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VH pairs (V, HV ) and (W, HW ) are in the same genus, then the module categories V-mod and W-mod are equivalent (Theorem 4.5). Hence, if V has a good representation theory (e.g., completely reducible), then W also has a good representation theory. This result suggests that the genus of VH pairs can be used to construct good vertex algebras from a good vertex algebra. The tensor product functor V is first considered by R. Borcherds and is an im- − ⊗ II1,1 portant step in his proof of the monstrous moonshine conjecture (see [Bo2]). It is also used by G. H¨ohn and N. Scheithauer in their constructions of holomorphic VOAs [HS1]. In the process of the constructions, they show that some non-isomorphic 17 holomorphic VOAs of central charge 24 are isomorphic to each other after taking the tensor product V , that is, in the same genus in our sense [HS1]. They also study other examples − ⊗ II1,1 of holomorphic VOAs 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 algebras based 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 def- inition, whereas they are not so in our sense. Following the pattern suggested by the equivalence of categories in Theorem 4.5, we believe that if VOAs are in the same genus in our sense, 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 even lattices, L V , is not a functor from the category of lattices to the category of ver- 7→ L tex algebras (see Remark 2.13). We focus on a twisted group algebra C L considered { } in [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 an AH pair (A, H), which is a pair of an associative algebra A satisfying some properties and 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, : AH pair VH pair, (A, H) (V , H). In the special case where an AH pair V → 7→ A,H is the twisted group algebra (C L , C Z L) constructed from an even lattice L, its image { } ⊗ by this functor coincides with the lattice vertex algebra VL. We call a class of AH pairs 3 which axiomatizes twisted group algebras a lattice pair, and the vertex algebra associ- ated with a lattice pair a H-lattice vertex algebra. Lattice pairs are classified by using the cohomology of an abelian group. In Section 3, we construct a right adjoint functor of , denoted by Ω : VH pair V → AH pair, (V, H) (Ω , H) and, in particular, construct an associative algebra from a 7→ V,H vertex algebra (Theorem 3.10). In the case that a VH pair (V, H) is a simple as a vertex algebra, the structure of the AH pair (ΩV,H, H) is studied by using the results in the previ- ous section. We prove that there exists a maximal H-lattice vertex subalgebra (Theorem 3.21), which implies we can classify H-lattice vertex subalgebras in a VH pair (V, H). This maximal algebra is described by a lattice L H, called a maximal lattice of the V,H ⊂ good VH pair (V, H). In Section 4, we define the notion of a genus of VH pairs. We remark that the category of VH pairs (resp. AH pairs) naturally admits a monoidal category structure and any lattice vertex algebra is naturally a VH pair. Two VH pairs (V, HV ) and (W, HW ) are said tobe inthe samegenus ifthe VH pairs(V V , H H ) and (W V , H H ) are ⊗ II1,1 V ⊕ II1,1 ⊗ II1,1 W ⊕ II1,1 isomorphic. Roughly speaking, classifying VH pairs in the genus of (V, HV ) is equivalent to classifying subalgebras in V V which are isomorphic to V and compatible with ⊗ II1,1 II1,1 the VH pair structure; We prove that such subalgebras are completely described by the maximal lattice LV VII HV HII1,1 . In this way the classification of genera of VH pairs ⊗ 1,1 ⊂ ⊕ is reduced to the study of maximal lattices. For a VH pair (V, HV ), a subgroup of the automorphism group of the lattice, G Aut L , is defined from the automorphism V,HV ⊂ V,HV group of a vertex algebra. The difference between a genus of lattices and a genus of

VH pairs is measured by this subgroup GV,HV . The mass of the VH pairs in the genus 1 which contains (V, HV) is defined by mass(V, HV ) = , where the sum is (W,HW ) GW,H | W | over all the VH pairs (W, HW ) in the genus, which isP finite if LV,HV is positive-definite and the index of groups [Aut LV,HV II1,1 : GV VII ,Hv HII ] is finite. In this case, we ⊕ ⊗ 1,1 ⊕ 1,1 prove that the ratio of the masses mass(V, HV )/mass(LV,HV ) is equal to the index of groups

[Aut LV,HV II1,1 : GV VII ,Hv HII ] (Theorem 4.16). ⊕ ⊗ 1,1 ⊕ 1,1 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 the = α Heisenberg Lie algebra, V α H,n Z Vn . We assume here technical conditions, e.g., α = (α,α) L ∈ ∈ Vn 0 if n < 2 (see Definition 5.5), which are satisfied in interesting cases, e.g., extensions of rational affine vertex algebras. Those conformal vertex algebras are closed under the tensor product, which means that it is possible to study their genus. The main results of this section are (Theorem 5.7, Theorem 5.8)

α = (α,α) (1) dim Vn 0, 1 if n 2 > n 1; ≥ − α (2) The maximal lattice LV,H is equal to α H V (α,α) , 0 ; { ∈ | 2 } α , α α ffi (3) If V1 0 for (α, α) > 0, then V1 and V1− generate the simple a ne vertex algebra L (k, 0) with k = 2 Z . sl2 (α,α) ∈ >0 4

The results suggest that a vector in Vα with n (α,α) > n 1 has interesting structure, n ≥ 2 − which is beyond the scope of this paper. As an application, we study the genus of some holomorphic VOAs. A holomorphic VOA (or more generally a holomorphic conformal vertex algebra) is a VOA such that any module is completely reducible and it has a unique irreducible module. The holomorphic VOA Vhol constructed in [LS1] is uniquely char- E8,2 B8,1 acterized as a holomorphic VOA of central charge 24 whose Lie algebra, the degree one subspace of the VOA, is the semisimple Lie algebra g g [LS2]. Let H be a Cartan E8 ⊕ B8 subalgebra of the degree one subspace. Then, the maximal lattice LVhol ,H is √2E8 D8, E8,2B8,1 ⊕ 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 a holomorphic conformal vertex algebra of central charge 26 satisfies our condition and its hol maximal lattice is √2E8 D8 II1,1, then it is isomorphic to V VII (Theorem ⊕ ⊕ E8,2 B8,1 ⊗ 1,1 5.38).

Contents Introduction 1 1. Preliminaries 4 1.1. Vertex algebras and their modules 5 1.2. Lattice 6 2. AH pairs and vertex algebras 6 2.1. Twisted group algebras 7 2.2. AHpairs 9 2.3. Functor 10 V 3. VH pairs and Associative Algebras 12 3.1. The functor Ω 12 3.2. Adjoint functor 15 3.3. Good VH pairs and Maximal lattices 17 4. GeneraofVHpairsandMassFormula 18 4.1. Coset constructions 19 4.2. Mass formulae for VH pairs 22 5. VH pair with coformal vector 24 5.1. Structure of (V,ω, H) 27 5.2. Application to extensions of affineVOAs 30 Acknowledgements 35 References 35

1. Preliminaries We denote the sets of all integers, positive integers, real numbers and complex numbers by Z, Z>0, R and C respectively. This section provides definitions and notations we need 5 for what follows. Most of the contents in subsection 1.1 and 1.2 are taken from the literature [Bo2, LL, Li1, FHL, FLM, CS]. Throughout this paper, we will work over the field C of complex numbers.

1.1. Vertex algebras and their modules. Following [Bo2], a vertex algebra is a C- vector space V equipped with a linear map

n 1 Y( , z): V End(V)[[z±]], a Y(a, z) = a(n)z− − − → 7→ Xn Z ∈ and a non-zero vector 1 V satisfying the following axioms: ∈ V1) For any a, b V, there exists N Z such that a(n)b = 0 for any n N; ∈ ∈ ≥ V2) For any a V, ∈ 0, (n 0), a(n)1 =  ≥  a, (n = 1),  −  holds;  V3) Borcherds identity: For any a, b V and p, q, r Z, ∈ ∈ ∞ p ∞ r (a(r + i)b)(p + q i) = ( 1)i a(p + r i)b(q + i) ( 1)rb(q + r i)a(p + i) i ! − − i! − − − − Xi=0 Xi=0   holds. For a, b V, a(n)b is called the n-th product of a and b. Let T (or simply T) denote ∈ V the endomorphism of V defined by T a = a( 2)1 for a V. V − ∈ The following properties follow directly from the axioms of a vertex algebra (see for example [LL]):

(1) Y(1, z) = idV ; (2) For any a, b V and n Z, T1 = 0 and ∈ ∈ (Ta)(n) = na(n 1), − − T(a(n)b) = (Ta)(n)b + a(n)Tb;

(3) Skew-symmetry: For any a, b V and n Z, ∈ ∈ T i a(n)b = ( 1)n+1+i (b(n + i)a); − i! Xi 0 ≥ (4) Associativity: For any a, b V and n, m Z, ∈ ∈ n (a(n)b)(m) = ( 1)i(a(n i)b(m + i) ( 1)nb(n + m i)a(i)); i! − − − − − Xi 0 ≥ (5) Commutativity: For any a, b V and n, m Z, ∈ ∈ n [a(n), b(m)] = (a(i)b)(n + m i). i! − Xi 0 ≥ We denote by V-mod the category of V-modules. 6

1.2. Lattice. A lattice of rank n N is a rank n free abelian group L equipped with a ∈ Z-valued symmetric bilinear form

( , ): L L Q. × → A lattice L is said to be even if

(α, α) 2Z for any α L, ∈ ∈ integral if (α, β) Z for any α, β L, ∈ ∈ and positive-definite if (α, α) > 0 for any α L 0 . ∈ \{ } 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 R is ⊗ ⊗ non-degenerate. The dual of L is the set

L∨ = α L Z R (α, L) Z . { ∈ ⊗ | ⊂ }

The integral lattice L is said to be unimodular if L = L∨. Two integral lattices L and M are said to be equivalent or in the same genus if their base changes are isomorphic as lattices:

L Z R M Z R, L Z Z M Z Z , ⊗ ≃ ⊗ ⊗ p ≃ ⊗ p for all the prime integers p, where Zp 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 genus mass(L) Q is defined by ∈ 1 (1) mass(L) = , Aut L L genus(X L) ′ ′∈ | | where Aut L is the automorphism group of L. The Smith-Minkowski-Siegel mass formula is a formula which computes mass(L) (see [CS, Ki]). Lattices over R are completely determined by the signature. Similarly, lattices over Zp 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 √ +10 2E8D8 is denoted by II16,0(2II ) (see [CS] for the precise definition).

2. AH pairs and vertex algebras In this section, we generalize the famous construction of vertex algebras from even lattices. In Section 2.1, we recall some fundamental results from the cohomology of abelian groups. Much of the material in this subsection is based on [FLM, Chapter 5]. We focus on a twisted group algebras considered in [FLM] and generalize their result. In Section 2.2, we introduce the concept of an AH pair which is a generalization of a twisted group 7 algebra, while in Section 2.3, we construct a vertex algebra associated with an AH pair generalizing the construction of lattice vertex algebras.

2.1. Twisted group algebras. Let A be an abelian group. A (normalized) 2-cocycle of

A with coefficients in C× is a map f : A A C× such that: × → (1) f (0, a) = f (a, 0) = 1 (2) f (a, b) f (a + b, c) = f (a, b + c) f (b, c). 2 We denote by Z (A, C×) the set of the 2-cocyles of A with coefficients in C×. For a map h : A C×, the map → 1 1 dh : A A C×, (a, b) h(a + b)h(a)− h(b)− × → 7→ 2 is a 2-cocycle, called a coboundary. The subset of Z (A, C×) consisted of coboundaries is 2 2 2 2 2 denoted by B (A, C×), which isa subgroupof 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, 0) = 1 for any a A; ∈ (2) c(a + b, c) = c(a, c)c(b, c) and c(a, b + c) = c(a, b)c(a, c) for any a, b, c A; ∈ (3) c(a, a) = 1 for any a A. ∈ The set of alternative bilinear forms on A is denoted by Alt2(A). For a 2-cocycle f 2 1 ∈ Z (A, C×), define the map c : A A C× by setting c (a, b) = f (a, b) f (b, a)− . It f × → f is easy to prove that the map c f is an alternative bilinear form on A, which is called a 2 2 commutator map [FLM]. Hence, we have a map Z (A, C×) Alt (A), which induces 2 2 → c : H (A, C×) Alt (A). → 2 2 Lemma 2.1. The map c : H (A, C×) Alt (A) is injective. → 2 Proof. For any 2-cocycle f Z (A, C×), there exists a central extension of A by C×, ∈

1 C× A˜ A 1 → → → → and 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 sequence in the category of abelian groups. Since C× is injective, the exact sequence splits. 

According to [FLM], we have the following:

Proposition 2.2 ([FLM, Proposition 5.2.3]). If A is a free abelian group of finite rank, 2 2 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 a A Ra satisfying the following conditions: ∈ (1) 1 RL; ∈ 0 (2) dimC Ra = 1; (3) e e , 0 for any e R 0 and e R 0 . a b a ∈ a \{ } b ∈ b \{ } 8

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 ) = S for any a A. → a a ∈ Let R be a twisted group algebra of A. Let us choose a nonzero element ea of Ra for all a A. Then, define the map f : A A C× by eaeb = f (a, b)ea+b. Clearly, the co- ∈ 2 × → homology class f H (A, C×) is independent of the choice of ea Ra a A. Furthermore, ∈ { ∈ } ∈ the cohomology class of f uniquely determines the isomorphism class. The alternative bilinear form c Alt2(A) is called an associated commutator map of the twisted group f ∈ algebra, denoted by cR. Hence, we have:

Lemma 2.3. There exists a bijection between the isomorphism classes of twisted group 2 algebras of an abelian group A and H (A, C×).

(α,β) Let L be an even lattice. Define the alternative bilinear form on L by cL(α, β) = ( 1) 2 − for α, β L. By Proposition 2.2, there exists a 2-cocycle f Z (L, C×) such that the ∈ ∈ associated commutator map c f is coincides with cL. The corresponding twisted group algebra is denoted by C L , which plays an important role in the construction of the lattice { } vertex algebra attached to L. Hereafter, we generalize this construction to some abelian group L, called an even H- lattice, which is not always free. Let H be a finite-dimensional vector space over C equipped with a non-degenerate symmetric bilinear form ( , ). An even H-lattice is a − − subgroup L H such that (α, β) Z and (α, α) 2Z for any α, β L, denoted by (L, H). ⊂ ∈ ∈ ∈ Define an alternative bilinear form on L by

c (α, β) = ( 1)(α,β) L − for α, β L. ∈ Let (L, H) be an even H-lattice. Since the abelian group L is not always free, we cannot directly apply Proposition 2.2 to the alternative bilinear form cL. We can, however, prove 2 that there exists a 2-cocycle f Z (L, C×) such that the associated alternative bilinear L ∈ form c fL coincides with cL. We construct fL by pulling back a cocycle on a quotient lattice of L, where the kernel of the quotient map is the radical of the bilinear form on L.

Let E be the C-vector subspace of H spanned by L. Set E⊥ = v E (v, E) = 0 . { ∈ | } Let π : E E/E⊥ be the canonical projection and L¯ the image of L under π. Then, the → bilinear form on H induces a bilinear form on E/E⊥ and L¯ and the bilinear form on E/E⊥ is non-degenerate. Since L¯ is an even E/E⊥-lattice and spans E/E⊥, L¯ is a free abelian group of rank dimC E/E⊥. By Proposition 2.2, there exits a 2-cocycle g : L¯ L¯ C× 1 (¯a,b¯) × → such that g(¯a, b¯)g(b¯, a¯)− = ( 1) for anya ¯, b¯ L¯. Then, the 2-cocycle fL defined = − ∈ = by fL(a, b) g(π(a), π(b)) for a, b L satisfies c fL cL. The twisted group algebra 2∈ constructed by the 2-cocycle fL Z (L, C×) is denoted by AL,H, which is independent of ∈2 the choice of the 2-cocycle f Z (L, C×). Hence, we have: L ∈ Proposition 2.4. For even H-lattice (L, H), there exits a unique twisted group algebra

AL,H such that the associated commutator map is cL. 9

Remark 2.5. By the construction, if L is non-degenerate, then AL,H is isomorphic to the twisted group algebra C L . { } 2.2. AH pairs. Let H be a finite-dimensional vector space over C equipped with a non- degenerate symmetric bilinear form ( , ) and A an associative algebra over C with a − − = α non-zero unit element 1. Assume that A is graded by H as A α H A . We willsaythat sucha pair(A, H)isan AH pair if the followingL conditions∈ are satisfied: AH1) 1 A0 and AαAβ Aα+β for any α, β H ; ∈ ⊂ ∈ AH2) AαAβ , 0 implies (α, β) Z; ∈ AH3) For v Aα, w Aβ, vw = ( 1)(α,β)wv. ∈ ∈ − For AH pairs (A, HA) and (B, HB), a homomorphism of AH pairs is a pair ( f, f ′) of

maps f : A B and f ′ : H H such that f is an algebra homomorphism and f ′ an −→ A −→ B isometry such that f (Aα) B f ′(α) for all α H . We denote by AH pair the category of ⊂ ∈ A AH pairs. For an AH pair (A, H), we set M = α H Aα , 0 . A,H { ∈ | } A good AH pair is an AH pair (A, H) such that: GAH1) A0 is a one-dimensional vector space with the basis 1 ; { } GAH2) vw , 0 for any α, β M , v Aα 0 and w Aβ 0 . ∈ A,H ∈ \{ } ∈ \{ } By isolating the leading terms with respect to some linear order on H, we have:

Proposition 2.6. For a good AH pair (A, H), 0 is the only zero divisor in A.

Let (A, H) be a good AH pair. Then, (GAH1) and (GAH2) directly imply MA,H is a submonoid of H. Furthermore, (AH2) and (AH3) imply (α, β) Z, (α, α) 2Z for any ∈ ∈ α, β M . ∈ A,H A lattice pair is a good AH pair (A, H) such that: LP) M is an abelian group, that is, α M for any α M . A,H − ∈ A,H ∈ A,H Lemma 2.7. If (A, H) is a lattice pair, then A is a twisted group algebra of the abelian (α,β) group M whose commutator map is defined by M M C×, (α, β) ( 1) . A,H A,H × A,H → 7→ −

Proof. It suffices to show that dim Aα = 1 for any α MA,H. Let α MA,H. By (LP), α α ∈ ∈ α A− , 0. Let a, a′ be non-zero vectors in A and b a non-zero vector in A− . By Definition (GAH1) and (GAH2), ab A0 = C1 is non zero. We may assume that ab = 1. Then, ∈ α a′ = (a b) a′ = a (b a′) Ca. Hence, A = Ca.  · · · · ∈ By Lemma 2.1 and Lemma 2.3, Proposition 2.4, Lemma 2.7, we have the following classification result of lattice pairs:

Proposition 2.8. For any even H-lattice (L, H), there exists a unique lattice pair (AL,H, H) = such that MAL,H,H L. Furthermore, the lattice pairs (A, H) and (A′, H′) are isomorphic iff there exists an isometry f : H H′ such that f (M ) = M . → A,H A′,H′ 10

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 , set A = Aα. Then, we have: ⊂ A,H N α N L ∈ Lemma 2.9. For a submonoid N M , (A , H) is a subalgebra of (A, H) as an AH ⊂ A,H N pair.

Set

α α (2) L = α H A , A− , 0 = M ( M ). A,H { ∈ | } A,H ∩ − A,H lat lat We denote ALA,H by A . Then, (A , H) is a lattice pair, which is maximal in the following sense:

Proposition 2.10. Let (B, H ), (A, H ) be good AH pairs and ( f, f ′) : (B, H ) (A, H ) B A B → A an AH pair homomorphism. If (B, HB) is a lattice pair, then f is injective and f (B) lat ⊂ A . In particular, there is a natural bijection between Hom AH pair((B, HB), (A, HA)) and lat Hom AH pair((B, HB), (A , HA)).

Proof. Since f is an AH pair homomorphism, ker f is an HB-graded ideal. By the proof of Lemma 2.7, every nonzero element in Bα is invertible for any α M . Hence, ∈ B,HB α f ′(α) α f ′(α) ker f = 0. For α MB,H, we have 0 , f (B ) A and 0 , f (B− ) A− . Thus, ∈ lat ⊂ ⊂ f ′(α) L and f (B) A .  ∈ A,HA ⊂ 2.3. Functor . Let H be a vector subspace of a vertex algebra V. We will say that H is V a Heisenberg subspace of V if the following conditions are satisfied:

HS1) h(1)h′ C1 for any h, h′ H; ∈ ∈ HS2) h(n)h′ = 0 if n 2 for any h, h′ H; ≥ ∈ HS3) The bilinear form ( , ) on H defined by h(1)h′ = (h, h′)1 for h, h′ H is non- − − ∈ degenerate.

We will call such a pair (V, H) a VH pair. For VH pairs (V, HV ) and (W, HW ), a VH pair homomorphism from (V, H )to(W, H ) is a vertex algebra homomorphism f : V W V W → such that f (H ) = H . Since f (h(1)h′) = f (h)(1) f (h′) for any h, h′ H and the bilinear V W ∈ V form on H is non-degenerate, f : H H is an isometric isomorphism of vector V |HV V → W spaces. A vertex subalgebra W of a VH pair (V, H) is said to be a subVH pair if H is a subset of W. VH pair is a category whose objects are VH pairs and morphisms are VH pair homomorphisms.

Let (A, H) be an AH pair and consider the Heisenberg vertex algebra MH(0) associated with (H, ( , )). Let us identify the degree 1 subspace of M (0) with H. Then, the vertex − − H algebra M (0) is generated by H and the actions of h(n) for h H and n Z give rise to H ∈ ∈ an action of the Heisenberg Lie algebra H on MH(0). Consider the vector space b V = M (0) A = M (0) Aα. A,H H ⊗ H ⊗ Mα H ∈ 11

We let H act on this space by setting, for h H, v M (0) and a Aα, ∈ ∈ H ∈ b (h, α)v a, n = 0, h(n)(v a) =  ⊗ ⊗  (h(n)v) a, n , 0.  ⊗  α  Let α H and a A . Denote by la End A the left multiplication by a and define α ∈ ∈ α (α,β) ∈ β l z : A A[z±] by l z b = z ab for b A , where we used that ab , 0 implies a → a · ∈ (α, β) Z. Then, set ∈

α( n) n α(n) n α Y(1 a, z) = exp − z exp z− l z EndV [[z±]]. ⊗ n n ⊗ a ∈ A,H Xn 1  Xn 1  ≥ ≥ −

= n 1 The series Y(h 1, z) n Z h(n)z− − with h H and Y(1 a, z) with a A form a set ⊗ ∈ ∈ ⊗ ∈ of mutually local fields onP VA,H and generate a structure of a vertex algebra on it. Since

MH(0) is a vertex subalgebra of VA,H, (VA,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 is nothing else but the lattice vertex algebra V . ⊗ A,H L Proposition 2.11. The above construction gives a functor from the category of AH pairs to the category of VH pairs.

We denote by this functor, thus, (A, H)isaVHpair(V , H). V V A,H In order to prove the above proposition, we need the following result from [LL, Propo- sition 5.7.9]:

Proposition 2.12 ([LL]). Let f be a linear map from a vertex algebra V to a vertex algebra W such that f (1) = 1 and such that

f (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 2.11. Let (A, H ) and (B, H ) be AH pairs and ( f, f ′) : (A, H ) A B A → (B, HB) an AH pair homomorphism. It is clear that there is a unique C-linear map F : V V such that: A,HA → B,HB

(1) F(h(n) ) = f ′(h)(n)F( ) for any h H ; − − ∈ A (2) F(1 a) = 1 f (a) for any a A. ⊗ ⊗ ∈

It is easy to prove that the restriction of F gives an isomorphism from MHA (0) to MHB (0) and satisfies F(v a) = F(v) f (a) for any v M (0) and a A. Itsuffices to show that ⊗ ⊗ ∈ HA ∈ F is a vertex algebra homomorphism. We will apply Proposition 2.12 with S = H A. A ⊕ 12

Let a Aα and b Aβ for α, β H and v M (0). Since ∈ ∈ ∈ A ∈ HA F(Y(1 a, z)v b) ⊗ ⊗ (α,β) α(n) n α(n) n = z F(exp z− exp z− v ab) − n − n ⊗  nX1   Xn 1  ≤− ≥ ( f (α), f (β)) f ′(α)(n) n f ′(α)(n) n = z ′ ′ exp z− exp z− f (a)F(v b) − n − n ⊗  nX1   Xn 1  ≤− ≥ = Y(1 f (a), z)F(v b), ⊗ ⊗ F is a vertex algebra homomorphism. 

For an even H-lattice (L, H), the VH pair VAL,H,H is called H-lattice vertex algebra. We denote it by VL,H.

Remark 2.13. The construction of the lattice vertex algebra VL from an even lattice L does not form a functor, since there is no natural homomorphism from the automorphism group of the lattice L to the automorphism group of the twisted group algebra C L (The { } automorphism group of the twisted group algebra is an extension of the automorphism group of the lattice, but it is not necessarily split). This is one reason we introduce AH pairs.

3. VH pairs and Associative Algebras In the previous section, we constructed a functor : AH pair VH pair. In this V → section, we construct a right adjoint functor Ω : VH pair AH pair of (see Theorem → V 3.10). That is, we construct a ‘universal’ associative algebra from a VH pair. Section 3.1 is devoted to constructing the functor Ω, while in 3.2, we prove that Ω and form an V adjoint pair. In subsection 3.3, we combine the results in this section and previous section and classify H-lattice vertex subalgebras of a VH pair.

3.1. The functor Ω. Let (V, H) be a VH pair and ωH be the canonical conformal vector of M (0) given by the Sugawara construction, that is, ω = 1 h ( 1)hi, where h is H H 2 i i − { i}i a basis of H and hi is the dual basis of h with respect to theP bilinear form on H. For { }i { i}i α H, we let Ωα be the set of all vectors v V such that: ∈ V,H ∈ VS1) TV v = ωH(0)v. VS2) h(n)v = 0 for any h H and n 1. ∈ ≥ VS3) h(0)v = (h, α)v for any h H. ∈ Here, T is the canonical derivation of V defined by T v = v( 2)1. Set V V − Ω = Ωα V,H V,H. Mα H ∈ A vector of a Heisenberg module which satisfies the condition (VS2) and (VS3) is called an H-vacuum vector. 13

Remark 3.1. Since ω (0) T is a derivationon V, ker(ω (0) T ) is a vertex subalgebra H − V H − V of V. Furthermore, since H ker(ω (0) T ), (ker(ω (0) T ), H) is a VH pair and the ⊂ H − V H − V map (V, H) (ker(ω (0) T ), H) defines a functor from VH pair to itself. 7→ H − V Lemma 3.2. For v Ωα , ω (0)v = α( 1)v and ω (1)v = (α,α) v. Furthermore, Ω0 = ∈ V,H H − H 2 V,H ker TV .

Proof. Let h be a basis of H and hi the dual basis with respect to the bilinear form on { i}i { }i = 1 i = 1 i + i = H. Then, ωH(0)v 2 i(hi( 1)h )(0)v 2 i n 0(hi( 1 n)h (n) h ( n 1)hi(n))v − ≥ − − − − 1 (hi, α)h ( 1)v + (hP, α)hi( 1)v = α( 1)Pv. PSimilarly, ω (1)v = 1 (h ( 1)hi)(1)v = 2 i i − i − − H 2 i i − 1 i + + i = 1 i = (α,α) Ω0 2 Pi n 0(hi( 1 n)h (n 1) h ( n)hi(n))v 2 i(h , α)(hi, α)v P2 v. If v V,H, ≥ − − − ∈ thenP PTV v = ωH(0)v = 0. If v ker TV , then a(n)vP= 0 for any n 0 and a V, which ∈ ≥ ∈ implies v Ω0 . Thus, Ω0 = ker T .  ∈ V,H V,H V The following simple observation is fundamental:

Ωα Ωβ , + Lemma 3.3. Let v V,H and w V,H. If Y(v, z)w 0, then (α, β) Z and v( (α, β) ∈ ∈ ∈ − + i)w = 0 for any i 0 and 0 , v( (α, β) 1)w = ( 1)(α,β)w( (α, β) 1)v Ωα β. In ≥ − − − − − ∈ V,H particular, if (α, β) < Z, then Y(v, z)w = 0.

Proof. Let k be an integer such that v(k)w , 0 and v(k + i)w = 0 for any i > 0. We claim Ωα+β = n + = + that v(k)w V . Since [h(n), v(k)] i 0 i (h(i)v)(n k i) (h, α)v(n k), we ∈ ≥ − have h(n)v(k)w = 0 and h(0)v(k)w = (h, αP+ β)v(k)w for any h H and n 1. Further- ∈ ≥ more, ωH(0)v(k)w = (ωH(0)v)(k)w + v(k)ωH(0)w = (TV v)(k)w + v(k)TV w = TV (v(k)w), + + + which implies v(k)w Ωα β. By Lemma 3.2, ω (1)v(k)w = (α β,α β) v(k)w. Since ∈ V,H H 2 = + = 1 + + = ωH(1)v(k)w [ωH(1), v(k)]w v(k)ωH(1)w i 0 i (ωH(i)v)(k 1 i)w v(k)ωH(1)w ≥ − ((α, α)/2 + (β,β)/2)v(k)w + (T v)(k + 1)w = ((Pα, α)/2 + (β,β)/2 k 1)v(k)w, we have V − − k = (α, β) 1. Hence, (α, β) Z. By applying skew-symmetry, v( (α, β) 1)w = − − ∈ − − i+(α,β) i + = (α,β)  i 0( 1) TV /i!w( (α, β) 1 i)v ( 1) w( (α, β) 1)v. ≥ − − − − − − P We may equip ΩV,H with a structure of an associative algebra in the following way: For v Ωα and w Ωβ , define the product vw by ∈ V,H ∈ V,H v( (α, β) 1)w, if Y(v, z)w , 0, vw =  − −  0, otherwise.   In order to show that the product is associative, we need the following lemma.

Lemma 3.4. Let a, b, c V, p, q, r Z satisfy a(r+i)b = 0,b(q+i)c = 0 and a(p 1+i)c = ∈ ∈ − 0 for any i 1. Then, (a(r)b)(p + q)c = a(r + p)(b(q)c). ≥ + = p + + Proof. Applying the Borcherds identity, we have (a(r)b)(p q))c i 0 i (a(r i)b)(p ≥ = i r + + r + + P=  +  q i)c i 0( 1) i (a(r p i)b(q i)c ( 1) b(r q i)a(p i)c) a(r p)b(q)c. − ≥ − − − − − P   Lemma 3.5. (ΩV,H, H) is an AH pair. 14

Proof. Let a Ωα , b Ωβ and c Ωγ for α,β,γ H. By Lemma 3.3, it re- ∈ V,H ∈ V,H ∈ V,H ∈ mains to show that the product is associative, i.e., (ab)c = a(bc). Suppose that one of (α, β), (β,γ), (α, γ)is not integer. By Lemma 3.3, 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) 1)b)( (α + ∈ − − − β,γ) 1)c and a(bc) = a( (α, β + γ) 1)(b( (β,γ) 1)c). By applying Lemma 3.4 with − − − − − (p, q, r) = ( (α, γ), (β,γ) 1, (α, β) 1), we have (ab)c = a(bc).  − − − − − Proposition 3.6. The map Ω : VH pair AH pair, (V, H) (Ω , H) is a functor. → 7→ V,H Proof. Let (V, H ) and (W, H ) be VH pairs and f : (V, H ) (W, H ) a VH pair V W V → W homomorphism. Set f = f : H H . Let e Ωα . For h H , since ′ HV V W α V,HV W 1 | 1 → ∈ ∈ h(n) f (eα) = f ( f ′− (h))(n) f (eα) = f ( f ′− (h)(n)eα), we have h(n) f (eα) = 0 and h(0) f (eα) = 1 1 ( f ′− (h), α) f (e ) for any n 1. Since f ′ is an orthogonal transformation, ( f ′− (h), α) = α ≥ (h, f ′(α)). Since f (ω ) = ω , ω (0) f (e ) = f (ω (0)e ) = f (e ( 2)1) = f (e )( 2)1. HV HW HW α HV α α − α − Hence, f (Ωα ) Ω f ′(α). The rest is obvious.  V ⊂ W

We will often abbreviate the AH pair (and the functor) (ΩV,H, H) as Ω(V, H). Since Ω is a functor, we have:

Lemma 3.7. Let f Aut V suchthat f (H) = H. Then, f induces an automorphism of the ∈ AH pair Ω(V, H).

Let (V, H ) and (W, H ) be VH pairs. Then, V W is a vertex algebra, and H H is V W ⊗ V ⊕ W a Heisenberg subspace of V W. Hence, we have: ⊗ Lemma 3.8. The category of VH pairs and the category of AH pairs are strict symmetric monoidal categories.

It is easy to confirm that the functor : AH pair VH pair is a monoidal functor, V → that is, ((A, H ) (B, H )) is naturally isomorphic to (A, H ) (B, H ). However, V A ⊗ B V A ⊗V B Ω is not strict monoidal functor, since ker(ω (0) T) does not preserve the tensor product H − (see Remark 3.1).

Lemma 3.9. Let (V, H ) and (W, H ) be VH pairs. If ω (0) = T , then Ω(V W, H V W HW W ⊗ V ⊕ H ) is isomorphic to Ω(V, H ) Ω(W, H ) as an AH-pair. W V ⊗ W Ω Ω Ω Ω Proof. It is clear that V,HV W,HW V W,HV HW . Let i vi wi V W,HV HW . We ⊗ ⊂ ⊗ ⊕ ⊗ ∈ ⊗ ⊕ can assume that vi i and wi i are both linearly independent.P For h HV and n 1, = { } { } = ∈ = ≥ h(n) i vi wi i h(n)vi wi 0. Since wi are linearly independent, h(n)vi 0 for any ⊗ = ⊗ = + i. Similarly,P h′(n)Pwi 0 for any i and h′ HW . Since ωHV HW ωHV 1 1 ωHW , we = ∈ =⊕ ⊗ ⊗ = have ωHV (0) i vi wi (ωHV HW (0) ωHW (0)) i vi wi (TV W 1 TW ) i vi wi ⊗ ⊕ − = ⊗ ⊗ −Ω⊗ ⊗ (TV 1) i vPi wi, where we used ωHW (0) TPW and i vi wi V W,HV PHW . Hence, ⊗ ⊗ ⊗ ∈ ⊗ ⊕ ω (0)v P= T v , which implies that v Ω , w Ω P .  HV i V i i ∈ V,HV i ∈ W,HW 15

3.2. Adjoint functor. In this subsection, we will prove the following theorem:

Theorem 3.10. The functor Ω : VH pair AH pair is a right adjoint to the functor → : AH pair VH pair. V → The following observation is important:

Lemma 3.11. Let (A, H) be an AH pair. Then, the following conditions hold for the VH pair (VA,H, H): = (1) TVA,H ωH(0); (2) The AH pair Ω(VA,H, H) is isomorphic to (A, H); Ω = (3) The vertex algebra VA,H is generated by the subspaces H and VA,H ,H A as a vertex algebra. In particular, the composite functor Ω : AH pair AH pair is isomorphic to the ◦V → identity functor.

Hereafter, we will construct a natural transformation from Ω : VH pair VH pair V◦ → to the identity functor and prove the above Theorem. We start from the observations in

Lemma 3.11. Let (V, HV ) be a VH pair and (A, HA) an AH pair and f : (A, HA) = V → (V, HV ) a VH pair homomorphism. Then, by Lemma 3.11 and f (ωHA ) ωHV , the image of f satisfies the condition (VS1) and is generated by HV and the subset of HV -vacuum vectors as a vertex algebra. So, let W be the vertex subalgebra of V generated by HV and Ω V,HV . Then, there is a natural isomorphism Hom (( , H ), (V, H )) Hom (( , H ), (W, H )). VH pair VA,HA A V → VH pair VA,HA A V Hence, the proof of Theorem 3.10 is divided into two parts. First, we prove that the subalgebra W is isomorphic to VΩ ,H = Ω(V, HV ) as a VH pair, which gives us a V,HV V V◦ natural transformation Ω(V, H ) ֒ (V, H ). Second, we prove that there is a natural V◦ V → V isomorphism Hom ((A, H ), (B, H )) Hom ( (A, H ), (B, H )), AH pair A B → VH pair V A V B for AH pairs (A, HA) and (B, HB).

Let us prove the first step. Since MHV (0) is a subalgebra of W, W is an MHV (0)-module.

Lemma 3.12. The subspace W V is isomorphic to M (0) C Ω as an M (0)- ⊂ HV ⊗ V,HV HV module. Ω Proof. Let W′ be an MHV (0)-submodule of V generated by V,HV . Then, according to the representation theory of Heisenberg Lie algebras (see [FLM, Theorem 1.7.3]), W′ 

M (0) C Ω as an M (0)-module. Hence, it suffices to show that W′ is closed HV ⊗ V,HV HV under the products of the vertex algebra. Let v, w W′. We may assume that v = ∈ α β h1(i1) ... hk(ik)eα and w = h′ ( j1) ... h′( jl)eβ where hi, h′ HV and eα Ω , eβ Ω . 1 l j ∈ ∈ V,HV ∈ V,HV = i n + n + Since (h(n)v)(m)w i 0( 1) i h(n i)v(m i)w ( 1) v(n m i)h(i)w, we may assume ≥ α − − − − − that v = eα Ω .P Since eα(n)h(m)w = [eα(n), h(m)]w + h(m)eα(n)w = (h, α)eα(n + ∈ V,HV − 16

β m)w + h(m)eα(n)w, we may assume that w = eβ Ω . If Y(eα, z)eβ = 0, then there is ∈ V,HV nothing to prove. Assume that Y(eα, z)eβ , 0. Then, by Lemma 3.3, eα( (α, β) + k)eβ = 0 + − for any k 0 and e ( (α, β) 1)e Ωα β. We will show that e ( (α, β) k)e ≥ α − − β ∈ V α − − β ∈ W′ for k 2. By Lemma 3.2, ω (0)e = β( 1)e . Then, β( 1)e ( (α, β) k)e = ≥ HV β − β − α − − β [β( 1), e ( (α, β) k)]e +e ( (α, β) k)β( 1)e = (α, β)e ( (α, β) k 1)e +e ( (α, β) − α − − β α − − − β α − − − β α − − k)ω (0)e = ω (0)e ( (α, β) k)e ke ( (α, β) k 1)e . Since k 1, we have HV β H α − − β − α − − − β ≥ 1 (3) e ( (α, β) k 1)e = (ω (0) β( 1))e ( (α, β) k)e α − − − β k HV − − α − − β 1 = (ω (0) β( 1))ke ( (α, β) 1)e . k! HV − − α − − β

Since ω (0)W′ W′, we have e ( (α, β) k 1)e W′.  HV ⊂ α − − − β ∈ Remark 3.13. Equation (3) was proved by [Ro] for lattice vertex algebras.

Lemma 3.14. The subalgebra W is isomorphic to Ω(V, H ) as a VH pair. V◦ V

Proof. By Lemma 3.12, we have an isomorphism f : W VΩ ,H = MH (0) ΩV,H → V,HV V V ⊗ V as an M (0)-module. We will apply Proposition 2.12 with S = H Ω . Itsuffices to HV V ⊕ V,HV show that f (a(n)v) = f (a)(n) f (v) for any a Ω and v W and n Z, which can be ∈ V,HV ∈ ∈ proved by the similar arguments as Lemma 3.12. 

As discussed above, the subalgebra VΩ ,H V is universal as follows: V,HV V ⊂ Proposition 3.15. Let (A, H ) beanAH pairand f : (V , H ) (V, H ) be a VH pair A A,HA A → V homomorphism. Then, f (VA,H ) VΩ ,H . In particular, there is a natural bijection, A ⊂ V,HV V Hom (( (A, H ), (V, H ))  Hom ( (A, H ), Ω(V, H )). VH pair V A V VH pair V A V◦ V Let us prove the second claim. Let (B, H ) be an AH pair. Since and Ω are functors, B V we have natural maps

φ : Hom ((A, H ), (B, H )) Hom ( (A, H ), (B, H )) AH pair A B → VH pair V A V B and

ψ : Hom ( (A, H ), (B, H )) Hom (Ω (A, H ), Ω (B, H )). VH pair V A V B → AH pair ◦V A ◦V B By Lemma 3.11,

Hom (Ω (A, H ), Ω (B, H ))  Hom ((A, H ), (B, H )). AH pair ◦V A ◦V B AH pair A B It is easy to prove that ψ is an inverse of φ. Hence, we have:

Lemma 3.16. The map φ is a bijection.

Theorem 3.10 follows immediately from the combination of Proposition 3.15 and Lemma 3.16. 17

3.3. Good VH pairs and Maximal lattices. We will say that a VH pair (V, H)isa good VH pair if the AH pair Ω(V, H) is a good AH pair, i.e., the following conditions are satisfied: Ω0 = 1) V,H C1. 2) vw , 0 for all α, β H and v Ωα 0 and w Ωβ 0 . ∈ ∈ V,H \{ } ∈ V,H \{ } The following proposition follows from [DL, Proposition 11.9]:

Proposition 3.17. If V is a simple vertex algebra, then for any nonzero elements a, b V, ∈ Y(a, z)b , 0, i.e., there exists an integer n Z such that a(n)b , 0. ∈ By the above proposition and Lemma 3.3 and Lemma 3.2, we have the following lemma:

Lemma 3.18. For a VH pair (V, HV ), if V is simple and Ker TV = C1, then (V, HV) is a good VH pair.

The kernel of TV is called the center of V and it is well-known that Ker TV = C1 if V is simple and has countable dimension over C (see for example, [M, Section 6]). Hereafter, we will study good VH pairs.

Lemma 3.19. For an even H-lattice (L, H) and a good VH pair (V, H ), (V V , H H) V ⊗ L,H V ⊕ is a good VH pair. Furthermore, Ω(V V , H H) is isomorphic to Ω(V, H ) (A , H) ⊗ L,H V ⊕ V ⊗ L,H as an AH pair. = Proof. For an H-lattice vertex algebra VL,H, we have ωH(0) TVL,H . Hence, by Lemma 3.9, we have Ω  Ω Ω . Since V and V are good VH pairs, Ω0 = Ω0 V VL,H V VL,H L,H V VL,H V ⊗ ⊗+ + ⊗ ⊗ Ω0 = C1 1. Let v Ωα β and v Ωα′ β′ be nonzero elements, where α, α H VL,H V VL,H ′ V VL,H ′ V ⊗ ∈ ⊗ ∈ ⊗ β β ∈ and β,β′ H. Since VL,H is an H-lattice vertex algebra, dim Ω = dim A = 1 for any ∈ VL,H L,H = = Ωα Ωα′ β L. Hence, we may assume that v a b and v′ a′ b′, where a V and a′ V , ∈ β β′ ⊗ ⊗ ∈ ∈ b A , b′ A . Then, vw = aa′ bb′ , 0. Hence, (V V , H H) is a good VH ∈ L,H ∈ L,H ⊗ ⊗ L,H V ⊕ pair.  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 following proposition follows from the Theorem 3.10:

Corollary 3.20. The adjoint functors in Theorem 3.10 induce adjoint functors between good VH pair and good AH pair.

lat Let (V, HV ) be a good VH pair. Then, by Proposition 2.10, Ω(V, HV ) is a lattice pair.

Denote the even H-lattice (LΩ(V,HV ), HV ) by LV,HV , which we call a maximal lattice of the lat good VH pair (V, HV ) (see Definition 2). By Proposition 2.8, the AH pair Ω(V, HV ) is isomorphic to the AH pair (A , H ). By Lemma 3.14, the H-lattice vertex algebra LV,HV ,HV V lat VL ,H = (Ω(V, HV ) ) isa subVHpairof(V, HV ), which is a maximal H-lattice vertex V,HV V V algebra as follows: 18

Theorem 3.21. Let (V, H) be a good VH pair. Then, the subVH pair (Ω(V, H )lat) can V V be characterized by the following properties: (1) (Ω(V, H )lat) is an H-lattice vertex algebra. V V (2) For any even H-lattice (L, H) and any VH pair homomorphism f : (V , H) L,H → (V, H ),theimageof f isin (Ω(V, H )lat). V V V Furthermore, there is a natural bijection between Hom VH pair((VL,H, H), (V, HV )) and Hom ((A , H), (A )). AH pair L,H LV,HV ,HV Proof. By Theorem 3.10 and Proposition 2.10,

Hom VH pair((VL,H, H), (V, HV ))  Hom good AH pair((AL,H, H), Ω(V, HV )) lat  Hom good AH pair((AL,H, H), Ω(V, HV ) )  Hom ((V , H), (Ω(V, H )lat)). VH pair L,H V V 

By Theorem 3.21 and Lemma 3.19, we have:

Corollary 3.22. For a good VH pair (V, HV ) and an even H-lattice (L, H), the maximal lattice of the good VH pair (V V , H H) is (L L, H H). ⊗ L,H V ⊕ V,HV ⊕ V ⊕ We end this section by showing the existence of lattice vertex algebras in a good VH pair. Let (V, H) be a good VH pair and (LV,H, H) be the maximal lattice. Lemma 3.23. If a subgroup M L spans a non-degenerate subspace of H, then there ⊂ V,H exists a vertex subalgebra of V which is isomorphic to the lattice vertex algebra VM.

lat Proof. It is clear that M isafree abelian groupof finiterank. By Lemma2.9, (Ω(V, H) )M  (A ) is isomorphic to the twisted group algebra C M associated with the even lat- LV,H ,H M { } tice M. Set H′ = h H (h, α) = 0 for any α M . Then, the vertex algebra VC M ,H is { ∈ | ∈ } { } isomorphic to M (0) V as a VH pair. Thus, V is a subalgebra of V.  H′ ⊗ M M 4. Genera of VH pairs and Mass Formula In this section, we introduce the notionof a genus of VH pairs and prove a Mass formula (Theorem 4.16) which is an analogous result of that for lattices.

Recall that two lattices L1 and L2 are said to be equivalent or in the same genus if their base changes are isomorphic as lattices:

L Z R L Z R, L Z Z L Z Z , 1 ⊗ ≃ 2 ⊗ 1 ⊗ p ≃ 2 ⊗ p for all the prime integers p. Consider the unique even unimodular lattice II1,1 of signature (1, 1). The proof of the following lemma can be found in [KP]:

Lemma 4.1. The lattices L1 and L2 are in the same genus if and only if L II L II 1 ⊗ 1,1 ≃ 2 ⊗ 1,1 as lattices. 19

We will define an equivalence relation on VH pairs that is analogous to the equivalence relation on lattices given in this lemma. For the rest of this paper, we always assume VH pairs to be good and denote a VH pair by V instead of (V, HV ) and the associated AH pair Ω ΩHV by V instead of ( V , HV ) for simplicity. Note that tensor products of good VH pairs and H-lattice vertex algebras are again good VH pairs by Lemma 3.19.

Let VII1,1 denote the lattice vertex algebra associated with the lattice II1,1 and consider it as a VH pair (V , H ) with H = C Z II . Two (good) VH pairs V and W are said II1,1 II1,1 II1,1 ⊗ 1,1 to be equivalent if there exists an isomorphism f : V V W V ⊗ II1,1 ≃ ⊗ II1,1 of VH pairs, that is, a vertex algebra isomorphism f which satisfies = f (HV VII ) HV VII . ⊗ 1,1 ⊗ 1,1 We call an equivalence class of VH pairs, a genus of VH pairs, and denote by genus(V) the equivalence class of the VH pair V. Before going into details, we briefly describe how to use the methods developed in the previous section to study the genera. Assume that V V  W V as a VH pair, that is, ⊗ II1,1 ⊗ II1,1 theyare in the same genus of VH pairs. We observe that V is obtained as a coset of V V ⊗ II1,1  by VII1,1 , that is, V CosetV VII (VII1,1 ), and so is W (see Section 4.1). Furthermore, it is ⊗ 1,1 clear that the subVH pair (M (0) V , H H ) (V V , H H )isan H-lattice HV ⊗ II1,1 V ⊕ II1,1 ⊂ ⊗ II1,1 V ⊕ II1,1 vertex algebra, whose maximal lattice is (II , H H ). Conversely, in Section 4.1, we 1,1 V ⊕ II1,1 will show that for each subVH pair of V V which is isomorphic to the H-lattice vertex ⊗ II1,1 algebra M (0) V , we can obtain a VH pair W such that W V  V V by HV ⊗ II1,1 ⊗ II1,1 ⊗ II1,1 using the coset construction. Thus, in order to determine genus(V), it suffices to classify the subVH pairs which are isomorphic to the H-lattice vertex algebra M (0) V . HV ⊗ II1,1 This is possible because we can classify all H-lattice vertex subalgebras of a VH pair by Lemma 4.10. Section 4.1 is devoted to studying the coset constructions of vertex algebras and VH pairs. In section 4.2, a mass formula will be proved.

4.1. Coset constructions. Let V and W be vertex algebras. For each V-module M, the tensor product W M becomes a W V-module. We denote by T the functor which ⊗ ⊗ W assigns the W V-module W M to each V-module M: ⊗ ⊗ T : V-mod W V-mod, M W M. W → ⊗ 7→ ⊗ For a vertex algebra V˜ and a vertex subalgebra W of V˜ and a V˜ -module M˜ , the subspace

C ˜ (W) = v M˜ w(n)v = 0 for any w W and n 0 M { ∈ | ∈ ≥ } = is called a coset (or commutant). If M˜ V˜ , viewed as a V˜ -module, then CV˜ (W) is a ˜ ˜ ˜ vertex subalgebra of V. It is clear that CM˜ (W)isa CV˜ (W)-module for any V-module M.

Consider the functor RW which assigns the CV˜ (W)-module CM˜ (W) to each V˜ -module M˜ ,

R : V˜ -mod C ˜ (W)-mod, M˜ C ˜ (W). W → V 7→ M 20

In this subsection, we consider the composition of the coset functor RW and the tensor product functor T . We first consider the case of R T . If C (W) = C1, then R W W ◦ W W W ◦ TW (M) = CW M (W) = C1 M for any V-module M. Hence, we have: ⊗ ⊗ Lemma 4.2. If C (W) = C1, then the composite functor R T : V-mod V-mod is W W ◦ W → naturally isomorphic to the identity functor.

Remark 4.3. We note that CW (W) is the center of W and the assumption CW (W) = C1 is satisfied for almost all vertex algebras. In particular, if W is a conformal vertex algebra with the conformal vector ω, then TW = ω(0) and thus CW (W) = ker TW .

We now consider the case of TW RW . Let V˜ be a vertex algebra and W a vertex = ◦ subalgebra. Set W′ CV˜ (W). Since the vertex subalgebras W and W′ commute with each other (in V˜ ), by the universality of the tensor product in the category of vertex algebras, there is a vertex algebra homomorphism i : W W′ V˜ , (w, w′) w( 1)w′, which W ⊗ → 7→ − implies that M˜ is a W W′-module. Consider this pullback functor, ⊗ i ∗ : V˜ -module W W′-module, M˜ M˜ . W → ⊗ 7→ We let f ˜ denote the linear map W C ˜ (W) M˜ defined by setting f ˜ (w m) = M ⊗ M → M ⊗ w( 1)m for w W and m C ˜ (W). − ∈ ∈ M

Lemma 4.4. The family of linear maps f ˜ gives a natural transformation from T R M W ◦ W to the pullback functor iW ∗.

Proof. It suffices to show that f ˜ : W C ˜ (W) M˜ is a W W′-module homomorphism M ⊗ M → ⊗ (The naturality of f ˜ is clear). Let w , w W and w′ W′, m C ˜ (W) and k M 1 2 ∈ ∈ ∈ M ∈ Z. Itsuffices to show that f ˜ (w w′)(k)(w m) = (w w′)(k) f ˜ (w m). Since M 1 ⊗ 2 ⊗ 1 ⊗ M 2 ⊗ [w (p), w′( j)] = [w (p), w′( j)]= 0 and w (q)m = w (
q)m = 0 for any j, p Z and q 0, 1 2 1 2 ∈ ≥ we have f ˜ (w (i)w w′( j)m)) = (w (i)w )( 1)(w′( j)m) M 1 2 ⊗ 1 2 − i l i = ( 1) w1(i l)w2( 1 + l) ( 1) w2(i 1 l)w1(l) w′( j)m l! − − − − − − − Xl 0 ≥
= w (i)w′( j)w ( 1)m 1 2 − for any i, j Z. Hence, ∈ f ˜ (w w′)(k)(w m) = f ˜ (w (l)w w′(k l 1)m) M 1 ⊗ 2 ⊗ M 1 2 ⊗ − − Xl Z
∈ = w (l)w′(k l 1)w ( 1)m 1 − − 2 − Xl Z ∈ = ( w ( 1 l)w′(k + l) + w′(k l 1)w (l))w ( 1)m 1 − − − − 1 2 − Xl 0 ≥ = (w ( 1)w′)(k) f ˜ (w m) = (w w′)(k) f ˜ (w m). 1 − M 2 ⊗ 1 ⊗ M 2 ⊗  21

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 consider the functor RW and TW 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

CW (W) = C1. Since W is a holomorphic vertex algebra, as a W-module, M˜ is a direct sum of copies of W. Since CW (W) = C1, the natural transformation given in Lemma 4.4,

(4) f ˜ : W C ˜ (W) M˜ , (w, m) w( 1)m, M ⊗ M → 7→ − is an isomorphism. In particular, V˜ is isomorphic to W C ˜ (W) as a vertex algebra. ⊗ V Combining Lemma 4.2 and Lemma 4.4 and the above discussion, we have:

Theorem 4.5. Let V˜ be a vertex algebra and W be a vertex subalgebra of V such that

C (W) = C1. If W is holomorphic, then V˜ is isomorphic to W C ˜ (W) as a vertex W ⊗ V algebra and the functors TW and RW are mutually inverse equivalences between V˜ -mod and CV˜ (W)-mod.

Corollary 4.6. If W is holomorphic and CW (W) = C1, then a vertex algebra V is simple (resp. completely reducible, holomorphic) if and only if V W is simple (resp. completely ⊗ reducible, holomorphic).

Remark 4.7. Let M be an irreducible module of an associative C-algebra A and N an irreducible module of an associative C-algebra B. Then, it is well-known that M N is ⊗ an irreducible module of A B if both M and N are finite dimensional, which is not true ⊗ in general. For example, let A = B = M = N = C(x). Then, the claim fails, since A B is ⊗ not a field.

For our application, we are interested in a coset of a VH pair by a holomorphic lattice ˜ vertex algebra. Let (V, H) be a good VH pair and (LV˜ ,H, H) be a maximal lattice. Let M be a sublattice of LV˜ ,H such that M is unimodular. By Proposition 3.23, the lattice vertex algebra VM, which is holomorphic, is a vertex subalgebra of V˜ . Let H′ be the canonical

Heisenberg subspace of the lattice vertex algebra V and set H′′ = h H (h, h′) = M { ∈ | 0 for any h′ H′ . By Theorem 4.5, V˜  V R (V˜ ). Hence, we have: ∈ } M ⊗ VM

Lemma 4.8. R (V˜ ) = v V˜ h(n)v = 0 for any n 0 and h H′ . VM { ∈ | ≥ ∈ }

Since M is non-degenerate, H = H′ H′′ and H′′ R (V˜ ). In particular, (R (V˜ ), H′′) ⊕ ⊂ VM VM is canonically a VH pair. Hence, we have:

Lemma 4.9. If M is a unimodular sublattice of L ˜ , then (V˜ , H ) is isomorphic to (V V,H V M ⊗ R (V˜ ), H′ H′′) as a VH pair. VM ⊕

Let II1,1 be the unique even unimodular lattice of signature (1, 1); that is, II1,1 has a basis z, w such that (z, z) = (w, w) = 0 and (z, w) = 1. Then, we have: − 22

Lemma 4.10. For a good VH pair V,˜ there is a one-to-one correspondence between sub- ˜ lattices of LV˜ which are isomorphic to II1,1 and decompositions of V into tensor products V˜  V V′ as VH pairs. II1,1 ⊗ 4.2. Mass formulae for VH pairs. By Corollary 3.22, if V and W are in the same genus, then L II  L II , where L and L are the maximal lattices of V and W, V ⊕ 1,1 W ⊕ 1,1 V W respectively. By Lemma 4.1, LV and LW are in the same genus of lattices. Let f Aut V such that f (H ) = H . By Lemma 3.7, f induces an automorphism ∈ V V on Ω . Furthermore, f induces an automorphism on the maximal lattice L H . The V V ⊂ V image of such automorphisms forms a subgroup of Aut LV , denoted by GV . For a lattice L, set L˜ = L II ⊕ 1,1 and S (L˜) = M L˜ M is a rank 2 sublattice which is isomorphic to II . II1,1 { ⊂ | 1,1} The group Aut L˜ naturally acts on S (L˜). LetusdenotethesetoforbitsbyAut L˜ S (L˜). II1,1 \ II1,1 Then, there is a one-to-one correspondence between Aut L˜ S (L˜) and the isomorphism \ II1,1 classes in genus(L).

Since the maximal lattice of V˜ = V V is L = L II , the group G ˜ acts on ⊗ II1,1 V V ⊕ 1,1 V S (L ). By analogy with the case of lattices, we have: II1,1 V f

Propositionf 4.11. There is a one-to-one correspondence between G ˜ S (L ) and the V\ II1,1 V isomorphism classes in genus(V). f

Proof. Let S, S ′ S (L ). By Lemma 4.10, there exist the corresponding lattice vertex ∈ II1,1 V subalgebras V V˜ and V V˜ . Set C = C ˜ (V ) and C = C ˜ (V ). Then, V˜  V C S ⊂ fS ′ ⊂ S V S S ′ V S ′ S ⊗ S as a VH pair. Hence, C genus(V). S ∈ Suppose that there exists f G ˜ such that f (S ) = S ′. Then, f (V ) = V in V˜ . Thus, ∈ V S S ′ f induces an isomorphism C C . Thus, we have a map S → S ′ G ˜ S (L ) the VH pair isomorphism classes in genus(V) , S C . V\ II1,1 V →{ } 7→ S Let g : C Cf be an isomorphism of VH pairs. Take an isomorphism of VH pairs S → S ′ i : C C . Then, i g is an isomorphism V C V C as VH pairs. Since 0 S → S ′ 0 ⊗ S ⊗ S → S ′ ⊗ S ′ both V C and V C are canonically isomorphic to V˜ as VH pairs, the composite of S ⊗ S S ′ ⊗ S ′ them is an element of GV˜ that takes S to S ′. Hence, the map is injective. The surjectivity of it follows from Lemma 4.10. 

In general, GV˜ 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 a sublattice of L˜. Set

S ⊥ = α L˜ (α, β) = 0 for any β S , { ∈ | ∈ } which is a sublattice of L˜, and set

St ˜ (S ) = f Aut L˜ f (S ) = S . Aut L { ∈ | } 23

Lemma 4.12. LetV beagoodVHpairandL be the maximal lattice. Let L genus(L ) V 0 ∈ V and choose S S (L ) with S ⊥  L . Then, there is a one-to-one correspondence 0 ∈ II1,1 V 0 0 between the isomorphism classes of VH pairs in genus(V, H ) whose maximal lattice is f V isomorphic to L0 and the double coset G ˜ Aut(LV )/St (S 0). Furthermore, Aut LV = V \ Aut LV G ˜ if and only if G ˜ Aut(LV )/St (S 0) = 1 and St f(S 0) G ˜ . In this case, there V | V\ Aut LV | f Aut LV ⊂ V f exists a one-to-one correspondence between genus(L ) and genus(V, H ). In particular, f f V f V a VH pair in genus(V, HV ) is uniquely determined by its maximal lattice.

Proof. Set X = S S (L ) L  S ⊥ . Since Aut L acts on X transitively, X  { ∈ II1,1 V | 0 } V Aut LV /St (S 0)asaAut LV -set. Then, the number of the isomorphism class of VH pairs LV f f in genus(V) whose maximal lattice is isomorphic to L is equal to f f f 0

G ˜ X = G ˜ Aut(LV )/St (S 0) . | V \ | | V \ Aut LV | f If G ˜ X = 1, then G ˜ St (S 0) = Aut(LV ).f Thus, the assertion follows.  | V \ | V Aut LV f We need the following lemmas: f

Lemma 4.13. Let L0 genus(LV ) and S 0 S II (LV ) with S ⊥  L0. Then, St (S 0) is ∈ ∈ 1,1 0 Aut LV isomorphic to Aut L Aut II . 0 × 1,1 f f

Proof. Let f St (S 0). Since f (S 0) = S 0, we have f (S ⊥) S ⊥. Hence, f induces a ∈ Aut LV 0 ⊂ 0 lattice automorphism of S ⊥  L0. Thus, we have a group homomorphism St (S 0) f 0 Aut LV → Aut L Aut II , which is clearly an isomorphism.  0 × 1,1 f Lemma 4.14. Let S S (L ) and C genus(V) be the corresponding VH pair con- ∈ II1,1 V S ∈ structed in Proposition 4.11. Then, St (S ) G ˜ is isomorphic to GC Aut II1,1. f Aut LV ∩ V S × f Proof. Let VS be the lattice vertex algebra constructed in Lemma 4.10 associated with S and f St (S ) G ˜ . Then, by the definition of G ˜ , there exists f Aut V˜ such Aut LV V V ′ ∈ = ∩ ∈ that f ′(HV˜ ) fHV˜ and the induced automorphism of the maximal lattice LV from f ′ is equal to f . Then, f (V ) = V and f (C ) = f (C ˜ (V )) = C ˜ (V ) = C . Furthermore, ′ S S ′ S ′ V S V S S f f ′ induces VH pair automorphisms on VS and CS . Thus, f ′ induces automorphisms on the maximal lattices S and S ⊥, and the image in the lattice automorphism groups are in  GCS and GVS Aut II1,1, respectively. Hence, we have a group homomorphism FS :

St (S ) G ˜ GC Aut II1,1, which is clearly an isomorphism.  Aut LV ∩ V → S × f Suppose that LV is positive-definite and [Aut (LV ): GV˜ ] is finite. Then, GV and Aut LV are finite groups and G ˜ Aut(LV )/St (S 0) is finite for any L0 genus(LV ). We define | V \ Aut LV | f ∈ the mass of genus(V, H ) tobe V f f 1 mass(V, H ) = . V G W genus(X V) W ∈ | | By the following proposition, we can calculate the number of isomorphism classes of VH pairs in a genus with a fixed maximal lattice: 24

Proposition 4.15. Suppose that LV is positive-definite and [Aut (LV ): GV˜ ] is finite. Then, for L genus(L ), the following equation holds: 0 ∈ V f [Aut L : G ˜ ] 1 V V = . Aut L0 GW W genus(XV), LW L0 | f | ∈ | |

Proof. Take S S (L ) with S ⊥  L . Then, by the elementary group theory, 0 ∈ II1,1 V 0 0 1 [Aut (LV ): G ˜ ] =f [St (S 0):St (S 0) g− G ˜ g]. V Aut LV Aut LV ∩ V g G ˜ Aut (LXV )/St (S 0) f ∈ V \ Aut LV f f f g Let g Aut(L ) and C be the VH pair corresponding to gS S (L ). Then by ∈ V gS 0 0 ∈ II1,1 V Lemma 4.13 and Lemma 4.14, f f 1 1 1 [St (S 0):St (S 0) g− G ˜ g] = [gSt (S 0)g− : gSt (S 0)g− G ˜ ] Aut LV Aut LV ∩ V Aut LV Aut LV ∩ V f f = [St f(gS 0):St (gSf0) G ˜ ] Aut LV Aut LV ∩ V = [Aut (gSf 0)⊥ Aut IIf1,1 : GC Aut II1,1] × gS 0 × Aut L = | 0|. GC | gS 0 | Hence, by Lemma 4.12, we have

[Aut (LV ): GV˜ ] = 1 Aut L G 0 CgS 0 f g G ˜ Aut (LXV )/St (S 0) | | ∈ V \ Aut LV | | f g1 = . GW W genus(XV),LW L0 ∈ | | 

Summing up the equation in Proposition 4.15 over L genus(L ), we have: 0 ∈ V

Theorem 4.16. Let (V, HV ) be a good VH pair. If LV is positive-definite and [Aut (LV ) : G ˜ ] is finite, then V f = mass(V, HV ) mass(LV )[Aut LV : GV˜ ].

5. VH pair with coformalf 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 the ∈ following conditions: (1) There exists a scalar c C such that ∈ m3 m [L(m), L(n)] = (m n)L(m + n) + − δ + c − 12 m n,0 holds for any n, m Z, where L(n) = ω(n + 1); ∈ (2) L( 1) = T ; − V (3) L(0) is semisimple on V and all its eigenvalues are integers. 25

The scalar c of a conformal vertex algebra is called the central charge of the conformal vertex algebra. Let (V,ω) be a conformal vertex algebra and set V = v V L(0)v = nv n { ∈ | } for n Z. A conformal vertex algebra (V,ω) is called a vertex operator algebra of CFT ∈ type (hereafter called a VOA) if it satisfies the following conditions: VOA1) If n 1, then V = 0; ≤ − n VOA2) V0 = C1; VOA3) V is a finite dimensional vector space for any n 0. n ≥ In this paper, a conformal vertex algebra (resp. a VOA) of central charge l is called a c = l conformal vertex algebra (resp. a c = l VOA). = Let (V,ω) be a VOA. Set V∨ n 0 Vn∗, where Vn∗ is the dual vector space of Vn. For ≥ f V∨ and a V, set L ∈ ∈ L(1)z 2 L(0) 1 Y (a, z) f ( ) = f (Y(e ( z− ) a, z− ) ). V∨ − − − The following theorem was proved in [FHL, Theorem 5.2.1]:

Theorem 5.1. (V∨, YV∨ ) is a V-module.

The V-module V∨ in Theorem 5.1 is called a dual module of V. A VOA (V,ω) issaidto

be self-dual if V is isomorphic to V∨ as a V-module. The following theorem was proved in [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)V1 = 0.

If (V,ω) is self-dual, then there exists a non-degenerate bilinear form (, ): V V C × → such that L(1)z 2 L(0) 1 (Y(a, z)b, c) = (b, Y(e ( z− ) a, z− )c) − for any a, b, c V. A bilinear form on V with the above property is called an invariant ∈ bilinear 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 vertex algebra with a Heisenberg subspace H, which is an analogue of a self-dual VOA (of CFT type). Let (V,ω) be a conformal vertex algebra with a Heisenberg subspace H V. Assume ⊂ that L(n)h = 0 and L(0)h = h and h(0) is semisimple on V for any h H and n 1. Then, ∈ ≥ h(0) commutes with L(0). For α H, set ∈ Vα = v V h(0)v = (h, α)v for any h H n { ∈ n | ∈ } and M = α H Vα , 0 for some n Z V,H { ∈ | n ∈ } 26 and = α VH∨ (Vn )∗, n ZM,α MV,H ∈ ∈ which is a restricted dual space. We would like to define a V-module structure on VH∨ in analogy with Theorem 5.1. The difficulty in defining the dual module structure on a conformal vertex algebra lies in the fact that it does not always satisfy the following condition: L(1) is a locally nilpotent operator. • We assume that, for any α M , Vα = 0 for sufficiently small n. Then, since L(1) ∈ V,H n commutes with h(0) for any h H, L(1) is locally nilpotent. By using Vα(n)Vβ Vα+β, ∈ ⊂ we can define a V-module structure on VH∨ similarly to Theorem 5.1. Hence, we have: Lemma 5.3. If a conformal vertex algebra (V,ω) and a Heisenberg subspace H V ⊂ satisfy the above two assumptions, then VH∨ is a V-module by the formula for YV∨ given just before Theorem 5.1.

Remark 5.4. In [HLZ], Huang, Lepowsky and Zhang consider essentially the same con- dition for a module of a M¨obius vertex algebra graded by an abelian group.

Definition 5.5. Given a conformal vertex algebra (V,ω) and a Heisenberg subspace H ⊂ V, we say that the triple (V,ω, H) satisfies assumption (A) if the following conditions are satisfied: A1) L(n)h = 0 and L(0)h = h for any n 1 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 (α,α) α = A4) If 2 > n, then Vn 0; 0 = 0 = A5) V0 C1 and V1 H; α A6) Vn is a finite dimensional vector space;  A7) VH∨ V as a V-module. Remark 5.6. Let (V,ω, H) satisfy Assumption (A). By (A7), there exists a non-degenerate invariant bilinear form. Normalize the bilinear form so that (1, 1) = 1. Let h, h′ H. By − ∈ the invariance of the form, (h, h′) = ( 1)(1, h(1)h′). Hence, our choice of normalization − makes this bilinear form coincide with the form given in the definition of Heisenberg subspace.

In this section, we study a conformal vertex algebra and a Heisenberg subspace satis- fying Assumption (A), which is denoted by (V,ω, H). Assumption (A) is used in order to show the following Theorems, which are proved in Section 5.1:

Theorem 5.7. Set L = α H Vα , . Then, Ω lat = Vα . In V′ ,H α2/2 0 ( V,H) α L α2/2 { ∈ | } ∈ V′ ,H particular, LV′ ,H is equal to the maximal lattice LV,H of the VH pair (V, HL). 27 2 α , α α Theorem 5.8. Let α MV,H such that α > 0 and V1 0. Then, V1 and V1− generate ∈ α the simple affine vertex algebra L (k, 0) of level k = 2/(α, α) Z and dim V± = 1, sl2 ∈ >0 1 dim Vkα = 0 for any k < 1, 0, 1 . Furthermore, dim Vβ 0, 1 for any β in M with 1 {− } n ∈ { } V,H n (β,β) > n 1. ≥ 2 − Those theorems assert that the existence of non-zero vectors in Vα for α M and n ∈ V,H n Z with 1 > n α2/2 0 is very important. It determines the maximal lattice and the ∈ − ≥ existence of affine vertex subalgebras. Examples and applications of those results are studied in subsection 5.2; There, we compute the maximal lattice for a VOA which is an extension of an affine VOA at positive integer level, together with the mass of the genus for some c = 24 holomorphic VOAs.

5.1. Structure of (V,ω, H). Let (V,ω, H) satisfy Assumption (A). In this subsection, we study the structure of (V,ω, H).

Lemma 5.9. If (V,ω, H) satisfies Assumption (A), then V is a simple vertex algebra and ker T = C1. In particular, (V, H) is a good VH pair.

Proof. Let I be a non-zero ideal of V. Set Iα = I Vα. Then, I = Iα since I n ∩ n n Z,α MV,H n is a submodule for a diagonalizable action of an abelian Lie algebra.L Hence∈ ∈ there exists α M and n Z such that Iα , 0 and Iα = 0 for any k 1. Let v Iα be a non-zero ∈ V,H ∈ n n k ≥ ∈ n − α = vector. Since V is self-dual, there exists v′ Vn− such that (v, v′) 1. ∈ 0 Combining L(1)v = 0 and v(2n 1)v′ V = C1, (v, v′) = 1, we have 1 I. Hence, − ∈ 0 ∈ I = V. Since v(n)a = 0 for any a ker T and v V and n 0, ker T V0 = C1.  ∈ ∈ ≥ ⊂ 0 Corollary 5.10. For any α, β M , α + β, α M . ∈ V,H − ∈ V,H

Proof. Let α, β MV,H. Since the invariant bilinear form on V induces a non-degenerate α ∈α α β pairing, V V− C, we have α M . Let v V and w V be nonzero elements. n × n → − ∈ V,H ∈ ∈ Then, by Lemma 3.17, we have 0 , v(k)w Vα+β for some k Z. Thus, α+β M .  ∈ ∈ ∈ V,H Ω Ωlat By Remark 3.18 and Lemma 5.9, V,H is a good AH pair and V,H is a lattice pair. Set α L′ = α H V , 0 . Before proving Theorem 5.7, we show the following lemma: V,H { ∈ | α2/2 } α α Lemma 5.11. For any α L′ , Ω = V . ∈ V,H V,H α2/2 α Proof. Let α L′ and v V be a nonzero element. Then h(0)v = (α, h)v. By ∈ V,H ∈ α2/2 (A4) h(n)v = 0 and and L(n)v = 0 for any n 1 and h H. It suffices to show ≥ ∈ that ωH(0)v = v( 2)1. Since the invariant bilinear form on V induces a non-degenerate α − α α α2/2+1 pairing on V V− C, there exists v′ V− such that (v, v′) = ( 1) . By n × n → ∈ α2/2 − α2/2 the invariance of the form, we have 1 = (v, v′)( 1) = (1, v((α, α) 1)v′). Since 0 − − − 0 v((α, α) 1)v′ V = C1, we have v((α, α) 1)v′ = 1. Since v((α, α) 2)v′ V = H and − ∈ 0 − − ∈ 1 (h, v((α, α) 2)v′) = ( 1)(1, h(1)v((α, α) 2)v′ = ( 1)(h, α)(1, v((α, α) 1)v′) = (h, α), − − − − − we have v((α, α) 2)v′ = α. − 28

+ 0 = + 2α = Since v′((α, α) k)v V k 1 0 and v( (α, α) k)v V2(α,α) k 1 0 for any k 0, by ∈ − − − ∈ − − ≥ applying the Borcherds identity with (p, q, r) = ( (α, α), (α, α) 1, (α, α) 2), we have − − − (α, α) α( 1)v = − (v((α, α) 2 + i)v′)( 1 i)v − i ! − − − Xi 0 ≥ i (α, α) 2 = ( 1) − (v( 2 i)v′((α, α) 1 + i)v v′(2(α, α) 3 i)v( (α, α) + i))v − i ! − − − − − − − Xi 0 ≥ = v( 2)1. − Hence, we have Vα Ωα . By Lemma 2.7, dim Ωα = 1. Thus, we have Ωα = α2/2 ⊂ V,H V,H V,H Vα  α2/2. Ωα Ω α , proof of Theorem 5.7. Let α LV,H, that is, V,H, V−,H 0. By the above lemma, LV′ ,H ∈ α α ⊂ L . Thus, it suffices to show that α L′ . Let v Ω and v′ Ω− be nonzero V,H ∈ V,H ∈ V,H ∈ V,H = α elements. Then, v i vi where vi Vi . Since h(n)Vi Vi n and TVi Vi+1, we − α α ∈ α α⊂ α ⊂α have vi Ω . SinceP dim Ω = 1, we have Ω V and Ω− V− for some ∈ V,H V,H V,H ⊂ k V,H ⊂ k′ = Ω0 = k, k′ Z. By Lemma 3.3, we can assume that 1 v((α, α) 1)v′ V,H C1. Since ∈ 2 − 2 ∈ + = =  v((α, α) 1)v′ Vk k′ (α,α) and k, k′ α /2, we have k k′ α /2. Thus, α LV′ ,H. − ∈ − ≥ ∈ = We now show that (α, β) Z and (α, α) 2Z for any α, β LV′ ,H. Recall that MV,H α ∈ ∈ 2 ∈ α α H V , 0 for some n Z . Let α L such that α , 0. Then, V± generates { ∈ | n ∈ } ∈ V,H α2/2 the lattice vertex algebra VZα and V is a direct sum of irreducible VZα-modules. Hence, we have:

Corollary 5.12. Let α L such that α2 , 0. Then, (α, β) Z for any β M . ∈ V,H ∈ ∈ V,H

Set L∨ = α H (α, β) Z for any β L . V,H { ∈ | ∈ ∈ V,H}

Corollary 5.13. If L is non-degenerate, then M L∨ . V,H V,H ⊂ V,H We will use the following lemma:

Lemma 5.14. Suppose that LV,H is positive-definite and spans H. Then, V is a VOA (of

CFT type) and MV,H is a subgroup of LV∨,H.

Proof. Since L is non-degenerate and spans H, M L∨ span L , which im- V,H V,H ⊂ V,H ⊂ R V,H plies that M is positive-definite. If 0 , α M , Vα = 0 for any n 0. Hence, V,H ∈ V,H n ≤ V = V and V = C1. Since α M α2 < n is a finite set for any n > 0, dim V n 0 n 0 { ∈ V,H | } n is finite.L ≥  Now, we will show Theorem 5.8:

α α 2 proof of Theorem 5.8. Let 0 , e V and f V− such that (e, f ) = 1. Since α > 0, ∈ 1 ∈ 1 α = 0 = = = L(n)e V1 n 0 for any n 1. Since e(0) f V1 H and (h, e(0) f ) (h, α)(e, f ) ∈ − ≥ ∈ = 2α = (h, α), we have e(0) f α. Since V n 0 for any n 0, by skew-symmetry of vertex − ≥ algebras, we have e(0)e = e(0)e + Te(1)e + = e(0)e. Thus, e, 2 f, 2 α is a sl - − ··· − { α2 α2 } 2 triple under the 0-th product. Similarly, we have e(n)e = f (n) f = 0, h(n)e = h(n) f = 0 29 for any n 1. Since 1 = (e, f ) = (1, e(1) f ), we have e(1) f = 1. Thus, e, 2 f, α ≥ − { α2 } generates an affine vertex algebra W of level k = 2/(α, α). Since e( 1)n1 Vnα = 0 for − ∈ n 2 2 ffi any n α > n, W is isomorphic to the simple a ne vertex algebra Lsl2 (k, 0) and the level α α k is a positive integer [FZ, Theorem 3.1.2]. We will show that dim V− = 1. Let f ′ V− 1 ∈ 1 such that (e, f ′) = 0. Then, e(0) f ′ = 0 follows from the computation presented above. nα = ffi n = = Since V1± 0 for su ciently large n, f (0) f ′ 0. Hence, f ′ 0 follows from the α = α = representation theory of the Lie algebra sl2. Hence, dim V1− 1. Since dim V1± 1, for any k Z with k , 1, 0, dim Vkα = 0 follows from the representation theory of the Lie ∈ ± 1 algebra sl again. Finally, suppose Vβ , 0 for β in M with n (β,β) > n 1. Then, by 2 n V,H ≥ 2 − Lemma 5.17, V V satisfies Assumption (A). Let γ II such that (γ,γ) = n + 1. ⊗ II1,1 ∈ 1,1 2 − , β γ (β,γ) ((β,γ),(β,γ)) = (β,β) + (γ,γ) Since 0 Vn VII1,1 n+1 (V VII1,1 )1 and 1 2 2 2 > 0, by the above ⊗ − ⊂ ⊗ ≥ result, we have dim(V V )(β,γ) = 1. Thus, dim Vβ = 1.  ⊗ II1,1 1 n For α M with α2 > 0 and Vα , 0, define r : H H by ∈ V,H 1 α → (h, α) r (h) = h 2 α, α − α2 and set 2 2 r˜ = exp( f (0)) exp( e(0)) exp( f (0)), α α2 − α2 where e and f is the same in the proof of Theorem 5.8. The following Corollary follows from the representation theory of affine sl2 and Theorem 5.8:

Corollary 5.15. For α M with α2 > 0 and Vα , 0, r˜ is a vertex algebra automor- ∈ V,H 1 α phism of V satisfying the following conditions:

(1) r˜α(ω) = ω;

(2) r˜α(H) = H; (3) r˜ = r . α|H α In particular, the reflection rα is an automorphism of the maximal lattice LV,H.

Lemma 5.16. If α2 > 0 and Vα , 0, then 2 α L . 1 α2 ∈ V Proof. Let e Vα be a nonzero element and set k = 2/α2. By the representation theory of ∈ 1 the Lie algebra sl ,0 , e( 1)k1 Vkα. Since (kα)2/2 = k, we have kα L .  2 − ∈ k ∈ V We end this subsection 5.1 by mentioning a generalization of the theory of genera of VH 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 5.17. Let (V,ω , H ) and (W,ω , H ) satisfy Assumption (A). Then, (V W,ω + V V W W ⊗ V ω , H H ) satisfy Assumption (A). W V ⊕ W (α,α ) ′ = α α′ Proof. Let α HV and α′ HW . Then, (V W)n = + V W . Hence, ∈ ∈ ⊗ n k k′ k ⊗ k′ Definition (A1),..., (A6) is easy to verify. Since (V W)(α,α′L) is finite dimensional, (V ⊗ n ⊗ W)  V W  V W.  ∗HV HW H∗ V H∗ W ⊕ ⊗ ⊗ The following lemma is clear from the definition: 30

Proposition 5.18. Let L be a non-degenerate even lattice and VL the lattice conformal vertex algebra. Set H = L Z C (V ) . Then, (V ,ω , H ) satisfy Assumption (A). L ⊗ ⊂ L 1 L HL L Results in Section 4 are obtained as follows. Hereafter, we assume that any triple (V,ω, H) satisfy Assumption (A). By Lemma 5.17, the category of triples (V,ω, H) forms a symmetric monoidal category, where a morphism from (V,ωV , HV ) to (W,ωW, HW ) is a vertex algebra homomorphism f : V W such that f (ω ) = ω and f (H ) = H . → V W V W We also remark that Lemma 4.10 can be generalized to triples (V,ω, H). Hence, we can define a genus of (V,ω, H) similarly to Section 4. More precisely, triples (V,ωV, HV ) and (W,ωW , HW ) are in the same genus if there exists a vertex algebra isomorphism = = f : V VII1,1 W VII1,1 such that f (ωV VII ) ωW VII and f (HV VII ) HW VII . ⊗ → ⊗ ⊗ 1,1 ⊗ 1,1 ⊗ 1,1 ⊗ 1,1 Denote by genus(V,ωV , HV ) the genus of (V,ωV, HV ). Denote by G(V,ωV ,HV ) the image of automorphisms f Aut(V) such that f (ω ) = ω and f (H ) = H in Aut L . If L ∈ V V V V V,HV V is positive-definite and [Aut L : G ˜ ] is finite, then the mass of genus(V,ω , H ) is V (V,ω˜ ),HV˜ V V defined to be f 1 mass(V,ωV, HV ) = . G(V ,ω V ,HV ) (V ,ω ,H )Xgenus(V,ωV ,HV ) ′ ′ ′ ′ ′ ′V′ V′ ∈ | | Then, we have:

Theorem 5.19. mass(L )[Aut L : G ˜ ] = mass(V,ω , H ). V V (V,ω˜ ),HV˜ V V We remark on the genus off VOAs. Let V be a VOA. Assume that two Heisenberg subspaces H and H′ satisfy Assumption (A). Then, H and H′ are split Cartan subalge- bras of Lie algebra V . Hence, there exists a ,..., a V such that f (H) = H′, where 1 1 k ∈ 1 f = exp(a (0)) exp(a (0)) (see, for example, [Hu]). Since exp(a (0)) is a VOA auto- 1 ··· n i morphism of V, we have:

Lemma 5.20. All Heisenberg subspaces of a VOA which satisfies Assumption (A) are conjugate under the VOA automorphism group.

The lemma asserts that the genus of a VOA is independent of the choice of a Heisenberg subspace H. Similarly to the proof of Lemma 5.14 and by Theorem 4.5, we have:

Lemma 5.21. Let (V,ω, H) satisfy Assumption (A). Suppose that LV,H is positive-definite and spans H. Then, all conformal vertex algebras in genus(V,ω, HV ) are VOAs.

5.2. Application to extensions of affine VOAs. In this subsection, we prove that many important conformal vertex algebras satisfy Assumption (A) and study the maximal lat- tices for those vertex algebras. The following Proposition follows from Theorem 5.2:

Proposition 5.22. Let (V,ω) be a conformal vertex algebra. Then, (V,ω, 0) satisfies As- sumption (A) if and only if V is a simple self-dual VOA with V1 = 0. 31

In order to prove extensions of a simple affine VOAs at positive integer level satisfy Assumption (A), we recall some results on simple affine VOA at positive integer level. Let g be a simple Lie algebra and h a Cartan subalgebra and ∆ the root system of g. Let

Qg be the sublattice of the weight lattice spanned by long roots and α1,...,αl a set of { = l } simple roots. Let θ be the highest root and a1,..., al integers satisfying θ i=1 aiαi. Let , be an invariant bilinear form on g, which we normalize by θ,θ = 2. P h i h i The following lemma easily follows:

2 ∆ Lemma 5.23. The sublattice Qg h is generated by α2 α α . If g is An, Dn, En (resp. ⊂ { | ∈ } n Bn, Cn, F4, G2), then Qg is isomorphic to the root lattice (resp. Dn,A1,D4,A2).

Let Lg(k, 0) denote the simple affine VOA of level k C associated with g [FZ]. Then, ∈ the VOA Lg(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 θ,λ k. Then, the highest weight h i ≤ module Lg(k,λ) is an irreducible module of Lg(k, 0). Conversely, any irreducible module + of Lg(k, 0) is isomorphic to the module of this form. Let Pg,k be the set of all dominant α weights λ satisfying θ,λ k. For α h and n Z, set Lg(k,λ) = v Lg(k,λ) h(0)v = h i ≤ ∈ ∈ n { ∈ | = = α = h, α v and L(0)v nv for any h h . Then, Lg(k,λ) α h,n Z Lg(k,λ)n . Set Ig h i ∈ } ∈ ∈ 0 Λi ai = 1 and i 1,..., n , where Λi i 1,...,n isL the fundamental weights of { }∪{ | ∈ { }} { } ∈{ } g, α ,...,α . The following lemma follows from [DR]: { 1 l} + α Lemma 5.24. Let λ P . If Lg(k,λ) , 0, then n α, α /2k and the equality holds if ∈ g,k n ≥ h i and only if λ kIg and α λ + kQg. ∈ ∈

Remark 5.25. Non-zero weights in Ig are called cominimal and Lg(k, kλ) is known to be a simple current for λ Ig (see [Li2, Proposition 3.5]). In fact, Ig form a group by the ∈ fusion product. Let Rg be a root lattice of g and Rg∨ the dual lattice of the root lattice.

Then, there is a natural group isomorphism between Ig and Rg∨/Rg. By Lemma 5.24, we have:

Proposition 5.26. Let (V,ω) be a VOA. If Lg(k, 0) isasubVOAofV andV1 = Lg(k, 0)1 = g, then (V,ω, H) satisfy Assumption (A), where H is a Cartan subalgebra of g = V1.

Combining Lemma 5.24 with Theorem 5.7, we have:

Corollary 5.27. For k Z and simple Lie algebra g, the maximal lattice of the simple ∈ >0 affine VOA Lg(k, 0) is √kQg.

Remark 5.28. This corollary implies that a level one simple affine vertex algebra associ- ated with a simply laced 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 an affine VOA at positive integer level. Let (V,ω)beaVOAand gi be a finite dimensional semisimple Lie N algebra and k Z for i = 1,..., N. Set g = gi and let H be a Cartan subalgebra i ∈ >0 i=1 L 32

g N = g of . Assume that i=1 Lgi (ki, 0) is a subVOA of (V,ω) such that V1 . Then, it is clear that (V,ω, H) satisfyN Assumption (A). We will describe the maximal lattice of (V, H) using Lemma 5.24. We simply write L (~k, ~λ) for N L (k ,λ ), where ~λ = (λ ,...,λ ) P = N P+ . g i=1 gi i i 1 N i=1 gi,k n ∈ ~ N = ~ ~ ~λ L Since Lg(k, 0) is completely reducible, V ~ Lg(k, λ) , where n~λ Z 0 is the mul- λ P ∈ ≥ ~ ~ L ∈ N tiplicity of the module Lg(k, λ). Let QV be the sublattice of H generated by i=1 √kiQgi and all 1 ~λ satisfying the following conditions: L √k (1) (λ ,...,λ ) P; 1 N ∈ (2) n~λ > 0; (3) λ k Ig for any i = 1,..., N. i ∈ i i Then, we have:

Proposition 5.29. The maximal lattice LV,H is equal to QV . = N We will consider two examples whose genera are non-trivial. Set Ag,~k i=1 Igi , which is a group by Remark 5.25. Consider the quadratic form on Ag,~k defined byL

1 N A Q/Z, ~λ k (λ ,λ ), g,~k → 7→ 2 i i i Xi=1 for ~λ Ag . A subgroup N A is called isotropic if the square norm of any vector in N ∈ ,k ⊂ g,~k is zero. Let N be an isotropic subgroup of Ag,~k. Set

Lg(~k, N) = Lg(~k, k~λ). M~λ N ∈

Then, by [Ca, Theorem 3.2.14], Lg(~k, N) inherits a unique simple vertex operator alge- bra structure. The assumption of the evenness in the theorem follows from the fact that

Lg(~k, 0) is unitary [CKL]. ~ N Proposition 5.30. The maximal lattice of Lg(k, N) is generated by i=1 √kiQgi and N. L √ For example, the maximal lattice of LB12 (2, 0) is 2D12, where D12 is a root lattice √ = 10 2 of type D12. The genus( 2D12) II12,0(2−II 4−II ) (see Section 1.2) contains two isomor- √ √ phism classes of lattices, 2D12 and 2E8D4. Hence, thegenus of a VOAgenus(LB12 (2, 0)) contains at least two non-isomorphic VOAs. Hereafter, we give another class of examples for which we can completely determine the genera of VOAs. We first recall c = 24 holomorphic VOAs.

Remark 5.31. Our definition of “holomorphic” is slightly different from the usual one. The usual definition of a holomorphic VOA requires completely reducibility for admissible modules, rather than all weak modules. Our definition together with the assumption that a VOA is of CFT type corresponds to strongly holomorphic in the literature. 33

For a VOA V, the 0-th product gives a Lie algebra structure on V1. The rank of a finite dimensional Lie algebra is the dimension of a Cartan subalgebra of the Lie algebra. The following theorem was proved in [DM1, Sc, EMS, DM2]:

Theorem 5.32. Let V bea holomorphicc = 24 VOA. Then, the Lie algebra V1 is reductive, and exactly one of the following holds:

(1) V1 = 0;

(2) V1 is abelian of rank 24 and V is isomorphic to the lattice VOA of the Leech lattice;

(3) V1 is one of 69 semisimple Lie algebras listed in the table below.

Furthermore, if V1 , 0, then the vertex subalgebra generated by V1 is a subVOA of V.

rank Lie algebra 24 6 2 2 24 8 6 4 3 4 24 U(1) , (D4,1) , (A7,1) (D5,1) , (A1,1) , (A3,1) , (A4,1) , (A6,1) , (A8,1) , (D6,1) 4 2 2 2 (E6,1) , A11,1D7,1E6,1, (A12,1) , A15,1D9,1, D10,1(E7,1) , A17,1E7,1, (D12,1) , A24,1 3 4 2 3 12 (E8,1) , D16,1E8,1, (A5,1) D4,1, (A9,1) D6,1, (D8,1) , D24,1, A2,1 4 4 2 4 2 2 2 2 2 16 (A3,2) (A1,1) , (D4,2) (C2,1) , (A5,2) C2,1(A2,1) , (D5,2) (A3,1) , A7,2(C3,1) A3,1 4 2 (C4,1) , D6,2C4,1(B3,1) , A9,2A4,1B3,1, E6,2C5,1A5,1, C10,1B6,1, E8,2B8,1 2 2 2 16 D8,2(B4,1) , (C6,1) B4,1, D9,2A7,1, C8,1(F4,1) , E7,2B5,1F4,1, A1,2 12 2 3 4 6 2 4 12 (A1,4) , B12,2, (B6,2) , B4,2, (B3,2) , (B2,2) , A8,2F4,2, C4,2(A4,2) , D4,4(A2,2) 6 3 2 3 12 (A2,3) , E7,3A5,1, A5,3D4,3(A1,1) , A8,3(A2,1) , E6,3(G2,1) , D7,3A3,1G2,1 3 2 3 10 A7,4(A1,1) , D5,4C3,2(A1,1) , E6,4C2,1A2,1, C7,2A3,1, A3,4A1,2 2 2 8 A4,5, D6,5(A1,1) 8 A5,6C2,3A1,2, C5,3G2,2A1,1

6 A6,7

6 F4,6A2,2, D4,12A2,6

6 D5,8A1,2

4 C4,10

hol For a Lie algebra g listed above, let Vg denote a holomorphic VOA whose Lie algebra is g. Recently, the existence and uniqueness of those holomorphic VOAs are proved except for the case that V1 = 0. In this paper, we will use the following result:

Theorem 5.33 ([LS2, LS1]). There exists a unique holomorphic VOA of central charge

24 whose Lie algebra is E8,2B8,1.

By Theorem 5.32 and Proposition 5.22 and Proposition 5.26, we have:

Corollary 5.34. Any c = 24 holomorphic VOA satisfy Assumption (A).

The following lemma follows from Theorem 4.5: 34

Lemma 5.35. Let (V,ω, H) bea c = l holomorphic conformal vertex algebra satisfying Assumption (A). Then, any W genus(V,ω, H) isac = l holomorphic conformal vertex ∈ algebra satisfying Assumption (A).

Since the maximal lattice of any c = 24 holomorphic VOA is positive-definite, we have:

Proposition 5.36. If (V,ω) is a c = 24 holomorphic VOA, then all conformal vertex algebras in genus(V,ω) are c = 24 holomorphic VOAs.

The above proposition gives us a effective method to construct holomorphic VOAs. In the rest of this paper, we calculate the mass of the VOA Vhol in Theorem 5.33. By E8,2 B8,1 Proposition 5.29, the maximal lattice of Vhol is √2E D and its genus is II (2+10). E8,2 B8,1 8 8 16,0 II +10 Since II16,0(2II ) contains 17 isomorphism classes of lattices, by lemma 5.36 and Theorem 5.19, there exist at least 17 c = 24 holomorphic VOAs whose maximal lattices are in +10 II16,0(2II ). The following lemma follows from Lemma 5.8:

Lemma 5.37. Let (V,ω, H) satisfy Assumption (A). If V is a VOA and V1 is a semisimple

Lie algebra, then there is a one-to-one correspondence between norm 2 vectors in LV,H and long roots of a level 1 component of V1.

Hence, the level 1 components of V1 are determined by the maximal lattice LV,H. Set

hol hol V + = V V , II (2 10) E8,2 B8,1 II1,1 17,1 II ⊗ and set II (2+10) = II √2E D . 17,1 II 1,1 ⊕ 8 8 hol We prove the following characterization result for V +10 . II17,1(2II ) Theorem 5.38. Let (V,ω, H) satisfy the following conditions: (1) (V,ω, H) satisfy Assumption (A); (2) (V,ω) isac = 26 holomorphic conformal vertex algebra; +10 (3) The maximal lattice of (V, H) is isomorphic to II17,1(2II ). hol Then, (V,ω, H) is isomorphic to V +10 . II17,1(2II ) Proof. Consider an isomorphism of lattices II (2+10) and √2E D II . Similarly 17,1 II 8 8 ⊕ 1,1 to the proof of Proposition 4.11, let C be the coset vertex algebra correspondence to this decomposition. The vertex algebra C is a c = 24 holomorphic VOA and its maximal lattice is √2E8D8. By Theorem 5.32 and Lemma 5.37 and Theorem 5.33, C is isomorphic hol hol to V as a VOA and also as a VH pair, by Lemma 5.20. Hence, V  V VII as E8,2 B8,1 E8,2 B8,1 ⊗ 1,1 a conformal vertex algebra and VH pair. 

+10 hol hol = = Lemma 5.39. GV + Aut II17,1(2 ) and #genus(V ) 17 holds. II (2 10) II E8,2 B8,1 17,1 II 35

Proof. By Lemma 4.12 and the proof of Theorem 5.38, it suffices to show that

Aut ( √2E8D8) GVhol . Clearly, Aut ( √2E8D8) is a semidirect product of Weyl group ⊂ E8,2B8,1 W W and the order 2 Dynkin diagram automorphism of D , which is isomorphic to E8 × D8 8 hol the Weyl group WE8 WB8 . By Corollary 5.15, GV contains it. Hence, the assertion × E8,2B8,1 holds. 

Let V be a c = 24 holomorphic VOA such that the maximal lattice is contained in II (2+10). Since the maximal lattice of V V is II (2+10), by Theorem 5.38, we 16,0 II ⊗ II1,1 17,1 II hol  hol have VII (2+10) V VII1,1 . Hence, V is contained in genus(VE B ). By Lemma 5.39, we 17,1 II ⊗ 8,2 8,1 have:

Proposition 5.40. Let V be a c = 24 holomorphic VOA. If the maximal lattice LV of V +10 = is contained in II16,0(2II ), then V is uniquely determined as a VOA by LV and GV,ω Aut L . In particular, there is a one-to-one correspondence between genus(Vhol ) and V E8,2 B8,1 genus( √2E8D8).

Remark 5.41. The one-to-one correspondence between genus(Vhol ) and genus( √2E D ) E8,2 B8,1 8 8 first appeared in [HS1], asitwaspointedoutin [HS2]. Our approachis motivated by their results.

Acknowledgements The author would like to express his gratitude to Professor Atsushi Matsuo, for his encouragement throughout this work and numerous advices to improve this paper. He is also grateful to Hiroki Shimakura and Shigenori Nakatsuka for careful reading of this manuscript and their valuable comments. This work was supported by the Program for Leading Graduate Schools, MEXT, Japan.

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