aa r X i v : . [ m a t h . QA ] A p r GENUS OF VERTEX ALGEBRAS AND MASS FORMULA
YUTO MORIWAKIA bstract . We introduce the notion of a genus and its mass for vertex algebras. For latticevertex algebras, their genera are the same as those of lattices, which plays an importantrole in the classification of lattices. We derive a formula relating the mass for vertexalgebras to that for lattices, and then give a new characterization of some holomorphicvertex operator algebras. I ntroduction Vertex operator algebra (VOA) is an algebraic structure that plays an important rolein two-dimensional conformal field theory. The classification of conformal field theoriesis an extremely interesting problem, of importance in mathematics, statistical mechanics,and string theory; Mathematically, it is related to the classification of vertex operatoralgebras, which is a main theme of this work. An important class of vertex operatoralgebras can be constructed from positive-definite even integral lattices, called latticevertex algebras [Bo1, LL, FLM]. In this paper, we propose a new method to constructand classify vertex algebras by using a method developed in a study of lattices.A lattice is a finite rank free abelian group equipped with a Z -valued symmetric bilinearform. Two such lattices are equivalent (or in the same genus ) if they are isomorphic overthe ring of p -adic integers Z p for each prime p and also equivalent over the real numberfield R . Lattices in the same genus are not always isomorphic over Z . For example,there are two non-isomorphic rank 16 positive-definite even unimodular lattices D + and E ⊕ E which are in the same genus. A mass of the genus of a lattice L is defined bymass( L ) = P M / | Aut M | , where the sum is over all isomorphism classes of lattices in thegenus. The mass can be calculated by the Smith-Minkowski-Siegel mass formula (see forexample [CS, Ki]). The concept of genera of lattices and the mass formula are importantfor the classification of lattices. In fact, rank 24 positive-definite even unimodular lattices,called Niemeier lattices, could be classified by using the mass formula.The following result is important to define the notion of genus of vertex algebras; Twointegral lattices L and L are in the same genus if and only if L ⊕ II , and L ⊕ II , areisomorphic, where II , is the unique even unimodular lattice of signature (1 ,
1) ([CS]).Motivated by this fact, we would like to define that two vertex algebras V and V arein the same genus if V ⊗ V II , and V ⊗ V II , are isomorphic. Little is known abouta structure of V ⊗ V II , , more generally, Z -graded vertex algebra whose homogeneoussubspaces are infinite dimensional. We overcome this di ffi culty by using a Heisenbergvertex subalgebra as an additional structure. To be more precise, we consider a pair of a vertex algebra V and its Heisenberg vertex subalgebra H , called a VH pair ( V , H ), anddefine a genus and a mass of VH pairs, and prove a formula relating the mass of VHpairs and that of lattices, which we call here a mass formula for vertex algebras (Theorem4.2). The important point to note here is that if VH pairs ( V , H V ) and ( W , H W ) are in thesame genus, then the module categories V -mod and W -mod are equivalent (Theorem 4.1).Hence, if V has a good representation theory (e.g., completely reducible), then W also hasa good representation theory. This result suggests that the genus of VH pairs can be usedto construct good vertex algebras from a good vertex algebra.The tensor product functor − ⊗ V II , is first considered by R. Borcherds and is an impor-tant step in his proof of the moonshine conjecture (see [Bo2]). It is also used by G. H¨ohnand N. Scheithauer in their constructions of holomorphic VOAs [HS1]. In the process ofthe constructions, they show that some non-isomorphic 17 holomorphic VOAs of centralcharge 24 are isomorphic to each other after taking the tensor product − ⊗ V II , , that is,in the same genus in our sense [HS1]. They also study other examples of holomorphicVOAs with non-trivial genus (see [Ho2], [HS2]), which motivates us to define the genus.We remark that G. H¨ohn gives another definition of a genus of vertex operator algebrasbased on the representation theory of VOAs (more precisely, modular tensor category)[Ho1]. All holomorphic VOAs of central charge 24 are in the same genus in their defini-tion, whereas they are not so in our sense. As suggested by the equivalence of categoriesin the same genus (Theorem 4.1), we believe that if VOAs are in the same genus in oursense, then they are in the same genus in the sense of [Ho1]. Organization of the paper
Let us explain our basic idea and the contents of this pa-per. It is worth pointing out that the construction of lattice vertex algebras from evenlattices, L V L , is not a functor from the category of lattices to the category of ver-tex algebras (see Remark 2.2). We focus on a twisted group algebra C { L } consideredin [FLM] which plays an important role in the construction of lattice vertex algebras.In Section 2, we generalize the twisted group algebra and introduce the notion of anAH pair ( A , H ), which is a pair of an associative algebra A satisfying some propertiesand a finite dimensional vector space H equipped with a bilinear form (see Section 2.2).Then, we construct a functor from the category of AH pairs to the category of VH pairs, V : AH pair → VH pair , ( A , H ) ( V A , H , H ). In the special case where an AH pair is thetwisted group algebra ( C { L } , C ⊗ Z L ) constructed from an even lattice L , its image by thisfunctor coincides with the lattice vertex algebra V L . We call a class of AH pairs whichaxiomatizes twisted group algebras a lattice pair , and the vertex algebra associated witha lattice pair a generalized lattice vertex algebra. Lattice pairs are classified by using acohomology of an abelian group.In Section 3, we construct an adjoint functor of V , denoted by Ω : VH pair → AH pair , ( V , H ) ( Ω V , H , H ) and, in particular, construct an associative algebra froma vertex algebra (Theorem 3.1). In the case that a VH pair ( V , H ) is a simple as a vertex ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 3 algebra, the structure of the AH pair ( Ω V , H , H ) is studied by using the results in the pre-vious section. We prove that there exists a maximal generalized lattice vertex subalgebra(Theorem 3.2), which implies we can classify generalized lattice vertex subalgebras in aVH pair ( V , H ). This algebra is described by a lattice L V , H ⊂ H , called a maximal latticeof the good VH pair ( V , H ).In Section 4, we define the notion of a genus of VH pairs. We remark that the categoryof VH pairs (resp. AH pairs) naturally admits a monoidal category structure and anylattice vertex algebra is naturally a VH pair. Two VH pairs ( V , H V ) and ( W , H W ) are saidto be in the same genus if VH pairs ( V ⊗ V II , , H V ⊕ H II , ) and ( W ⊗ V II , , H W ⊕ H II , )are isomorphic as a VH pair. Roughly speaking, classifying VH pairs in the genus of( V , H V ) is equivalent to classify subalgebras in V ⊗ V II , which is isomorphic to V II , andcompatible with the VH pair structure; We prove that such subalgebras are completelydescribed by the maximal lattice L V ⊗ V II , ⊂ H V ⊕ H II , . In this way the classification of agenus of VH pairs is reduced to the study of maximal lattices. For a VH pair ( V , H V ), asubgroup of the automorphism group of the lattice, G V , H V ⊂ Aut L V , H V , is defined from theautomorphism group of a vertex algebra. The di ff erence between a genus of lattices anda genus of VH pairs is measured by this subgroup G V , H V . The mass of the VH pairs in thegenus which contains ( V , H V ) is defined by mass( V , H V ) = P ( W , H W ) 1 | G W , HW | , where the sumis over all the VH pairs ( W , H W ) in the genus, which is finite if L V , H V is positive-definiteand the index of groups [Aut L V , H V ⊕ II , : G V ⊗ V II , , H v ⊕ H II , ] is finite. In this case, weprove that the ratio of the masses mass( V , H V ) / mass( L V , H V ) is equal to the index of groups[Aut L V , H V ⊕ II , : G V ⊗ V II , , H v ⊕ H II , ] (Theorem 4.2).In the final section, a conformal vertex algebra ( V , ω ) with a Heisenberg vertex subal-gebra H is studied. This algebra is graded by the action of the Virasoro algebra and theHeisenberg Lie algebra, V = L α ∈ H , n ∈ Z V α n . We assume here technical conditions, e.g., V α n = n < ( α,α )2 (see Definition 1), which are satisfied in interesting cases, e.g., ex-tensions of a rational a ffi ne vertex algebras. Those conformal vertex algebras are closedunder the tensor product, which means that it is possible to study their genus. The mainresults of this section is (Theorem 5.3, Theorem 5.4)(1) dim V α n = , n ≥ ( α,α )2 > n − L V , H is equal to { α ∈ H | V α ( α,α )2 , } ;(3) If V α , α, α ) >
0, then V α and V − α generate the simple a ffi ne vertex algebra L sl ( k ,
0) with k = α,α ) ∈ Z > .The results suggest that a vector in V α n with n ≥ ( α,α )2 > n − holomorphic VOA (or more generally a holomorphic conformalvertex algebra ) is a VOA such that any module is completely reducible and it has a uniqueirreducible module. The holomorphic VOA V holE , B , constructed in [LS1] is uniquely char-acterized as a holomorphic VOA of central charge 24 whose Lie algebra, the degree onesubspace of the VOA, is the semisimple Lie algebra g E ⊕ g B [LS2]. Let H be a Cartan YUTO MORIWAKI subalgebra of the degree one subspace. Then, the maximal lattice L V holE , B , , H is √ E ⊕ D ,and we determine the genus of this VOA, which gives another proof of the above men-tioned result of [HS1]. Furthermore, by using the characterization, we prove that if aholomorphic conformal vertex algebra of central charge 26 satisfies our condition and itsmaximal lattice is √ E ⊕ D ⊕ II , , then it is isomorphic to V holE , B , ⊗ V II , (Theorem5.8). C ontents Introduction 11. Preliminaries 41.1. Vertex algebras and their modules 51.2. Lattice 62. AH pairs and vertex algebras 62.1. Twisted group algebras 72.2. AH pairs 92.3. Functor V Ω V , ω, H ) 265.2. Application to extensions of a ffi ne VOAs 296. Appendix 34Acknowledgements 34References 341. P reliminaries We denote the sets of all integers, positive integers, real numbers and complex numbersby Z , Z > , R and C respectively. This section provides definitions and notations we needfor what follows. Most of the contents in subsection 1.1 and 1.2 are taken from theliteratures [Bo2, LL, Li1, FHL, FLM, CS]. Throughout this paper, we will work over thefield C of complex numbers. ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 5
Vertex algebras and their modules.
Following [Bo2], a vertex algebra is a C -vector space V equipped with a linear map Y ( − , z ) : V → End( V )[[ z ± ]] , a Y ( a , z ) = X n ∈ Z a ( n ) z − n − and an element ∈ V satisfying the following axioms:V1) For any a , b ∈ V , there exists N ∈ Z such that a ( n ) b = n ≥ N ;V2) For any a ∈ V , a ( n ) = , ( n ≥ , a , ( n = − , holds;V3) Borcherds identity: For any a , b ∈ V and p , q , r ∈ Z , ∞ X i = pi ! ( a ( r + i ) b )( p + q − i ) = ∞ X i = ( − i ri !(cid:16) a ( p + r − i ) b ( q + i ) − ( − r b ( q + r − i ) a ( p + i ) (cid:17) holds.For a , b ∈ V , a ( n ) b is called the n -th product of a and b . Let T V (or simply T ) denotethe endomorphism of V defined by T V a = a ( − for a ∈ V .The following properties follow directly from the axioms of a vertex algebra (see forexample [LL]):(1) Y ( , z ) = id V ;(2) For any a , b ∈ V and n ∈ Z , T = T a )( n ) = − na ( n − , T ( a ( n ) b ) = ( T a )( n ) b + a ( n ) T b ;(3) Skew-symmetry: For any a , b ∈ V and n ∈ Z , a ( n ) b = X i ≥ ( − n + + i T i i ! ( b ( n + i ) a );(4) Associativity: For any a , b ∈ V and n , m ∈ Z ,( a ( n ) b )( m ) = X i ≥ ni ! ( − i ( a ( n − i ) b ( m + i ) − ( − n b ( n + m − i ) a ( i ));(5) Commutativity: For any a , b ∈ V and n , m ∈ Z ,[ a ( n ) , b ( m )] = X i ≥ ni ! ( a ( i ) b )( n + m − i ) . We denote by V -mod the category of V -modules. YUTO MORIWAKI
Lattice. A lattice of rank n ∈ N is a rank n free abelian group L equipped with a Q -valued symmetric bilinear form ( , ) : L × L → Q . A lattice L is said to be even if ( α, α ) ∈ Z for any α ∈ L , integral if ( α, β ) ∈ Z for any α, β ∈ L , and positive-definite if ( α, α ) > α ∈ L \ { } . For an integral lattice L and a unital commutative ring R , we extend the bilinear form ( , )bilinearly to L ⊗ Z R and L is said to be non-degenerate if the bilinear form on L ⊗ Z C isnon-degenerate. The dual of L is the set L ∨ = { α ∈ L ⊗ Z C | ( α, L ) ⊂ Z } . The lattice L is said to be unimodular if L (cid:27) L ∨ as a lattice.Two integral lattices L and M are said to be equivalent or in the same genus if their basechanges are isomorphic as lattices: L ⊗ Z R ≃ M ⊗ Z R , L ⊗ Z Z p ≃ M ⊗ Z Z p , for all the prime integers p , where Z p is the ring of p -adic integers. Denote by genus( L )the genus of lattices which contains L . If L is positive-definite, then a mass of its genusmass( L ) ∈ Q is defined by mass( L ) = X L ′ ∈ genus( L ) | Aut L ′ | , (1)where Aut L is the automorphism group of L . The Smith-Minkowski-Siegel’s mass for-mula is a formula which computes mass( L ) (see [CS, Ki]). Lattices over R are completelydetermined by the signature. Similarly, lattices over Z p are determined by some invariant,called p -adic signatures (If p =
2, we have to consider another invariant, called oddity).Conventionally, a genus of lattices is denoted by using those invariants, e.g., the genus of √ E D is denoted by II , (2 + II ) (see [CS] for the precise definition).2. AH pairs and vertex algebras In this section, we generalize the famous construction of vertex algebras from evenlattices.In Sections 2.1, we recall some fundamental results of a cohomology of an abeliangroup. Much of the material in this subsection is based on [FLM, Chapter 5]. We focuson a (twisted) group algebras considered in [FLM] and generalize their result. In Section2.2, we introduce the concept of AH pairs which generalizes twisted group algebra, while
ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 7 in Section 2.3, we construct a vertex algebra associated with an AH pair generalizing theconstruction of lattice vertex algebras.2.1.
Twisted group algebras.
Let A be an abelian group. A (normalized) 2-cocycle of A with coe ffi cients in C × is a map f : A × A → C × such that:(1) f (0 , a ) = f ( a , = f ( a , b ) f ( a + b , c ) = f ( a , b + c ) f ( b , c ) . We denote by Z ( A , C × ) the set of the 2-cocyles of A with coe ffi cients in C × . For a map h : A → C × , the map dh : A × A → C × , ( a , b ) h ( a + b ) h ( a ) − h ( b ) − is a 2-cocycle, called a coboundary. The subset of Z ( A , C × ) consisted of coboundaries isdenoted by B ( A , C × ), which is a subgroup of Z ( A , C × ). Set H ( A , C × ) = Z ( A , C × ) / B ( A , C × ).An alternative bilinear form on an abelian group A is a map c : A × A → C × such that:(1) c (0 , a ) = c ( a , = a ∈ A ;(2) c ( a + b , c ) = c ( a , c ) c ( b , c ) for any a , b , c ∈ A ;(3) c ( a , a ) = a ∈ A .The set of alternative bilinear forms on A is denoted by Alt ( A ). For a 2-cocycle f ∈ Z ( A , C × ), define the map c f : A × A → C × by setting c f ( a , b ) = f ( a , b ) f ( b , a ) − . Itis easy to prove that the map c f is an alternative bilinear form on A , which is called a commutator map [FLM]. Hence, we have a map Z ( A , C × ) → Alt ( A ), which induces c : H ( A , C × ) → Alt ( A ). Lemma 2.1.
The map c : H ( A , C × ) → Alt ( A ) is injective.Proof. For any 2-cocycle f ∈ Z ( A , C × ), there exists a central extension of A by C × ,1 → C × → ˜ A → A → f is a coboundary if and only if the extension splits (see [FLM]). If c f =
1, that is, f ( a , b ) = f ( b , a ), then ˜ A is an abelian group and the above sequence is an exact sequencein the category of abelian groups. Since C × is injective, the exact sequence splits. (cid:3) According to [FLM], we have the following:
Proposition 2.1 ([FLM, Proposition 5.2.3]) . If A is a free abelian group of finite rank,then the map c : H ( A , C × ) → Alt ( A ) is an isomorphism.A twisted group algebra of an abelian group A is an A -graded unital associative C -algebra R = L a ∈ A R a satisfying the following conditions:(1) 1 ∈ R ;(2) dim C R a = e a e b , e a ∈ R a \ { } and e b ∈ R b \ { } . YUTO MORIWAKI
Twisted group algebras R , S of A are isomorphic if there is a C -algebra isomorphism h : R → S which preserves the A -grading, that is, h ( R a ) = S a for any a ∈ A .Let R be a twisted group algebra of A . Let us choose a nonzero element e a of R a for all a ∈ A . Then, define the map f : A × A → C × by e a e b = f ( a , b ) e a + b . Clearly,the cohomology class f ∈ H ( A , C × ) is independent of the choice of { e a ∈ R a } a ∈ A , fur-thermore, the isomorphism class of the twisted group algebra R . The alternative bilinearform c f ∈ Alt ( A ) is called an associated commutator map of the twisted group algebra ,denoted by c R . Hence, we have: Lemma 2.2.
There exists a bijection between the isomorphism classes of twisted groupalgebras of an abelian group A and H ( A , C × ) . Let L be an even lattice. Define the alternative bilinear form on L by c L ( α, β ) = ( − ( α,β ) for α, β ∈ L . By Proposition 2.1, there exists a 2-cocycle f ∈ Z ( L , C × ) such that theassociated commutator map c f is coincides with c L . The correspondent twisted groupalgebra is denoted by C { L } , which plays an important role in the construction of a latticevertex algebra.Hereafter, we generalize this construction to some abelian group L , called even H - lat-tice, which is not always free. Let H be a finite-dimensional vector space over C equippedwith a non-degenerate symmetric bilinear form ( − , − ). A even H-lattice is a subgroup L ⊂ H such that ( α, β ) ∈ Z and ( α, α ) ∈ Z for any α, β ∈ L , denoted by ( L , H ). Define analternative bilinear form on L by c L ( α, β ) = ( − ( α,β ) for α, β ∈ L .Let ( L , H ) be an even H -lattice. Since the abelian group L is not always free, we cannotdirectly apply Proposition 2.1 to the alternative bilinear form c L . We can, however, provethat there exists a 2-cocycle f L ∈ Z ( L , C × ) such that the associated alternative bilinearform c f L coincides with c L .Let E be the C -vector subspace of H spanned by L . Set E ⊥ = { v ∈ E | ( v , E ) = } .Let π : E → E / E ⊥ be the canonical projection and ¯ L the image of L under π . Then, thebilinear form on H induces a bilinear form on E / E ⊥ and ¯ L . Since ¯ L is an even E / E ⊥ -lattice and spans E / E ⊥ , ¯ L is a free abelian group of rank dim C E / E ⊥ . By Proposition 2.1,there exits a 2-cocycle g : ¯ L × ¯ L → C × such that g (¯ a , ¯ b ) g (¯ b , ¯ a ) − = ( − (¯ a , ¯ b ) for any ¯ a , ¯ b ∈ ¯ L .Then, the 2-cocycle f L defined by f L ( a , b ) = g ( π ( a ) , π ( b )) for a , b ∈ L satisfies c f L = c L .The twisted group algebra constructed by the 2-cocycle f L ∈ Z ( L , C × ) is denoted by A L , H ,which is independent of the choice of the 2-cocycle f L ∈ Z ( L , C × ). Hence, we have: Proposition 2.2.
For even H-lattice ( L , H ) , there exits a unique twisted group algebraA L , H such that the associated commutator map is c L . Remark 2.1.
By the construction, if L is non-degenerate, then A L , H is isomorphic to thetwisted group algebra C { L } . ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 9
AH pairs.
Let H be a finite-dimensional vector space over C equipped with a non-degenerate symmetric bilinear form ( − , − ) and A a unital associative algebra over C withthe unity 1. Assume that A is graded by H as A = L α ∈ H A α .We will say that such a pair ( A , H ) is an AH pair if the following conditions are satisfied:AH1) 1 ∈ A and A α A β ⊂ A α + β for any α, β ∈ H ;AH2) A α A β , α, β ) ∈ Z ;AH3) For v ∈ A α , w ∈ A β , vw = ( − ( α,β ) wv .For AH pairs ( A , H A ) and ( B , H B ), a homomorphism of AH pairs is a pair ( f , f ′ ) ofmaps f : A −→ B and f ′ : H A −→ H B such that f is an algebra homomorphism and f ′ anisometry such that f ( A α ) ⊂ B f ′ ( α ) for all α ∈ H A . We denote by AH pair the category ofAH pairs. For an AH pair ( A , H ), we set M A , H = { α ∈ H | A α , } . A good AH pair is an AH pair ( A , H ) such that:GAH1) A = C vw , α, β ∈ M A , H , v ∈ A α \ { } and w ∈ A β \ { } .Since any torsion-free abelian group admits a linear order, we have: Proposition 2.3.
For a good AH pair ( A , H ) , is the only zero divisor in A. Let ( A , H ) be a good AH pair. For α, β ∈ M A , H , by Definition (AH2) and (AH3),(GAH), ( α, β ) ∈ Z , ( α, α ) ∈ Z and α + β ∈ M A , H . Hence, M A , H is a submonoid of H .A lattice pair is a good AH pair ( A , H ) such that:LP) M A , H is a abelian group, that is, − α ∈ M A , H for any α ∈ M A , H . Lemma 2.3. If ( A , H ) is a lattice pair, then A is a twisted group algebra of the abeliangroup M A , H whose commutator map is defined by M A , H × M A , H → C × , ( α, β ) ( − ( α,β ) .Proof. It su ffi ces to show that dim A α = α ∈ M A , H . Let α ∈ M A , H . By (LP), A − α ,
0. Let a , a ′ be non-zero vectors in A α and b a non-zero vector in A − α . By Definition(GAH1) and (GAH2), ab ∈ A = C is non zero. We may assume that ab =
1. Then, a ′ = ( a · b ) · a ′ = a · ( b · a ′ ) ∈ C a . Hence, A α = C a . (cid:3) By Lemma 2.1 and Lemma 2.2, Proposition 2.2, Lemma 2.3, we have the followingclassification result of lattice pairs:
Proposition 2.4.
For any even H-lattice ( L , H ) , there exists a unique lattice pair ( A L , H , H ) such that M A L , H , H = L. Furthermore, Lattice pairs ( A , H ) and ( A ′ , H ′ ) is isomorphic i ff there exists an isometry f : H → H ′ such that f ( M A , H ) = M A ′ , H ′ . We end this subsection by defining a maximal lattice pair of a good AH pair. Let ( A , H )be a good AH-pair. For a submonoid N ⊂ M A , H , set A N = L α ∈ N A α . Then, we have: Lemma 2.4.
For a submonoid N ⊂ M A , H , ( A N , H ) is a subalgebra of ( A , H ) as an AHpair. Set L A , H = { α ∈ H | A α , A − α , } = M A , H ∩ ( − M A , H ) . (2)We denote A L A , H by A lat . Then, ( A lat , H ) is a lattice pair, which is maximal in the followingsense: Proposition 2.5.
Let ( B , H B ) , ( A , H A ) be good AH pairs and ( f , f ′ ) : ( B , H B ) → ( A , H A ) an AH pair homomorphism. If ( B , H B ) is a lattice pair, then f is injective and f ( B ) ⊂ A lat . In particular, there is a natural bijection between Hom
AH pair (( B , H B ) , ( A , H A )) and Hom
AH pair (( B , H B ) , ( A lat , H A )) .Proof. Since f is an AH pair homomorphism, ker f is an H B -graded ideal. According tothe proof of Lemma 2.3, every nonzero element in B α is invertible for any α ∈ M B , H B .Hence, ker f =
0. For α ∈ M B , H , we have 0 , f ( B α ) ⊂ A f ′ ( α ) and 0 , f ( B − α ) ⊂ A − f ′ ( α ) .Thus, f ′ ( α ) ∈ L A , H A and f ( B ) ⊂ A lat . (cid:3) Functor V . Let H be a vector subspace of a vertex algebra V . We will say that H isa Heisenberg subspace of V if the following conditions are satisfied:HS1) h (1) h ′ ∈ C for any h , h ′ ∈ H ;HS2) h ( n ) h ′ = n ≥ h , h ′ ∈ H ;HS3) The bilinear form ( − , − ) on H defined by h (1) h ′ = ( h , h ′ ) for h , h ′ ∈ H is non-degenerate.We will call such a pair ( V , H ) a VH pair . For VH pairs ( V , H V ) and ( W , H W ), a VH pairhomomorphism from ( V , H V ) to ( W , H W ) is a vertex algebra homomorphism f : V → W such that f ( H V ) = H W . Since f ( h (1) h ′ ) = f ( h )(1) f ( h ′ ) for any h , h ′ ∈ H V and the bilinearform on H V is non-degenerate, f | H V : H V → H W is an isometric isomorphism of vectorspaces. A vertex subalgebra W of a VH pair ( V , H ) is said to be a subVH pair if H is asubset of W . VH pair is a category whose objects are VH pairs and morphisms are VHpair homomorphisms.Let ( A , H ) be an AH pair and consider the Heisenberg vertex algebra M H (0) associatedwith ( H , ( − , − )). Let us identify the degree 1 subspace of M H (0) with H . Then, the vertexalgebra M H (0) is generated by H and the actions of h ( n ) for h ∈ H and n ∈ Z give rise toan action of the Heisenberg Lie algebra b H on M H (0).Consider the vector space V A , H = M H (0) ⊗ A = M α ∈ H M H (0) ⊗ A α . We let b H acts on this space by setting, for h ∈ H , v ∈ M H (0) and a ∈ A α , h ( n )( v ⊗ a ) = ( h , α ) v ⊗ a , n = , ( h ( n ) v ) ⊗ a , n , . ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 11
Let α ∈ H and a ∈ A α . Denote by l a ∈ End A the left multiplication by a and define l a z α : A → A [ z ± ] by l a z α · b = z ( α,β ) ab for b ∈ A β , where we used that ab , α, β ) ∈ Z . Then, set Y ( ⊗ a , z ) = exp (cid:18)X n ≥ α ( − n ) n z n (cid:19) exp (cid:18)X n ≥ α ( n ) − n z − n (cid:19) ⊗ l a z α ∈ End V A , H [[ z ± ]] . The series Y ( h ⊗ , z ) = P n ∈ Z h ( n ) z − n − with h ∈ H and Y (1 ⊗ a , z ) with a ∈ A form a setof mutually local fields on V A , H and generate a structure of a vertex algebra on it. Since M H (0) is a subalgebra of V A , H , ( V A , H , H ) is canonically a VH pair.In the case that A = C { L } is the twisted group algebra associated with an even lattice L and H = C ⊗ Z L , the vertex algebra V A , H is nothing else but the lattice vertex algebra V L . Proposition 2.6.
The above construction gives a functor from the category of AH pairs tothe category of VH pairs.
We denote by V this functor, thus, V ( A , H ) is a VH pair ( V A , H , H ).In order to prove the above proposition, we need the following result from [LL, Propo-sition 5.7.9]: Proposition 2.7 ([LL]) . Let f be a linear map from a vertex algebra V to a vertex algebraW such that f ( ) = and such thatf ( s ( n ) v ) = f ( s )( n ) f ( v ) for any s ∈ S , v ∈ V and n ∈ Z , where S is a given generating subset of V. Then, f is a vertex algebra homomorphism.proof of Proposition 2.6. Let ( A , H A ) and ( B , H B ) be AH pairs and ( f , f ′ ) : ( A , H A ) → ( B , H B ) an AH pair homomorphism. It is clear that there is a unique C -linear map F : V A , H A → V B , H B such that:(1) F ( h ( n ) − ) = f ′ ( h )( n ) F ( − ) for any h ∈ H A ;(2) F ( ⊗ a ) = ⊗ f ( a ) for any a ∈ A .It is easy to prove that the restriction of F gives an isomorphism from M H A (0) to M H B (0)and satisfies F ( v ⊗ a ) = F ( v ) ⊗ f ( a ) for any v ∈ M H A (0) and a ∈ A . It su ffi ces to show that F is a vertex algebra homomorphism. We will apply Proposition 2.7 with S = H A ⊕ A .Let a ∈ A α and b ∈ A β for α, β ∈ H A and v ∈ M H A (0). Since F ( Y ( ⊗ a , z ) v ⊗ b ) = z ( α,β ) F (exp (cid:18) − X n ≤− α ( n ) n z − n (cid:19) exp (cid:18) − X n ≥ α ( n ) n z − n (cid:19) v ⊗ ab ) = z ( f ′ ( α ) , f ′ ( β )) exp (cid:18) − X n ≤− f ′ ( α )( n ) n z − n (cid:19) exp (cid:18) − X n ≥ f ′ ( α )( n ) n z − n (cid:19) f ( a ) F ( v ⊗ b ) = Y ( ⊗ f ( a ) , z ) F ( v ⊗ b ) . (cid:3) For an even H -lattice ( L , H ), the VH pair V A L , H , H is called generalized lattice vertexalgebra . We denote it by V L , H . Remark 2.2.
The construction of the lattice vertex algebra V L from an even lattice L doesnot form a functor, since there is no natural homomorphism from the automorphism groupof the lattice L to the automorphism group of the twisted group algebra C { L } (It dependson a choice of a coboundary). This is one reason we introduce AH pairs.
3. VH pairs and A ssociative A lgebras In the previous section, we construct a functor V : AH pair → VH pair. In this section,we construct a right adjoint functor Ω : VH pair → AH pair of V (see Theorem 3.1). Thatis, we construct a ‘universal’ associative algebra from a VH pair. Section 3.1 is devotedto constructing the functor Ω , while in 3.2, we prove that Ω and V is an adjoint pair. Insubsection 3.3, we combine the results in this section and previous section and classifygeneralized lattice vertex subalgebras of a VH pair.3.1. Functor Ω . Let ( V , H ) be a VH pair and ω H be the canonical conformal vector of M H (0) given by the Sugawara construction, that is, ω H = P i h i ( − h i , where { h i } i is abasis of H and { h i } i is the dual basis of { h i } i with respect to the bilinear form on H . For α ∈ H , we let Ω α V , H be the set of all vectors v ∈ V such that:VS1) T V v = ω H (0) v .VS2) h ( n ) v = h ∈ H and n ≥ h (0) v = ( h , α ) v for any h ∈ H .Here, T V is the canonical derivation of V defined by T V v = v ( − . Set Ω V , H = M α ∈ H Ω α V , H . A vector of a Heisenberg module which satisfies the condition (VS2) and (VS3) is calleda
H-vacuum vector . Remark 3.1.
Since ω H (0) − T is a derivation on V, ker( ω H (0) − T ) is a vertex subalgebraof V. Furthermore, since H ⊂ ker( ω H (0) − T ) , (ker( ω H (0) − T ) , H ) is a VH pair and themap ( V , H ) (ker( ω H (0) − T ) , H ) defines a functor from VH pair to itself. Lemma 3.1.
For v ∈ Ω α V , H , ω H (0) v = α ( − v and ω H (1) v = ( α,α )2 v. Furthermore, Ω V , H = ker T V .Proof. Let { h i } i be a basis of H and { h i } i the dual basis with respect to the bilinear form on H . Then, ω H (0) v = P i ( h i ( − h i )(0) v = P i P n ≥ ( h i ( − − n ) h i ( n ) + h i ( − n − h i ( n )) v = P i ( h i , α ) h i ( − v + ( h i , α ) h i ( − v = α ( − v . Similarly, ω H (1) v = P i ( h i ( − h i )(1) v = P i P n ≥ ( h i ( − − n ) h i ( n + + h i ( − n ) h i ( n )) v = P i ( h i , α )( h i , α ) v = ( α,α )2 v . If v ∈ Ω V , H ,then T V v = ω H (0) v =
0. If v ∈ ker T V , then a ( n ) v = n ≥ a ∈ V , whichimplies v ∈ Ω V , H . Thus, Ω V , H = ker T V . (cid:3) The following simple observation is fundamental:
ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 13
Lemma 3.2.
Let v ∈ Ω α V , H and w ∈ Ω β V , H . If Y ( v , z ) w , , then ( α, β ) ∈ Z and v ( − ( α, β ) + i ) w = for any i ≥ and , v ( − ( α, β ) − w = ( − ( α,β ) w ( − ( α, β ) − v ∈ Ω α + β V , H . Inparticular, if ( α, β ) < Z , then Y ( v , z ) w = .Proof. Let k be an integer such that v ( k ) w , v ( k + i ) w = i >
0. We claimthat v ( k ) w ∈ Ω α + β V . Since [ h ( n ) , v ( k )] = P i ≥ (cid:16) ni (cid:17) ( h ( i ) v )( n + k − i ) = ( h , α ) v ( n + k ), we have h ( n ) v ( k ) w = h (0) v ( k ) w = ( h , α + β ) v ( k ) w for any h ∈ H and n ≥
1. Furthermore, ω H (0) v ( k ) w = ( ω H (0) v )( k ) w + v ( k ) ω H (0) w = ( T V v )( k ) w + v ( k ) T V w = T V ( v ( k ) w ), whichimplies v ( k ) w ∈ Ω α + β V , H . According to Lemma 3.1, ω H (1) v ( k ) w = ( α + β,α + β )2 v ( k ) w . Since ω H (1) v ( k ) w = [ ω H (1) , v ( k )] w + v ( k ) ω H (1) w = P i ≥ (cid:16) i (cid:17) ( ω H ( i ) v )( k + − i ) w + v ( k ) ω H (1) w = (( α, α ) / + ( β, β ) / v ( k ) w + ( T V v )( k + w = (( α, α ) / + ( β, β ) / − k − v ( k ) w , we have k = − ( α, β ) −
1. Hence, ( α, β ) ∈ Z . By applying the skew-symmetry, v ( − ( α, β ) − w = P i ≥ ( − i + ( α,β ) T iV / i ! w ( − ( α, β ) − + i ) v = ( − ( α,β ) w ( − ( α, β ) − v . (cid:3) We may equip Ω V , H with a structure of an associative algebra in the following way: For v ∈ Ω α V , H and w ∈ Ω β V , H , define the product vw by vw = v ( − ( α, β ) − w , if Y ( v , z ) w , , , otherwise . In order to show that the product is associative, we need the following lemma.
Lemma 3.3.
Let a , b , c ∈ V, p , q , r ∈ Z satisfy a ( r + i ) b = , b ( q + i ) c = and a ( p − + i ) c = for any i ≥ . Then, ( a ( r ) b )( p + q ) c = a ( r + p )( b ( q ) c ) .Proof. Applying the Borcherds identity, we have ( a ( r ) b )( p + q )) c = P i ≥ (cid:16) pi (cid:17) ( a ( r + i ) b )( p + q − i ) c = P i ≥ ( − i (cid:16) ri (cid:17) ( a ( r + p − i ) b ( q + i ) c − ( − r b ( r + q − i ) a ( p + i ) c ) = a ( r + p ) b ( q ) c . (cid:3) Lemma 3.4. ( Ω V , H , H ) is an AH pair.Proof. Let a ∈ Ω α V , H , b ∈ Ω β V , H and c ∈ Ω γ V , H for α, β, γ ∈ H . According to Lemma 3.2,it remains to show that the product is associative, i.e., ( ab ) c = a ( bc ). Suppose that oneof ( α, β ) , ( β, γ ) , ( α, γ ) is not integer. According to Lemma 3.2, it is easy to show that( ab ) c = a ( bc ) =
0. Thus, we may assume that ( α, β ) , ( β, γ ) , ( α, γ ) ∈ Z . Then, ( ab ) c = ( a ( − ( α, β ) − b )( − ( α + β, γ ) − c and a ( bc ) = a ( − ( α, β + γ ) − b ( − ( β, γ ) − c ). By applyingLemma 3.3 with ( p , q , r ) = ( − ( α, γ ) , − ( β, γ ) − , − ( α, β ) − ab ) c = a ( bc ). (cid:3) Proposition 3.1.
The map Ω : VH pair → AH pair , ( V , H ) ( Ω V , H , H ) is a functor.Proof. Let ( V , H V ) and ( W , H W ) be VH pairs and f : ( V , H V ) → ( W , H W ) a VH pairhomomorphism. Set f ′ = f | H V : H V → H W . Let e α ∈ Ω α V , H V . For h ∈ H W , since h ( n ) f ( e α ) = f ( f ′− ( h ))( n ) f ( e α ) = f ( f ′− ( h )( n ) e α ), we have h ( n ) f ( e α ) = h (0) f ( e α ) = ( f ′− ( h ) , α ) f ( e α ) for any n ≥
1. Since f ′ is an orthogonal transformation, ( f ′− ( h ) , α ) = ( h , f ′ ( α )). Since f ( ω H V ) = ω H W , ω H W (0) f ( e α ) = f ( ω H V (0) e α ) = f ( e α ( − ) = f ( e α )( − .Hence, f ( Ω α V ) ⊂ Ω f ′ ( α ) W . The rest is obvious. (cid:3) we will often abbreviate the AH pair (and the functor) ( Ω V , H , H ) as Ω ( V , H ).Since Ω is a functor, we have: Lemma 3.5.
Let f ∈ Aut
V such that f ( H ) = H. Then, f induces an automorphism of theAH pair Ω ( V , H ) . Let ( V , H V ) and ( W , H W ) be VH pairs. Then, V ⊗ W is a vertex algebra, and H V ⊕ H W isa Heisenberg subspace of V ⊗ W . Hence, we have: Lemma 3.6.
The category of VH pairs and the category of AH pairs are strict symmetricmonoidal categories.
It is easy to confirm that the functor V : AH pair → VH pair is a monoidal functor,that is, V (( A , H A ) ⊗ ( B , H B )) is naturally isomorphic to V ( A , H A ) ⊗ V ( B , H B ). However, Ω is not strict monoidal functor, since ker( ω H (0) − T ) do not preserve the tensor product(see Remark 3.1). Lemma 3.7.
Let ( V , H V ) and ( W , H W ) be VH pairs. If ω H W (0) = T W , then Ω ( V ⊗ W , H V ⊕ H W ) is isomorphic to Ω ( V , H V ) ⊗ Ω ( W , H W ) as an AH-pair.Proof. It is clear that Ω V , H V ⊗ Ω W , H W ⊂ Ω V ⊗ W , H V ⊕ H W . Let P i v i ⊗ w i ∈ Ω V ⊗ W , H V ⊕ H W . Wecan assume that { v i } i and { w i } i are both linearly independent. For h ∈ H V and n ≥ h ( n ) P i v i ⊗ w i = P i h ( n ) v i ⊗ w i =
0. Since w i are linearly independent, h ( n ) v i = i . Similarly, h ′ ( n ) w i = i and h ′ ∈ H W . Since ω H V ⊕ H W = ω H V ⊗ + ⊗ ω H W , wehave ω H V (0) P i v i ⊗ w i = ( ω H V ⊕ H W (0) − ω H W (0)) P i v i ⊗ w i = ( T V ⊗ W − ⊗ T W ) P i v i ⊗ w i = ( T V ⊗ P i v i ⊗ w i , where we used ω H W (0) = T W and P i v i ⊗ w i ∈ Ω V ⊗ W , H V ⊕ H W . Hence, ω H V (0) v i = T V v i , which implies that v i ∈ Ω V , H V , w i ∈ Ω W , H W . (cid:3) Adjoint functor.
In this subsection, we will prove the following theorem:
Theorem 3.1.
The functor Ω : VH pair → AH pair is a right adjoint to the functor V : AH pair → VH pair.
The following observation is important:
Lemma 3.8.
Let ( A , H ) be an AH pair. Then, the following conditions hold for the VHpair ( V A , H , H ) :(1) T V A , H = ω H (0) ;(2) The AH pair Ω ( V A , H , H ) is isomorphic to ( A , H ) ;(3) The vertex algebra V A , H is generated by the subspaces H and Ω V A , H , H = A as avertex algebra.In particular, the composite functor Ω ◦ V : AH pair → AH pair is isomorphic to theidentity functor.
Hereafter, we will construct a natural transformation from
V ◦ Ω : VH pair → VH pairto the identity functor and prove the above Theorem. We start from the observation,
ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 15
Lemma 3.8. Let ( V , H V ) be a VH pair and ( A , H A ) an AH pair and f : V ( A , H A ) → ( V , H V )a VH pair homomorphism. Then, by Lemma 3.8 and f ( ω H A ) = ω H V , the image of f satisfies the condition (VS1) and is generated by H V and the subset of H V -vacuum vectorsas a vertex algebra. So, let W be the vertex subalgebra of V generated by H V and Ω V , H V .Then, there is a natural isomorphismHom VH pair (( V A , H A , H A ) , ( V , H V )) → Hom
VH pair (( V A , H A , H A ) , ( W , H V )) . Hence, the proof of Theorem 3.1 is divided into two parts. First, we prove that the sub-algebra W is isomorphic to V Ω V , HV , H V = V ◦ Ω ( V , H V ) as a VH pair, which gives us anatural transformation V ◦ Ω ( V , H V ) ֒ → ( V , H V ). Second, we prove that there is a naturalisomorphismHom AH pair (( A , H A ) , ( B , H B )) → Hom
VH pair ( V ( A , H A ) , V ( B , H B )) , for AH pairs ( A , H A ) and ( B , H B ).Let us prove the first step. Since M H V (0) is a subalgebra of W , W is an M H V (0)-module. Lemma 3.9.
The subspace W ⊂ V is isomorphic to M H V (0) ⊗ C Ω V , H V as an M H V (0) -module.Proof. Let W ′ be an M H V (0)-submodule of V generated by Ω V , H V . Then, according tothe representation theory of Heisenberg Lie algebras (see [FLM, Theorem 1.7.3]), W ′ (cid:27) M H V (0) ⊗ C Ω V , H V as an M H V (0)-module. Hence, it su ffi ces to show that W ′ is closedunder the products of the vertex algebra. Let v , w ∈ W ′ . We may assume that v = h ( i ) . . . h k ( i k ) e α and w = h ′ ( j ) . . . h ′ l ( j l ) e β where h i , h ′ j ∈ H V and e α ∈ Ω α V , H V , e β ∈ Ω β V , H V . Since ( h ( n ) v )( m ) w = P i ≥ ( − i (cid:16) ni (cid:17) h ( n − i ) v ( m + i ) w − ( − n v ( n + m − i ) h ( i ) w , wemay assume that v = e α ∈ Ω α V , H V . Since e α ( n ) h ( m ) w = [ e α ( n ) , h ( m )] w + h ( m ) e α ( n ) w = − ( h , α ) e α ( n + m ) w + h ( m ) e α ( n ) w , we may assume that w = e β ∈ Ω β V , H V . If Y ( e α , z ) e β = Y ( e α , z ) e β ,
0. Then, according to Lemma3.2, e α ( − ( α, β ) + k ) e β = k ≥ e α ( − ( α, β ) − e β ∈ Ω α + β V . We will showthat e α ( − ( α, β ) − k ) e β ∈ W ′ for k ≥
2. According to Lemma 3.1, ω H V (0) e β = β ( − e β .Then, β ( − e α ( − ( α, β ) − k ) e β = [ β ( − , e α ( − ( α, β ) − k )] e β + e α ( − ( α, β ) − k ) β ( − e β = ( α, β ) e α ( − ( α, β ) − k − e β + e α ( − ( α, β ) − k ) ω H V (0) e β = ω H (0) e α ( − ( α, β ) − k ) e β − ke α ( − ( α, β ) − k − e β . Since k ≥
1, we have e α ( − ( α, β ) − k − e β = k ( ω H V (0) − β ( − e α ( − ( α, β ) − k ) e β (3) = k ! ( ω H V (0) − β ( − k e α ( − ( α, β ) − e β . Since ω H V (0) W ′ ⊂ W ′ , we have e α ( − ( α, β ) − k − e β ∈ W ′ . (cid:3) Remark 3.2.
Equation (3) was proved by [Ro] for lattice vertex algebras.
Lemma 3.10.
The subalgebra W is isomorphic to
V ◦ Ω ( V , H V ) as a VH pair.Proof. According to Lemma 3.9, we have an isomorphism f : W → V Ω V , HV , H V = M H V (0) ⊗ Ω V , H V as an M H V (0)-module. We will apply Proposition 2.7 with S = H V ⊕ Ω V , H V . It su ffi ces to show that f ( a ( n ) v ) = f ( a )( n ) f ( v ) for any a ∈ Ω V , H V and v ∈ W and n ∈ Z ,which can be proved by the similar arguments as Lemma 3.9. (cid:3) As discussed above, the subalgebra V Ω V , HV , H V ⊂ V is universal as follows: Proposition 3.2.
Let ( A , H A ) be an AH pair and f : ( V A , H A , H A ) → ( V , H V ) be a VH pairhomomorphism. Then, f ( V A , H A ) ⊂ V Ω V , HV , H V . In particular, there is a natural bijection, Hom
VH pair (( V ( A , H A ) , ( V , H V )) (cid:27) Hom
VH pair ( V ( A , H A ) , V ◦ Ω ( V , H V )) . let us prove the second step. Let ( B , H B ) be an AH pair. Since V and Ω is a functor, wehave a natural map φ : Hom AH pair (( A , H A ) , ( B , H B )) → Hom
VH pair ( V ( A , H A ) , V ( B , H B ))and ψ : Hom VH pair ( V ( A , H A ) , V ( B , H B )) → Hom
AH pair ( Ω ◦ V ( A , H A ) , Ω ◦ V ( B , H B )) . By Lemma 3.8,Hom
AH pair ( Ω ◦ V ( A , H A ) , Ω ◦ V ( B , H B )) (cid:27) Hom
AH pair (( A , H A ) , ( B , H B )) . It is easy to prove that ψ is an inverse of φ . Hence, we have: Lemma 3.11.
The map φ is a bijection. Combining Proposition 3.2 and Lemma 3.11, we prove Theorem 3.1.3.3.
Good VH pairs and Maximal lattices.
We will say that a VH pair ( V , H ) is a goodVH pair if the AH pair Ω ( V , H ) is a good AH pair, i.e., the following conditions aresatisfied:1) Ω V , H = C .2) vw , α, β ∈ H and v ∈ Ω α V , H \ { } and w ∈ Ω β V , H \ { } .The following proposition follows from [DL, Proposition 11.9]: Proposition 3.3.
If V is a simple vertex algebra, then for any nonzero elements a , b ∈ V,Y ( a , z ) b , , i.e., there exists an integer n ∈ Z such that a ( n ) b , . By the above proposition and Lemma 3.2 and Lemma 3.1, we have the followinglemma:
Lemma 3.12.
For a VH pair ( V , H V ) , if V is simple and Ker T = C , then ( V , H V ) is agood VH pair. For a vertex algebra V and v , w ∈ ker T V , define the product on ker T V by vw = v ( − w . Then, ker T V is a unital commutative associative C -algebra. The following proposition isinteresting in its own right, which is related to a physical assumption in the quantum fieldtheory. It says any field which does not depend on its position, i.e., ddz Y ( a , z ) =
0, is a
ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 17 scalar, called ‘c-number’. We do not use it in the rest of this paper. The proof is given inthe appendix.
Proposition 3.4.
If V is simple, then ker
T is a field. In particular, if V is simple and hascountable dimension over C , then ker T = C . Hereafter, we will study good VH pairs.
Lemma 3.13.
For an even H-lattice ( L , H ) and a good VH pair ( V , H V ) , ( V ⊗ V L , H , H V ⊕ H ) is a good VH pair. Furthermore, Ω ( V ⊗ V L , H , H V ⊕ H ) is isomorphic to Ω ( V , H V ) ⊗ ( A L , H , H ) as an AH pair.Proof. For a generalized lattice vertex algebra V L , H , we have ω H (0) = T V L , H . Hence,according to Lemma 3.7, we have Ω V ⊗ V L , H (cid:27) Ω V ⊗ Ω V L , H . Since V and V L , H are good VHpairs, Ω V ⊗ V L , H = Ω V ⊗ Ω V L , H = C ⊗ . Let v ∈ Ω α + β V ⊗ V L , H and v ′ ∈ Ω α ′ + β ′ V ⊗ V L , H be nonzeroelements, where α, α ′ ∈ H V and β, β ′ ∈ H . Since V L , H is a generalized lattice vertexalgebra, dim Ω β V L , H = dim A β L , H = β ∈ L . Hence, we may assume that v = a ⊗ b and v ′ = a ′ ⊗ b ′ , where a ∈ Ω α V and a ′ ∈ Ω α ′ V , b ∈ A β L , H , b ′ ∈ A β ′ L , H . Then, vw = aa ′ ⊗ bb ′ , V ⊗ V L , H , H V ⊕ H ) is a good VH pair. (cid:3) The category of good VH pairs (resp. good AH pairs) are full subcategory of VH pair(resp. AH pair) whose objects are good VH pairs (reps. good AH pairs). The followingproposition follows from the Theorem 3.1:
Corollary 3.1.
The adjoint functors in Theorem 3.1 induce adjoint functors betweengood VH pair and good AH pair.
Let ( V , H V ) be a good VH pair. Then, by Lemma 2.5, Ω ( V , H V ) lat is a lattice pair. Denotethe even H-lattice ( L Ω ( V , H V ) , H V ) by L V , H V , which we call a maximal lattice of the good VHpair ( V , H V ) (see Definition 2). By Proposition 2.4, the AH pair Ω ( V , H V ) lat is isomorphicto the AH pair ( A L V , HV , H V , H V ). By Lemma 3.10, the generalized lattice vertex algebra V L V , HV , H V = V ( Ω ( V , H V ) lat ) is a subVH pair of ( V , H V ), which is a maximal generalizedlattice vertex algebra as follows: Theorem 3.2.
Let ( V , H ) be a good VH pair. Then, the subVH pair V ( Ω ( V , H V ) lat ) can becharacterized by the following properties:(1) V ( Ω ( V , H V ) lat ) is a generalized lattice vertex algebra:(2) For any even H-lattice ( L , H ) and any VH pair homomorphism f : ( V L , H , H ) → ( V , H V ) , the image of f is in V ( Ω ( V , H V ) lat ) .Furthermore, there is a natural bijection between Hom
VH pair (( V L , H , H ) , ( V , H V )) and Hom
AH pair (( A L , H , H ) , ( A L V , HV , H V )) . Proof.
By Theorem 3.1 and Lemma 2.5,Hom
VH pair (( V L , H , H ) , ( V , H V )) (cid:27) Hom good AH pair (( A L , H , H ) , Ω ( V , H V )) (cid:27) Hom good AH pair (( A L , H , H ) , Ω ( V , H V ) lat ) (cid:27) Hom
VH pair (( V L , H , H ) , V ( Ω ( V , H V ) lat )) . (cid:3) By Theorem 3.2 and Lemma 3.13, we have:
Corollary 3.2.
For a good VH pair ( V , H V ) and an even H-lattice ( L , H ) , the maximallattice of the good VH pair ( V ⊗ V L , H , H V ⊕ H ) is ( L V , H V ⊕ L , H V ⊕ H ) . We end this section by showing the existence of lattice vertex algebras in a good VHpair. Let ( V , H ) be a good VH pair and ( L V , H , H ) be the maximal lattice. Lemma 3.14.
If a subgroup M ⊂ L V , H spans a non-degenerate subspace of H, then thereexists a vertex subalgebra of V which is isomorphic to the lattice vertex algebra V M .Proof. It is clear that M is a free abelian group of finite rank. By Lemma 2.4, ( Ω ( V , H ) lat ) M (cid:27) ( A L V , H , H ) M is isomorphic to the twisted group algebra C { M } associated with the even lat-tice M . Let H ′ be the subspace of H spanned by M and set H ′′ = { h ∈ H | ( h , h ′ ) = h ′ ∈ H ′ } . Then, the vertex algebra V C { M } , H is isomorphic to M H ′ (0) ⊗ V M as a VHpair. Thus, V M is a subalgebra of V . (cid:3)
4. G enera of VH pairs and M ass F ormula In this section, we introduce the notion of a genera of VH pairs and prove a Massformula (Theorem 4.2) which is an analogous result of that for lattices.Recall that two lattices L and L are said to be equivalent or in the same genus if theirbase changes are isomorphic as lattices: L ⊗ Z R ≃ L ⊗ Z R , L ⊗ Z Z p ≃ L ⊗ Z Z p , for all the prime integers p . Consider the unique even unimodular lattice II , of signature(1 , Lemma 4.1.
The lattices L and L are in the same genus if and only ifL ⊗ II , ≃ L ⊗ II , as lattices. We will use this characterization of the equivalence to define that for VH pairs. In therest of this paper, we always assume VH pairs to be good and denote a VH pair by V instead of ( V , H V ) and the associated AH pair by Ω V instead of ( Ω H V V , H V ) for simplicity.Note that tensor products of good VH pairs and generalized lattice vertex algebras areagain good VH pairs by Lemma 3.13. ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 19
Let V II , denote the lattice vertex algebra associated with the lattice II , and consider itas a VH pair ( V II , , H II , ) with H II , = C ⊗ Z II , . Two (good) VH pairs V and W are saidto be equivalent if there exists an isomorphism f : V ⊗ V II , ≃ W ⊗ V II , of a VH pair, that is, a vertex algebra isomorphism f which satisfies f ( H V ⊗ V II , ) = H V ⊗ V II , . We call an equivalent class of VH pairs, an genus of VH pairs, and denote by genus( V )the equivalent class of a VH pair V .Before going into details, we briefly describe how to use the methods developed in theprevious section to study the genera. Assume that V ⊗ V II , (cid:27) W ⊗ V II , as a VH pair, that is,they are in the same genus of VH pairs. We observe that V is obtained as a coset of V ⊗ V II , by V II , , that is, V (cid:27) Coset V ⊗ V II , ( V II , ), and so is W (see Section 4.1). Furthermore, itis clear that the subVH pair ( M H V (0) ⊗ V II , , H V ⊕ H II , ) ⊂ ( V ⊗ V II , , H V ⊕ H II , ) is ageneralized lattice vertex algebra, whose maximal lattice is ( II , , H V ⊕ H II , ). By choosinga subVH pair which is isomorphic to the generalized lattice vertex algebra M H V (0) ⊗ V II , ,we obtain vertex subalgebras V , W and all the VH pair in genus( V ). A subVH pair whichis isomorphic to M H V (0) ⊗ V II , can be classified by Lemma 4.6.Section 4.1 is devoted to studying the coset constructions of vertex algebras and VHpairs. In section 4.2, a mass formula will be proved.4.1. Coset constructions.
Let V and W be a vertex algebra. For each V -module M , thetensor product W ⊗ M becomes a W ⊗ V -module. We denote by T W the functor whichassigns the W ⊗ V -module W ⊗ M to each V -module M : T W : V -mod → W ⊗ V -mod , M W ⊗ M . For a vertex algebra ˜ V and a vertex subalgebra W of ˜ V and a ˜ V -module ˜ M , the subspace C ˜ M ( W ) = { v ∈ ˜ M | w ( n ) v = w ∈ W and n ≥ } is called a coset. If ˜ M = ˜ V , viewed as a ˜ V -module, then C ˜ V ( W ) is a vertex subalgebra of˜ V . It is clear that C ˜ M ( W ) is a C ˜ V ( W )-module for any ˜ V -module ˜ M . Consider the functor R W which assigns the C ˜ V ( W )-module C ˜ M ( W ) to each ˜ V -module ˜ M , R W : ˜ V -mod → C ˜ V ( W )-mod , ˜ M C ˜ M ( W ) . In this subsection, we consider the composition of the coset functor R W and the tensorproduct functor T W . We first consider the case of R W ◦ T W . If C W ( W ) = C , then R W ◦ T W ( M ) = C W ⊗ M ( W ) = C ⊗ M for any V -module M . Hence, we have: Lemma 4.2.
If C W ( W ) = C , then the composite functor R W ◦ T W : V-mod → V-mod isnaturally isomorphic to the identity functor.
We now consider the case of T W ◦ R W . Let ˜ V be a vertex algebra and W a vertexsubalgebra. Set W ′ = C ˜ V ( W ). Since the vertex subalgebras W and W ′ are commutewith each other (in ˜ V ), by the universality of the tensor product in the category of vertexalgebras, there is a vertex algebra homomorphism i W : W ⊗ W ′ → ˜ V , ( w , w ′ ) w ( − w ′ ,which implies that ˜ M is a W ⊗ W ′ -module. Consider this pullback functor, i W ∗ : ˜ V -module → W ⊗ W ′ -module , ˜ M ˜ M . We let f ˜ M denote the linear map W ⊗ C ˜ M ( W ) → ˜ M defined by setting f ˜ M ( w ⊗ m ) = w ( − m for w ∈ W and m ∈ C ˜ M ( W ). Lemma 4.3.
The family of linear maps f ˜ M gives a natural transformation from T W ◦ R W to the pullback functor i W ∗ .Proof. It su ffi ces to show that f ˜ M : W ⊗ C ˜ M ( W ) → ˜ M is a W ⊗ W ′ -module homomorphism(The naturality of f ˜ M is clear). Let w , w ∈ W and w ′ ∈ W ′ , m ∈ C ˜ M ( W ) and k ∈ Z . It su ffi ces to show that f ˜ M (( w ⊗ w ′ )( k ) w ⊗ m ) = ( w ⊗ w ′ )( k ) f ˜ M ( w ⊗ m ). Since[ w ( p ) , w ′ ( j )] = [ w ( p ) , w ′ ( j )] = w ( q ) m = w ( q ) m = p ∈ Z and q ≥ f ˜ M ( w ( i ) w ⊗ w ′ ( j ) m )) = ( w ( i ) w )( − w ′ ( j ) m ) = X l ≥ il ! ( − l ( w ( i − l )) w ( − + l ) + w ( i − − l ) w ( l )) w ′ ( j ) m = w ( i ) w ′ ( j ) w ( − m for any i , j ∈ Z . Hence, f ˜ M (( w ⊗ w ′ )( k ) w ⊗ m ) = X l ∈ Z f ˜ M ( w ( l ) w ⊗ w ′ ( k − l − m ) = X l ∈ Z w ( l ) w ′ ( k − l − w ( − m = ( X l ≥ w ( − − l ) w ′ ( k + l ) + w ′ ( k − l − w ( l )) w ( − m = ( w ( − w ′ )( k ) f ˜ M ( w ⊗ m ) = ( w ⊗ w ′ )( k ) f ˜ M ( w ⊗ m ) . (cid:3) We will say that a vertex algebra V is completely reducible if every V -module is com-pletely reducible. If a completely reducible vertex algebra V has a unique simple V -module up to isomorphism, we will say that V is holomorphic . Hereafter, we considerthe functor R W and T W for a holomorphic vertex algebra W . Let W be a vertex subalge-bra of ˜ V and ˜ M a ˜ V -module. Suppose that W be a holomorphic vertex algebra such that C W ( W ) = C . Since W is a holomorphic vertex algebra, as a W -module, ˜ M is a directsum of copies of W . For each copy of W , since C W ( W ) = C , there is a unique vector ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 21 which commutes with W (called a vacuum like vector ). Hence, the natural transformationgiven in Lemma 4.3, f ˜ M : W ⊗ C ˜ M ( W ) → ˜ M , ( w , m ) w ( − m , (4)is an isomorphism. In particular, ˜ V is isomorphic to W ⊗ C ˜ V ( W ) as a vertex algebra.Combining Lemma 4.2 and Lemma 4.3 and the above discussion, we have: Theorem 4.1.
Let ˜ V be a vertex algebra and W be a vertex subalgebra of V such thatC W ( W ) = C . Then, ˜ V is isomorphic to W ⊗ C ˜ V ( W ) as a vertex algebra and the functorsT W and R W are mutually inverse equivalences between ˜ V -mod and C ˜ V ( W )-mod . Corollary 4.1.
If W is holomorphic and C W ( W ) = C , then a vertex algebra V is simple(resp. completely reducible, holomorphic) if and only if V ⊗ W is simple (resp. completelyreducible, holomorphic).
Remark 4.1.
Let M be an irreducible module of an associative C -algebra A and N anirreducible module of an associative C -algebra B. Then, it is well-known that M ⊗ N isan irreducible module of A ⊗ B if both M and N are finite dimensional, which is not truein general. For example, let A = B = M = N = C ( x ) . Then, the claim fails, since A ⊗ B isnot a field.
For our application, we are interested in a coset of a VH pair by a holomorphic latticevertex algebra. Let ( ˜ V , H ) be a good VH pair and ( L ˜ V , H , H ) be a maximal lattice. Let M be a sublattice of L V , H such that M is unimodular. According to Proposition 3.14, thelattice vertex algebra V M , which is holomorphic, is a vertex subalgebra of ˜ V . Let H ′ bethe canonical Heisenberg subspace of the lattice vertex algebra V M and set H ′′ = { h ∈ H | ( h , h ′ ) = h ′ ∈ H ′ } . By Theorem 4.1, ˜ V (cid:27) V M ⊗ R V M ( ˜ V ). Hence, we have: Lemma 4.4. R V M ( ˜ V ) = { v ∈ ˜ V | h ( n ) v = for any n ≥ and h ∈ H ′ } . Since M is non-degenerate, H = H ′ ⊕ H ′′ and H ′′ ⊂ R V M ( ˜ V ). In particular, ( R V M ( ˜ V ) , H ′′ )is canonically a VH pair. Hence, we have: Lemma 4.5.
If M is a unimodular sublattice of L ˜ V , H . then ( ˜ V , H V ) is isomorphic to ( V M ⊗ R V M ( ˜ V ) , H ′ ⊕ H ′′ ) as a VH pair. Let II , be the unique even unimodular lattice of signature (1 , II , has abasis z , w such that ( z , z ) = ( w , w ) = z , w ) = −
1. Then, we have:
Lemma 4.6.
For a good VH pair ˜ V, there is a one-to-one correspondence between a sub-lattice of L ˜ V which is isomorphic to II , and a decomposition of ˜ V into a tensor product ˜ V (cid:27) V II , ⊗ V ′ as a VH pair. Mass formulae for VH pairs.
By Corollary 3.2, if V and W are in the same genus,then L V ⊕ II , (cid:27) L W ⊕ II , , where L V and L W are the maximal lattice of V and W . ByLemma 4.1, L V and L W are in the same genus of lattices. Let f ∈ Aut V such that f ( H V ) = H V . According to Lemma 3.5, f induces the automor-phism on Ω V . Furthermore, f induces an automorphism on the maximal lattice L V ⊂ H V .The image of such automorphisms forms a subgroup of Aut L V , denoted by G V .For a lattice L , set ˜ L = L ⊕ II , and S II , ( ˜ L ) = { M ⊂ ˜ L | M is a rank 2 sublattice which is isomorphic to II , } . The group Aut ˜ L naturally acts on S II , ( ˜ L ). Let us denote the set of orbits by Aut ˜ L \ S II , ( ˜ L ).Then, there is a one-to-one correspondence between Aut ˜ L \ S II , ( ˜ L ) and the isomorphismclasses of genus( L ).Since the maximal lattice of ˜ V = V ⊗ V II , is ˜ L V = L V ⊕ II , , the group G ˜ V acts on S II , ( ˜ L V ). In analogy with the case of lattices, we have: Proposition 4.1.
There is a one-to-one correspondence between G ˜ V \ S II , ( ˜ L V ) and theisomorphism classes of genus( V ) .Proof. Let S , S ′ ∈ S II , ( ˜ L V ). According to Lemma 4.6, there exist the correspondentlattice vertex subalgebras V S ⊂ ˜ V and V S ′ ⊂ ˜ V . Set C S = C ˜ V ( V S ) and C S ′ = C ˜ V ( V S ′ ).Then, ˜ V (cid:27) V S ⊗ C S as a VH pair. Hence, C S ∈ genus( V ).Suppose that there exists f ∈ G ˜ V such that f ( S ) = S ′ . Then, f ( V S ) = V S ′ in ˜ V . Thus, f induces an isomorphism C S → C S ′ . Thus, we have a map G ˜ V \ S II , ( ˜ L V ) → { the VH pair isomorphism classes of genus( V ) , S C S . For an isomorphism g : C S → C S ′ as a VH pair, id ⊗ g is an isomorphism V S ⊗ C S → V S ′ ⊗ C S ′ , that is, an automorphism of ˜ V , which maps S to S ′ . Hence, the map is injective.The surjectivity of it follows from Lemma 4.6. (cid:3) In general, G ˜ V is a proper subgroup of Aut ˜ L . Hence, it is possible that there are non-isomorphic vertex algebras in a genus whose maximal lattices are the same. Let S be asublattice of ˜ L . Set S ⊥ = { α ∈ ˜ L | ( α, β ) = β ∈ S } , which is a sublattice of ˜ L , and setSt Aut ˜ L ( S ) = { f ∈ Aut ˜ L | f ( S ) = S } . Lemma 4.7.
Let V be a good VH pair and L V be the maximal lattice. Let L ∈ genus( L V ) and choose S ∈ S II , ( ˜ L V ) with S ⊥ (cid:27) L . Then, there is a one-to-one correspondencebetween the isomorphism classes of VH pairs in genus( V , H V ) whose maximal lattice isisomorphic to L and the double coset G ˜ V \ Aut ( ˜ L V ) / St Aut ˜ L V ( S ) . Furthermore, Aut ˜ L V = G ˜ V if and only if | G ˜ V \ Aut ( ˜ L V ) / St Aut ˜ L V ( S ) | = and St Aut ˜ L V ( S ) ⊂ G ˜ V . In this case, thereexists a one-to-one correspondence between genus( L V ) and genus( V , H V ) . In particular,a VH pair in genus( V , H V ) is uniquely determined by its maximal lattice. ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 23
Proof.
Set X = { S ∈ S II , ( ˜ L V ) | L (cid:27) S ⊥ } . Since Aut ˜ L V acts on X transitively, X (cid:27) Aut ˜ L V / St ˜ L V ( S ) as a Aut ˜ L V -set. Then, the number of the isomorphism class of VH pairsin genus( V ) whose maximal lattice is isomorphic to L is equal to | G ˜ V \ X | = | G ˜ V \ Aut ( ˜ L V ) / St Aut ˜ L V ( S ) | . If | G ˜ V \ X | =
1, then G ˜ V St Aut ˜ L V ( S ) = Aut ( ˜ L V ). Thus, the assertion follows. (cid:3) Suppose that L V is positive-definite and [Aut ( ˜ L V ) : G ˜ V ] is finite. Then, G V and Aut L V are finite groups and | G ˜ V \ Aut ( ˜ L V ) / St Aut ˜ L V ( S ) | is finite for any L ∈ genus( L V ). A massof genus( V , H V ) is defined to bemass( V , H V ) = X W ∈ genus( V ) | G W | . By the formula for the number of double cosets, we have:
Proposition 4.2.
Suppose that L V is positive-definite and [Aut ( ˜ L V ) : G ˜ V ] is finite. Then,for L ∈ genus( L V ) , the following equation holds: [Aut ˜ L V : G ˜ V ] | Aut L | = X W ∈ genus( V ) , L W (cid:27) L | G W | . Summing up the equation in Proposition 4.2 on L ∈ genus( L V ), we have: Theorem 4.2.
Let ( V , H V ) be a good VH pair. If L V is positive-definite and [Aut ( ˜ L V ) : G ˜ V ] is finite, then mass( V , H V ) = mass( L V )[Aut ˜ L V : G ˜ V ] .
5. VH pair with coformal vector
In this section, VH pairs with a conformal vector are studied. We first recall the defini-tion of a conformal vertex algebra and a vertex operator algebra and its dual module.A conformal vertex algebra is a vertex algebra V with a vector ω ∈ V satisfying thefollowing conditions:(1) There exists a scalar c ∈ C such that[ L ( m ) , L ( n )] = ( m − n ) L ( m + n ) + m − m δ m + n , c holds for any n , m ∈ Z , where L ( n ) = ω ( n + L ( − = T V ;(3) L (0) is semisimple on V and all its eigenvalues are integers.The scalar c of a conformal vertex algebra is called a central charge of the conformalvertex algebra. Let ( V , ω ) be a conformal vertex algebra and set V n = { v ∈ V | L (0) v = nv } for n ∈ Z . A conformal vertex algebra ( V , ω ) is called a vertex operator algebra of CFTtype (hereafter called a VOA ) if it satisfies the following conditions:
VOA1) If n ≤ −
1, then V n = V = C ;VOA3) V n is a finite dimensional vector space for any n ≥ l is called a c = lconformal vertex algebra (resp. a c = l VOA) .Let ( V , ω ) be a VOA. Set V ∨ = L n ≥ V ∗ n , where V ∗ n is the dual vector space of V n . For f ∈ V ∨ and a ∈ V , set Y V ∨ ( a , z ) f ( − ) = f ( Y ( e L (1) z ( − z − ) L (0) a , z − ) − ) . The following theorem was proved in [FHL, Theorem 5.2.1]:
Theorem 5.1. ( V ∨ , Y V ∨ ) is a V-module. The V -module V ∨ in Theorem 5.1 is called a dual module of V . A VOA ( V , ω ) is said tobe self-dual if V is isomorphic to V ∨ as a V -module. The following theorem was provedin [Li1, Corollary 3.2]: Theorem 5.2.
Let ( V , ω ) be a simple VOA. Then, ( V , ω ) is self-dual if and only if L (1) V = . If ( V , ω ) is self-dual, then there exists a non-degenerate bilinear form ( , ) : V × V → C such that ( Y ( a , z ) b , c ) = ( b , Y ( e L (1) z ( − z − ) L (0) a , z − ) c )for any a , b , c ∈ V . A bilinear form on V with the above property is called an invariantbilinear form on ( V , ω ). We remark that ( V , ω ) is self-dual if and only if it has a non-degenerate invariant bilinear form. A subspace W ⊂ V is called a subVOA of ( V , ω ) if W is a vertex subalgebra and ω ∈ W .In this section, we will introduce some technical conditions for a conformal vertexalgebra with a Heisenberg subspace H , which is an analogue of a self-dual VOA (of CFTtype).Let ( V , ω ) be a conformal vertex algebra with a Heisenberg subspace H ⊂ V . Assumethat ω ( n ) h = ω (1) h = h and h (0) is semisimple on V for any h ∈ H and n ≥ h (0) commutes with L (0). For α ∈ H , set V α n = { v ∈ V n | h (0) v = ( h , α ) v for any h ∈ H } and M V , H = { α ∈ H | V α n , n ∈ Z } and V ∨ H = M n ∈ Z ,α ∈ L ( V α n ) ∗ , which is a restricted dual space. We would like to define a V -module structure on V ∨ H in analogy with Theorem 5.1. The di ffi culty in defining a dual module structure on aconformal vertex algebra lies in the fact that it does not always satisfy the followingconditions: ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 25 • V n is bounded below, that is, V n = ffi ciently small n ; • ω (2) is a locally nilpotent operator.We assume that, for any α ∈ M V , H , V α n = ffi ciently small n . Then, since ω (2)commutes with h (0) for any h ∈ H , ω (2) is locally nilpotent. By using V α ( n ) V β ⊂ V α + β ,we can define a V -module structure on V ∨ H similarly to Theorem 5.1. Hence, we have: Lemma 5.1.
If a conformal vertex algebra ( V , ω ) and a Heisenberg subspace H ⊂ Vsatisfy the above conditions, then V ∨ H is a V-module similarly to Theorem 5.1. Definition 1.
Suppose that a conformal vertex algebra ( V , ω ) and a Heisenberg subspaceH ⊂ V satisfy the following assumption (A):A1) ω ( n ) h = and ω (1) h = h for any n ≥ and h ∈ H;A2) For any h ∈ H, h (0) is semisimple on V;A3) ( α, β ) ∈ R , for any α, β ∈ M V , H ;A4) If ( α,α )2 > n, then V α n = ;A5) V = C and V = H;A6) V α n is a finite dimensional vector space;A7) V ∨ H (cid:27) V as a V-module.
Remark 5.1.
Let ( V , ω, H ) satisfy Assumption (A). By (A7), there exists a non-degenerateinvariant bilinear form (see Section 3.2). Normalize the bilinear form so that ( , ) = − .Let h , h ′ ∈ H. By the invariance, ( h , h ′ ) = ( − , h (1) h ′ ) . The bilinear form on theHeisenberg subspace H is defined by h (1) h ′ = ( h , h ′ ) (see Section 2.3). Hence, twobilinear form on H coincide. In this section, a conformal vertex algebra and a Heisenberg subspace satisfying As-sumption (A), which is denoted by ( V , ω, H ), is studied.Assumption (A) is used in order to show the following Theorems, which are proved inSection 5.1: Theorem 5.3.
Set L ′ V , H = { α ∈ H | V αα / , } . Then, ( Ω V , H ) lat = L α ∈ L ′ V V αα / . Inparticular, L ′ V , H is equal to the maximal lattice L V , H of the VH pair ( V , H ) . Theorem 5.4.
Let α ∈ M V , H such that α > and V α , . Then, V α and V − α generatethe simple a ffi ne vertex algebra L sl ( k , of level k = / ( α, α ) ∈ Z > and dim V ± α = , dim V k α = for any , ± , k ∈ Z . Furthermore, dim V β n = , for any β in M V , H withn ≥ ( β,β )2 > n − . Those theorems assert that the existence of non-zero vectors in V α n for α ∈ M V , H and n ∈ Z with 1 > n − α / ≥ ffi ne vertex subalgebras.Examples and applications of those results are studied in subsection 5.2; There, wecompute the maximal lattice for a VOA which is an extension of an a ffi ne VOA at positiveinteger level, together with the mass of the genus for some c =
24 holomorphic VOAs.
Structure of ( V , ω, H ) . Let ( V , ω, H ) satisfy Assumption (A). In this subsection, westudy the structure of ( V , ω, H ). Lemma 5.2. If ( V , ω, H ) satisfy Assumption (A), then V is a simple vertex algebra and ker T = C . In particular, ( V , H ) is a good VH pair.Proof. Let I be a non-zero ideal of V . Set I α n = I ∩ V α n . Then, I = L n ∈ Z ,α ∈ M V , H I α n . Thereexists α ∈ M V , H and n ∈ Z such that I α n , I α n − k = k ≥
1. Let v ∈ I α n be anon-zero vector. Since V is self-dual, there exists v ′ ∈ V − α n such that ( v , v ′ ) = L (1) v = v (2 n − v ′ ∈ V = C , ( v , v ′ ) =
1, we have ∈ I . Hence, I = V . Since v ( n ) a = a ∈ ker T and v ∈ V and n ≥
0, ker T ⊂ V = C . (cid:3) Corollary 5.1.
For any α, β ∈ M V , H , α + β, − α ∈ M V , H .Proof. Let α, β ∈ M V , H . Since the invariant bilinear form on V induces a non-degeneratepairing, V α n × V − α n → C , we have − α ∈ M V , H . Let v ∈ V α and w ∈ V β be nonzeroelements. Then, according to Lemma 3.3, we have 0 , v ( k ) w ∈ V α + β for some k ∈ Z .Thus, α + β ∈ M V , H . (cid:3) According to Remark 3.12 and Lemma 5.2, Ω V , H is a good AH pair and Ω lat V , H is a latticepair. Set L ′ V , H = { α ∈ H | V αα / , } . Before proving Theorem 5.3, we show the followinglemma: Lemma 5.3.
For any α ∈ L ′ V , H , Ω α V , H = V αα / .Proof. Let α ∈ L ′ V , H and v ∈ V αα / be a nonzero element. By (A4), h ( n ) v = h (0) v = ( α, h ) v and L ( n ) v = n ≥ h ∈ H . It su ffi ces to show that ω H (0) v = v ( − . Since the invariant bilinear form on V induces a non-degenerate pairing on V α n × V − α n → C , there exists v ′ ∈ V − αα / such that ( v , v ′ ) = ( − α / + . By the invariance, − = ( v , v ′ )( − α / = ( , v (( α, α ) − v ′ ). Since v (( α, α ) − v ′ ∈ V = C , we have v (( α, α ) − v ′ = . Since v (( α, α ) − v ′ ∈ V = H and ( h , v (( α, α ) − v ′ ) = ( − , h (1) v (( α, α ) − v ′ = ( − h , α )( , v (( α, α ) − v ′ ) = ( h , α ), we have v (( α, α ) − v ′ = α .Since v ′ (( α, α ) + k ) v ∈ V − k − = v ( − ( α, α ) + k ) v ∈ V α α,α ) − k − = k ≥
0, byapplying the Borcherds identity with ( p , q , r ) = ( − ( α, α ) , ( α, α ) − , ( α, α ) − α ( − v = X i ≥ − ( α, α ) i ! ( v (( α, α ) − + i ) v ′ )( − − i ) v = X i ≥ ( − i ( α, α ) − i ! ( v ( − − i ) v ′ (( α, α ) − + i ) v − v ′ (2( α, α ) − − i ) v ( − ( α, α ) + i )) v = v ( − . Hence, we have V αα / ⊂ Ω α V , H . According to Lemma 2.3, dim Ω α V , H =
1. Thus, we have Ω α V , H = V αα / . (cid:3) proof of Theorem 5.3. Let α ∈ L V , H , that is, Ω α V , H , Ω − α V , H ,
0. By the above lemma, L ′ V , H ⊂ L V , H . Thus, it su ffi ces to show that α ∈ L ′ V , H . Let v ∈ Ω α V , H and v ′ ∈ Ω − α V , H be nonzero ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 27 elements. Then, v = P i v i where v i ∈ V α i . Since h ( n ) V i ⊂ V i − n and T V i ⊂ V i + , we have v i ∈ Ω α V , H . Since dim Ω α V , H =
1, we have Ω α V , H ⊂ V α k and Ω − α V , H ⊂ V − α k ′ for some k , k ′ ∈ Z .According to Lemma 3.2, we can assume that = v (( α, α ) − v ′ ∈ Ω V , H = C . Since v (( α, α ) − v ′ ∈ V k + k ′ − ( α,α ) and k , k ′ ≥ α /
2, we have k = k ′ = α /
2. Thus, α ∈ L ′ V , H . (cid:3) In particular, we prove that ( α, β ) ∈ Z and ( α, α ) ∈ Z for any α, β ∈ L ′ V , H . Recall that M V , H = { α ∈ H | V α n , n ∈ Z } . Let α ∈ L V , H such that α ,
0. Then, V ± αα / generates the lattice vertex algebra V Z α and V is a direct sum of irreducible V Z α -modules.Hence, we have: Corollary 5.2.
Let α ∈ L V , H such that α , . Then, ( α, β ) ∈ Z for any β ∈ M V , H . Set L ∨ V , H = { α ∈ H | ( α, β ) ∈ Z for any β ∈ L V , H } . Corollary 5.3.
If L V , H is non-degenerate, then M V , H ⊂ L ∨ V , H . We will use the following lemma:
Lemma 5.4.
Suppose that L V , H is positive-definite and spans H. Then, V is a VOA (ofCFT type) and M V , H is a subgroup of L ∨ V , H .Proof. Since L V , H is non-degenerate and spans H , M V , H ⊂ L ∗ V ⊂ span R L V , H , which impliesthat M V , H is positive-definite. If 0 , α ∈ M V , H , V α n = n ≤
0. Hence, V = L n ≥ V n and V = C . Since { α ∈ M V , H | α < n } is a finite set for any n >
0, dim V n is finite. (cid:3) Now, we will show Theorem 5.4: proof of Theorem 5.4.
Let 0 , e ∈ V α and f ∈ V − α such that ( e , f ) =
1. Since α > L ( n ) e ∈ V α − n = n ≥
1. Since e (0) f ∈ V = H and ( h , e (0) f ) = ( h , α )( e , f ) = ( h , α ), we have e (0) f = α . Since V α − n = n ≥
0, according to the skew-symmetryof vertex algebras, we have e (0) e = − e (0) e + T e (1) e + · · · = − e (0) e . Thus, { e , α f , α α } is a sl -triple under the 0-th product. Similarly, we have e ( n ) e = f ( n ) f = , h ( n ) e = h ( n ) f = n ≥
1. Since 1 = ( e , f ) = − ( , e (1) f ), we have e (1) f = . Thus, { e , α f , α } generates an a ffi ne vertex algebra W of level k = / ( α, α ). Since e ( − n ∈ V n α n = n α > n , W is isomorphic to the simple a ffi ne vertex algebra L sl ( k ,
0) and the level k is a positive integer. We will show that dim V − α =
1. Let f ′ ∈ V − α such that ( e , f ′ ) = e (0) f ′ = V ± n α = ffi ciently large n , f (0) n f ′ =
0. Hence, f ′ = sl . Hence, dim V − α =
1. Since dim V ± α =
1, for any k ∈ Z with k , ± ,
0, dim V k α = sl again. Finally, suppose V β n , β in M V , H with n ≥ ( β,β )2 > n −
1. Then, by Lemma5.7, V ⊗ V II , satisfy Assumption (A). Let γ ∈ II , such that ( γ,γ )2 = − n +
1. Since0 , V β n ⊗ V II , γ − n + ⊂ ( V ⊗ V II , ) ( β,γ )1 and 1 ≥ (( β,γ ) , ( β,γ ))2 = ( β,β )2 + ( γ,γ )2 >
0, by the above result,we have dim( V ⊗ V II , ) ( β,γ )1 =
1. Thus, dim V β n = (cid:3) For α ∈ M V , H with α > V α ,
0, define r α : H → H by r α ( h ) = h − h , α ) α α, and set ˜ r α = exp( 2 α f (0)) exp( − e (0)) exp( 2 α f (0)) , where e and f is the same in the proof of Theorem 5.4. The following lemma followsfrom the representation theory of the Lie algebra sl and Theorem 5.4: Lemma 5.5.
For α ∈ M V , H with α > and V α , , ˜ r α is a vertex algebra automorphismof V satisfying the following conditions:(1) ˜ r α ( ω ) = ω ;(2) ˜ r α ( H ) = H;(3) ˜ r α | H = r α .In particular, the reflection r α is an automorphism of the maximal lattice L V , H . Lemma 5.6. If α > and V α , , then α α ∈ L V .Proof. Let e ∈ V α be a nonzero element and set k = /α . According to the representationtheory of the Lie algebra sl , 0 , e ( − k ∈ V k α k . Since ( k α ) / = k , we have k α ∈ L V . (cid:3) We end this subsection 5.1 by mentioning a generalization of the theory of genera ofVH pairs to the triples ( V , ω, H ). The results in Section 4 are also valid for the triple( V , ω, H ) with minor changes. Lemma 5.7.
Let ( V , ω V , H V ) and ( W , ω W , H W ) satisfy Assumption (A). Then, ( V ⊗ W , ω V + ω W , H V ⊕ H W ) satisfy Assumption (A).Proof. Let α ∈ H V and α ′ ∈ H W . Then, ( V ⊗ W ) ( α,α ′ ) n = L n = k + k ′ V α k ⊗ W α ′ k ′ . Hence,Definition ( A , . . . , ( A
6) is easy to verify. Since ( V ⊗ W ) ( α,α ′ ) n is finite dimensional, ( V ⊗ W ) ∗ H V ⊕ H W (cid:27) V ∗ H V ⊗ W ∗ H W (cid:27) V ⊗ W . (cid:3) The following lemma is clear from the definition:
Proposition 5.1.
Let L be a non-degenerate even lattice and V L the lattice conformalvertex algebra. Set H L = L ⊗ Z C ⊂ ( V L ) . Then, ( V L , ω H L , H L ) satisfy Assumption (A). Results in Section 4 are obtained as follows. Hereafter, we assume that any triple( V , ω, H ) satisfy Assumption (A). By Lemma 5.7, the category of triples ( V , ω, H ) formsa symmetric monoidal category, where a morphism from ( V , ω V , H V ) to ( W , ω W , H W ) isa vertex algebra homomorphism f : V → W such that f ( ω V ) = ω W and f ( H V ) = H W .We also remark that Lemma 4.6 can be generalized to triples ( V , ω, H ). Hence, we candefine a genus of ( V , ω, H ) similarly to Section 4. More precisely, triples ( V , ω V , H V )and ( W , ω W , H W ) are in the same genus if there exists a vertex algebra isomorphism f : V ⊗ V II , → W ⊗ V II , such that f ( ω V ⊗ V II , ) = ω W ⊗ V II , and f ( H V ⊗ V II , ) = H W ⊗ V II , .Denote by genus( V , ω V , H V ) the genus of ( V , ω V , H V ). Denote by G ( V ,ω V , H V ) the image of ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 29 automorphisms f ∈ Aut ( V ) such that f ( ω V ) = ω V and f ( H V ) = H V in Aut L V , H V . If L V is positive-definite and [Aut ˜ L V : G ( ˜ V , ˜ ω ) , H ˜ V ] is finite, then the mass of genus( V , ω V , H V ) isdefined to be mass( V , ω V , H V ) = X ( V ′ ,ω ′′ V , H V ′ ) ∈ genus( V ,ω V , H V ) | G ( V ′ ,ω ′ V ′ , H V ′ ) | . Then, we have:
Theorem 5.5. mass( L V )[Aut ˜ L V : G ( ˜ V , ˜ ω ) , H ˜ V ] = mass( V , ω V , H V ) . We remark on the genus of VOAs. Let V be a VOA. Assume that two Heisenbergsubspaces H and H ′ satisfy Assumption (A). Then, H and H ′ are split Cartan subalger-bas of Lie algebra V . Hence, there exists a , . . . , a k ∈ V such that f ( H ) = H ′ , where f = exp( a (0)) · · · exp( a n (0)) (see, for example, [Hu]). Since exp( a i (0)) is a VOA auto-morphism of V , we have: Lemma 5.8.
All Heisenberg subspace of a VOA which satisfy Assumption (A) are conju-gate under the VOA automorphism group.
The lemma asserts that the genus of a VOA is independent of the choice of a Heisenbergsubspace H .Similarly to the proof of Lemma 5.4 and according to Theorem 4.1, we have: Lemma 5.9.
Let ( V , ω, H ) satisfy Assumption (A). Suppose that L V , H is positive-definiteand spans H. Then, all conformal vertex algebras in genus( V , ω, H V ) are VOAs. Application to extensions of a ffi ne VOAs. In this subsection, we prove that manyimportant conformal vertex algebras satisfy Assumption (A) and study the maximal latticefor those vertex algebras.The following Proposition follows from Theorem 5.2:
Proposition 5.2.
Let ( V , ω ) be a conformal vertex algebra. Then, ( V , ω, satisfy Assump-tion (A) if and only if V is a simple self-dual VOA with V = . In order to prove extensions of a simple a ffi ne VOAs at positive integer level satisfyAssumption (A), we recall some results on simple a ffi ne VOA at positive integer level.Let g be a simple Lie algebra and h a Cartan subalgebra and ∆ the root system of g .Let Q g the sublattice of the weight lattice spanned by long roots and { α , . . . , α l } a set ofsimple roots. Let θ be the highest root and a , . . . , a l integers satisfying θ = P li = a i α i . Let h , i be an invariant bilinear form on g , which we normalize by h θ, θ i = Lemma 5.10.
The sublattice Q g ⊂ h is generated by { α α | α ∈ ∆ } . If g is A n , D n , E n (resp.B n , C n , F , G ), then Q g is isomorphic to the root lattice (resp. D n , A n , D , A ). Let L g ( k ,
0) denote the simple a ffi ne VOA of level k ∈ C associated with g [FZ]. Then,the VOA L g ( k ,
0) is completely reducible if and only if k is a non-negative integer [FZ,DLM]. We assume that k is a positive integer.Suppose that a dominant weight λ of g satisfies h θ, λ i ≤ k . Then, the highest weightmodule L g ( k , λ ) is an irreducible module of L g ( k , L g ( k ,
0) is isomorphic to the module of this form. Let P + g , k be the set of all dominantweights λ satisfying h θ, λ i ≤ k . For α ∈ h and n ∈ Z , set L g ( k , λ ) α n = { v ∈ L g ( k , λ ) | h (0) v = h h , α i v and L (0) v = nv for any h ∈ h } . Then, L g ( k , λ ) = L α ∈ h , n ∈ Z L g ( k , λ ) α n . Set I g = { } ∪ { Λ i | a i = i ∈ { , . . . , n }} , where { Λ i } i ∈{ ,..., n } is the fundamental weights of { g , α , . . . , α l } . The following lemma follows from [DR]: Lemma 5.11.
Let λ ∈ P + g , k . If L g ( k , λ ) α n , , then n ≥ h α, α i / k and the equality holds ifand only if λ ∈ kI g and α ∈ λ + kQ g . Remark 5.2.
Non-zero weights in I g is called cominimal and L g ( k , k λ ) is known to be asimple current for λ ∈ I g (see [Li2, Proposition 3.5] ). In fact, I g form a group by the fusionproduct. Let R g be a root lattice of g and R ∨ g the dual lattice of the root lattice. Then, thereis a natural group isomorphism between I g and R ∨ g / R g . According to Lemma 5.11, we have:
Proposition 5.3.
Let ( V , ω ) be a VOA. If L g ( k , is a subVOA of V and V = L g ( k , = g,then ( V , ω, H ) satisfy Assumption (A), where H is a Cartan subalgebra of g = V . Combining Lemma 5.11 with Theorem 5.3, we have:
Corollary 5.4.
For k ∈ Z > and simple Lie algebra g , the maximal lattice of the simplea ffi ne VOA L g ( k , is √ kQ g . Remark 5.3.
This corollary implies that a level one simple a ffi ne vertex algebra associ-ated with a simply-raced simple Lie algebra is isomorphic to the lattice vertex algebra. Herein, we concentrate on the case that a VOA ( V , ω ) is an extension of a ffi ne VOA atpositive integer level. Let ( V , ω ) be a VOA and g i be a finite dimensional semisimple Liealgebra and k i ∈ Z > for i = , . . . , N . Set g = L Ni = g i and let H be a Cartan subalgebraof g . Assume that N Ni = L g i ( k i ,
0) is a subVOA of ( V , ω ) such that V = g . Then, it is clearthat ( V , ω, H ) satisfy Assumption (A). We will describe the maximal lattice of ( V , H ) byLemma 5.11.We simply write L g ( ~ k , ~λ ) for N Ni = L g i ( k i , λ i ), where ~λ = ( λ , . . . , λ N ) ∈ P = L Ni = P + g i , k .Since L g ( ~ k ,
0) is completely reducible, V = L ~λ ∈ P L g ( ~ k , ~λ ) n ~λ , where n ~λ ∈ Z ≥ is the multi-plicity of the module L g ( ~ k , ~λ ). Let Q V be a sublattice of H generated by L Ni = √ k i Q g i andall √ k ~λ satisfying the following conditions:(1) ( λ , . . . , λ N ) ∈ P ;(2) n ~λ > λ i ∈ k i I g i for any i = , . . . , N . ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 31
Then, we have:
Proposition 5.4.
The maximal lattice L V , H is equal to Q V . We will consider two examples whose genera are non-trivial. Set A g ,~ k = L Ni = I g i , whichis a group by Remark 5.2. Consider the quadratic form on A g ,~ k defined by A g ,~ k → Q / Z , ~λ N X i = k i ( λ i , λ i ) , for ~λ ∈ A g , k . A subgroup N ⊂ A g ,~ k is called isotropic if the square norm of any vector in N is zero. Let N be an isotropic subgroup of A g ,~ k .Set L g ( ~ k , N ) = M ~λ ∈ N L g ( ~ k , k ~λ ) . Then, by [Ca, Theorem 3.2.14], L g ( ~ k , N ) inherits a unique simple vertex operator alge-bra structure. The assumption of the evenness in the theorem follows from the fact that L g ( ~ k ,
0) is unitary [CKL].
Proposition 5.5.
The maximal lattice of L g ( ~ k , N ) is generated by L Ni = √ k i Q g i and N. For example, the maximal lattice of L B (2 ,
0) is √ D , where D is a root latticeof type D . The genus( √ D ) = II , (2 − II − II ) (see Section 1.2) contains two isomor-phism classes of lattices, √ D and √ E D . Hence, the genus of a VOA genus( L B (2 , c =
24 holomorphic VOAs. For a VOA V , the 0-th productgives a Lie algebra structure on V . A rank of a finite dimensional Lie algebra is thedimension of a Cartan subalgebra of the Lie algebra. The following theorem was provedin [DM1, Sc, EMS, DM2]: Theorem 5.6.
Let V be a holomorphic c = VOA. Then, the Lie algebra V is reductive,and exactly one of the following holds:(1) V = ;(2) V is abelian of rank and V is isomorphic to the lattice VOA of the Leechlattice;(3) V is one of semisimple Lie algebras listed in the table below.Furthermore, if V , , then the vertex subalgerba generated by V is a subVOA of V. rank Lie algebra24 U (1) , ( D , ) , ( A , ) ( D , ) , ( A , ) , ( A , ) , ( A , ) , ( A , ) , ( A , ) , ( D , ) ( E , ) , A , D , E , , ( A , ) , A , D , , D , ( E , ) , A , E , , ( D , ) , A , ( E , ) , D , E , , ( A , ) D , , ( A , ) D , , ( D , ) , D , , A ,
16 ( A , ) ( A , ) , ( D , ) ( C , ) , ( A , ) C , ( A , ) , ( D , ) ( A , ) , A , ( C , ) A , ( C , ) , D , C , ( B , ) , A , A , B , , E , C , A , , C , B , , E , B , D , ( B , ) , ( C , ) B , , D , A , , C , ( F , ) , E , B , F , , A ,
12 ( A , ) , B , , ( B , ) , B , ) , ( B , ) , ( B , ) , A , F , , C , ( A , ) , D , ( A , )
12 ( A , ) , E , A , , A , D , ( A , ) , A , ( A , ) , E , ( G , ) , D , A , G , A , ( A , ) , D , C , ( A , ) , E , C , A , , C , A , A , , D , ( A , ) A , C , A , , C , G , A , A , F , A , , D , A , D , A , C , For a Lie algebra g listed above, let V hol g denote a holomorphic VOA whose Lie algebrais g . Recently, the existence and uniqueness of those holomorphic VOAs are provedexcept for the case that V =
0. In this paper, we will use the following result:
Theorem 5.7 ([LS2, LS1]) . There exists a unique holomorphic VOA of central charge whose Lie algebra is E , B , . According to Theorem 5.6 and Proposition 5.2 and Proposition 5.3, we have:
Corollary 5.5.
Any c = holomorphic VOA satisfy Assumption (A). The following lemma follows from Theorem 4.1:
Lemma 5.12.
Let ( V , ω, H ) be a c = l holomoprhic conformal vertex algebra satisfyingAssumption (A). Then, any W ∈ genus( V , ω, H ) is a c = l holomorphic conformal vertexalgebra satisfying Assumption (A). Since the maximal lattice of c =
24 holomoprhic VOA is positive-definite, we have:
Proposition 5.6. If ( V , ω ) is a c = holomoprhic VOA, then all conformal vertex alge-bras in genus( V , ω ) are c = holomorphic VOAs. The above proposition gives us a e ff ective method to construct holomorphic VOAs.In the rest of this paper, we calculate the mass of the VOA V holE , B , in Theorem 5.7. ByProposition 5.4, the maximal lattice of V holE , B , is √ E D and its genus is II , (2 + II ).Since II , (2 + II ) contains 17 isomorphism classes of lattices, by lemma 5.6 and Theorem5.5, there exists at least 17 c =
24 holomorphic VOAs whose maximal lattices are in II , (2 + II ). ENERA OF VERTEX ALGEBRAS AND MASS FORMULA 33
The following lemma follows from Lemma 5.4:
Lemma 5.13.
Let ( V , ω, H ) satisfy Assumption (A). If V is a VOA and V is semisimpleLie algebra, then there is a one-to-one correspondence between norm vectors in L V , H and long roots of a level component of V . Hence, the level 1 components of V is determined by the maximal lattice L V , H . Set V holII , (2 + II ) = V holE , B , ⊗ V II , , and set II , (2 + II ) = II , ⊕ √ E D . We prove the following characterization result for V holII , (2 + II ) . Theorem 5.8.
Let ( V , ω, H ) satisfy the following conditions:(1) ( V , ω, H ) satisfy Assumption (A);(2) ( V , ω ) is a c = holomorphic conformal vertex algebra;(3) The maximal lattice of ( V , H ) is isomorphic to II , (2 + II ) .Then, ( V , ω, H ) is isomorphic to V holII , (2 + II ) .Proof. Consider an isomorphism of lattices II , (2 + II ) and √ E D ⊕ II , . Similarlyto the proof of Proposition 4.1, let C be the coset vertex algebra correspondence to thisdecomposition. The vertex algebra C is a c =
24 holomorphic VOA and its maximallattice is √ E D . According to Theorem 5.6 and Lemma 5.13 and Theorem 5.7, C isisomorphic to V holE , B , as a VOA and also as a VH pair, by Lemma 5.8. Hence, V (cid:27) V holE , B , ⊗ V II , as a conformal vertex algebra and VH pair. (cid:3) Lemma 5.14. G V holII , + II = Aut II , (2 + II ) and genus ( V holE , B , ) = holds.Proof. According to Lemma 4.7 and the proof of Theorem 5.8, it su ffi ces to show that G V holE , B , ⊂ Aut ( √ E D ). Clearly, Aut ( √ E D ) is a semidirect product of Weyl group W E × W D and the order 2 Dynkin diagram automorphism of D , which is isomorphic tothe Weyl group W E × W B . According to Lemma 5.5, G V holE , B , contains it. Hence, theassertion holds. (cid:3) Let V be a c =
24 holomorphic VOA such that the maximal lattice is contained in II , (2 + II ). Since the maximal lattice of V ⊗ V II , is II , (2 + II ), according to Theorem5.8, we have V holII , (2 + II (cid:27) V ⊗ V II , . Hence, V is contained in genus ( V holE , B , ). Accordingto Lemma 5.14, we have: Proposition 5.7.
Let V be a c = holomorphic VOA. If the maximal lattice of V is con-tained in II , (2 + II ) , then V is uniquely determined as a VOA and G V ,ω = Aut L V . In par-ticular, there is a one-to-one correspondence between genus ( V holE , B , ) and genus ( √ E D ) . Remark 5.4.
The one-to-one correspondence between genus ( V holE , B , ) and genus ( √ E D ) first appeared in [HS1] , as it was pointed out in [HS2] . Our approach is motivated by theirresults.
6. A ppendix
In this appendix, we will prove Proposition 3.4. Let V be a vertex algebra and set R V = ker T . For x , y ∈ R V define the product on R V by x · y = x ( − y . It follows easily from the definition of a vertex algebra that R V is a unital commutativeassociative C -algebra. In this appendix, we will prove Proposition 3.4.The following lemma is an easy consequence of the Borcherds identity: Lemma 6.1.
For x ∈ R V , x ( n ) = for any − , n ∈ Z . Let v , w ∈ V and x ∈ R V . According to the Borcherds identity and the above lemma,( x ( − v )( n ) w = P i ≥ x ( − − i ) v ( n + i ) w + v ( n − − i ) x ( i ) w = x ( − v ( n ) w ). Define theaction of R V on V by x · v = x ( − v for x ∈ R V and v ∈ V . Then, the action is associative,that is, V is an R V -module. Hence, we have the following lemma: Lemma 6.2.
The map Y : V → End V [[ z ± ]] is R V -linear, that is, Y ( x ( − a , z ) = x ( − Y ( a , z ) for any x ∈ R V and a ∈ V.proof of Proposition 3.4.
Let x be a non-zero vector of R V , v ∈ V and k ∈ Z . Then, for k ≥ v ( k ) x = v ( − k − x = T k vk ! ( − x . Since V is simple, V is spanned by the vectors { v ( k ) x | v ∈ V and k ∈ Z } . Thus, by theabove equation, there exists y ∈ V such that y ( − x = . Since 0 = T = T ( y ( − x ) = T ( x ( − y ) = x ( − T y and y ( − x ( − T y = ( y ( − x )( − T y = ( − T y = T y , we have y ∈ ker T . Hence, x is invertible in R V , which implies that R V is a field. Finally, supposethat R V , C . Since R V is field, C ( t ), a function field of one variable over C is a subalgebraof R V , which has uncountable dimension over C . (cid:3) A cknowledgements The author would like to express his gratitude to Professor Atsushi Matsuo, for hisencouragement throughout this work and numerous advices to improve this paper. Heis also grateful to Hiroki Shimakura and Shigenori Nakatsuka for careful reading of thismanuscript and their valuable comments. 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