aa r X i v : . [ m a t h . QA ] J u l KAZAMA–SUZUKI COSET CONSTRUCTION AND ITSINVERSE
RYO SATO
Abstract.
We study the representation theory of the Kazama–Suzuki cosetvertex operator superalgebra associated with the pair of a complex simpleLie algebra and its Cartan subalgebra. In the case of type A , B.L. Feigin,A.M. Semikhatov, and I.Yu. Tipunin introduced another coset construction,which is “inverse” of the Kazama–Suzuki coset construction. In this paperwe generalize the latter coset construction to arbitrary type and establish acategorical equivalence between the categories of certain modules over an affinevertex operator algebra and the corresponding Kazama–Suzuki coset vertexoperator superalgebra. Moreover, when the affine vertex operator algebra isregular, we prove that the corresponding Kazama–Suzuki coset vertex operatorsuperalgebra is also regular and the category of its ordinary modules carries abraided monoidal category structure by the theory of vertex tensor categories. Contents
1. Introduction 12. Fermionization and Defermionization 43. Main Theorem 94. Proof of Main Theorem 135. Regular cases 21Appendix A. Vertex superalgebras and modules 25Appendix B. Twisted sector 27References 271.
Introduction
Representation theory of vertex operator superalgebras plays a fundamental rolein the mathematical study of the corresponding 2-dimensional conformal field the-ories, see e.g. [FZ92], [NT05], [Hua08]. When the symmetry of the Virasoro algebraon a given vertex operator superalgebra is extended to that of Z / Z -graded general-izations of the Virasoro algebra, known as superconformal algebras (e.g. [KvdL89]),the corresponding theory is naturally expected to provide an example of super-conformal field theories. Though such generalizations are originally motivatedby purely physical applications (e.g. [NS71], [Ram71]), the superconformal algebrasymmetry turns out to be non-trivially related with several areas of mathematics,see e.g. [KRW03], [BZHS08], [EOT11], [Wit12], [KW16]. Mathematics Subject Classification.
Key words and phrases.
Vertex operator superalgebra, Superconformal algebra, Kazama–Suzuki coset construction.
In this paper, we study the representation theory of a certain specific family ofvertex operator superalgebras with the N = 2 superconformal algebra symmetry,which is given by the Kazama–Suzuki coset construction [KS89] associated withthe pair of a complex simple Lie algebra g and its Cartan subalgebra h . Our maintool to study such vertex operator superalgebras is a generalization of the Feigin–Semikhatov–Tipunin coset construction introduced in [FST98] for g = sl . Roughlyspeaking, these two constructions are Heisenberg coset constructions (see [CKLR18]for the general theory and many examples), which are “mutually inverse” to eachother:(1) the Kazama–Suzuki coset construction (“fermionization”)( Affine VOAs ) → ( N = 2 VOSAs ) , (2) the Feigin–Semikhatov–Tipunin coset construction (“defermionization”)( N = 2 VOSAs ) → ( Affine VOAs ) . One of the most important features of the above bidirectional constructions isthat they relate the representation theory of Z / Z -graded objects to that of purelyeven objects. In fact, we establish a categorical equivalence between C -linearadditive categories of certain modules over affine vertex operator algebras and thecorresponding N = 2 superconformal vertex operator superalgebras.1.1. Main result.
In order to explain the main result, we first give a brief reviewabout the Kazama–Suzuki coset construction for the pair ( g , h ) as above.Let k ∈ C \ {− h ∨ } , where h ∨ is the dual Coxeter number of g . We denoteby V k ( g ) the universal affine vertex operator algebra (see Lemma A.1) associatedwith the invariant bilinear form kB , where B is the normalized symmetric invariantbilinear form on g . We simply write V af for V k ( g ) or its simple quotient L k ( g ).As is described in [HT91, § h -actions on g and on the orthogonalcomplement h ⊥ of h with respect to B give rise to an injective vertex superalgebrahomomorphism (see § V k + h ∨ ( h ) → V af ⊗ V + , where V k + h ∨ ( h ) is the Heisenberg vertex operator algebra associated with the bi-linear form ( k + h ∨ ) B | h × h and V + is the Clifford vertex operator superalgebraassociated with h ⊥ and B | h ⊥ × h ⊥ . In this paper, we write H + for the image of(1.1) and denote the corresponding commutant (usually referred to as coset ) vertexsuperalgebra by V sc := Com (cid:0) H + , V af ⊗ V + (cid:1) . As a special case of [HT91, Theorem 2.5] (see also [KS89, § V sc has the N = 2 superconformal algebra symmetry: Theorem 1.1 ([HT91]) . The coset vertex superalgebra V sc carries a structure ofan N = 2 superconformal vertex operator superalgebra of central charge c sc := k dim g k + h ∨ + 12 dim h ⊥ − dim h We say that a Z / Z -graded vector space V = V ¯0 ⊕ V ¯1 is purely even if V ¯1 = { } . Throughoutthis paper, we regard all the vertex algebras as purely even vertex superalgebras. It is clear that both C k ( g ) and C k ( g ) are of CFT type, that is, every L -eigenvalue is non-negative and the L -eigenspace of eigenvalue 0 is 1-dimensional. AZAMA–SUZUKI COSET CONSTRUCTION AND ITS INVERSE 3 in the sense of [HM02, Definition 1.1] (see also [Ada99, Definition 1.1]).Next we move on to the Feigin–Semikhatov–Tipunin coset construction. In[FST98], B.L. Feigin, A.M. Semikhatov, and I.Yu. Tipunin introduced a coset con-struction of the affine Lie algebra b sl from the N = 2 superconformal algebra andthe lattice vertex superalgebra associated with the negative-definite lattice √− Z .After a while, D. Adamovi´c reformulated their construction purely in terms of ver-tex superalgebras in [Ada99, § Proposition 1.2 (Proposition 2.11) . Assume that k ∈ C \ { , − h ∨ } . Let V − bethe lattice vertex superalgebra associated with the negative-definite integral lattice √− Z dim h . Then, for a certain Heisenberg vertex subalgebra H − of V sc ⊗ V − (seeLemma 2.8 for the definition), there exists an explicit isomorphism f FST : V af → Com (cid:0) H − , V sc ⊗ V − (cid:1) of purely even vertex operator superalgebras.For each coset [ λ ] ∈ h ∗ /Q , where Q is the root lattice of g , we introduce certain C -linear full subcategories C [ λ ] ( V af ) and C [ λ sc ] ( V sc ) of weak V af -modules and weak V sc -modules, respectively (see Definition 3.1 and Proposition 3.5). Now we canstate our main result as follows: Theorem 1.3 (Theorem 3.9) . Let k ∈ C \{ , − h ∨ } and λ ∈ h ∗ . Then the following C -linear functorsΩ + ( λ ) : C [ λ ] ( V af ) → C [ λ sc ] ( V sc ) , Ω − ( λ sc ) : C [ λ sc ] ( V sc ) → C [ λ ] ( V af )defined in § spectral flow automor-phisms (see § g = sl is essentially the same as[Sat16, Theorem 4.4, 7.7, and Corollary 6.3] and the original idea in this case goesback to [FST98, Theorem IV. 10]. It is worth noting that, when g = sl , the sameproof as [Sat16, Theorem 4.4] does not work due to the existence of a non-triviallattice vertex subsuperalgebra of V sc (see § Regular cases.
It is well-known that the simple affine vertex operator algebra L k ( g ) is regular if k is a positive integer (see [DLM97, Theorem 3.7]). In this case,we obtain further information about the simple vertex operator superalgebra C k ( g ) := Com (cid:0) H + , L k ( g ) ⊗ V + (cid:1) . We first determine the bicommutant of the Heisenberg vertex subalgebra H + and H − in L k ( g ) ⊗ V + and C k ( g ) ⊗ V − , respectively. Proposition 1.4 (Proposition 5.2 and 5.6) . Assume that k is a positive integer.(1) The bicommutant vertex superalgebra E + := Com (cid:16)
Com (cid:0) H + , L k ( g ) ⊗ V + (cid:1) , L k ( g ) ⊗ V + (cid:17) is purely even and isomorphic to the lattice vertex algebra associated with √ k + h ∨ Q l , RYO SATO where Q l is the even integral sublattice of the root lattice Q spanned bylong roots (cf. [DW11, Proposition 4.11]).(2) The bicommutant vertex superalgebra E − := Com (cid:16)
Com (cid:0) H − , C k ( g ) ⊗ V − (cid:1) , C k ( g ) ⊗ V − (cid:17) is isomorphic to the lattice vertex superalgebra associated with p − ( k + h ∨ ) Q l ⊕ Z dim h ⊥ − dim h . Next, by using [Miy15, Corollary 2] and [CKLR18, Theorem 4.12] (cf. [CM16,Theorem 5.24]), we prove the following.
Theorem 1.5 (Theorem 5.7) . When k is a positive integer, both C k ( g ) and itseven part C k ( g ) ¯0 are regular.At last, we obtain the following theorem as a special case of [CKM17, Theorem3.65], which is based on the theory of vertex tensor categories developed by Y.-Z. Huang and J. Lepowsky (see [HL95a], [HL95b], [HL95c], [Hua95], [Hua05], andreferences therein). Theorem 1.6 (Theorem 5.12) . When k is a positive integer, the semisimple C -linear abelian category of Z / Z -graded ordinary C k ( g )-modules carries a braidedmonoidal category structure induced by [Hua05, Theorem 3.7] and [CKM17, The-orem 3.65].We should note that Theorem 1.5 (resp. Theorem 1.6) for g = sl is proved byD. Adamovi´c in [Ada01, Theorem 8.1] (resp. by Y.-Z. Huang and A. Milas in [HM02,Theorem 4.8]).1.3. Structure of the paper.
We organize this paper as follows. In § V sc and its “Cartan subalgebra” h sc in anexplicit way. Our main tool, a generalization of the Feigin–Semikhatov–Tipunincoset construction, is introduced in § §
3, after introducing an appropriatecategory of modules and two families of functors Ω + and Ω − , we give the precisestatement of our main result. The proof of the main theorem is given in §
4. Atlast, in §
5, we discuss the case that the level k is a positive integer. Acknowledgments:
The author would like to express his gratitude to Shun-Jen Cheng, Ching Hung Lam, Masahiko Miyamoto, Hiromichi Yamada, and HiroshiYamauchi for helpful discussions and valuable advice. He also would like to expresshis appreciation to Tomoyuki Arakawa, Kenichiro Tanabe, Shintarou Yanagida,and many others for constructive comments.2.
Fermionization and Defermionization
Kazama–Suzuki coset construction.
In this subsection, we review thevertex superalgebraic formulation of the Kazama-Suzuki coset constructions. See[KS89] and [HT91] for details.Let g be a simple complex Lie algebra with a fixed Borel subalgebra b . Let h bethe corresponding Cartan subalgebra and ∆ (resp. ∆ + , Π) the set of (resp. positive, Note that the symmetric center (see [M¨ug03, Definition 2.9]) of this braided monoidal categorycontains at least two non-isomorphic simple objects, the monoidal unit C k ( g ) and its Z / Z -parityreversed object. In particular, it is not modular in the sense of [M¨ug03, Definition 3.1]. AZAMA–SUZUKI COSET CONSTRUCTION AND ITS INVERSE 5 simple) roots associated with ( g , b ). We write B = B (? , ?) : g × g → C for thenormalized symmetric invariant bilinear form on g .Define a positive-definite integral lattice L + of rank N := | ∆ + | by L + := M α ∈ ∆ + Z α + , h α + , β + (cid:11) := δ α,β ( α, β ∈ ∆ + )and take a bimultiplicative map ǫ + : L + × L + → {± } in the same way as ǫ : Z N × Z N → {± } in [Kac98, Example 5.5a] . Then we write V + for the lattice vertexsuperalgebra associated with ( L + , ǫ + ), which is isomorphic to the N -times tensorproduct of the charged free fermions. Unless otherwise specified, we follow thestandard notation of lattice vertex superalgebras used in [Kac98].Let h ∨ be the dual Coxeter number of g and k ∈ C \ {− h ∨ } . We denote by V af the universal affine vertex operator algebra V k ( g ) of level k or its simple quotientvertex operator algebra L k ( g ). Based on the setting of the Kazama–Suzuki cosetconstruction, we consider a Heisenberg vertex subalgebra of V af ⊗ V + as follows. Lemma 2.1.
For α ∈ ∆, we set H α := ( α, α )2 α ∨ ∈ h , where (? , ?) : h ∗ × h ∗ → C is the normalized bilinear form induced by the restrictionof B to h × h and α ∨ ∈ h is the coroot of α . We also set H + α := H α, − af ⊗ V + − af ⊗ α f , − V + ∈ ( V af ⊗ V + ) ¯0 , where af is the vacuum vector of V af and α f := X β ∈ ∆ + ( α, β ) β + ∈ L + ⊗ Z Q . Then the vertex subalgebra H + of V af ⊗ V + generated by the set { H + α | α ∈ Π } isthe Heisenberg vertex algebra of rank ℓ := dim h . Proof.
Since we have h α f , β f i = X γ ∈ ∆ + ( α, γ )( γ, β ) = h ∨ ( α, β )for α, β ∈ Π, there exists a unique vertex algebra isomorphism V k + h ∨ ( h ) → H + such that H α, − af H + α for α ∈ Π. (cid:3) Let ω af , ω f , and ω b be the standard conformal vectors of central charge c af := k dim g k + h ∨ , N , and ℓ of V af , V + , and H + , respectively (see e.g. [Kac98] for details). Lemma 2.2.
The coset vertex superalgebra V sc := Com (cid:0) H + , V af ⊗ V + (cid:1) together with ω sc := ω af ⊗ V + + af ⊗ ω f − ω b forms a vertex operator superalgebraof central charge c sc := c af + N − ℓ. Proof.
It follows from the general theory of coset vertex superalgebras (see e.g. [FZ92,Theorem 5.1]). (cid:3) Our discussion below does not depend on the choice of the lattice isomorphism L + ≃ Z N . RYO SATO
Proposition 2.3.
The coset vertex operator superalgebra C k ( g ) := Com (cid:0) H + , L k ( g ) ⊗ V + (cid:1) is a unique simple quotient vertex operator superalgebra of C k ( g ) := Com (cid:0) H + , V k ( g ) ⊗ V + (cid:1) . Proof.
Let I be the maximal proper ideal of V k ( g ). Since V k ( g ) ⊗ V + and I ⊗ V + are completely reducible as weak H + -modules by the same argument as in [FLM89, § C k ( g ) is a quotient of C k ( g ). The simplicity and uniquenessfollow from the same argument of [CKLR18, Proposition 3.2] (see also [ACKL17,Lemma 2.1]) and the fact that C k ( g ) is of CFT type, respectively. (cid:3) Lemma 2.4.
The following homomorphism ι + : V sc ⊗ H + → V af ⊗ V + ; A ⊗ B A − B of vertex operator superalgebras is injective. Proof.
Since H + is simple and V af ⊗ V + is completely reducible as a weak H + -module, the injectivity of ι + follows. (cid:3) Cartan subalgebra.
In this subsection, we introduce a Heisenberg vertexsubalgebra of V sc . From now on, we always assume that k ∈ C \ { , − h ∨ } . Lemma 2.5.
The following set (cid:26) J α := 1 k H α, − af ⊗ V + + af ⊗ α + − V + ∈ ( V af ⊗ V + ) ¯0 (cid:12)(cid:12)(cid:12)(cid:12) α ∈ ∆ + (cid:27) generates a vertex subalgebra H sc of V sc , which is isomorphic to the Heisenbergvertex algebra of rank N . Proof.
Since it is easy to verify that each J α lies in V sc , we only need to check thenon-degeneracy. By a direct computation, we have J α ( z ) J β ( w ) ∼ g α,β ( z − w ) := ( α,β ) k + δ α,β ( z − w ) , for any α, β ∈ ∆ + . By using the formula X α ∈ ∆ + ( λ, α ) α = h ∨ λ for λ ∈ h ∗ , we can verify that( g ∗ α,β ) α,β ∈ ∆ + := (cid:18) − ( α, β ) k + h ∨ + δ α,β (cid:19) α,β ∈ ∆ + is the inverse matrix of ( g α,β ) α,β ∈ ∆ + . This completes the proof. (cid:3)
By the inverse matrix in the above proof, we obtain the following.
Corollary 2.6.
We set J ∗ α := X β ∈ ∆ + g ∗ α,β J β = 1 k + h ∨ H + α + af ⊗ α + − V + ∈ H sc for α ∈ ∆ + . Then we have J ∗ α ( z ) J β ( w ) ∼ δ α,β ( z − w ) , J ∗ α ( z ) J ∗ β ( w ) ∼ g ∗ α,β ( z − w ) for any α, β ∈ ∆ + . Lemma 2.7.
We set P ∨ sc := M α ∈ ∆ + Z J ∗ α , h sc := M α ∈ ∆ + C J ∗ α , and Q sc := (cid:8) λ ∈ ( h sc ) ∗ (cid:12)(cid:12) λ ( H ) ∈ Z for any H ∈ P ∨ sc (cid:9) . Then V sc decomposes into adirect sum V sc = M γ ∈ Q sc ( V sc ) γ of h sc -weight subspaces, where( V sc ) γ := { A ∈ V sc | H A = γ ( H ) A for any H ∈ h sc } . Proof.
It follows from the fact that the operator J ∗ α, coincides with af ⊗ α +0 on V sc for any α ∈ ∆ + . (cid:3) Feigin–Semikhatov–Tipunin coset construction.
In [FST98], B.L. Feigin,A.M. Semikhatov, and I.Yu. Tipunin studied a coset construction of V k ( sl ) in thetensor product of C k ( sl ) and the lattice vertex superalgebra associated with √− Z (see also [Ada99, § L − of rank ℓ by L − := M α ∈ Π Z α − , h α − , β − i := − δ α,β and a bimultiplicative map ǫ − : L − × L − → {± } by ǫ − ( α − , β − ) := − ǫ + ( α + , β + )for α, β ∈ Π. We write V − for the lattice vertex superalgebra associated with( L − , ǫ − ) . Since the lattice L − is negative-definite, the vertex superalgebra V − hasa natural conformal structure of central charge ℓ and the set of the L -eigenvaluesis not bounded below. Lemma 2.8.
For each α ∈ ∆ + , we set H − α := ( J ∗ α ⊗ V − + sc ⊗ α −− V − if α ∈ Π ,J ∗ α ⊗ V − if α / ∈ Π , where sc is the vacuum vector of V sc . Then the vertex subalgebra H − of V sc ⊗ V − generated by { H − α | α ∈ ∆ + } is the Heisenberg vertex algebra of rank N . Proof.
By Corollary 2.6, we have H − α ( z ) H − β ( w ) ∼ G α,β ( z − w ) := g ∗ α,β − δ α,β ( z − w ) if α ∈ Π or β ∈ Π ,g ∗ α,β ( z − w ) if α, β / ∈ Π One can easily verify that ǫ − satisfies the condition ǫ − ( ζ, ζ ′ ) ǫ − ( ζ ′ , ζ ) = ( − h ζ,ζ ′ i + h ζ,ζ ih ζ ′ ,ζ ′ i for any ζ, ζ ′ ∈ L − in [Kac98, Theorem 5.5 (b)]. RYO SATO for α, β ∈ ∆ + . We define an N × N matrix ( G ∗ α,β ) α,β ∈ ∆ + by(2.1) G ∗ α,β := − kc α,β − δ α,β if α, β ∈ Π , − β ( α ∗ ) if α ∈ Π and β / ∈ Π , − α ( β ∗ ) if α / ∈ Π and β ∈ Π ,δ α,β if α, β / ∈ Π , where ( c α,β ) α,β ∈ Π is the inverse of the symmetrized Cartan matrix (cid:0) ( α, β ) (cid:1) α,β ∈ Π and α ∗ ∈ h is the fundamental coweight defined by β ( α ∗ ) = δ α,β for β ∈ Π. Thenwe can verify that ( G ∗ α,β ) α,β ∈ ∆ + is the inverse matrix of ( G α,β ) α,β ∈ ∆ + and thus thelatter is non-singular. (cid:3) Let Q be the root lattice of g . We define Z -linear maps f ± af : Q → L ± by f ± af ( γ ) := X α ∈ Π γ ( α ∗ ) α ± for γ ∈ Q . Then we obtain the next lemma as a generalization of the “anti-Kazama–Suzuki mapping” for g = sl in [FST98, Lemma III.5] and [Ada99, Lemma 5.1] toarbitrary type. Lemma 2.9.
There exists a unique vertex superalgebra homomorphism(2.2) f FST : V k ( g ) → Com (cid:0) H − , C k ( g ) ⊗ V − (cid:1) . such that X α, − af e X α := X α, − af ⊗ e f + af ( α ) ⊗ e f − af ( α ) ,H β, − af e H β := k (cid:0) J β ⊗ V − + sc ⊗ β −− V − (cid:1) , for any α ∈ ∆ and β ∈ Π, where X α ∈ g is a root vector associated with the root α and normalized by [ X α , X − α ] = α ∨ . Proof.
First we prove the above assignment induces a vertex superalgebra homo-morphism from V k ( g ) to V k ( g ) ⊗ V + ⊗ V − . It suffices to show that the aboveelements in the right-hand side obey the same OPE as in the left-hand side. Bythe general theory of lattice vertex superalgebras, we havee ξn e ξ ′ = n ≥ −h ξ, ξ ′ i ,ǫ ( ξ, ξ ′ )e ξ + ξ ′ if n = −h ξ, ξ ′ i − ,ǫ ( ξ, ξ ′ ) ξ − e ξ + ξ ′ if n = −h ξ, ξ ′ i − ξ, ξ ′ of a general non-degenerate integral lattice, where ǫ is an appro-priate 2-cocycle of the lattice (see e.g. [Kac98, § h f + af ( γ ) , f + af ( γ ′ ) i + h f − af ( γ ) , f − af ( γ ′ ) i = 0 for any γ, γ ′ ∈ Q , we obtain(2.4) (cid:16) e f + af ( γ ) ( z ) ⊗ e f − af ( γ ) ( z ) (cid:17) (cid:16) e f + af ( γ ′ ) ( w ) ⊗ e f − af ( γ ′ ) ( w ) (cid:17) ∼ . Then, by (2.3), (2.4), and some computations, we obtain e X α ( z ) e X β ( w ) ∼ ^ [ X α , X β ]( w ) z − w + δ α + β, k id ⊗ id ⊗ id ( z − w ) The root lattice Q with the normalized bilinear form (? , ?) is integral (resp. even) if and onlyif g is not of type G (resp. is of type A, D, E ). AZAMA–SUZUKI COSET CONSTRUCTION AND ITS INVERSE 9 for any α, β ∈ ∆. In addition, we also obtain e H α ( z ) e X β ( w ) ∼ ^ [ H α , X β ]( w ) z − w , e H α ( z ) e H β ( w ) ∼ k ( α, β ) id ⊗ id ⊗ id ( z − w ) . Thus we get a homomorphism from V k ( g ) to V k ( g ) ⊗ V + ⊗ V − .Next we need to prove that the image of the homomorphism is actually con-tained in the commutant of H − in C k ( g ) ⊗ V − . It suffices to check that ( H + α ( z ) ⊗ id V − ) e X β ( w ) ∼ H − α ( z ) e X β ( w ) ∼ α ∈ ∆ + and β ∈ ∆. Since they areverified by straightforward computations, we omit the detail. (cid:3) Corollary 2.10.
The homomorphism (2.2) gives rise to conformal vertex superal-gebra homomorphisms ι − : V k ( g ) ⊗ H − → C k ( g ) ⊗ V − and ι af : V k ( g ) ⊗ H + ⊗ H − → V k ( g ) ⊗ V + ⊗ V − . As a matter of fact, the following stronger statement holds.
Proposition 2.11.
The homomorphism (2.2) is an isomorphism. In particular, itdescends to their simple quotients and gives rise to conformal vertex superalgebrahomomorphisms ι − : L k ( g ) ⊗ H − → C k ( g ) ⊗ V − and ι af : L k ( g ) ⊗ H + ⊗ H − → L k ( g ) ⊗ V + ⊗ V − . The proof of Proposition 2.11 is postponed to Corollary 4.5.3.
Main Theorem
In this section, by using the notation defined in the previous section, we give theprecise statement of our main result (Theorem 3.9). Recall that the pair ( V af , V sc )stands for (cid:0) V k ( g ) , C k ( g ) (cid:1) or (cid:0) L k ( g ) , C k ( g ) (cid:1) . Throughout this section, we assumethat k ∈ C \ { , − h ∨ } and the letter V always stands for V af or V sc .3.1. Setting of the category of modules.
For V = V af (cid:0) resp. V sc (cid:1) , we set P ∨ V := P ∨ (resp. P ∨ sc ) , t V := h (resp. h sc ) , Q V := Q (resp. Q sc ) , where P ∨ is the coweight lattice in h . Definition 3.1.
Let λ ∈ t ∗ V and set[ λ ] := { λ } + Q V ∈ t ∗ V /Q V . Then we define C [ λ ] ( V ) to be a full subcategory of V - gMod (see § A.3 for the defi-nition) whose object M = M ¯0 ⊕ M ¯1 satisfies the following conditions:(1) For each ¯ i ∈ Z / Z , the Z / Z -homogeneous subspace M ¯ i decomposes intoa direct sum M ¯ i = M µ ∈ [ λ ] M ¯ iµ of t V -weight spaces, where M ¯ iµ := { v ∈ M ¯ i | H v = µ ( H ) v for any H ∈ t V } . Note that V − is not a vertex operator superalgebra. (2) For each ¯ i ∈ Z / Z and µ ∈ [ λ ], the t V -weight space M ¯ iµ further decomposesinto a direct sum M ¯ iµ = M ∆ ∈ C M ¯ iµ ( h )of finite-dimensional subspaces, where M ¯ iµ ( h ) := { v ∈ M ¯ iµ | ( L − h ) n v = 0 if n ≫ } . In addition, for any h ∈ C , we have M ¯ iµ ( h − r ) = { } if r ≫ M µ and M µ ( h ) for M ¯0 µ ⊕ M ¯1 µ and M ¯0 µ ( h ) ⊕ M ¯1 µ ( h ), respectively. Definition 3.2.
For an object M of C [ λ ] ( V ), we define the string function of M through µ ∈ [ λ ] by s µM ( q ) := X ∆ ∈ C dim C (cid:0) M µ (∆) (cid:1) q ∆ − c and the formal character of M by ch ( M ) := X µ ∈ [ λ ] s µM ( q )e µ . Definition of the functors Ω + and Ω − . In this subsection, we assumeProposition 2.11 whenever we consider the case of V = C k ( g ). Note that the othercases are proved independently of Proposition 2.11.We set t + := M α ∈ Π C H + α ( H + , t − := M β ∈ ∆ + C H − β ( H − . From now on, we fix the following linear isomorphisms t + ≃ −→ t V af = h ; H + α H α , t − ≃ −→ t V sc = h sc ; H − β J ∗ β and denote the induced linear isomorphisms by ν + : t ∗ V af = h ∗ ≃ −→ ( t + ) ∗ , ν − : t ∗ V sc = ( h sc ) ∗ ≃ −→ ( t − ) ∗ . Definition 3.3.
Let λ, µ ∈ t ∗ V and M be an object of C [ λ ] ( V ).(1) When V = V af , we define a Z / Z -graded weak V sc -module byΩ + ( µ )( M ) := { v ∈ M ⊗ V + | h n v = δ n, ν + ( µ )( h ) v for any h ∈ t + and n ≥ } . (2) When V = V sc , we define a Z / Z -graded weak V af -module byΩ − ( µ )( M ) := { v ∈ M ⊗ V − | h n v = δ n, ν − ( µ )( h ) v for any h ∈ t − and n ≥ } . For a morphism f : M → M of C [ λ ] ( V ), we define a linear mapΩ ± ( µ )( f ) : Ω ± ( µ )( M ) → Ω ± ( µ )( M )by the restriction of f ⊗ id V ± to the subspace Ω ± ( µ )( M ) of M ⊗ V ± .We denote the purely even Heisenberg Fock H ± -module of highest weight µ ± ∈ ( t ± ) ∗ by H ± µ ± and its even highest weight vector by | µ ± i . When V = C k ( g ), we use Proposition 2.11. AZAMA–SUZUKI COSET CONSTRUCTION AND ITS INVERSE 11
Lemma 3.4.
For V = V af , there exists a unique isomorphism(3.1) M µ ∈ [ λ ] Ω + ( µ )( M ) ⊗ H + ν + ( µ ) ≃ −→ ι ∗ + ( M ⊗ V + )of Z / Z -graded weak V sc ⊗ H + -modules such that v ⊗ | ν + ( µ ) i 7→ v for any v ∈ Ω + ( µ )( M ). Similarly, for V = V sc , there exists a unique isomorphism(3.2) M µ ∈ [ λ ] Ω − ( µ )( M ) ⊗ H − ν − ( µ ) ≃ −→ ι ∗− ( M ⊗ V − )of Z / Z -graded weak V af ⊗ H − -modules such that v ⊗ | ν − ( µ ) i 7→ v for any v ∈ Ω − ( µ )( M ). Proof.
Since ι ∗± ( M ⊗ V ± ) is completely reducible as a Z / Z -graded weak H ± -module by the same argument as in [FLM89, § ± ( µ )( M ) = { } holds forany µ ∈ t ∗ V \ [ λ ], we get the above isomorphisms. (cid:3) Proposition 3.5.
We have the following.(1) Let λ ∈ h ∗ and µ ∈ [ λ ]. Then the assignment Ω + ( µ ) gives rise to a C -linearfunctor Ω + ( µ ) : C [ λ ] ( V af ) → C [ µ sc ] ( V sc ) , where µ sc ∈ ( h sc ) ∗ is defined by µ sc ( J α ) := µ (cid:0) k − H α (cid:1) for α ∈ ∆ + .(2) Let λ ∈ ( h sc ) ∗ and µ ∈ [ λ ]. Then the assignment Ω − ( µ ) gives rise to a C -linear functor Ω − ( µ ) : C [ λ ] ( V sc ) → C [ µ af ] ( V af ) , where µ af ∈ h ∗ is defined by µ af ( H α ) := µ ( kJ α )for any α ∈ Π. Proof.
Since the proof of (1) and that of (2) are simliar, we only verify (1) here.Let M be an object of C [ λ ] ( V af ). As the functoriality is obvious, it suffices to showthat Ω + ( µ )( M ) is an object of C [ µ sc ] ( V sc ). By the definition of Ω + ( µ )( M ), theoperator H + α, acts on Ω + ( µ )( M ) as scalar ν + ( µ )( H + α ) = µ ( H α ). Therefore J α, acts on Ω + ( µ )( M ) as scalar µ sc ( J α ) modulo Z and every h sc -weight of Ω + ( µ )( M )lies in the coset [ µ sc ]. The other conditions of C [ µ sc ] ( V sc ) are easily verified by thecorresponding conditions of C [ λ ] ( V af ). This completes the proof. (cid:3) The next lemma is obvious by the definition.
Lemma 3.6.
For any λ ∈ h ∗ , we have ( λ sc ) af = λ . The sublattice K . We define a Z -linear map g + af : L + → Q by(3.3) g + af ( ξ ) := X α ∈ ∆ + h ξ, α + i α for ξ ∈ L + and regard its kernel K as a positive-definite sublattice of L + . Since wehave g + af ◦ f + af = id Q , it is easy to verify that the set (cid:8) ξ ( α ) := α + − f + af ( α ) ∈ L + (cid:12)(cid:12) α ∈ ∆ + \ Π (cid:9) forms a Z -basis of K . Lemma 3.7.
Let V K be the lattice vertex superalgebra associated with the lattice K . We identify V K with the vertex subsuperalgebra of V af ⊗ V + generated by { af ⊗ e ξ | ξ ∈ K } . Then V K is contained in V sc . In addition, we have(3.4) e ξ ( α ) := af ⊗ ξ ( α ) − V + = J α − X β ∈ Π α ( β ∗ ) J β ∈ h sc for any α ∈ ∆ + . Proof.
Since h α f , ξ ( β ) i = 0 holds for any α ∈ Π and β ∈ ∆ + \ Π, the set of generatorsis contained in V sc . The equality (3.4) follows from H α − X β ∈ Π α ( β ∗ ) H β = 0 . (cid:3) With the help of the vertex subsuperalgebra V K of V sc , we obtain the ‘converse’of Lemma 3.6 as follows. Proposition 3.8.
Let λ ∈ ( h sc ) ∗ . Assume that C [ λ ] ( V sc ) contains at least onenon-zero object. Then [( µ af ) sc ] = [ λ ] holds for any µ ∈ [ λ ]. Proof.
Take µ ∈ [ λ ]. When g = sl , we have ( µ af ) sc = µ and [( µ af ) sc ] = [ µ ] = [ λ ].Now we assume that g = sl . It suffices to show that µ ( J α ) − ( µ af ) sc ( J α ) liesin Z for any α ∈ ∆ + . Since K is non-trivial and V K is regular (see [DLM97,Theorem 3.16]), any non-zero object of C [ λ ] ( V sc ) decomposes into a non-emptydirect sum of simple ordinary V K -modules. Therefore we have µ (cid:0)e ξ ( α ) (cid:1) ∈ Z for any α ∈ ∆ + . On the other hand, by the definition of ( µ af ) sc and (3.4), we also have( µ af ) sc (cid:0)e ξ ( α ) (cid:1) = 0. Then, by using (3.4) and µ ( J β ) = ( µ af ) sc ( J β ) for any β ∈ Π, weobtain µ ( J α ) − ( µ af ) sc ( J α ) ∈ Z for any α ∈ ∆ + . We thus complete the proof. (cid:3) Main Theorem.
Our main result is as follows:
Theorem 3.9.
Let k ∈ C \ { , − h ∨ } .(1) For any [ λ ] ∈ h ∗ /Q and µ ∈ [ λ ], the following C -linear functorsΩ + ( µ ) : C [ λ ] ( V af ) → C [ µ sc ] ( V sc ) , Ω − ( µ sc ) : C [ µ sc ] ( V sc ) → C [ λ ] ( V af )are mutually quasi-inverse to each other. AZAMA–SUZUKI COSET CONSTRUCTION AND ITS INVERSE 13 (2) For any λ ∈ h ∗ and γ ∈ Q , there exist an isomorphism U γ sc : C [ λ sc ] ( V sc ) ≃ −→ C [( λ − γ ) sc ] ( V sc )of categories and an isomorphismΩ + ( λ − γ ) ≃ U γ sc ◦ Ω + ( λ )of functors from C [ λ ] ( V af ) to C [( λ − γ ) sc ] ( V sc ).(3) For any λ ∈ ( h sc ) ∗ and γ ∈ Q sc , there exist an isomorphism U γ af : C [ λ af ] ( V af ) ≃ −→ C [( λ + γ ) af ] ( V af )of categories and an isomorphismΩ − ( λ + γ ) ≃ U γ af ◦ Ω − ( λ )of functors from C [ λ ] ( V sc ) to C [( λ + γ ) af ] ( V af ). Remark 3.10.
Following the notation and terminology of [BE17], we give a com-ment on supercategory structures. Since the category V - gMod is a C -linear super-category (see Lemma A.2 for the definition), so is the full subcategory C [ λ ] ( V ). Itis clear by the definition that Ω + ( µ ) and Ω − ( µ sc ) are superfunctors in the sense of[BE17, Definition 1.1]. In addition, by the proof in the next section (see Remark 4.4and 4.12), the underlying categories C [ λ ] ( V af ) and C [ µ sc ] ( V sc ) in the sense of [BE17,Definition 1.1] are also categorical equivalent to each other.4. Proof of Main Theorem
In this subsection, we fix λ ∈ h ∗ and µ ∈ [ λ ].4.1. Faithfulness of Ω + ( µ ) . In this subsection, without using Proposition 2.11,we prove that Ω − ( µ sc ) ◦ Ω + ( µ ) : C [ λ ] (cid:0) V k ( g ) (cid:1) → C [ λ ] (cid:0) V k ( g ) (cid:1) is naturally isomorphic to the identity functor. As corollaries, we obtain the proofof Proposition 2.11 and the faithfulness of Ω + ( µ ).4.1.1. Character formula.
Similarly to (3.3), we define Z -linear maps g − af : L − → Q and g ± sc : L ± → Q C k ( g ) by g − af ( ζ ) := X α ∈ Π h ζ, α − i α,g + sc ( ξ )( J ∗ β ) := h ξ, β + i for β ∈ ∆ + ,g − sc ( ζ )( J ∗ β ) := ( h ζ, β − i if β ∈ Π , β ∈ ∆ + \ Πfor ξ ∈ L + and ζ ∈ L − . Proposition 4.1.
Let M be an object of C [ λ ] (cid:0) V k ( g ) (cid:1) . Then we have(4.1) ch (cid:0) Ω + ( µ )( M ) (cid:1) = X ξ ∈ L + s µ + g + af ( ξ ) M ( q ) q h ξ,ξ i− ∆ + µ η ( q ) N − ℓ e µ sc + g + sc ( ξ ) and(4.2) ch (cid:0) Ω − ( µ sc ) ◦ Ω + ( µ )( M ) (cid:1) = ch ( M ) , where ∆ + µ := k + h ∨ ( µ,µ )2 is the lowest L b -eigenvalue of H + ν + ( µ ) . Proof.
We first prove (4.1). One can verify that the restriction of the isomorphism(3.1) for V = V k ( g ) gives an even linear isomorphismΩ + ( µ )( M ) ⊗ H + ν + ( µ ) ≃ M ξ ∈ L + M µ + g + af ( ξ ) ⊗ V + ξ , where V + ξ := (cid:8) v ∈ V + (cid:12)(cid:12) ξ ′ v = h ξ, ξ ′ i v for any ξ ′ ∈ L + (cid:9) . It follows by direct calcula-tion that M µ + g + af ( ξ ) ⊗ V + ξ is the t C k ( g ) -eigenspace of eigenvalue µ sc + g + sc ( ξ ), the ( L af ⊗ id + id ⊗ L f )-graded dimension of M µ + g + af ( ξ ) ⊗ V + ξ is given by s µ + g + af ( ξ ) M ( q ) q h ξ,ξ i η ( q ) − N ,and the L b -graded dimension of H + ν + ( µ ) is given by q ∆ + µ η ( q ) − ℓ . By combining them,we obtain the required formula (4.1).Next we verify (4.2). For an arbitrary object M in C [ µ sc ] (cid:0) C k ( g ) (cid:1) , in the same wayas above, we can verify that the restriction of the isomorphism (3.2) for V = C k ( g )gives Ω − ( µ sc )( M ) ⊗ H − ν − ( µ sc ) ≃ M ζ ∈ L − M µ sc − g − sc ( ζ ) ⊗ V − ζ and(4.3) ch (cid:0) Ω − ( µ sc )( M ) (cid:1) = X ζ ∈ L − s µ sc − g − sc ( ζ ) M ( q ) q h ζ,ζ i +∆ + µ η ( q ) − N + ℓ e µ − g − af ( ζ ) , where V − ζ := (cid:8) v ∈ V − (cid:12)(cid:12) ζ ′ v = h ζ, ζ ′ i v for any ζ ′ ∈ L − (cid:9) . On the other hand, bysome computation, we obtain(4.4) X ( ξ,ζ ) ∈ S ( γ ) q h ξ,ξ i + h ζ,ζ i s µ + g + af ( ξ ) M ( q ) = s µ + γM ( q )for any γ ∈ Q , where S ( γ ) := (cid:8) ( ξ, ζ ) ∈ L + × L − (cid:12)(cid:12) γ = − g − af ( ζ ) , g + sc ( ξ ) = − g − sc ( ζ ) (cid:9) . Then, by using (4.3) for M = Ω + ( µ )( M ) together with (4.1) and (4.4), we canderive the character formula (4.2). (cid:3) Twisted embedding.
Let M be an object of C [ λ ] (cid:0) V k ( g ) (cid:1) . We define an evenlinear operator H on M ⊗ V + ⊗ V − by H := X α ∈ Π ∞ X n =1 n α ∗ n ⊗ (cid:0) α + − n ⊗ id + id ⊗ α −− n (cid:1) . Then, by the condition (2) of Definition 3.1, the formal sum exp ( H ) defines an evenlinear automorphism on M ⊗ V + ⊗ V − . Lemma 4.2.
The assignment(4.5) v ⊗ | ν + ( µ ) i ⊗ | ν − ( µ sc ) i 7→ e v := exp ( H ) (cid:0) v ⊗ e f + af ( γ ) ⊗ e f − af ( γ ) (cid:1) for v ∈ M µ + γ and γ ∈ Q uniquely extends to an injective morphism(4.6) ι µM : M ⊗ H + ν + ( µ ) ⊗ H − ν − ( µ sc ) → ι ∗ af ( M ⊗ V + ⊗ V − )of Z / Z -graded weak V k ( g ) ⊗ H + ⊗ H − -modules. AZAMA–SUZUKI COSET CONSTRUCTION AND ITS INVERSE 15
Proof.
First we prove that the assignment (4.5) defines a Z / Z -graded weak V k ( g )-module homomorphism from M ⊗| ν + ( µ ) i⊗| ν − ( µ sc ) i to ι ∗ af ( M ⊗ V + ⊗ V − ) . It sufficesto prove that e X α,m e v = exp ( H ) (cid:0) X α,m v ⊗ e f + af ( γ + α ) ⊗ e f − af ( γ + α ) (cid:1) (4.7)for any α ∈ Π ⊔ ( − Π), X α ∈ g α , and m ∈ Z . By direct calculation, we obtain ad ( α ∗ n ⊗ α + − n ) N ( X β,p ⊗ e β + q )= δ α,β N X N ′ =0 (cid:18) NN ′ (cid:19)(cid:0) X α,p +( N − N ′ ) n ⊗ ( α + − n ) N − N ′ (cid:1) ◦ (cid:0) ( α ∗ n ) N ′ ⊗ e α + q − N ′ n (cid:1) ad ( α ∗ n ⊗ α −− n ) N ( X β,p ⊗ e β − r )= δ α,β N X N ′ =0 ( − N ′ (cid:18) NN ′ (cid:19)(cid:0) X α,p +( N − N ′ ) n ⊗ ( α −− n ) N − N ′ (cid:1) ◦ (cid:0) ( α ∗ ) N ′ ⊗ e α − r − N ′ n (cid:1) for any α, β ∈ Π and p, q, r ∈ Z . By using them, for α ∈ Π, we haveΦ := exp ( − H ) e X α,m exp ( H ) (cid:0) v ⊗ e f + af ( γ ) ⊗ e f − af ( γ ) (cid:1) = X p + q + r = m − ∞ X N =0 ad ( − H ) N N ! ( X α,p ⊗ e α + q ⊗ e α − r ) (cid:0) v ⊗ e f + af ( γ ) ⊗ e f − af ( γ ) (cid:1) = X α,m v ⊗ e f + af ( γ + α ) ⊗ e f − af ( γ + α ) + X p + q + r = m − ∞ X N =1 ( − N N ! N X N ′ =0 (cid:18) NN ′ (cid:19) ∞ X n =1 X α,p +( N − N ′ ) n ( α ∗ n ) N ′ v ⊗ Ψ N,N ′ ,n ; q,r whereΨ N,N ′ ,n ; q,r := ( α + − n ) N − N ′ e α + q − N ′ n e f + af ( γ ) ⊗ e α − r e f − af ( γ ) + ( − N ′ e α + q e f + af ( γ ) ⊗ ( α −− n ) N − N ′ e α − r − N ′ n e f − af ( γ ) . It is clear that Φ lies in V k ( g ) ⊗ V + f + af ( γ + α ) ⊗ V − f − af ( γ + α ) . By straightforward com-putations, we have ( id ⊗ ( α ′ ) + n ⊗ id )Φ = ( id ⊗ id ⊗ ( α ′′ ) − n )Φ = 0 for any α ′ ∈ ∆ + , α ′′ ∈ Π, and n ∈ Z > . Therefore, by the uniquness of singular vector in theHeisenberg Fock module V + f + af ( γ + α ) ⊗ V − f − af ( γ + α ) , there exists Φ af ∈ V k ( g ) such thatΦ = Φ af ⊗ e f + af ( γ + α ) ⊗ e f − af ( γ + α ) . Then, by the explicit form of Ψ, we conclude thatΦ af = X α,m v . This proves the formula (4.7) for α ∈ Π. Since the proof for α ∈ − Πis the same, we omit it.Next we prove that the assignment (4.5) uniquely extends to (4.6). By a directcomputation, we obtain H + α,n e v = ( n > ,ν + ( µ )( H + α ) e v if n = 0 , H − β,n e v = ( n > ,ν − ( µ sc )( H − β ) e v if n = 0for α ∈ Π and β ∈ ∆ + . Hence (4.5) uniquely extends to (4.6).At last, as the injectivity of (4.5) follows from the bijectivity of the operator exp ( H ) on M ⊗ V + ⊗ V − , the induced homomorphism (4.6) is also injective. (cid:3) Natural isomorphism.
Proposition 4.3.
Let M be an object of C [ λ ] (cid:0) V k ( g ) (cid:1) . Then the injective homo-morphism (4.6) gives rise to an isomorphism(4.8) F µM : M ≃ −→ Ω − ( µ sc ) ◦ Ω + ( µ )( M ); v ι µM (cid:0) v ⊗ | ν + ( µ ) i ⊗ | ν − ( µ sc ) i (cid:1) of Z / Z -graded weak V k ( g )-modules, which is natural in M . Proof.
First we prove that F µM is a Z / Z -graded weak V k ( g )-module isomorphism.Since F µM is a well-defined injective morphism of Z / Z -graded weak V k ( g )-modulesby Lemma 4.2, it suffices to show that F µM is surjective. The surjectivity of F µM follows from its injectivity and the character formula (4.2).Next we prove that F µM is natural in M . Let f : M → M be a morphism of C [ λ ] (cid:0) V k ( g ) (cid:1) . By using the explicit form (4.5) of F µM , we have (cid:0) Ω − ( µ sc ) ◦ Ω + ( µ ) (cid:1) ( f ) (cid:0) F µM ( v ) (cid:1) = ( f ⊗ id V + ⊗ id V − ) (cid:0) F µM ( v ) (cid:1) = F µM (cid:0) f ( v ) (cid:1) for any v ∈ M . We thus complete the proof. (cid:3) Remark 4.4.
It is clear by the proof that the above natural isomorphism is an even supernatural transformation in the sense of [BE17, Definition 1.1].Now we give the proof of Proposition 2.11.
Corollary 4.5.
Proposition 2.11 holds.
Proof.
Since f FST = F V k ( g ) , it follows from Proposition 4.3. (cid:3) By using Proposition 2.11, we obtain the following.
Proposition 4.6.
The composed functor Ω − ( µ sc ) ◦ Ω + ( µ ) : C [ λ ] ( V af ) → C [ λ ] ( V af ) isnaturally isomorphic to the identity functor of C [ λ ] ( V af ). In particular, the functorΩ + ( µ ) : C [ λ ] ( V af ) → C [ µ sc ] ( V sc ) is faithful. Proof.
By Proposition 2.11, we can verify that Proposition 4.3 remains true if wereplace V k ( g ) by L k ( g ). Therefore (4.8) gives a desired natural isomorphism. (cid:3) Remark 4.7.
When g = sl , one can describe an explicit natural isomorphismfrom the identity functor of C µ sc ( V sc ) to Ω + ( µ ) ◦ Ω − ( µ sc ) in the same way as [Sat16, § g = sl , there exist certain “inner automorphisms”of V sc induced by the lattice K (see Lemma 4.16) and the same argument as [Sat16, § Essentially surjectivity and fullness of Ω + ( µ ) . In this subsection, weprove that Ω + ( µ ) : C [ λ ] ( V af ) → C [ µ sc ] ( V sc ) is essentially surjective and full.We first recall the theory of Li’s ∆-operators developed in [Li97]. Let ( V = V ¯0 ⊕ V ¯1 , Y, , ω ) be a conformal vertex superalgebra. We use the following notationintroduced in [Li97, (3.4)]: E ± ( h, z ) := exp ∞ X j =1 h ± j j z ∓ j ∈ End ( V )[[ z ∓ ]]for h ∈ V ¯0 . AZAMA–SUZUKI COSET CONSTRUCTION AND ITS INVERSE 17
Lemma 4.8 ([Li97]) . Let ( M , Y M ) be a Z / Z -graded weak V -module. We assumethat there exist h ∈ V ¯0 and c ∈ C \ { } satisfying the following conditions:(1) L n h = δ n, h holds for any n ≥ h n h = δ n, c holds for any n ≥ h on V is diagonalizable and has only integral eigenvalues,(4) E + ( − h, − z ) v ∈ V [ z, z − ] holds for any v ∈ V .Then Li’s ∆-operator∆( h, z ) := z h E + ( − h, − z ) ∈ End ( V )[[ z, z − ]]lies in the group G ( V ) introduced in [Li97, § twist the Z / Z -graded weak V -module structure Y M on M to obtain a new Z / Z -gradedweak V -module M h := ( M , Y M h ) defined by Y M h = Y M h (? , z ) := Y M (cid:0) ∆( h, z )?; z (cid:1) : V → End ( M )[[ z, z − ]] . Proof.
See [Li97, Proposition 2.1 and 3.2] for the proof. (cid:3)
We now focus on the case of V = C k ( g ) ⊗ H + . Lemma 4.9.
Let M be an object of C [ µ sc ] (cid:0) C k ( g ) (cid:1) . We consider the direct sum(4.9) e M = M γ ∈ Q e M ( γ ) := M γ ∈ Q Π h f + af ( γ ) ,f + af ( γ ) i (cid:16) M ⊗ H + ν + ( µ ) (cid:17) h ( γ ) of Z / Z -graded weak C k ( g ) ⊗ H + -modules, where Π is the Z / Z -parity reversingfunctor and h ( γ ) := X α ∈ Π γ ( α ∗ ) (cid:18) J ∗ α ⊗ + − sc ⊗ k + h ∨ H + α (cid:19) ∈ C k ( g ) ⊗ H + . Then the Z / Z -graded weak C k ( g ) ⊗ H + -module structure on e M can be extendedto a Z / Z -graded weak V k ( g ) ⊗ V + -module structure on e M . Proof.
First we extend the Z / Z -graded action of V K on e M to that of V + . Since wehave ι + (cid:0) h ( γ ) (cid:1) = af ⊗ f + af ( γ ), the action of the Heisenberg vertex algebra generatedby (cid:8) h ( α ) (cid:12)(cid:12) α ∈ Π (cid:9) on e M can be extended to that of V f + af ( Q ) along with the 2-cocycle ǫ + (cf. [LL12, Theorem 6.5.18]). More precisely, we define a family of Z / Z -homogeneous linear automorphisms { S h ( γ ) | γ ∈ Q } on e M by S h ( γ ) (cid:12)(cid:12)(cid:12) f M ( γ ′ ) := ǫ + (cid:0) f + af ( γ ) , f + af ( γ ′ ) (cid:1) id M ⊗ H + ν +( µ ) : e M ( γ ′ ) → e M ( γ ′ + γ )and then the following Z / Z -homogeneous mutually local fields n e f + af ( γ ) ( z ) := S h ( γ ) z h ( γ ) E − (cid:0) h ( γ ) , z (cid:1) E + (cid:0) − h ( γ ) , z (cid:1) (cid:12)(cid:12)(cid:12) γ ∈ Q o (4.10)on e M generate the Z / Z -graded weak V f + af ( Q ) -module structure. On the other hand,by direct calculation, we have Y f M (e ξ ⊗ + , z ) (cid:12)(cid:12)(cid:12) f M ( γ ) = Y M (e ξ , z ) ⊗ id H + ν +( µ ) ∈ End (cid:0) e M ( γ ) (cid:1) [[ z, z − ]] The Z / Z -parity of e f + af ( γ ) ( z ) is given by h f + af ( γ ) , f + af ( γ ) i mod Z . for any γ ∈ Q . By the general theory of regular lattice vertex superalgebra, thereexists a family of Z / Z -homogeneous linear automorphisms { S ξ | ξ ∈ K } of M suchthat Y M (e ξ , z ) = S ξ z ξ E − ( ξ, z ) E + ( − ξ, z )(4.11)for any ξ ∈ K . Then, by using the explicit description (4.10) and (4.11), one canverify that the following Z / Z -homogeneous fields n e f + af ( γ ) ( z ) , Y f M (e ξ ⊗ + , z ) (cid:12)(cid:12)(cid:12) γ ∈ Q, ξ ∈ K o are mutually local and generate the Z / Z -graded weak V + -module structure on e M (cf. [Li96, Theorem 3.2.10]).Next, we further extend the above V + -action on e M to a V k ( g ) ⊗ V + -actionwhich is compatible with the original C k ( g ) ⊗ H + -action . By using the residueproduct of mutually local fields (see e.g. [Kac98]), we define the following mutuallylocal even fields: X α ( z ) := ǫ + (cid:0) f + af ( α ) , f + af ( − α ) (cid:1) Y f M (Ψ α , z ) h f + af ( α ) ,f + af ( α ) i− e − f + af ( α ) ( z ) ,H α ( z ) := Y f M (cid:16) X β ∈ ∆ + ( α, β ) J ∗ β ⊗ + + sc ⊗ kk + h ∨ H + α , z (cid:17) for α ∈ ∆, where Ψ α := X α, − af ⊗ e f + af ( α ) ⊗ + . Then, by using the Borcherdsidentity for mutually local fields, we can verify that the set of mutually local fields (cid:8) X α ( z ) , H α ( z ) (cid:12)(cid:12) α ∈ ∆ (cid:9) generates a Z / Z -graded weak V k ( g )-module structure on e M , which commutes with the V + -action on e M . (cid:3) Since L + ≃ Z N is a positive-definite unimodular lattice, the canonical pairing Hom V + - Mod ( V + , e M ) ⊗ V + → e M ; f ⊗ v f ( v )gives an isomorphism of Z / Z -graded weak V k ( g ) ⊗ V + -modules. Lemma 4.10.
Let M and e M be as in Lemma 4.9. Then the Z / Z -graded weak V k ( g )-module Hom V + - Mod ( V + , e M ) is an object of C [ λ ] (cid:0) V k ( g ) (cid:1) . Proof.
Since M is an object of C [ µ sc ] (cid:0) C k ( g ) (cid:1) , by the same computation as in Propo-sition 4.1, we can verify that every h -weight of Hom V + - Mod ( V + , e M ) lies in [ λ ] andeach string function of Hom V + - Mod ( V + , e M ) has the lowest exponent in q . Thus Hom V + - Mod ( V + , e M ) is an object of C [ λ ] (cid:0) V k ( g ) (cid:1) . (cid:3) Proposition 4.11.
The functor Ω + ( µ ) : C [ λ ] ( V af ) → C [ µ sc ] ( V sc ) is essentially sur-jective and full. Proof.
By Proposition 4.6, it suffices to prove the case of ( V af , V sc ) = (cid:0) V k ( g ) , C k ( g ) (cid:1) .We first prove that Ω + ( µ ) is essentially surjective. It suffices to show that M is isomorphic to Ω + ( µ )( M ) for M := Hom V + - Mod ( V + , e M ) as a Z / Z -graded C k ( g )-module. By using Lemma 4.8, we obtain a Z / Z -graded H + -module isomorphism (cid:16) H + ν + ( µ ) (cid:17) k + h ∨ H + α ≃ −→ H + ν + ( µ + α ) ; | ν + ( µ ) i 7→ | ν + ( µ + α ) i Since C k ( g ) and V + ≃ af ⊗ V + generate the whole V k ( g ) ⊗ V + , such an extension is uniqueif it exists. AZAMA–SUZUKI COSET CONSTRUCTION AND ITS INVERSE 19 for α ∈ Π. Hence we can rewrite the Z / Z -graded decomposition (4.9) as(4.12) e M ≃ M γ ∈ Q (cid:16) Π h f + af ( γ ) ,f + af ( γ ) i M h sc [ γ ] (cid:17) ⊗ H + ν + ( µ − γ ) , where h sc [ γ ] := P α ∈ Π γ ( α ∗ ) J ∗ α . Therefore, by (3.1) and Schur’s lemma, we concludethat M ≃ Ω + ( µ )( M ) . Next we prove that Ω + ( µ ) is full. Let M i be an object of C [ µ sc ] (cid:0) C k ( g ) (cid:1) and e M i theextended Z / Z -graded weak V k ( g ) ⊗ V + -module as in Lemma 4.9 for i ∈ { , } .Then an arbitrary morphism f : M → M of weak C k ( g )-modules extends to amorphism of weak V k ( g ) ⊗ V + -modules defined by e f := M γ ∈ Q f ( γ ) : e M → e M , where f ( γ ) := f ⊗ id H + ν +( µ ) : e M ( γ ) → e M ( γ ). Then, by the Z / Z -graded linearisomorphisms Hom V k ( g ) ⊗ V + - Mod ( e M , e M ) ≃ Hom V k ( g )- Mod ( M , M ) ⊗ Hom V + - Mod ( V + , V + ) ≃ Hom V k ( g )- Mod ( M , M ) , we conclude that f = Ω + ( µ )( e f ). This completes the proof. (cid:3) Remark 4.12.
By the proof, the superfunctor Ω + ( µ ) is also evenly dense in thesense of [BE17, Definition 1.1].4.3. Spectral flow equivariance.
In this subsection, we prove (2) and (3) inTheorem 3.9.4.3.1.
Equivariance of Ω + . Lemma 4.13.
Let λ ∈ h ∗ and γ ∈ Q . For an object M and a morphism f : M → M of C [ λ sc ] ( V sc ), we set U γ sc ( M ) := Π h f + af ( γ ) ,f + af ( γ ) i M h sc [ γ ] and U γ sc ( f ) := f : U γ sc ( M ) → U γ sc ( M ) , respectively. Then the assignment U γ sc defines a C -linearfunctor U γ sc : C [ λ sc ] ( V sc ) → C [( λ − γ ) sc ] ( V sc ) , which is a categorical isomorphism with the inverse U − γ sc . Proof.
By the two isomorphisms (3.1) and (4.12) of Z / Z -graded V sc ⊗ H + -modules,the weak V sc -module U γ sc ( M ) turns out to lie in C [( λ − γ ) sc ] ( V sc ). The rest of the proofis straightforward and we omit it. (cid:3) Remark 4.14.
It is known that U γ sc induces the spectral flow automorphism of the N = 2 superconformal algebra. See [HT91, Proposition 4.2] for details. Theorem 4.15.
For any λ ∈ h ∗ and γ ∈ Q , we have an isomorphismΩ + ( λ − γ ) ≃ U γ sc ◦ Ω + ( λ )of functors from C [ λ ] ( V af ) to C [( λ − γ ) sc ] ( V sc ). Proof.
By using (3.1) and (4.12), we can verify that the natural isomorphism M ⊗ V + ≃ −→ M ⊗ Π h f + af ( γ ) ,f + af ( γ ) i ( V + ) f + af ( γ ) ; m ⊗ V + m ⊗ e − f + af ( γ )0 RYO SATO of Z / Z -graded weak V af ⊗ V + -modules for an object M of C [ λ ] ( V af ) gives rise toa natural isomorphism Ω + ( λ − γ )( M ) ≃ −→ U γ sc ◦ Ω + ( λ )( M ) of Z / Z -graded weak V sc -modules. (cid:3) Equivariance of Ω − . Lemma 4.16.
Let λ ∈ ( h sc ) ∗ and ξ ∈ K . For an object M and a morphism f : M → M of C [ λ ] ( V sc ), we set U ξ sc ( M ) := Π h ξ,ξ i M ξ and U ξ sc ( f ) := f : U ξ sc ( M ) → U ξ sc ( M ), respectively. Then the assignment U ξ sc defines a C -linear functor U ξ sc : C [ λ ] ( V sc ) → C [ λ ] ( V sc ) , which is naturally isomorphic to the identity functor of C [ λ ] ( V sc ). Proof.
By Proposition 3.8, we may assume that [ λ ] = [( λ af ) sc ]. For an object M of C [ λ af ] ( V af ), in the same way as the proof of Theorem 4.15, the natural isomorphism M ⊗ V + ≃ −→ M ⊗ Π h ξ,ξ i ( V + ) ξ ; m ⊗ V + m ⊗ e − ξ of Z / Z -graded weak V af ⊗ V + -modules gives rise to a natural isomorphismΩ + ( λ )( M ) ≃ −→ U ξ sc ◦ Ω + ( λ )( M )of Z / Z -graded weak V sc -modules. Then, by Theorem 3.9, we get the conclusion. (cid:3) Theorem 4.17.
Let λ ∈ ( h sc ) ∗ and γ ∈ Q sc .(1) For an object M and a morphism f : M → M of C [ λ af ] ( V af ), we set U γ af ( M ) := Π h f + sc ( γ ) ,f + sc ( γ ) i M h af [ γ ] and U γ af ( f ) := f : U γ af ( M ) → U γ af ( M ),respectively, where h af [ γ ] := X α ∈ Π γ ( J ∗ α ) α ∗− af , f + sc ( γ ) := X β ∈ ∆ + γ ( J ∗ β ) β + . Then the assignment U γ af defines a C -linear functor U γ af : C [ λ af ] ( V af ) → C [( λ + γ ) af ] ( V af ) , which is a categorical isomorphism with the inverse U − γ af .(2) We have an isomorphismΩ − ( λ + γ ) ≃ U γ af ◦ Ω − ( λ )of functors from C [ λ ] ( V sc ) to C [( λ + γ ) af ] ( V af ). Proof.
To prove (1) and (2), it suffices to construct a natural isomorphism fromΩ − ( λ + γ ) to U γ af ◦ Ω − ( λ ). For each α ∈ ∆ + , we set( H − α ) ∗ := X β ∈ ∆ + G ∗ α,β H − β ∈ H − . By using the formula α ∗ = P β ∈ Π c α,β H β and tedious but straightforward calcula-tion, we obtain ι − (cid:16) h af [ γ ] ⊗ − + af ⊗ h γ (cid:17) = ξ γ ⊗ V − + sc ⊗ ζ γ − V − , where h γ := X α ∈ ∆ + γ ( J ∗ α )( H − α ) ∗ , ξ γ := X α ∈ ∆ + γ ( J ∗ α ) ξ ( α ) , ζ γ := X α ∈ ∆ + γ ( J ∗ α ) f − af ( α ) . AZAMA–SUZUKI COSET CONSTRUCTION AND ITS INVERSE 21
Since we have a Z / Z -graded weak H − -module isomorphism (cid:16) H − ν − ( λ ) (cid:17) h γ ≃ −→ H − ν − ( λ + γ ) ; | ν − ( λ ) i 7→ | ν − ( λ + γ ) i , in the same way as Theorem 4.15, we can verify that the natural isomorphism M ⊗ V − ≃ −→ U ξ γ sc ( M ) ⊗ Π h ζ γ ,ζ γ i ( V − ) ζ γ − V − of Z / Z -graded weak V sc ⊗ V − -modules for an object M of C [ λ ] ( V sc ) gives rise toa natural isomorphism Ω − ( λ + γ )( M ) ≃ U γ af ◦ Ω − ( λ )( M ) of Z / Z -graded weak V af -modules. (cid:3) Regular cases
Throughout this section, we assume that k is a positive integer. In this section,we give additional information about the representation theory of C k ( g ).5.1. Bicommutant of H + . Let Q l be the even integral sublattice of the rootlattice Q spanned by long roots. By the normalization of (? , ?) on h ∗ , we have Q l = M α ∈ Π Z ˇ α (cid:18) ˇ α := 2 α ( α, α ) (cid:19) . We first recall that C. Dong and Q. Wang give a proof of the following fact (see also[DLY09, §
4] for the case of g = sl ). Proposition 5.1 ([DW11, Proposition 4.11]) . The vertex operator algebra exten-sion E := Com (cid:16)
Com (cid:0) H , L k ( g ) (cid:1) , L k ( g ) (cid:17) of the Heisenberg vertex subalgebra H generated by h in L k ( g ) is isomorphic to thelattice vertex operator algebra associated with √ kQ l .The next proposition is proved in a similar way as [DW11, Proposition 4.11] andwe omit the detail (see [CL19, § g = sl ). Proposition 5.2.
The vertex operator superalgebra extension E + := Com (cid:16)
Com (cid:0) H + , L k ( g ) ⊗ V + (cid:1) , L k ( g ) ⊗ V + (cid:17) of the Heisenberg vertex subalgebra H + in L k ( g ) ⊗ V + is purely even and isomorphicto the lattice vertex operator algebra associated with √ k + h ∨ Q l . Moreover, wehave E + = Com (cid:16)
Com (cid:0) H + , L k ( g ) ⊗ ( V + ) ¯0 (cid:1) , L k ( g ) ⊗ ( V + ) ¯0 (cid:17) . Proof.
Note that the even part ( V + ) ¯0 is the lattice vertex operator algebra associ-ated with the even sublattice { α ∈ L + | h α, α i ∈ Z } ≃ ( Z if g = sl , (the root lattice of type D N ) otherwiseof L + . Then we can apply a straightforward modification of the proof of [DW11,Proposition 4.11]. (cid:3) Bicommutant of H − . By (3.2) and Theorem 4.17, we have an isomorphism M γ ∈ Q sc U γ af (cid:0) L k ( g ) (cid:1) ⊗ H − ν − ( γ ) ≃ ι ∗− (cid:0) C k ( g ) ⊗ V − (cid:1) of Z / Z -graded weak L k ( g ) ⊗ H − -modules. We first study the structure of U γ af (cid:0) L k ( g ) (cid:1) by the following lemma. Lemma 5.3.
Let θ ∈ P ∨ . Then L k ( g ) θ := (cid:0) L k ( g ) (cid:1) θ − af is isomorphic to L k ( g ) asa weak L k ( g )-module if and only if θ ∈ Q ∨ . Proof.
Define a Lie algebra automorphism t θ of b g = b g B by t θ : b g → b g ; X α,n X α,n + α ( θ ) , H n H n + B ( H, θ ) Kδ n, , K K for α ∈ ∆, n ∈ Z , and H ∈ h . Then, by a direct computation, we obtain Y V af (cid:16) ∆ (cid:0) θ − af , z (cid:1) X − af , z (cid:17) = X n ∈ Z t θ ( X n ) z − n − for any X ∈ g . By the same argument in [HT91, (4-1)], the automorphism t θ isinner if θ ∈ Q ∨ . On the other hand, since t θ for θ ∈ P ∨ \ Q ∨ corresponds to anon-trivial Dynkin diagram automorphism and the induced P ∨ /Q ∨ -action on theset of special indices is simply transitive (see e.g. [Wak01, § L k ( g ) θ coincides with that of L k ( g ) if and onlyif θ ∈ Q ∨ . (cid:3) Remark 5.4.
Following [HT91, Proposition 4.1], we call the automorphism t θ thespectral flow automorphism of b g associated with θ ∈ P ∨ .We use the next lemma to determine the vertex superalgebra structure of thebicommutant of H − . Lemma 5.5 (cf. [CKLR18, Theorem 4.1]) . Let ( V = V ¯0 ⊕ V ¯1 , Y, , ω ) be a sim-ple conformal vertex superalgebra and (cid:0) L, h ? , ? i (cid:1) a non-degenerate integral lattice.Assume that there exists an injective Z -linear map i : L → V ¯0 such that(1) L n i ( h ) = δ n, i ( h ) for any h ∈ L and n ≥ i ( h ) n i ( h ′ ) = δ n, h h, h ′ i for any n ≥ M h ∈ L ( H L ) i ( h ) ≃ V of (not necessarily Z / Z -graded) weak H L -modules, where H L is the Heisen-berg vertex subalgebra generated by i ( L ) in V and ( H L ) i ( h ) is the weak H L -module defined in Lemma 4.8.Then V is isomorphic to the lattice vertex superalgebra associated with the lattice L as a vertex superalgebra. Proof.
It follows from a straightforward generalization of [LX95, Theorem 3.14] (see[Xu98, Theorem 6.3.1] for details). (cid:3)
Now we prove the following proposition.
AZAMA–SUZUKI COSET CONSTRUCTION AND ITS INVERSE 23
Proposition 5.6.
The conformal vertex supralgebra extension E − := Com (cid:16)
Com (cid:0) H − , C k ( g ) ⊗ V − (cid:1) , C k ( g ) ⊗ V − (cid:17) of the Heisenberg vertex subalgebra H − in C k ( g ) ⊗ V − is isomorphic to the latticevertex superalgebra associated with p − ( k + h ∨ ) Q l ⊕ Z N − ℓ . Proof.
In order to apply Lemma 5.5, we verify the assumptions in Lemma 5.5.We first prove that E − is simple. By Proposition 2.11 and the regularity of L k ( g ), the vertex superalgebra C k ( g ) ⊗ V − decomposes into a direct sum of simpleordinary Com (cid:0) H − , C k ( g ) ⊗ V − (cid:1) -modules. Since V − is strongly L − -graded in thesense of [HLZ07, Definition 2.23], we can define the contragredient V − -module inthe sense of [HLZ07, Definition 2.35]. Then, we can apply the same argument in[ACKL17, Lemma 2.1] to our case and conclude that E − is simple.Next we verify the conditions (1), (2), and (3) in Lemma 5.5. Define a Z -linearmap i : p − ( k + h ∨ ) Q l ⊕ Z N − ℓ → ( E − ) ¯0 by i (cid:0)p − ( k + h ∨ ) γ (cid:1) := X α ∈ ∆ + ( γ, α )( H − α ) ∗ , i ( ε β ) := ( H − β ) ∗ for γ ∈ Q l and β ∈ ∆ + \ Π, where { ε β | β ∈ ∆ + \ Π } is an orthonormal Z -basisof Z N − ℓ . Then the condition (1) is clear and the condition (2) follows from (2.1)and direct computations. At last, we verify the condition (3). It suffices to provethat, for γ ∈ Q sc , the Z / Z -graded weak L k ( g )-module U γ af (cid:0) L k ( g ) (cid:1) is isomorphic to L k ( g ) if and only if h γ lies in the image of the map i . This is verified by Lemma5.3 and some computations. We thus complete the proof. (cid:3) Regularity and Unitarity.
In this subsection, we prove that C k ( g ) is regularand has a unitary structure.We first prove the regularity. By Proposition 5.2, we obtain the following cosetrealization(5.1) C k ( g ) ¯0 = Com (cid:0) H + , L k ( g ) ⊗ ( V + ) ¯0 (cid:1) = Com (cid:0) E + , L k ( g ) ⊗ ( V + ) ¯0 (cid:1) . Then we obtain the following.
Theorem 5.7.
Both C k ( g ) ¯0 and C k ( g ) are regular. Proof.
We first prove the regularity of C k ( g ) ¯0 . By [ABD04, Theorem 4.5], it sufficesto prove that C k ( g ) ¯0 is rational and C -cofinite. Then, by (5.1) and Proposition5.2, the C -cofiniteness follows from [Miy15, Corollary 2] and the rationality followsfrom [CKLR18, Theorem 4.12].Next we prove the regularity of C k ( g ). By a straightforward generalization of[ABD04, Theorem 4.5], it suffices to prove that C k ( g ) is rational and C -cofinite.The rationality of C k ( g ) follows from [DH12, Proposition 3.6]. The C -cofinitenessof C k ( g ) follows from [Miy04, Lemma 2.4] and the fact that C k ( g ) is finitely gener-ated over the regular vertex operator algebra C k ( g ) ¯0 . (cid:3) Corollary 5.8.
We have C k ( g )- gmod = M [ λ ] ∈ P k sc /Q sc C [ λ ] (cid:0) C k ( g ) (cid:1) , where P k sc := (cid:8) µ sc ∈ ( h sc ) ∗ (cid:12)(cid:12) µ ∈ P (cid:9) . Proof.
The right hand side is a full subcategory of the left hand side by the defini-tion. Since the left hand side is a semisimple abelian category with finitely many(up to isomorphism) simple objects, it suffices to prove that every simple object inthe left hand side lies in the right hand side. Let M be a simple Z / Z -graded or-dinary C k ( g )-module. Since M decomposes into a direct sum of finite-dimensional L -eigenspaces, the abelian Lie algebra h sc acts diagonally on M . Therfore, byProposition 3.8, Theorem 3.9, and [FZ92, Theorem 3.1.3], there exists µ ∈ P suchthat M is a simple object of C [ µ sc ] (cid:0) C k ( g ) (cid:1) . (cid:3) We next prove the unitarity.
Lemma 5.9.
There exists an anti-linear involution φ of C k ( g ) such that (cid:0) C k ( g ) , φ (cid:1) forms a unitary vertex operator superalgebra in the sense of [AL17, § Proof.
By [DL14, Theorem 4.7] and [AL17, Proposition 2.4 and Theorem 2.9], both L k ( g ) ⊗ V + and E + have a unitary structure. Then, by [DL14, Corollary 2.8], theunitary structure of L k ( g ) ⊗ V + inherits to C k ( g ). (cid:3) We close this subsection with the following result.
Proposition 5.10.
Every simple Z / Z -graded ordinary C k ( g )-module appears as adirect summand of the tensor product of some simple Z / Z -graded ordinary L k ( g )-module and the lattice vertex operator superalgebra V + . In particular, every simple Z / Z -graded ordinary C k ( g )-module is unitarizable. Proof.
Let M be a simple Z / Z -graded ordinary C k ( g )-module. By Corollary 5.8,we may assume that M is a simple object of C [ µ sc ] (cid:0) C k ( g ) (cid:1) for some µ ∈ P . Then,by Theorem 3.9, we obtain M ≃ Ω + ( µ ) ◦ Ω − ( µ sc )( M ) as a direct summand ofΩ − ( µ sc )( M ) ⊗ V + . Therefore the unitarizability of M follows from [DL14, Theorem4.8] and the same discussion as the proof of [DL14, Corollary 2.8]. (cid:3) Braided monoidal structure.
In this subsection, owing to the recent de-velopments in the general theory of vertex superalgebra extensions by [HKL15],[CKL19], and [CKM17], we prove that the category of Z / Z -graded ordinary C k ( g )-modules carries a braided monoidal category structure.As a consequence of the theory of vertex tensor categories developed by Y.-Z. Huang and J. Lepowsky (see [HL95a], [HL95b], [HL95c], [Hua95], [Hua05], andreferences therein), we obtain the following. Proposition 5.11.
The semisimple C -linear abelian category of purely even Z / Z -graded ordinary C k ( g ) ¯0 -modules has a braided monoidal category structure inducedby [Hua05, Theorem 3.7] (see also [HL94, Theorem 4.2 and 4.4]). Proof.
Since C k ( g ) ¯0 is regular by Theorem 5.7, all the conditions in [Hua05, Theo-rem 3.5] are satisfied. Therefore it follows from [Hua05, Theorem 3.7]. (cid:3) We now arrive at the last result in this paper.
Theorem 5.12.
The semisimple C -linear abelian category C := C k ( g )- gmod of Z / Z -graded ordinary C k ( g )-modules has a braided monoidal category structureinduced by [CKM17, Theorem 3.65]. AZAMA–SUZUKI COSET CONSTRUCTION AND ITS INVERSE 25
Proof.
By [CKL19, Theorem 3.14], the category C is equivalent to the underlyingcategory of the supercategory Rep (cid:0) C k ( g ) (cid:1) defined in [CKL19, Definition 2.16] (seealso [KJO02, Definition 1.8] for details). Since these categories carry a naturalbraided monoidal category structure by [CKM17, Theorem 3.65] (see also [CKM17,Remark 3.56 and 3.67]), we complete the proof. (cid:3) Remark 5.13.
When g = sl , Theorem 5.12 is proved by Y.-Z. Huang and A. Milasin [HM02, Theorem 4.8]. Note that Proposition 2.11 for g = sl (see also [Ada99,Theorem 5.1]) plays a crucial role in their proof of the convergence and extensionproperty for products of intertwining operators. See [HM02, §
3] for details.
Example 5.14.
Assume that g is simply-laced and k = 1. By Proposition 5.2and the isomoprhism L ( g ) ≃ V Q known as the Frenkel–Kac construction, wehave C ( g ) ≃ Com (cid:0) V √ h ∨ Q , V Q ⊕ Z N (cid:1) . Then this coset vertex operator superalgebraturns out to be isomorphic to the lattice vertex superalgebra associated with Q ∨ sc := M α ∈ ∆ + Z J α , h J α , J β i := ( α, β ) + δ α,β ( α, β ∈ ∆ + ) . By a direct computation, we can verify that its dual quotient group is given by( Q ∨ sc ) ∗ (cid:14) Q ∨ sc ≃ (cid:0) Z / (1 + h ∨ ) Z (cid:1) ℓ . Appendix A. Vertex superalgebras and modules
A.1.
Universal affine VOAs.
Throughout this subsection, g stands for a finite-dimensional reductive Lie algebra over C . Let B : g × g → C be a symmetricinvariant bilinear form and b g B = g ⊗ C [ t, t − ] ⊕ C K the corresponding affinizationof g , i.e. the commutation relations are given by[ X n , Y m ] = [ X, Y ] n + m + B ( X, Y ) nδ n + m, K, [ b g , K ] = { } for X, Y ∈ g and n, m ∈ Z , where X n stands for X ⊗ t n ∈ b g B . The next lemma iswell-known. Lemma A.1.
Let C B be a 1-dimensional representation of b g ≥ := g ⊗ C [ t ] ⊕ C K defined by g ⊗ C [ t ] . { } and K. V ( g , B ) := U ( b g B ) ⊗ U ( b g ≥ ) C B which is strongly generated by X ( z ) := X n ∈ Z X n z − n − for X ∈ g , which is called the universal affine vertex algebra associated with ( g , B ).When g is an n -dimensional abelian Lie algebra and B is non-degenerate, we call V ( g , B ) the Heisenberg vertex algebra of rank n . It is well-known that Heisenbergvertex algebras of fixed rank are simple and isomorphic to each other. A.2.
Coset vertex superlagebras.
Let V be a vertex superalgebra and W itsvertex subsuperalgebra. Then the Z / Z -graded subspace Com ( W , V ) := { A ∈ V | [ A m , B n ] = 0 for any B ∈ W and m, n ∈ Z } gives a vertex subsuperalgebra of V , called the coset vertex superalgebra withrespect to the pair ( W , V ). By the general theory of vertex superalgebras (seee.g. [Kac98]), the even linear map Com ( W , V ) ⊗ W → V ; A ⊗ B → A − B gives a vertex superalgebra homomorphism.A.3. Weak modules.
Let ( V = V ¯0 ⊕ V ¯1 , Y, ) be a vertex superalgebra. Let M = M ¯0 ⊕ M ¯1 be a Z / Z -graded vector space and Y M = Y M (?; z ) : V → End C ( M )[[ z, z − ]]a not necessarily Z / Z -homogeneous (resp. even) linear map. We write Y M ( A ; z ) = X n ∈ Z A Mn z − n − ∈ End C ( M )[[ z, z − ]]for A ∈ V . Then the above pair ( M, Y M ) is called a (resp. Z / Z -graded ) weak V -module if the following conditions hold for any A ∈ V ¯ a , B ∈ V ¯ b , and v ∈ M : Field axiom: A Mn v = 0 holds for n ≫ Vacuum axiom: 1 Mn v = δ n, − v holds for any n ∈ Z , The Borcherds identity: ∞ X ℓ =0 (cid:18) pℓ (cid:19) ( A r + ℓ B ) Mp + q − ℓ v = ∞ X ℓ =0 ( − ℓ (cid:18) rℓ (cid:19) (cid:0) A Mp + r − ℓ B Mq + ℓ − ( − r + ab B Mq + r − ℓ A Mp + ℓ (cid:1) v holds for any p, q, r ∈ Z .Let ( M i , Y M i ) be (resp. Z / Z -graded) weak V -modules for i ∈ { , } . A notnecessarily Z / Z -homogeneous (resp. even) linear map f : M → M is a morphismof (resp. Z / Z -graded ) weak V -modules if f ◦ A M n = A M n ◦ f ∈ Hom C ( M , M )for any A ∈ V and n ∈ Z .Let V - Mod be the C -linear abelian category of weak V -modules and V - gMod thefull subcategory of V - Mod whose objects are Z / Z -graded weak V -modules. Lemma A.2.
The categories V - Mod and V - gMod are C -linear supercategories , thatis, each space of morphisms is a Z / Z -graded vector space over C and compositionof morphisms induces an even C -linear map. In addition, the underlying category V - gMod of the supercategory V - gMod in the sense of [BE17, Definition 1.1] coincideswith the C -linear abelian category of Z / Z -graded weak V -modules. We note that the C -linear additive category V - gMod is not abelian in general. AZAMA–SUZUKI COSET CONSTRUCTION AND ITS INVERSE 27
A.4.
Ordinary modules.
In this subsection, we assume that ( V = V ¯0 ⊕ V ¯1 , Y, )is a vertex operator superalgebra with respect to a conformal vector ω ∈ V ¯0 . Thena (resp. Z / Z -graded) weak V -module ( M, Y M ) is called a (resp. Z / Z -graded ) or-dinary V -module if M decomposes into a direct sum M = M h ∈ C M h of (resp. Z / Z -homogeneous) finite-dimensional subspaces M h := { v ∈ M | L v := ω M v = hv } such that M h + n = { } for any h ∈ C and n ≪ V - mod (resp. V - gmod ) for thefull subsupercategory of V - Mod whose objects are (resp. Z / Z -graded) ordinary V -modules. Then the underlying category V - gmod of the C -linear supercategory V - gmod is the C -linear abelian category of Z / Z -graded ordinary V -modules. Appendix B. Twisted sector
In this section, we give a remark on the twisted sector of V sc for k ∈ C \ { , − h ∨ } .We set J ∗ := 12 X α ∈ ∆ + J ∗ α ∈ P ∨ sc . Then the parity involution σ : V sc → V sc , which is defined by σ | V ¯ i sc = ( − i id V ¯ i sc for i ∈ { , } , satisfies that σ = exp (2 π √− J ∗ ). Lemma B.1.
The category of (resp. Z / Z -graded) Z -twisted positive energy V sc -modules is isomorphic to that of (resp. Z / Z -graded) Z -twisted positive energy V sc -modules. Here we refer the reader to [DSK06, Definition 2.21] for the definitionof Γ-twisted positive energy modules, where Γ = Z or Z . Proof.
For a (resp. Z / Z -graded) Z -twisted positive energy V sc -module ( M , Y M ), wedefine Y σ M = Y σ M (? , z ) : V sc → End ( M )[[ z , z − ]]by Y σ M ( v, z ) := Y M (cid:0) ∆( − J ∗ , z ) v, z (cid:1) for v ∈ V sc . Then M σ := ( M , Y σ M ) is a (resp. Z / Z -graded) Z -twisted positive energy V sc -module and the assignment M M σ givesrise to a desired categorical isomorphism. (cid:3) Remark B.2.
By [HT91, Theorem 2.5], every Z -twisted (resp. Z -twisted) weak V sc -module admits an action of the N = 2 superconformal algebra in the Neveu–Schwarz sector (resp. in the Ramond sector) of central charge c sc . References [ABD04] T. Abe, G. Buhl, and C. Dong. Rationality, regularity, and C -cofiniteness. Trans.Amer. Math. Soc. , 356(8):3391–3402, 2004.[ACKL17] T. Arakawa, T. Creutzig, K. Kawasetsu, and A.R. Linshaw. Orbifolds and Cosets ofMinimal W -Algebras. Commun. Math. Phys. , 355(1):339–372, 2017.[Ada99] D. Adamovi´c. Representations of the N = 2 superconformal vertex algebra. Int. Math.Res. Not., IMRN , 2(2):61–79, 1999.[Ada01] D. Adamovi´c. Vertex algebra approach to fusion rules for N = 2 superconformalminimal models. J. Algebra , 239(2):549–572, 2001. Every Z -twisted positive energy V sc -module is a weak V sc -module by the definition. See[DSK06, Example 2.14] for details. [AL17] C. Ai and X. Lin. On the unitary structures of vertex operator superalgebras. J. Al-gebra , 487:217–243, 2017.[BE17] J. Brundan and A.P. Ellis. Monoidal supercategories.
Commun. Math. Phys. ,351(3):1045–1089, 2017.[BZHS08] D. Ben-Zvi, R. Heluani, and M. Szczesny. Supersymmetry of the chiral de Rham com-plex.
Compos. Math. , 144(2):503–521, 2008.[CKL19] T. Creutzig, S. Kanade, and A.R. Linshaw. Simple current extensions beyond semi-simplicity.
Commun. Contemp. Math. , page 1950001, 2019.[CKLR18] T. Creutzig, S. Kanade, A.R. Linshaw, and D. Ridout. Schur–Weyl duality for Heisen-berg cosets.
Transform. Groups , pages 1–54, 2018.[CKM17] T. Creutzig, S. Kanade, and R. McRae. Tensor categories for vertex operator superal-gebra extensions. arXiv preprint arXiv:1705.05017 , 2017.[CL19] T. Creutzig and A.R. Linshaw. Cosets of affine vertex algebras inside larger structures.
J. Algebra , 517:396 – 438, 2019.[CM16] S. Carnahan and M. Miyamoto. Regularity of fixed-point vertex operator subalgebras. arXiv preprint arXiv:1603.05645 , 2016.[DH12] C. Dong and J. Han. Some finite properties for vertex operator superalgebras.
PacificJ. Math. , 258(2):269–290, 2012.[DL14] C. Dong and X. Lin. Unitary vertex operator algebras.
J. Algebra , 397:252–277, 2014.[DLM97] C. Dong, H. Li, and G. Mason. Regularity of rational vertex operator algebras.
Adv.Math. , 132(1):148–166, 1997.[DLY09] C. Dong, C.H. Lam, and H. Yamada. W-algebras related to parafermion algebras.
J.Algebra , 322(7):2366–2403, 2009.[DSK06] A. De Sole and G. Kac, V. “Finite vs affine W -algebras”. Japan. J. Math. , 1(1):137–261, 2006.[DW11] C. Dong and Q. Wang. Parafermion vertex operator algebras.
Front. Math. China ,6(4):567–579, 2011.[EOT11] T. Eguchi, H. Ooguri, and Y. Tachikawa. Notes on the k3 surface and the mathieugroup M . Exper. Math. , 20(1):91–96, 2011.[FLM89] I. Frenkel, J. Lepowsky, and A. Meurman.
Vertex operator algebras and the Monster ,volume 134. Academic press, 1989.[FST98] B. L. Feigin, A. M. Semikhatov, and I. Yu. Tipunin. Equivalence between chain cate-gories of representations of affine sl (2) and N = 2 superconformal algebras. J. Math.Phys. , 39(7):3865–3905, 1998.[FZ92] I. Frenkel and Y. Zhu. Vertex operator algebras associated to representations of affineand Virasoro algebras.
Duke Math. J. , 66(1):123–168, 1992.[HKL15] Y.-Z. Huang, A. Kirillov, and J. Lepowsky. Braided tensor categories and extensionsof vertex operator algebras.
Commun. Math. Phys. , 337(3):1143–1159, 2015.[HL94] Y.-Z. Huang and J. Lepowsky. Tensor products of modules for a vertex operator algebraand vertex tensor categories.
Lie Theory and Geometry, in honor of Bertram Kostant ,pages 349–383, 1994.[HL95a] Y.-Z. Huang and J. Lepowsky. A theory of tensor products for module categories fora vertex operator algebra, I.
Selecta Math.(N.S.) , 1(4):699, 1995.[HL95b] Y.-Z. Huang and J. Lepowsky. A theory of tensor products for module categories fora vertex operator algebra, II.
Selecta Math.(N.S.) , 1(4):757, 1995.[HL95c] Y.-Z. Huang and J. Lepowsky. A theory of tensor products for module categories fora vertex operator algebra, III.
J. Pure Appl. Algebra , 100(1-3):141–171, 1995.[HLZ07] Y.-Z. Huang, J. Lepowsky, and L. Zhang. Logarithmic tensor product theory for gener-alized modules for a conformal vertex algebra. arXiv preprint arXiv:0710.2687 , 2007.[HM02] Y.-Z. Huang and A. Milas. Intertwining operator superalgebras and vertex tensor cate-gories for superconformal algebras, II.
Trans. Amer. Math. Soc. , pages 363–385, 2002.[HT91] S. Hosono and A. Tsuchiya. Lie algebra cohomology and N = 2 SCFT based on theGKO construction. Commun. Math. Phys. , 136(3):451–486, 1991.[Hua95] Y.-Z. Huang. A theory of tensor products for module categories for a vertex operatoralgebra, IV.
J. Pure Appl. Algebra , 100(1-3):173–216, 1995.[Hua05] Y.-Z. Huang. Differential equations and intertwining operators.
Commun. Contemp.Math. , 7(03):375–400, 2005.
AZAMA–SUZUKI COSET CONSTRUCTION AND ITS INVERSE 29 [Hua08] Y.-Z. Huang. Vertex operator algebras and the Verlinde conjecture.
Commun. Con-temp. Math. , 10(01):103–154, 2008.[Kac98] V. Kac.
Vertex algebras for beginners,
University Lecture Series .AMS, Providence, RI, 1998.[KJO02] A. Kirillov Jr and V. Ostrik. On a q -analogue of the McKay correspondence and theADE classification of ˆ sl conformal field theories. Adv. Math. , 171(2):183–227, 2002.[KRW03] V. Kac, S.-S. Roan, and M. Wakimoto. Quantum reduction for affine superalgebras.
Commun. Math. Phys. , 241(2):307–342, 2003.[KS89] Y. Kazama and H. Suzuki. New N = 2 superconformal field theories and superstringcompactification. Nucl. Phys. B , 321(1):232–268, 1989.[KvdL89] V.G. Kac and J.W. van de Leur. On classification of superconformal algebras. In : S.J.Gates, et al. (Eds.), Strings 88 , pages 77–106. World Scientific, Singapore, 1989.[KW16] V. G. Kac and M. Wakimoto. Representations of affine superalgebras and mock thetafunctions II.
Adv. Math. , 300:17–70, 2016.[Li96] H. Li. Local systems of vertex operators, vertex superalgebras and modules.
J. PureAppl. Algebra , 109(2):143–195, 1996.[Li97] H. Li. The physics superselection principle in vertex operator algebra theory.
J. Algebra ,196(2):436–457, 1997.[LL12] J. Lepowsky and H. Li.
Introduction to vertex operator algebras and their representa-tions , volume 227. Springer Science & Business Media, 2012.[LX95] H. Li and X. Xu. A characterization of vertex algebras associated to even lattices.
J.Algebra , 173(2):253–270, 1995.[Miy04] M. Miyamoto. Modular invariance of vertex operator algebras satisfying C -cofiniteness. Duke Math. J. , 122(1):51–91, 2004.[Miy15] M. Miyamoto. C -cofiniteness of cyclic-orbifold models. Commun. Math. Phys. ,335(3):1279–1286, 2015.[M¨ug03] M. M¨uger. On the structure of modular categories.
Proc. London Math. Soc. ,87(2):291–308, 2003.[NS71] A. Neveu and J.H. Schwarz. Factorizable dual model of pions.
Nucl. Phys. B , 31(1):86–112, 1971.[NT05] K. Nagatomo and A. Tsuchiya. Conformal field theories associated to regular chiral ver-tex operator algebras, I: Theories over the projective line.
Duke Math. J. , 128(3):393–471, 2005.[Ram71] P. Ramond. Dual theory for free fermions.
Phys. Rev. D , 3(10):2415, 1971.[Sat16] R. Sato. Equivalences between weight modules via N = 2 coset constructions. arXivpreprint arXiv:1605.02343 , 2016.[Wak01] M. Wakimoto. Lectures on infinite-dimensional Lie algebra . World scientific, 2001.[Wit12] E. Witten. Notes on super riemann surfaces and their moduli. arXiv preprintarXiv:1209.2459 , 2012.[Xu98] X. Xu.
Introduction to vertex operator superalgebras and their modules , volume 456 of
Mathematics and Its Applications . Kluwer Academic Publishers, Dordrecht, 1998.
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