , Ker Z : e − k h ∨ R A n ( z ) : dz = Ker Z : e R A n ( z ) : dz. Therefore, we have(Ψ ⊗ id) (cid:16) Com (cid:16) π e H , W k ( g ) ⊗ V Z √− (cid:17)(cid:17) ≃ Ker Z : e R X ( z ) : dz ∩ Ker Z : e R ( A + X + Y )( z ) : dz ∩ n \ i =2 Ker Z : e R A i ( z ) : dz. Now (1) follows from the above equation with (4.3) and the isomorphism V X + Y ⊗ π A ∼ −→ V x + y ⊗ π k + h ∨ h X ( z ) x ( z ) , Y ( z ) y ( z ) , A i ( z )
7→ − k + h ∨ α i ( z ) , i = 1 , . . . , n. Next, we will show (2) in the same way as the proof of (1). By (4.2), we have avertex superalgebra embeddingΥ ⊗ id : W k ( g , f sub ) ⊗ V Z ֒ → V x + y ⊗ π k + h ∨ h ⊗ V Z . Let V e Z be the vertex superalgebra generated by the fields : e R ± e φ ( z ) : where e φ ( z ) = x ( z ) + y ( z ) + φ ( z ) , and π B the Heisenberg vertex subalgebra generated by the fields B ( z ) = − y ( z ) − φ ( z ) , B ( z ) = α ( z ) − ( k + h ∨ )( x + y )( z ) ,B i ( z ) = α i ( z ) , ( i = 2 , . . . , n − , B n ( z ) = r α n ( z ) . W -algebras and principal W -superalgebras Then we have Com (cid:16) π e H , V x + y ⊗ π k + h ∨ h ⊗ V Z (cid:17) = V e Z ⊗ π B and thusΥ ⊗ id : Com (cid:16) π e H , W k ( g , f sub ) ⊗ V Z (cid:17) ֒ → V e Z ⊗ π B , whose image for generic k coincides withKer Z : e R ( B + e φ )( z ) : dz ∩ n − \ i =1 Ker Z : e − k h ∨ R B i ( z ) : dz ∩ Ker Z : e − r k h ∨ R B n ( z ) : dz = Ker Z : e R ( B + e φ )( z ) : dz ∩ n \ i =1 Ker Z : e R B i ( z ) : dz by (4.3) and the Feigin-Frenkel duality for the Virasoro vertex algebras. Now (2)follows from the above equation with (4.7) and the isomorphism V e Z ⊗ π B ∼ −→ V Z ⊗ π k + h ∨ h : e R ± e φ ( z ) : : e R ± φ ( z ) : , B i ( z )
7→ − k + h ∨ β i ( z ) , i = 0 , . . . , n. (cid:3) Applications
Here we discuss some important consequences of the theory of Heisenberg cosets[CKLR], for earlier literature see [Li3].5.1.
Simplicity of Heisenberg cosets.
We begin with simple observations forvertex superalgebras. Let B be a simple vertex superalgebra and { B λ } λ ∈ I a setof countably many inequivalent simple B -modules which contains B as B . Let C be a (not necessarily simple) vertex superalgebra and { C λ } λ ∈ I a set of C -modules,which contains C as C , with the same index set I . Assume that the B ⊗ C -module A := M λ ∈ I B λ ⊗ C λ has the structure of a vertex superalgebra extending the B ⊗ C -module structure.Assume the following conditions:(1) For each λ ∈ I , B λ and C λ are of countable dimension.(2) The B -modules { B λ } λ ∈ I are objects in a semi-simple abelian fullsub cat-egory B of the category of weak B -modules. B is closed under direct sumover any countable index set.Then we have the following Lemma. Proposition 5.1. (cf. [CL3, Thm 8.1, Rem 8.3] ) Any quotient A s of the vertexsuperalgebra A is an object of B . The coset vertex superalgebra D := Com( B, A s ) is a quotient of C . For each λ ∈ I , there exists some quotient D λ of the C -module C λ , which admits a D -module structure such that A s ≃ M λ ∈ I B λ ⊗ D λ as B ⊗ D -modules. uality of subregular W -algebras and principal W -superalgebras 29 Proof.
The assertions are obvious, but we include the proof for the completeness ofthe paper. Since A is of countable dimension, A is an object of B by the assumptionson B . Since B is abelian, any quotient of A as a vertex superalgebra is an object of B . Since B is semi-simple, we have A s ≃ M λ ∈ I B λ ⊗ D λ , D λ := Hom B ( B λ , A s )as B -modules. Then we have a natural surjection C λ ։ D λ for each λ ∈ I as vectorspaces. Since D ≃ Com(
B, A s ), it is a vertex superalgebra obtained as a quotientof C . Since A s is a D -module, the same is true for D λ = Hom B ( B λ , A s ). (cid:3) Next, we consider a criterion for the simplicity of the coset vertex superalgebraof a pair of simple vertex superalgebras.
Lemma 5.2.
Let V be a vertex superalgebra and M a V -module. Let m ∈ M be avacuum-like vector, i.e., satisfies a ( n ) m = 0 for all a ∈ V , n ≥ . Then the C -linearmap F m : V → M, a a ( − m is a V -module homomorphism.Proof. We need to show ( a ( n ) b ) ( − m = a ( n ) ( b ( − m ) for all a, b ∈ V and n ∈ Z ,which are special cases of [LLi, Prop 4.5.6]. (cid:3) Now, let W ⊂ V be simple vertex superalgebras of countable dimension. Supposethat V is semi-simple as a W -module: V = M λ ∈ Λ c W λ , where Λ is an index set that labels the inequivalent simple W -modules W λ thatappear as submodules of V , and c W λ is the W -submodule spanned by all the simple W -submodules isomorphic to W λ . We assume 0 ∈ Λ and W = W . Lemma 5.3.
The subspace c W is isomorphic to W ⊗ Com(
W, V ) as a W -module.In particular, c W has the structure of a vertex superalgebra.Proof. Since c W is semi-simple as a W -module, we have simple submodules { M α } α ∈ I such that c W = L α ∈ I M α . Note that M α is isomorphic to W as a W -module. Since W is simple and of countable dimension over C , we have End W ( W ) = C by Schur’slemma. Thus Com( W, W ) = C | i and so the space of vacuum-like vectors in M α is one dimensional. Note that the vacuum-like vectors in V with respect to W isnothing but Com( W, V ). Therefore, the linear map W ⊗ Com(
W, V ) → c W , w ⊗ u w ( − u is an isomorphism of W -modules by Lemma 5.2. (cid:3) Now we have a criterion for the simplicity of Com(
W, V ). Proposition 5.4.
Let V be a simple vertex superalgebra of countable dimensionand W a simple vertex (super)subalgebra. Suppose that V is semi-simple as a W -module and W ⊗ Com(
W, V ) has a conformal vector ω which is also a conformalvector of V . Then Com(
W, V ) is also simple. W -algebras and principal W -superalgebras Proof.
Take a non-zero ideal
I ⊂
Com(
W, V ) and let b I ⊂ V denote the idealgenerated by I in V . We show b I ∩ ( W ⊗ Com(
W, V )) = W ⊗ I . (5.1)Indeed, by [LLi, Prop 4.5.6], we have b I = Span C { a ( n ) u | a ∈ V, u ∈ I , n ∈ Z } . (5.2)Then by using the conformal field Y ( ω, z ) = P n ∈ Z L n z − n − and the skew-symmetry: Y ( a, z ) u = − ( − ¯ a ¯ u e zL − Y ( u, − z ) a, we have b I = Span C { u ( n ) a | u ∈ I , a ∈ V, n ∈ Z } = X λ ∈ Λ X n ∈ Z I ( n ) c W λ . Since vertex operators u ( n ) commute with w ( m ) for all u ∈ Com(
W, V ) and w ∈ W , the actions u ( n ) on W λ are in Hom W ( W λ , V ). By Schur’s lemma, we haveHom W ( W λ , W µ ) = 0 if λ = µ . Thus, I ( n ) c W λ ⊂ ( W ⊗ Com(
W, V )) ( n ) c W λ ⊂ c W λ so that we have b I ∩ ( W ⊗ Com(
W, V )) = Span C { u ( n ) a | u ∈ I , a ∈ W ⊗ Com(
W, V ) , n ∈ Z } = W ⊗ I . Since V is simple, b I = V and thus I = Com( W, V ). This shows that Com(
W, V )is simple. (cid:3)
Consider the case when V is a simple vertex operator superalgebra and W isa Heisenberg vertex subalgebra generated by primary fields of conformal degree 1.Let H ⊂ W be a subspace of primary fields of conformal degree 1 with OPEs h ( z ) h ( w ) ∼ ℓ ( h , h )( z − w ) , h , h ∈ H (5.3)for some symmetric bilinear form ℓ : H × H → C , generating W . Suppose that ℓ is non-degenerate (i.e. W is simple) and the conditions (1)(2) in Corollary 5.5.Then the assumption in Proposition 5.4 for the conformal vector is automaticallysatisfied, and we have c W = \ h ∈H Ker( h (0) ) . The conformal vector ω lies in this subspace since h (0) ω = 0 for h ∈ H . Theseimply Proposition 3.2 and Theorem 2.9 in [CKLR]: Corollary 5.5 ([CKLR, Proposition 3.2]) . Let V be a vertex operator superalgebraof countable dimension and π a non-degenerate Heisenberg vertex subalgebra gen-erated by a subspace H spanned by primary fields of conformal degree 1. Supposethe following conditions: (1) The action
H × V → V , (( h, a ) h (0) a ) , is semi-simple, (2) Each π -submodule spanned by all a ∈ V satisfying that h (0) a = λ ( h ) a for h ∈ H with fixed λ ∈ H ∗ is bounded from below by the conformal degree.Then a simple quotient V ։ V s induces a simple quotient Com( π, V ) ։ Com( π, V s ) . uality of subregular W -algebras and principal W -superalgebras 31 Proof.
By the assumption, V is semi-simple as a π -module, cf [FLM, Thm1.7.3].Thus we have V ≃ M λ ∈H ∗ π λ ⊗ C λ , C λ := (cid:8) a ∈ V | h ( n ) a = δ n, λ ( h ) a, ∀ h ∈ H , n ≥ (cid:9) (5.4)as π ⊗ Com( π, V )-modules, where π λ is the highest weight π -module with the highestweight λ . Then Proposition 5.1 applies and we obtain a quotientCom( π, V ) ։ Com( π, V s ) , induced by V → V s . By the argument just after Proposition 5.4, Proposition 5.4applies to conclude that Com( π, V s ) is simple. (cid:3) Corollary 5.6 ([CKLR, Theorem 2.9]) . Suppose that V is simple in addition tothe assumptions in Corollary 5.5. Then each C λ appearing in the decomposition (5.4) is a simple Com( π, V ) -module.Proof. First, observe that operator product algebra of V respects Heisenberg weight,since every vertex operator of V is especially an intertwiner for the Heisenberg sub-algebra. This means that v ( n ) w ∈ c W λ + µ = π λ + µ ⊗ C λ + µ for v ∈ c W λ = π λ ⊗ C λ and w ∈ c W µ = π µ ⊗ C µ . Now, fix a c W ν appearing in the decomposition of V andlet π ν ⊗ I ⊂ c W ν be a π ⊗ Com( π, V )-submodule of c W ν . Denote by b I the idealgenerated by π ν ⊗ I in V . Then by the same argument and same notation as inthe proof of Proposition 5.4, we have b I = Span C { u ( n ) a | u ∈ π ν ⊗ I , a ∈ V, n ∈ Z } = X λ ∈H ∗ X n ∈ Z ( π ν ⊗ I ) ( n ) c W λ . But we just explained that( π ν ⊗ I ) ( n ) c W λ ⊂ ( c W ν ) ( n ) c W λ ⊂ c W ν + λ . Therefore, b I ∩ c W λ = Span C { u ( n ) a | u ∈ π ν ⊗ I , a ∈ π ⊗ C, n ∈ Z } = π ν ⊗ I . Since V is simple, we have b I = V and thus I = C λ , that is, C λ is simple as aCom( π, V )-module. (cid:3) C -cofiniteness of Kazama-Suzuki cosets. The aim of this subsection isto study if the C -cofiniteness is inherited via a Kazama-Suzuki type coset con-struction. The proof follows essentially from results in [CKLR]. The main issue isto understand whether double commutants of Heisenberg cosets are lattice vertexoperator algebras.Recall the decomposition (5.4) as a π ⊗ Com( π, V )-module. We remark thatthe set L = { λ ∈ H ∗ | C λ = 0 } is a lattice if V is simple, see [LX, Proposition 3.6](statements in [LX] are stated for vertex operator algebras, but exactly the sameproofs apply for vertex operator superalgebras). Assumption 5.7.
Under the set-up of Corollary 5.5 for a simple vertex operatorsuperalgebra V , let L = { λ ∈ H ∗ | C λ = 0 } be the lattice of weights appearing in V .Then we assume the following conditions for a basis B = { h , . . . , h n } of H : (1) The bilinear form ℓ in (5.3) satisfies ℓ ( h i , h j ) ∈ R for all i, j . (2) λ ( h i ) ∈ R for all i and λ ∈ L . (3) L ⊗ Z C = H ∗ . W -algebras and principal W -superalgebras This reality assumption insures the existence of vertex tensor category of highestweight π -modules. Invoking the proof of [CKLR, Theorem 3.5], we obtain thefollowing theorem: Lemma 5.8.
Let V be a simple vertex operator superalgebra of countable dimensionwith a rank n Heisenberg subalgebra π acting semi-simply on V and denote by C = Com( π, V ) the coset. If C is a C -cofinite vertex operator algebra of CFT-type and Assumption 5.7 holds, then Com(
C, V ) is a rank n lattice vertex operatorsuperalgebra V N . Moreover, if the lattice N is positive definite, V is C -cofinite.Proof. Since C is a C -cofinite vertex operator algebra of CFT-type, [Hu3, Proposi-tion 4.1, Theorem 4.13] insure the assumption just after [CKLR, Proposition 3.2] forthe existence of the vertex tensor category of (generalized) C -modules. By [CKLR,Theorem 2.3], we also have a vertex tensor category generated by { π λ } λ ∈ L . By[CKM2, Theorem 5.5], the vertex tensor category of the tensor product π ⊗ C isthe Deligne product of these two categories. Thus the same arguments in Section 3in [CKLR] for the proof of [CKLR, Theorem 3.5] may apply. This guarantees that N = { λ ∈ L | C λ ≃ C as C -modules } is a sublattice of L and C µ ≃ C µ + λ for all λ ∈ N and µ ∈ L . By setting C [ µ ] := C µ for [ µ ] ∈ L/N , we have the following decomposition [CKLR, (3.7)]: V ≃ M [ µ ] ∈ L/N V [ µ ] ⊗ C [ µ ] , V [ µ ] = M λ ∈ N π µ + λ . (5.5)Especially, Com( C, V ) = V [0] = V N is the lattice vertex operator superalgebra ofthe lattice N . Again by the same arguments for the proof of [CKLR, Thm 3.5],the C -modules C [ µ ] , ([ µ ] ∈ L/N ), are mutually inequivalent and simple. Since C is C -cofinite, C has only finitely many inequivalent simple ordinary modules by [Li2,Thm 3.13]. Thus L/N is a finite set. Therefore N is a full rank sublattice of L .Now we assume that N is positive definite so that V N is especially C -cofiniteand of CFT-type. Then V is an extension (5.5) of the C -cofinite vertex operatorsuperalgebra C ⊗ V N of CFT-type by finitely many simple C ⊗ V N -modules. By[ABD, Prop 5.2], each V [ µ ] ⊗ C [ µ ] is already C -cofinite as a simple C ⊗ V N -moduleand thus V itself is C -cofinite. (cid:3) To apply this lemma to our setting, let us introduce a few Heisenberg vertexoperator algebras. Let c = C α ⊕ C β be a C -vector space equipped with a symmetricbilinear form (- | -) : c ⊗ c → C determined by ( α | α ) = q ∈ Q > , ( β | β ) = r ∈ Z \{ } and ( α | β ) = 0. We assume q + r = 0. Then the elements γ = ( α + β ) / ( q + r ) and µ = ( rα − qβ ) / ( q + r ) span c and satisfy( γ | γ ) = 1 q + r , ( µ | µ ) = qrq + r , ( γ | µ ) = 0 . For η ∈ { α, β, γ, µ } , let π η denote the Heisenberg vertex subalgebra of π c generatedby η , and π η,n the π η -module π η,nη for simplicity. Then we have π α,n ⊗ π β,m ≃ π γ,qn + rm ⊗ π µ,n − m (5.6)as π γ ⊗ π µ -modules. Now, let W be a simple vertex superalgebra containing π α such that W ≃ M n ∈ Z π α,n ⊗ D n , uality of subregular W -algebras and principal W -superalgebras 33 as π α ⊗ D -modules. Here D = D = Com( π α , W ) and D n is a D -module. Let V Z β denote the lattice vertex operator superalgebra associated with the lattice Z β andconsider W ⊗ V Z β and the coset C = Com( π γ , W ⊗ V Z β ) , where we consider an embedding π γ ֒ → π α ⊗ π β ⊂ W ⊗ V Z β . Then D = Com( π α ⊗ π β , W ⊗ V Z β ) = Com( π µ ⊗ π γ , W ⊗ V Z β )= Com( π µ , Com( π γ , W ⊗ V Z β )) = Com( π µ , C ) . If C is a C -cofinite vertex operator algebra of CFT-type, the assumptions ofLemma 5.8 for V = W ⊗ V Z β and C = C are satisfied. Thus, we have W ⊗ V Z β ≃ M [ λ ] ∈ L/N V [ λ ] ⊗ C [ λ ] (5.7)with L = Z ( q + r ) γ and N = Z a ( q + r ) γ for some a ∈ Z > . Lemma 5.9.
Let C , D be vertex algebras given in the above and suppose that C is a C -cofinite vertex operator algebra of CFT-type. Then Com(
D, C ) is a latticevertex algebra of some positive-definite even lattice with rank one.Proof. Since C is a vertex operator algebra and D = Com( π µ , C ), the doublecommutant Com( D, C ) is either π µ or a larger vertex operator algebra that extends π µ . We have W ⊗ V Z β ≃ M n,m ∈ Z π α,n ⊗ π β,m ⊗ D n ≃ M n,m ∈ Z π γ,qn + rm ⊗ π µ,n − m ⊗ D n . On the other hand, we have the decomposition (5.7) withCom( C , W ⊗ V Z β ) = V N = M s ∈ Z π γ,a ( q + r ) s . For any s ∈ Z , π γ,qn + rm ≃ π γ,a ( q + r ) s if and only if qn + rm = a ( q + r ) s . Thus, itfollows that C ≃ M n,m ∈ Z qn + rm = a ( q + r ) s π µ,n − m ⊗ D n ≃ M n,m ∈ Z qn + rm =0 π µ,n − m ⊗ D n − as for all s ∈ Z and hence D n ≃ D n − as for all s ∈ Z . This implies thatCom( D, C ) ≃ M n ∈ a Z ,m ∈ Z qn + rm =0 π µ,n − m ≃ M s,m ∈ Z aqs + rm =0 π µ,as − m ≃ M s ∈ Z π µ, a ( q + r ) r s . This must be an even lattice vertex operator algebra, since it is a vertex algebraextension of a Heisenberg algebra. From the assumption, we have ( µ, µ ) ∈ Q \{ } If ( µ, µ ) <
0, then the lattice has negative signature and thus conformal weights ofthe lattice vertex operator algebra are not bounded from below. This is impossiblefor the coset of a vertex operator algebra of CFT-type. Hence, ( µ, µ ) must bepositive. (cid:3)
Corollary 5.10.
With the same set-up as in Lemma 5.9, D is C -cofinite.Proof. The assertion is immediate from [Mi, Corollary 2]. (cid:3) W -algebras and principal W -superalgebras Rationality of Heisenberg cosets.
Heisenberg cosets of strongly rationalvertex operator algebras are strongly rational in the following sense:
Theorem 5.11.
Let V be a simple, rational, C -cofinite vertex operator algebra ofCFT-type. Suppose that U ⊂ H ⊂ V where H is a Cartan subalgebra of V and U is a nondegenerate subspace, so that π U is a simple Heisenberg vertex operatoralgebra of rank the dimension of U . Set C = Com( π U , V ) . Then (1) C is simple and of CFT-type. (2) C is C -cofinite. (3) Every grading-restricted generalized C -module is completely reducible.Proof. First, we show (1). Let ω U , ω V be the conformal vectors of π U , V respec-tively. Then ω V − ω U is the conformal vector of C . Since ( ω V ) ( n ) = ( ω V − ω U ) ( n ) on C for n ∈ Z ≥ , each homogeneous subspace of C with conformal degree m is inthat of V with the same degree. This proves that C is of CFT-type. The simplicityof C is immediate from Corollary 5.5. Then (1) is proven.Next, we show (2) and (3). By [Ma, Theorem 1], there is a unique maximal vertexsubalgebra W ⊂ V with the conformal vector ω U , which is indeed isomorphic to alattice vertex operator algebra V Λ for a positive-definite even lattice Λ ⊂ U withdim U = rank Λ. This implies that Com( C, V ) = V Λ . Then (2), (3) follow from [Mi,Corollary 2], [CKLR, Theorem 4.12] respectively. This completes the proof. (cid:3) Corollary 5.12.
Let V be a simple vertex superalgebra of CFT-type such that theeven subalgebra V is simple, rational, C -cofinite and of CFT-type. Then Theorem5.11 also holds for V .Proof. First, (1) is clear by using the same proof of (1) in Theorem 5.11. We needto show (2) and (3). Let C = C ⊕ C be the decomposition of C into the evenand odd parts. Then by Theorem 5.11, C = Com( π U , V ) is simple, C -cofiniteand of CFT-type. Consider Z -action on C with respect to the decomposition C = C ⊕ C . Then C = C Z . By [DM, Theorem 3] (where the assertion isproved for simple vertex operator algebras, but the same proof applies for simplevertex operator superalgebras), it follows that C is a simple C -module. Then C isalready C -cofininte as a C -module by [ABD, Prop 5.2]. This proves (2). Finally,we show (3), but this follows from Theorem 5.13. (cid:3) To state Theorem 5.13, let us introduce several categories consisting of V -modules. Let V be a vertex algebra and C be the vertex tensor category of V -module, that is, assume that C exists. Let V be a simple vertex superalgebrawith V = V ⊕ V the decomposition into even and odd part. Suppose that C is semisimple and V is an object in C . The vertex superalgebra V in Corollary5.12 satisfies these assumptions: The category of V -module forms a vertex tensorcategory by [Hu1, Hu2] and is in fact a modular tensor category C . The proof of[CKLR, Thm 3.1] also applies to the vertex superalgebra V and so V is a simplecurrent for V .By [CKL], V is a commutative associative superalgebra object in C . This al-gebra is an order two simple current extensions and these are always seperable[DMNO, Example 2.8] (special Frobenius algebras in the language of [FRS]). Let C V be the supercategory of modules for the commutative associative superalgebraobject V and C loc V be the full subcategory of C V consisting of local modules for V . Then objects in C V consist of pairs ( M, µ M ) of objects M = M ⊕ M with uality of subregular W -algebras and principal W -superalgebras 35 M , M ∈ C and even morphisms µ M : V ⊠ M → M satisfying some conditions.An object ( M, µ M ) in C V is called local if µ W ◦ R M,V ◦ R
V,M = µ M for the braid-ing isomorphism R N,N ′ : N ⊠ N ′ → N ′ ⊠ N . See [CKM1] for the definitions andproperties of C V and C loc V . By [CKM1, Theorem 1.3], C loc V is equivalent to the su-percategory of V -modules in C as braided tensor supercategories, and there is aninduction functor F whose right-adjoint is the restriction functor G : F : C → C V , M ( V ⊠ M, µ V ⊠ M ); G : C V → C , ( M, µ M ) M. See [CKM1] for details on these functors.
Theorem 5.13.
Let V be a simple vertex algebra, V be a simple vertex superalgebrawith V = V ⊕ V the decomposition into even and odd parts and C be a vertextensor category of modules of the vertex algebra V . Suppose that V is an object in C , and C is semisimple and rigid. Then the category C loc V of local V -modules thatlie in C is semisimple as well.Proof. Recall that the supercategory is an enriched category of the category ofsuperspaces [BE], and have the underlying category whose objects are the samebut morphisms consist of even parts. Denote by D the underlying category of asupercategory D . Then we have a forgetful functor H D : D → D that just forgetsthe parities on objects and morphisms.Let H = H C V : C V → C V which restricts to a functor on the subcategories of localmodules H = H C loc V : C loc V → C loc V . Since V is a simple vertex superalgebra that liesin a semisimple vertex tensor category C of the simple vertex algebra V , the proofof [CKLR, Thm 3.1] applies to V and so V is a simple current for V . Then V isseparable by [DMNO, Example 2.8]. Then the category C V of the separable algebraobject V in the fusion category C must be semisimple by [DMNO, Proposition 2.7].Moreover, by [CKM1, Prop 4.5, Rem 2.64], every simple object in C loc V is of the form H ◦ F ( X ) for some simple X in C .Let F = H ◦ F and G : C V → C be the restriction functor. Then by definition, G = G ◦ H . Let (
X, µ X ) be any object in C loc V . Then H ( X, µ X ) ≃ M i ∈ I ( Y i , µ Y i )as an object in C loc V with simple objects ( Y i , µ Y i ) in C V for some finite index set I .Let Y i ⊕ Y i be the decompositon of Y i into even and odd parts. By [CKM1, Cor.4.22] we then have that Y i is simple in C and ( Y i , µ Y i ) ≃ F ( Y i ) in C V . Thus, H ( X, µ X ) ≃ F ( X ) , X = M i ∈ I Y i ∈ C . Let (
Z, µ Z ) be an arbitrary object in C loc V that is simple in C V . Then by [CKM1,Prop 4.5], ( Z, µ Z ) ≃ F ( Z ) for some simple Z in C . Since G is the right-adjointof F , we haveHom C loc V (( Z, µ Z ) , ( X, µ X )) ≃ Hom C V ( F ( Z ) , ( X, µ X )) ≃ Hom C ( Z , G ( X, µ X )) ≃ Hom C ( Z , G ◦ F ( X )) ≃ Hom C loc V (( Z, µ Z ) , F ( X )) , where we use the equations G ( X, µ X ) = G ◦ H ( X, µ X ) ≃ G ◦ F ( X ) = G ◦ H ◦ F ( X ) = G ◦ F ( X ) . W -algebras and principal W -superalgebras But since F ( X ) is completely reducible in C loc V , this can only be true for all ob-jects ( Z, µ Z ) in C loc V that are simple in C V if F ( X ) ֒ → ( X, µ X ) in C loc V , . Since G ( F ( X )) ∼ = G ( X, µ X ), these two modules have the same graded dimensions andso the embedding must be an isomorphism, i.e. F ( X ) is isomorphic to ( X, µ X ) in C loc V . (cid:3) Application to the main Theorem.
We strengthen Theorem 4.2 and The-orem 4.4 from generic levels to all the “non-critical” levels, descend them to thesimple quotients, and obtain a rationality result.Let V be a finite dimensional vector space over C . A family of vector subspaces { W α } α ∈ C is called continuous if they are of the same dimension d ∈ Z ≥ and theinduced map C → Gr( d, V ) to the Grassmannian manifold is continuous [T]. Fora Z -graded vector space V = L n ∈ Z V n such that dim V n < ∞ , ( n ∈ Z ), a family ofgraded vector subspaces { W α } α ∈ C , ( W α = L n ∈ Z W αn ), is called continuous if thehomogeneous subspaces { W αn } α ∈ C , ( n ∈ Z ), are continuous families. Lemma 5.14.
Let V = L n ∈ Z V n be a Z -graded vertex superalgebra with dim V n < ∞ , ( n ∈ Z ). Let { W α } α ∈ C , { W α } α ∈ C be Z -graded vertex (super)subalgebras whichform continuous families as vector spaces. If W α = W α on some open dense subset U ⊂ C , then W α = W α for all α ∈ C as vertex superalgebras.Proof. Since W iα is a vertex subalgebra of V , we have embeddings ι iα : W iα ֒ → V for all α ∈ C with i = 1 ,
2. Let W iα = L n ∈ Z W iα,n be the decomposition withrespect to the Z -grading. By the assumption that { W iα,n } α ∈ C forms a continuousfamily for all n ∈ Z with i = 1 , W α = W α for α ∈ U , it follows thatdim W α,n = dim W α,n =: d n for all n ∈ Z , and the maps p i : C ∋ α ι iα ( W iα,n ) ∈ Gr( d n , V n ) , i = 1 , , define continuous curves in Gr( d n , V n ). Then, by the assumption that U is opendense in C , the curve p densely coincides with the curve p , and thus is equal to p by the continuity of p and p . This implies that ι α ( W α,n ) = ι α ( W α,n ) for all n ∈ Z and α ∈ C . Summing up all these equations, we have ι α ( W α ) = ι α ( W α ) =: W α .Since ι iα is the embedding of vertex (super)algebras into V for all α ∈ C and i = 1 , W α is a vertex subalgebra of V for all α ∈ C . Therefore W α = W α for all α ∈ C asvertex (super)algebras. This completes the proof. (cid:3) We note that it was used in [AFO] to prove the Feigin-Frenkel duality for principal W -algebras at all levels.Set the rational numbers x i , ( i = 1 , x , x ) = ((cid:16) − n + n , − n n +1 (cid:17) , if ( g , g ) = ( sl n +1 , sl (1 | n + 1)) , (cid:0) − n + 2 , − n + (cid:1) , if ( g , g ) = ( so n +1 , osp (2 | n )) , (5.8)and, set K i := {− h ∨ i } , S i := {− h ∨ i , x i } . Then we have the following. Corollary 5.15.
Let ( g , g ) = ( sl n +1 , sl (1 | n + 1)) , or ( so n +1 , osp (2 | n )) , and ( k , k ) satisfy (4.9) . Then the following isomorphisms hold: (1) For k ∈ C \ S and k ∈ C \ S , Com (cid:0) π H , W k ( g , f sub ) (cid:1) ≃ Com (cid:0) π H , W k ( g ) (cid:1) , uality of subregular W -algebras and principal W -superalgebras 37 (2) For k ∈ C \ K and k ∈ C \ K , W k ( g , f sub ) ≃ Com (cid:16) π e H , W k ( g ) ⊗ V Z √− (cid:17) , (3) For k ∈ C \ K and k ∈ C \ K , W k ( g ) ≃ Com (cid:16) π e H , W k ( g , f sub ) ⊗ V Z (cid:17) . Proof. (1) First, note that the pair ( x , x ) in (5.8) satisfies the relation (4.9).Recall that non-degenerate Heisenberg vertex algebras are all isomorphic if and onlyif their ranks are equal and that they have the conformal gradings by the Segal-Sugawara vectors with each homogeneous subspace is of finite dimension. Next,note that the excluded level ( k , k ) = ( x , x ) is exactly when the Heisenbergvertex algebras π H π H degenerate. Then { Com( π H , W k ( g , f sub )) } k ∈ C \ S isa continuous family of vertex algebras inside a non-degenerate Heisenberg vertexalgebra of rank n + 1 by (4.4) and so is { Com( π H , W k ( g ) } k ∈ C \ S by (4.8). Theyare isomorphic if a generic level ( k , k ) with (4.9), i.e., all values for k ∈ C \ S ,(equivalently k ∈ C \ S by (4.9)), except for countably many values. Thus wemay apply Lemma 5.14 with C replaced by C \ S . This completes the proof. (2)and (3) are proved in the same way. (cid:3) Since the W -superalgebras W k ( g , f sub ) and W k ( g ) as above are of CFT type,their simple quotients exist uniquely, which we denote by W k ( g , f sub ) and W k ( g )respectively. It follows from the construction [KRW, KW] of W -superalgebras thatthe Heisenberg subalgebras π H ⊂ W k ( g , f sub ) and π H ⊂ W k ( g ) act semi-simply. Then we see that the Heisenberg vertex subalgebras π e H ⊂ W k ( g , f sub ) ⊗ V Z and π e H ⊂ W k ( g ) ⊗ V Z √− act semi-simply. By applying Corollary 5.5 to theabove Corollary, we obtain the following. Corollary 5.16.
Let ( g , g ) = ( sl n +1 , sl (1 | n + 1)) , or ( so n +1 , osp (2 | n )) , and ( k , k ) satisfy (4.9) . Then the following isomorphisms hold: (1) For k ∈ C \ S and k ∈ C \ S , Com ( π H , W k ( g , f sub )) ≃ Com ( π H , W k ( g ))(2) For k ∈ C \ K and k ∈ C \ K , W k ( g , f sub ) ≃ Com (cid:16) π e H , W k ( g ) ⊗ V Z √− (cid:17) , (3) For k ∈ C \ K and k ∈ C \ K , W k ( g ) ≃ Com (cid:16) π e H , W k ( g , f sub ) ⊗ V Z (cid:17) . The following is a special case of a C -cofiniteness result [Ar2, Thm. 5.9.1] anda rationality result [AvE, Thm 9.4] of simple W -algebras at the admissible levels. Theorem 5.17 ([Ar2, AvE]) . (1) W k ( sl n +1 , f sub ) is C -cofinite and rational for k = − ( n + 1) + un , u ∈ Z >n , ( u, n ) = 1 . (5.9)(2) W k ( so n +1 , f sub ) is C -cofinite for k = − (2 n −
1) + uv , u ∈ Z >v , v ∈ { n − , n } , ( u, v ) = 1 . (5.10) W -algebras and principal W -superalgebras Corollary 5.18. (1) Com( π H , W k ( sl n +1 , f sub )) is C -cofinite and rational for (5.9) . (2) Com( π H , W k ( so n +1 , f sub )) is C -cofinite for (5.10) .Proof. (1) follows from Theorem 5.11 and Theorem 5.17 (1). We show (2). First,by Corollary 5.16 (2), W k ( so n +1 , f sub ) = Com (cid:16) π e H , W ℓ ( osp (2 | n )) ⊗ V Z √− (cid:17) with ℓ = − n + v u . Recall that e H = H + ψ , where H ∈ W ℓ ( osp (2 | n )) and ψ ∈ V Z √− ,see (4.11). Then we may apply Corollary 5.10 for W = W ℓ ( osp (2 | n )), α = H and β = ψ since C = W k ( so n +1 , f sub ) is a C -cofinite vertex operator algebra ofCFT-type by Theorem 5.17 (2). Here q = 1 − vu and r = −
1. As a consequence, D = Com( π H , W k ( so n +1 , f sub )) is C -cofinite. This proves (2). (cid:3) Corollary 5.19. (1)
Let ℓ = − n + nu with u ∈ Z >n and ( u, n ) = 1 . Then W ℓ ( sl (1 | n + 1)) is C -cofinite and rational. (2) Let ℓ = − n + uv with u ∈ Z >v , v ∈ { n − , n } and ( u, v ) = 1 . Then W ℓ ( osp (2 | n )) is C -cofinite.Proof. First, we show (1). Since V Z is simple, rational, C -cofinite and of CFT type,so are W k ( sl n +1 , f sub ) ⊗ V Z with k as (5.9) and its even part ( W k ( sl n +1 , f sub ) ⊗ V Z ) = W k ( sl n +1 , f sub ) ⊗ V Z by Theorem 5.17 (1). Since the Heisenberg subalgebra π e H acts semisimply, (1) follows from Corollary 5.12 and Corollary 5.16 (3).Next, we show (2). Let C = Com (cid:0) π H , W ℓ ( osp (2 | n )) (cid:1) . Then C is isomorphicto Com (cid:0) π H , W k ( so n +1 , f sub ) (cid:1) with k as (5.10) by Corollary 5.16 (1), and is C -cofinite by Corollary 5.18 (2). Now, by Lemma 5.8, the double coset Com (cid:0) C, W ℓ ( osp (2 | n )) (cid:1) is a rank one lattice vertex superalgebra V N with some lattice N that is positivedefinite since Com (cid:0) C, W ℓ ( osp (2 | n )) (cid:1) is of CFT-type. Therefore W ℓ ( osp (2 | n )) is C -cofinite by Lemma 5.8. (cid:3) References [Ad1] D Adamovic, Rationality of Neveu-Schwarz vertex operator superalgebras. Int. Math.Res. Not., 1997:865-874, 1997.[Ad2] D. Adamovic, Representations of the N = 2 superconformal vertex algebra, Int. Math.Res. Not. (1999), 61-79.[Ad3] D. Adamovic, Vertex algebra approach to fusion rules for N = 2 superconformal minimalmodels, J. Algebra 239 (2001), 549-572.[Ar1] T. Arakawa, A remark on the C -cofiniteness condition on vertex algebras, Math. Z.,Mathematische Zeitschrift, , 2012, 1-2, 559–575.[Ar3] T. Arakawa, Rationality of Bershadsky-Polyalov vertex algebras, Commun. Math.Phys. 323 (2013), no. 2, 627-633.[Ar2] T. Arakawa, Associated varieties of modules over Kac-Moody algebras and C -cofiniteness of W-algebras. Int. Math. Res. Notices, 2015(22):11605-11666, 2015.[Ar4] T. Arakawa, Rationality of W-algebras: principal nilpotent cases, Ann. Math. 182(2015), 565-604.[Ar5] T. Arakawa, Introduction to W-algebras and their representation theory, Perspectivesin Lie theory, Springer INdAM Ser., , 179–250, Springer, Cham, 2017.[ABD] T. Abe, G. Buhl and C. Dong, Rationality, Regularity, and C -cofiniteness, Trans.Amer. Math. Soc. 356 (2004), 3391-3402.[ACGY] D. Adamovic, T. Creutzig, N. Genra and J. Yang, The vertex algebras R ( p ) and V ( p ) ,arXiv:2001.08048 [math.RT]. uality of subregular W -algebras and principal W -superalgebras 39 [ACKR] J. Auger, T. Creutzig, S. Kanade and M. Rupert, Braided Tensor Categories related to B p Vertex Algebras, to appear in Commun. Math. Phys. arXiv:1906.07212 [math.QA].[ACL1] T. Arakawa, T. Creutzig and A. R. Linshaw, Cosets of Bershadsky-Polyakov algebrasand rational W-algebras of type A, Selecta Math. New Series, 23, No. 4 (2017), 2369-2395.[ACL2] T. Arakawa, T. Creutzig and A. R. Linshaw, W-algebras as coset vertex algebras.Invent. math. 218, 145-195 (2019).[AFO] M. Aganagic, E. Frenkel, and A. Okounkov, Quantum q -Langlands correspondence,Trans. Moscow Math. Soc., , 2018, 1–83.[AGT] L. F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. (2010) 167[AP] D. Adamovic and V. Pedic, On fusion rules and intertwining operators for the Weylvertex algebra, J. Math. Phys. (2019) no.8, 081701.[AvE] T. Arakawa and J. van Ekeren, ‘Rationality and Fusion Rules of Exceptional W-Algebras, arXiv:1905.11473 [math.RT].[BFM] M. Bershtein, B. Feigin, and G. Merzon, Plane partitions with a “pit”: generatingfunctions and representation theory, Selecta Math. (N.S.) 24 (2018), no. 1, 21-62.[BFST] P. Bowcock, B. L. Feigin, A. M. Semikhatov and A. Taormina, Affine sl(2 |
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