Orbifold construction of holomorphic vertex operator algebras associated to inner automorphisms
aa r X i v : . [ m a t h . QA ] A ug ORBIFOLD CONSTRUCTION OF HOLOMORPHIC VERTEXOPERATOR ALGEBRAS ASSOCIATED TO INNERAUTOMORPHISMS
CHING HUNG LAM AND HIROKI SHIMAKURA
Abstract.
In this article, we construct three new holomorphic vertex operator alge-bras of central charge 24 using the Z -orbifold construction associated to inner auto-morphisms. Their weight one subspaces has the Lie algebra structures D , A , G , , E , A , , and A , A , . In addition, we discuss the constructions of holomorphic vertexoperator algebras with Lie algebras A , C , A , and D , A , from holomorphic vertexoperator algebras with Lie algebras C , G , A , and A , , respectively. Introduction
The classification of holomorphic vertex operator algebras (VOAs) of central charge 24is one of the fundamental problems in vertex operator algebras and mathematical physics.In 1993, Schellekens [Sc93] obtained a partial classification by determining possible Liealgebra structures for the weight one subspaces of holomorphic VOAs of central charge24. There are 71 cases in his list but only 39 of the 71 cases were known explicitly at thattime. It is also an open question if the Lie algebra structure of the weight one subspacewill determine the VOA structure uniquely when the central charge is 24.In [La11, LS12, LS15], a special class of holomorphic VOAs, called framed VOAs, ofcentral charge 24 were studied and classified. In particular, it was shown in [LS15] thatthere exist exactly 56 holomorphic framed VOAs of central charge 24 and they are uniquelydetermined by the Lie algebra structures of their weight one subspaces. On the otherhand, a Z -orbifold theory associated to lattice VOAs has been developed by Miyamoto[Mi13] and as an application, a holomorphic VOA whose weight one subspace has theLie algebra structure E , G , was constructed. By using the similar methods, severalother holomorphic VOAs have been constructed in [SS]. Recently, van Ekeren, M¨ollerand Scheithauer [EMS] announced that they have obtained a mathematically rigorousproof for Schellekens’ list. They also claimed that they have established the Z n -orbifoldconstruction for general elements of arbitrary orders. In particular, they claimed that Mathematics Subject Classification.
Primary 17B69.C. H. Lam was partially supported by NSC grant 100-2628-M-001005-MY4 of Taiwan.H. Shimakura was partially supported by JSPS KAKENHI Grant Numbers 23540013 and 26800001,and by Grant for Basic Science Research Projects from The Sumitomo Foundation. hey can construct two holomorphic VOAs of central charge 24 such that their weight onesubspaces have the Lie algebras structures E , C , A , and A , . By the results and theannouncement above, there are 10 remaining Lie algebras in Schellekens’ list for whichthe corresponding holomorphic VOAs of central charge 24 have not been constructed yet.The main purpose of this article is to construct new holomorphic VOAs of centralcharge 24 by using the Z -orbifold construction associated to inner automorphisms. Moreprecisely, three new VOAs are constructed, and two new VOAs can be constructed fromunestablished holomorphic VOAs of central charge 24. The main theorem is the following(see Theorems 6.9, 7.7, 8.6, 9.6 and 10.10 for details): Theorem 1.1. (1)
There exist strongly regular, holomorphic VOAs of central charge with Lie algebras D , A , G , , E , A , and A , A , . (2) If there exists a strongly regular, holomorphic VOA of central charge with Liealgebra C , G , A , , then there exists a strongly regular, holomorphic VOA of centralcharge with Lie algebra A , C , A , . (3) If a strongly regular, holomorphic VOA of central charge with Lie algebra A , can be obtained by applying the Z -orbifold construction to the lattice VOA associatedto the Niemeier lattice with root lattice A as in § with Lie algebra D , A , . In order to prove this theorem, we choose a holomorphic VOA V and its inner auto-morphism σ h of order 2 carefully. Then, applying the Z -orbifold construction to V and σ h , we obtain a new holomorphic VOA ˜ V with the desired Lie algebra. We summarize theLie algebra structures of V , ( V σ h ) and ˜ V in Table 1, where V σ h is the set of fixed-pointsof σ h . Table 1.
Lie algebra structures of V , ( V σ h ) and ˜ V (Original) Lie algebra V (Fixed point) Lie subalgebra ( V σ h ) (New) Lie algebra ˜ V E , G , D , A , A , G , U (1) D , A , G , D , A , G , D , A , A , A , U (1) E , A , E , A , A , A , U (1) A , A , C , G , A , A , A , A , U (1) A , C , A , A , A , U (1) D , A , We note that van Ekeren, M¨oller and Scheithauer announced that the assumptionin Theorem 1.1 (3) is true [EMS]. We also notice that a holomorphic VOA with Liealgebra C , G , A , would be constructed by using Z -orbifold theory to the lattice VOAassociated to the Niemeier lattice with root lattice E , which will be discussed in ourfuture article. y the result in this article, there are remaining 5 cases in Schellekens’ list which havenot been constructed yet. The corresponding Lie algebras have the type C , , D , A , , A , , F , A , , and C , G , A , .Let us explain our construction in more detail. First, we recall the Z -orbifold con-struction associated to an inner automorphism. Let V be a strongly regular, holomorphicVOA and let h ∈ V . Assume that h (0) is semisimple on V and the associated inner auto-morphism σ h = exp( − π √− h (0) ) of V has order 2. Using Li’s ∆-operator introduced in[Li96], we construct the (unique) irreducible σ h -twisted V -module V ( h ) explicitly. It wasshown in [DLM96] that V ⊕ V ( h ) has an abelian intertwining algebra structure. Henceone can see that the subspace ˜ V = V σ h ⊕ ( V ( h ) ) Z has a VOA structure as a simple currentextension of V σ h , where V σ h is the set of fixed-points of σ h and ( V ( h ) ) Z is the subspaceof V ( h ) with integral L (0)-weights. If V and h satisfy some conditions (see Theorem 5.4for detail), then ˜ V is of CFT-type. By a similar argument as in [Li97], we see that ˜ V is C -cofinite and holomorphic. In addition, we prove that the Lie algebras V and ˜ V sharea common Cartan subalgebra under some assumptions.Next we check that our choices of V and h fit the Z -orbifold construction above. Let V be a strongly regular, holomorphic VOA of central charge 24 such that the Lie algebrastructure of V is one of the Lie algebra structures in column one of Table 1. Then we caneasily find h ∈ V so that ( V σ h ) has the Lie subalgebra structure in the correspondingcolumn two in Table 1 and h satisfies the necessary conditions. Clearly, the order of σ h on V is 2; however, we shall show that the order of σ h on V is 2, also. For this purpose,we consider the subVOA U generated by V , which is a full subVOA of V ([DM04a]). Itsuffices to show that the order of σ h is 2 on every irreducible U -submodule of V . Recallthat U is the tensor product of simple affine VOAs L g i ( k i ,
0) associated with simple Liealgebras g i at positive integral levels k i ([DM06a]). Hence any irreducible U -module isthe tensor product of irreducible L g i ( k i , L (0)-weights of V are integral, there are not so many possibilities for irreducible U -submodules of V . We can check that the order of σ h is 2 for each possibility. Hence σ h isof order 2 on V . Unfortunately, this argument does not work for the case (3) of Theorem1.1. For this case, we directly check this assertion by using the explicit description of thelattice VOA and its irreducible σ h -twisted module ([Le85, DL96]).Finally, we explain how to determine the Lie algebra structure of ˜ V . A key tool is thedimension formula mentioned in [Mo94]. We prove it by the following way: We checkthat the character of V σ h converges to a modular function of weight 0 on the congruencesubgroup Γ (2), and we express it as a Laurent polynomial of a Hauptmodul of Γ (2).Substituting it to certain equations about the characters of V , ˜ V and V σ h and comparingsome coefficients of the q -expansions, we obtain the formula. By this formula, dim ˜ V can e determined by dim( V ( h ) ) / . Indeed, in all our cases, ( V ( h ) ) / = 0. It follows fromthe dimension formula directly or by establishing that ( M ( h ) ) / = 0 for each possibleirreducible U -submodule M of V . In the case (3) of Theorem 1.1, we also check itdirectly. By [DM04a], ˜ V is semisimple, and the ratio of the dual Coxeter number andthe level for every simple ideal of ˜ V is determined by dim ˜ V . By using the fact that V and ˜ V share a common Cartan subalgebra H , any simple Lie subalgebra of ˜ V spannedby weight vectors for H is contained in a unique simple ideal of ˜ V , and the level of thesimple ideal can be determined. For example, any simple ideal of ( V σ h ) (cf. Table 1) isspanned by weight vectors for H since σ h is an inner automorphism. Thus we have enoughdata of ˜ V to determine its Lie algebra structure in each case.The organization of the article is as follows. In Section 2, we recall some preliminaryresults about strongly regular, holomorphic VOAs. In Section 3, we recall the definitionof Li’s ∆-operator and the associated construction of the σ h -twisted module for h ∈ V .We also discuss some basic properties of simple affine VOAs and their twisted modulesconstructed by the ∆-operator. In Section 4, we prove the dimension formula for the Z -orbifold construction associated to an inner automorphism. In Section 5, we discuss the Z -orbifold construction associated to inner automorphisms. We also discuss some prop-erties of the resulting VOAs. In Sections 6,7 and 8, we apply the construction successivelyand obtain three holomorphic VOAs of central charge 24 with Lie algebras D , A , G , , E , A , , and A , A , . Finally, in Sections 9 and 10, we will discuss the constructions ofholomorphic VOAs of central charge 24 with Lie algebras A , C , A , and D , A , fromholomorphic VOAs with Lie algebras C , G , A , and A , using the similar methods,respectively. otations ( ·|· ) the normalized Killing form on a semisimple Lie algebraso that ( α | α ) = 2 for long roots α . h·|·i the normalized symmetric invariant bilinear form on a VOAso that h | i = −
1, equivalently, h a | b i = a (1) b for a, b ∈ V . a ( h )( n ) the n -th mode of an element a ∈ V on the σ h -twisted V -module M ( h ) . α i a simple root of a root system. E α a root vector in a simple Lie algebra with respect to root α . h ∨ the dual Coxeter number of a simple Lie algebra. h (0) the subspace of fixed-points of τ in h = C ⊗ Z N ( A ). θ the highest root with respect to a fixed set of simple roots. L (0) the weight operator ω (1) . L ( h ) (0) the weight operator ω ( h )(1) on a σ h -twisted module M ( h ) . L g ( k,
0) the simple affine VOA associated with simple Lie algebra g at level k . L g ( k, λ ) the irreducible L g ( k, λ .Λ i the fundamental weight with respect to simple root α i . M ( h ) the σ h -twisted V -module constructed from a V -module M by Li’s ∆-operator. N = N ( A ) a Niemeier lattice with root lattice A .Π( λ, X n ) the set of all weights of the irreducible module with highest weight λ over the simple Lie algebra of type X n . ρ half of the sum of all positive roots. σ h the inner automorphism exp( − π √− h (0) ) of a VOA V associated to h ∈ V . τ the order 5 automorphism of the lattice VOA V N ( A ) defined in § h (0) the set of spectra of h (0) for a semisimple element h ∈ V . U (1) a 1-dimensional abelian Lie algebra. V σ h the set of fixed-points of σ h , which is a full subVOA of V . X n (the type of) a root system, a simple Lie algebra or a root lattice. X n,k (the type of) a simple Lie algebra whose type is X n and level is k . Preliminary
In this section, we will review some fundamental results about VOAs.2.1.
Vertex operator algebras.
Throughout this article, all VOAs are defined over thefield C of complex numbers. We recall the notion of vertex operator algebras (VOAs) andmodules from [Bo86, FLM88, FHL93].A vertex operator algebra (VOA) ( V, Y, , ω ) is a Z -graded vector space V = L m ∈ Z V m equipped with a linear map Y ( a, z ) = X i ∈ Z a ( i ) z − i − ∈ (End( V ))[[ z, z − ]] , a ∈ V nd the vacuum vector and the conformal vector ω satisfying a number of conditions([Bo86, FLM88]). We often denote it by V . For a ∈ V and n ∈ Z , we often call a ( n ) the n -th mode of a . Note that L ( n ) = ω ( n +1) satisfy the Virasoro relation:[ L ( m ) , L ( n ) ] = ( m − n ) L ( m + n ) + 112 ( m − m ) δ m + n, c id V , where c is a complex number, called the central charge of V .A linear automorphism of V is called an automorphism of V if it satisfies gω = ω and gY ( v, z ) = Y ( gv, z ) g for all v ∈ V. A vertex operator subalgebra (or a subVOA ) is a graded subspace of V which has a structureof a VOA such that the operations and its grading agree with the restriction of those of V and that they share the vacuum vector. When they also share the conformal vector,we will call it a full subVOA . For an automorphism g of a VOA V , let V g denote the setof fixed-points of g . Clearly V g is a full subVOA of V .An (ordinary) V -module ( M, Y M ) is a C -graded vector space M = L m ∈ C M m equippedwith a linear map Y M ( a, z ) = X i ∈ Z a ( i ) z − i − ∈ (End( M ))[[ z, z − ]] , a ∈ V satisfying a number of conditions ([FHL93]). We often denote it by M . For an automor-phism g of V , we also consider a g -twisted V -module. For the detail, see [Li96, DLM00]and references therein. Note that a g -twisted V -module is an (untwisted) V g -module.The L (0) -weight of a homogeneous vector v ∈ M k is k , where L (0) = ω (1) . Note that L (0) v = kv if v ∈ M k .A VOA is said to be rational if any module is completely reducible. A rational VOAis said to be holomorphic if it itself is the only irreducible module up to isomorphism. AVOA is said to be of CFT-type if V = C (note that V n = 0 for all n < V = C [DM06b, Lemma 5.2]), and is said to be C -cofinite if the codimension in V of the subspacespanned by the vectors of form u ( − v , u, v ∈ V , is finite. A module is said to be self-dual if its contragredient module is isomorphic to itself. It is obvious that a holomorphic VOAis simple and self-dual. A VOA is said to be strongly regular if it is rational, C -cofinite,self-dual and of CFT-type.Let V be a VOA of CFT-type. Then, the 0-th mode gives a Lie algebra structure on V . Moreover, the n -th modes v ( n ) , v ∈ V , n ∈ Z , define an affine representation of theLie algebra V on V . For a simple algebra a of V , the level of a is defined to be the scalarby which the canonical central element acts on V as the affine representation. When thetype of the root system of a is X n and the level of a is k , we denote the type of a by X n,k .Assume that V is self-dual. Then there exists a symmetric invariant bilinear form h·|·i on , which is unique up to scalar ([Li94]). We normalize it so that h | i = −
1. Then for a, b ∈ V , we have h a | b i = a (1) b . For an element a ∈ V , exp( a (0) ) is an automorphismof V , called an inner automorphism . For a semisimple element h ∈ V , we often considerthe inner automorphism σ h = exp( − π √− h (0) ) associated to h .Assume that V is semisimple. Let H be a Cartan subalgebra of V . Let ( ·|· ) be theKilling form on V . We identify H ∗ with H via ( ·|· ) and normalize ( ·|· ) so that ( α | α ) = 2 forany long root α ∈ H . In this article, weights for H are defined via ( ·|· ), that is, the weightof a vector v ∈ V for H is λ ∈ H if x (0) v = ( x | λ ) v for all x ∈ H . Remark that for h ∈ H , σ h acts on a vector with weight λ as the scalar multiple by exp( − π √− h | λ )). Thefollowing lemma is immediate from the commutator relations of n -th modes (cf. [DM06a,(3.2)]). Lemma 2.1.
If the level of a simple algebra of V is k , then h·|·i = k ( ·|· ) on it. Let us recall some results related to the Lie algebra V . Proposition 2.2 ([DM06a, Theorem 1.1, Corollary 4.3]) . Let V be a strongly regular,simple VOA. Then V is reductive. Let s be a simple Lie subalgebra of V . Then V is anintegrable module for the affine representation of s on V , and the subVOA generated by s is isomorphic to the simple affine VOA associated with s at positive integral level. Proposition 2.3 ([DM04b, Theorem 1] and [Ma14, Section 3.3]) . Let V be a stronglyregular, simple VOA and let M be a V -module. Then for any element x in a Cartansubalgebra of V , the -th mode x (0) acts semisimply on M . Proposition 2.4 ([DM04a, (1.1), Theorem 3 and Proposition 4.1]) . Let V be a stronglyregular, holomorphic VOA of central charge . If the Lie algebra V is neither { } norabelian, then V is semisimple, and the conformal vectors of V and the subVOA generatedby V are the same. In addition, for any simple ideal of V at level k , the identity h ∨ k = dim V − holds, where h ∨ is the dual Coxeter number.
3. ∆ -operator, simple affine VOAs and twisted modules
In this section, we recall the twisted module constructed by Li’s ∆-operator. Moreover,we discuss the lowest L (0)-weight of such a twisted module over simple affine VOAs.3.1. Twisted modules constructed by Li’s ∆ -operator. Let V be a vertex operatoralgebra of CFT-type. Let σ be a finite order automorphism of V and let h ∈ V with σ ( h ) = h . We assume that h (0) acts semisimply on V and that there exists a positiveinteger T ∈ Z > such that Spec h (0) , the set of spectra of h (0) on V , is contained in /T ) Z . Then σ h = exp( − π √− h (0) ) is an automorphism of V with σ Th = 1. Note that σσ h = σ h σ since σ ( h ) = h .Let ∆( h, z ) be Li’s ∆-operator defined in [Li96], i.e.,∆( h, z ) = z h (0) exp ∞ X n =1 h ( n ) − n ( − z ) − n ! . Proposition 3.1 ([Li96, Proposition 5.4]) . Let σ be an automorphism of V of finite orderand let h ∈ V be as above such that σ ( h ) = h . Let ( M, Y M ) be a σ -twisted V -module anddefine ( M ( h ) , Y M ( h ) ( · , z )) as follows: M ( h ) = M as a vector space; Y M ( h ) ( a, z ) = Y M (∆( h, z ) a, z ) for any a ∈ V. Then ( M ( h ) , Y M ( h ) ( · , z )) is a σ h σ -twisted V -module. Furthermore, if M is irreducible, thenso is M ( h ) . For a σ -twisted V -module M and a ∈ V , we denote by a ( h )( i ) the operator which corre-sponds to the coefficient of z − i − in Y M ( h ) ( a, z ), i.e., Y M ( h ) ( a, z ) = X i ∈ Z a ( h )( i ) z − n − for a ∈ V. Now we will review the action of some elements of V on the σ h σ -twisted V -module M ( h ) . The 0-th mode of an element x ∈ V on M ( h ) is given by(3.1) x ( h )(0) = x (0) + h h | x i id . Let us denote by L ( h ) ( n ) the ( n + 1)-th mode of the conformal vector ω ∈ V on M ( h ) .Then the L (0)-weights on M ( h ) are given by(3.2) L ( h ) (0) = L (0) + h (0) + h h | h i . The following lemma is immediate from the equation above.
Lemma 3.2.
Let M be a σ -twisted V -module whose L (0) -weights are half-integral. Let h ∈ V such that h (0) is semisimple on M and h h | h i ∈ Z . Assume that the spectra of h (0) on M are half-integral. Then the L (0) -weights of the σ h σ -twisted V -module M ( h ) are alsohalf-integral. Simple affine VOAs and irreducible twisted modules.
In this subsection, werecall some properties of simple affine VOAs and their modules from [Ka90, FZ92]. More-over, we study σ h -twisted modules constructed by Li’s ∆-operator.Let g be a simple Lie algebra with Cartan subalgebra H . Let Φ be the set of roots of g .Let ( ·|· ) be the Killing form on g . We identify H ∗ with H by ( ·|· ) and normalize the formso that ( α | α ) = 2 for any long root α ∈ Φ. Let { α i | ≤ i ≤ n } ⊂ H be a set of simple oots and { Λ i | ≤ i ≤ n } ⊂ H the set of the fundamental weights so that j | α i )( α i | α i ) = δ ij .For every β ∈ Φ, fix a root vector E β in g associated to β .Let V = L g ( k,
0) be the simple affine VOA associated with g at positive integral level k . It was proved in [FZ92] that all irreducible V -modules are given by L g ( k, λ ), where λ ranges over dominant integral weights with ( θ | λ ) ≤ k for the highest root θ . By [Ka90,Corollary 12.8], the lowest L (0)-weight of L g ( k, λ ) is given by(3.3) ( λ + 2 ρ | λ )2( k + h ∨ ) , where ρ = P ni =1 Λ i and h ∨ is the dual Coxeter number. The following facts on L g ( k, λ ) iswell-known. Lemma 3.3 ([FZ92, § . Let ℓ be the lowest L (0) -weight of L g ( k, λ ) . Then the followinghold: (1) L g ( k, λ ) ℓ is an irreducible g -module with highest weight λ , where V ∼ = g via the -thmode; (2) Let v ∈ L g ( k, λ ) ℓ . Then the L (0) -weight of E β ( − n ) . . . E β m ( − n m ) v is ℓ + P mj =1 n j ; (3) L g ( k, λ ) is spanned by { E β ( − n ) . . . E β m ( − n m ) v | β i ∈ Φ , n i ∈ Z > , m ∈ Z ≥ , v ∈ L g ( k, λ ) ℓ } . From now on, let h be an element in H with Spec h (0) ⊂ (1 /T ) Z on V for some T ∈ Z > .Note that h ∈ L ni =1 Q α i and that the restriction of the normalized Killing form ( ·|· ) to L ni =1 Q α i is positive-definite. In addition, we assume(3.4) ( h | α ) ≥ − α ∈ Φ . Lemma 3.4.
Let M be a V -module. Let v be a vector in M ( h ) with L (0) -weight p ,i.e., L ( h ) (0) v = pv . Let u = E β ( − n ) . . . E β m ( − n m ) v ∈ M ( h ) be a non-zero vector, where n i ∈ Z > . Then u is a homogeneous vector in M ( h ) and its L (0) -weight is greater thanor equal to p . Moreover the equality holds if and only if n i = 1 and ( h | β i ) = − for all ≤ i ≤ m .Proof. By (3.2) and [ L ( h ) (0) , E β ( − n ) ] = ( n + ( h | β )) id , we have L ( h ) (0) u = m X i =1 ( n i + ( h | β i )) + p ! u. By the assumption (3.4), n i + ( h | β i ) ≥ i , and hence the L (0)-weight of u in M ( h ) is greater than or equal to p . The latter assertion is obvious. (cid:3) Lemma 3.5.
The lowest L (0) -weight of the irreducible σ h -twisted V -module L g ( k, λ ) ( h ) isnon-negative. If the lowest L (0) -weight of L g ( k, λ ) ( h ) is , then λ = k Λ j and h = − Λ j forsome fundamental weight Λ j , or λ = h = 0 . roof. Let v be a lowest L (0)-weight vector in L g ( k, λ ) with weight µ ∈ H . By Lemmas3.3 (3) and 3.4, we may assume that v has also the lowest L (0)-weight in L g ( k, λ ) ( h ) . If λ = 0, then v ∈ C and µ = 0, which proves the assertion by (3.2). We assume that λ = 0. By Lemma 2.1, h h | h i = k ( h | h ). Hence by (3.2), the L (0)-weight of v in L g ( k, λ ) ( h ) is(3.5) ( λ + 2 ρ | λ )2( k + h ∨ ) + ( h | µ ) + k ( h | h )2 , which is equal to the lowest L (0)-weight of L g ( k, λ ) ( h ) . Applying 2 k ( λ | ρ ) ≥ h ∨ ( λ | λ )([Ka90, Theorem 13.11]) and ( λ | λ ) ≥ ( µ | µ ) ([Ka90, Proposition 11.4]) to (3.5), we seethat the lowest L (0)-weight of L g ( k, λ ) ( h ) is non-negative since(3.6) ( λ + 2 ρ | λ )2( k + h ∨ ) + ( h | µ ) + k ( h | h )2 ≥ ( µ | µ )2 k + ( h | µ ) + k ( h | h )2 = ( µ + kh ) k ≥ . If the L (0)-weight of v in L g ( k, λ ) ( h ) is 0, then the all equalities hold in (3.6). The formerequality holds if and only if µ = λ = k Λ j for some fundamental weight Λ j , and the latterequality holds when µ + kh = 0. Combining them, we obtain λ = k Λ j and h = − Λ j . (cid:3) Lemma 3.6.
Let X n be the type of the simple Lie algebra g . The lowest L (0) -weight of L g ( k, λ ) ( h ) is equal to (3.7) ( λ + 2 ρ | λ )2( k + h ∨ ) + min { ( h | µ ) | µ ∈ Π( λ, X n ) } + k ( h | h )2 , where Π( λ, X n ) is the set of all weights for H of the irreducible module with the highestweight λ over the simple Lie algebra of type X n .Proof. By Lemmas 3.3 (3) and 3.4, it suffices to consider the lowest L ( h ) (0)-weight of thelowest L (0)-weight space of L g ( k, λ ). Hence, this lemma follows from (3.5) and Lemma3.3 (1). (cid:3) Later, we will use the following lemma:
Lemma 3.7.
Let v be a vector in L g ( k, λ ) with weight µ for H . Then v is also a weightvector in L g ( k, λ ) ( h ) for H and its weight is µ + kh .Proof. This lemma follows from (3.1) and Lemma 2.1. (cid:3)
Lowest L (0) -weight of a twisted module. In this subsection, we give a sufficientcondition so that the lowest L (0)-weight of the σ h -twisted V -module V ( h ) is positive. Proposition 3.8.
Let V be a strongly regular, simple VOA. Assume that the Lie algebra g = V is semisimple. Let g = L ti =1 g i be the decomposition into the direct sum of t simpleideals g i . Let U be the subVOA of V generated by V . Let h be an element in a (fixed) artan subalgebra H of g such that Spec h (0) ⊂ (1 /T ) Z on V for some T ∈ Z > . Let h i bethe image of h under the canonical projection from H to H ∩ g i . We further assume that (1) the conformal vectors of V and U are the same, i.e., U is a full subVOA of V ; (2) ( h | α ) ≥ − for all roots α ∈ H of g , where ( ·|· ) is the normalized Killing form on g so that ( β | β ) = 2 for any long root β ; (3) for some i , − h i is not a fundamental weight.Then the lowest L (0) -weight of V ( h ) is positive.Proof. By Proposition 2.2, there exist k i ∈ Z > such that U ∼ = N ti =1 L g i ( k i ,
0) as VOAs.Moreover, by (1) and the rationality of U , V is a direct sum of finitely many irreducible U -submodules.Let M be an irreducible U -submodule of V . It suffices to show that the lowest L (0)-weight of M ( h ) is positive. It follows from U ∼ = N ti =1 L g i ( k i ,
0) that M ∼ = N ti =1 L g i ( k i , λ i )for some dominant integral weight λ i of g i (cf. [FZ92]). Let ω i be the conformal vector of L g i ( k i ,
0) and let L g i (0) = ( ω i ) (1) . Since h = P ti =1 h i , we have L ( h ) (0) = t X i =1 L ( h i ) g i (0) . Clearly, (2) shows that the assumption (3.4) holds for g i and h i . Hence by Lemma 3.5the lowest L g i (0)-weight of L g i ( k i , λ i ) ( h i ) is non-negative, and by (3), it is positive for atleast one i . Thus the lowest L (0)-weight of M ( h ) is positive. (cid:3) Dimension formula associated to the Z -orbifold construction In this section, we prove the dimension formula mentioned in [Mo94], which will playimportant roles in determining the Lie algebra structures of holomorphic VOAs.4.1.
Characters and trace functions.
Let U = L ∞ n =0 U n be a VOA of central charge c . Let f be an automorphism of U of finite order T . Let W = L ∞ n =0 W λ + n/T be anirreducible U -module or an irreducible f -twisted U -module, where λ ∈ C . The characterof W is defined by the formal series Z W ( q ) = q λ − c/ ∞ X n =0 dim W n q n/T , and the trace function of f on U is defined by the formal series Z U ( f, q ) = q − c/ ∞ X n =0 Tr f | U n q n , where q is a formal variable. ow, we assume that U is strongly regular and consider Z W ( q ) and Z U ( f, q ) less for-mally. Take q to be the usual local parameter at infinity in the upper half-plane H = { τ ∈ C | Im( τ ) > } , i.e, q = e π √− τ . Since U is strongly regular, Z W ( q ) and Z U ( f, q ) converge to holomorphicfunctions in H ([Zh96, Theorem 4.4.1] and [DLM00, Theorem 1.3]). We often denote Z W ( q ) and Z U ( f, q ) by Z W ( τ ) and Z U ( f, τ ), respectively.4.2. Montague’s dimension formula.
Let V be a strongly regular, holomorphic VOAof central charge 24. Let g be an inner automorphism of V of order 2. Note that g = exp( − π √− h (0) ) for some semisimple element h ∈ V . It was proved in [DLM00,Theorem 1.2] that V possesses a unique irreducible g -twisted V -module V ( g ) up to iso-morphism.Let S = −
11 0 ! and T = ! be the standard generators of SL (2 , Z ). Noticethat A = a bc d ! ∈ SL (2 , Z ) acts on H by A : τ aτ + bcτ + d . By [Zh96, Theorem 5.3.3], Z V ( Aτ ) = Z V ( τ ) for all A ∈ SL (2 , Z ). Since g is an inner automorphism, we can apply[KM12, Theorem 1.4] to our setting and obtain(A1) Z V ( g, Sτ ) = Z V ( g ) ( τ ).In this section, we also assume the following:(A2) Z V ( g ) ( τ ) ∈ q − / Z [[ q / ]], i.e., the L (0)-weights of V ( g ) are positive half-integral.Note that (A2) holds if h h | h i ∈ Z and the assumptions of Proposition 3.8 hold.Recall that V g is a subVOA of V , which is the set of fixed-points of g . Proposition 4.1.
The character Z V g ( q ) of V g converges to a holomorphic function in H .Moreover, it is a modular function of weight on the congruence subgroup Γ (2) .Proof. By definition, we obtain(4.1) Z V g ( q ) = 12 ( Z V ( q ) + Z V ( g, q )) . Since both Z V ( q ) and Z V ( g, q ) converge to holomorphic functions in H , so does Z V g ( q ).We denote the holomorphic function in (4.1) by Z V g ( τ ). Next we will show that Z V g ( Aτ ) = Z V g ( τ ) for all A ∈ Γ (2) . It is well-known (e.g. see [Ap90, Theorem 4.3]) that Γ (2) is generated by T and ST S − .It follows from Z V g ( q ) ∈ q − Z [[ q ]] that(4.2) Z V g ( T τ ) = Z V g ( τ ) . n addition, by (A2),(4.3) Z V ( g ) ( T τ ) = Z V ( g ) ( τ ) . Notice that for any
A, B ∈ SL (2 , Z ) and a meromorphic function f ( τ ) in H , we have f ( ABτ ) = g ( Bτ ) where g ( τ ) = f ( Aτ ) is a meromorphic function in H . By (A1), (4.2)and (4.3) , we have Z V ( g, ST S − τ ) = Z V ( g ) ( T S − τ ) = Z V ( g ) ( S − τ ) = Z V ( g, τ ) . Hence Z V g ( ST S − τ ) = 12 (cid:0) Z V ( ST S − τ ) + Z V ( g, ST S − τ ) (cid:1) = 12 ( Z V ( τ ) + Z V ( g, τ )) = Z V g ( τ )and Z V g ( τ ) is invariant under Γ (2).In order to complete the proof, it suffices to check that Z V g ( τ ) is meromorphic atcusps of Γ (2). It is well-known (e.g. see [Ha10, Proposition 1.23]) that Γ (2) has twocusps, represented by i ∞ and 0. It is clear that Z V g ( τ ) is meromorphic at i ∞ since Z V g ( q ) ∈ q − Z [[ q ]]. By (A1) and (4.1), we have Z V g ( Sτ ) = 12 (cid:0) Z V ( τ ) + Z V ( g ) ( τ ) (cid:1) and this function belongs to q − Z [[ q / ]] by (A2). Since S sends i ∞ to 0, the function Z V g ( τ ) is meromorphic at 0. (cid:3) Let V ( g ) Z be the subspace of V ( g ) with integral L (0)-weights and set ˜ V = V g ⊕ V ( g ) Z .We now assume that ˜ V has a strongly regular, holomorphic VOA structure. Note that(4.4) Z ˜ V ( τ ) = Z V g ( τ ) + Z V ( g ) Z ( τ ) . First, we prove the following equation, which was already mentioned in [Mo94, (8)]:
Proposition 4.2. Z V ( τ ) + Z ˜ V ( τ ) = Z V g ( τ ) + Z V g ( Sτ ) + Z V g ( ST τ ) . Proof.
By definition,(4.5) Z V g ( τ ) = 12 ( Z V ( τ ) + Z V ( g, τ )) . Since V is holomorphic, Z V ( τ ) is invariant under the action of SL (2 , Z ). Hence, by (A1),we obtain(4.6) Z V g ( Sτ ) = 12 (cid:0) Z V ( τ ) + Z V ( g ) ( τ ) (cid:1) . t follows from Z V ( T τ ) = Z V ( τ ) that(4.7) Z V g ( ST τ ) = 12 (cid:0) Z V ( τ ) + Z V ( g ) ( T τ ) (cid:1) . On the other hand, by (A2), we have(4.8) Z V ( g ) Z ( τ ) = 12 (cid:0) Z V ( g ) ( τ ) + Z V ( g ) ( T τ ) (cid:1) . Thus by (4.4), (4.6), (4.7) and (4.8), we have Z V g ( τ ) + Z V g ( Sτ ) + Z V g ( ST τ ) = Z V g ( τ ) + Z V ( τ ) + 12 (cid:0) Z V ( g ) ( τ ) + Z V ( g ) ( T τ ) (cid:1) = Z V ( τ ) + Z ˜ V ( τ ) . (cid:3) We now prove the following dimension formula described in [Mo94, (10),(11)].
Theorem 4.3.
The following equations hold: (1) dim V ( g ) / = dim( V g ) − ;(2) dim V + dim ˜ V = 3 dim( V g ) + 24(1 − dim V ( g ) / ) . Proof.
It is well-known (e.g. see [Ha10, Exercise 3.18]) that a hauptmodul for Γ (2) isgiven by f ( τ ) := η ( τ ) η (2 τ ) = q − −
24 + (cid:18) (cid:19) q + · · · , where η ( τ ) = q / Q ∞ n =1 (1 − q n ), q = e π √− τ , is the Dedekind eta function. By Lemma4.1, Z V g ( τ ) is a modular function of weight 0 on Γ (2) and holomorphic in H . In particular, Z V g ( τ ) is a rational function of f ( τ ). In addition, since the set of all cusps of Γ (2) is { , i ∞} and that f ( τ ) → τ → Z V g ( τ ) is a Laurent polynomialof f ( τ ), i.e., Z V g ( τ ) = X n ∈ Z c n f n ( τ ) , where only finitely many coefficients c n ∈ C are non-zero. It follows from Z V g ( τ ) ∈ q − + Z [[ q ]] that c n = 0 if n >
1, and c = 1. Since η ( Sτ ) = ( − iτ ) / η ( τ ), we have f ( Sτ ) = 2 η ( τ ) η ( τ / = 2 (cid:0) q / + 24 q + · · · (cid:1) , (4.9) f − ( Sτ ) = 12 η ( τ / η ( τ ) = 12 (cid:18) q − / −
24 + (cid:18) (cid:19) q / + · · · (cid:19) , (4.10) f − ( Sτ ) = 12 η ( τ / η ( τ ) = 12 (cid:18) q − − q − / + (cid:18) (cid:19) + · · · (cid:19) , (4.11) hich shows that f n ( Sτ ) ∈ q n/ Q [[ q / ]]. By (4.6) and the assumption (A2), we have Z V g ( Sτ ) ∈ q − + q − / Z [[ q / ]]. Hence c n = 0 if n < −
2, and c − = 2 . Thus(4.12) Z V g ( τ ) = f ( τ ) + c + c − f − ( τ ) + 2 f − ( τ ) = q − + dim( V g ) + · · · . Comparing the constant terms of the equation above, we have(4.13) dim( V g ) = c − . By (4.9), (4.10), (4.11) and (4.12), we have Z V g ( Sτ ) = 12 q − + (cid:16) c − − (cid:17) q − / + (cid:18) c − c − + 12 (cid:18) (cid:19)(cid:19) + · · · . Hence, comparing the coefficients of q − / in (4.6), we have(4.14) c − −
24 = 12 dim V ( g ) / . Combining (4.12) and (4.14) and comparing the coefficients of q , we obtaindim( V g ) = (cid:18) (cid:19) + 2 dim V ( g ) / + 24 × , which is equivalent to the equation (1).Note that T ( q n/ ) = ( − n q n/ for n ∈ Z . Comparing the constant terms of both sidesof the equation in Proposition 4.2, we obtaindim V + dim ˜ V = dim( V g ) + 2 (cid:18) c − c − + 12 (cid:18) (cid:19)(cid:19) . Hence by (4.13) and (4.14), we obtain the equation (2). (cid:3) Z -orbifold construction associated to inner automorphisms In this section, we establish the Z -orbifold construction of holomorphic VOAs associ-ated to inner automorphisms based on [DLM96].Let us recall the setting of [DLM96, § V be a strongly regular, holomorphic VOA.Then V is a reductive Lie algebra by Proposition 2.2. Let h ∈ V such that h h | h i ∈ Z and h (0) is semisimple on V . We further assume that Spec h (0) ⊂ Z / V but Spec h (0) Z on V . Then σ h = exp( − π √− h (0) ) ∈ Aut V is of order 2.Set L = Z h . For r ∈ Z and ν ∈ C h , we set V ( rh,ν ) := { v ∈ V ( rh ) | h ( rh )(0) v = h h | rh + ν i v } , and we set P = { ν ∈ C h | V (0 ,ν ) = 0 } . Since V is simple, there exists s ∈ Q h such that P = Z s . It follows from | σ h | = 2 that h h | s i ∈ Z + 1 /
2. For i, j ∈ { , } , we set U ( ih,js ) := M β ∈ P + js V ( ih,β ) . ote that U (0 , = V σ h = { v ∈ V | σ h ( v ) = v } , U ( h, = ( U (0 , ) ( h ) ,U (0 ,s ) = { v ∈ V | σ h ( v ) = − v } , U ( h,s ) = ( U (0 ,s ) ) ( h ) (5.1)and that V = U (0 , ⊕ U (0 ,s ) , V ( h ) = U ( h, ⊕ U ( h,s ) . Let ( V ( h ) ) Z be the subspace of V ( h ) with integral L (0)-weights. By (3.2), we have( V ( h ) ) Z = M n ∈ Z ( V ( h ) ) n = U ( h, if h h | h i ∈ Z ,U ( h,s ) if h h | h i ∈ Z + 1 . (5.2)Since V is strongly regular, so is U (0 , (= V σ h ) by [Mi, Mi15]. In addition, by [DM97], U (0 , is simple, and U (0 ,s ) is an irreducible U (0 , -module. For ν ∈ { , s } it follows from( U (0 ,ν ) ) ( h ) = U ( h,ν ) and Proposition 3.1 that U ( h,ν ) is an irreducible U (0 , -module. Thefollowing classification of irreducible U (0 , -modules is a consequence of [Mi, Mi15]: Proposition 5.1.
There exist exactly non-isomorphic irreducible U (0 , -modules U ( ih,js ) , i, j ∈ { , } . Simple current modules.
In this subsection, we prove that irreducible U (0 , -modules, U ( ih,js ) , i, j ∈ { , } , are simple current modules using the Verlinde formulaproved in [Hu08]. By (4.6), we have Z U (0 , ( Sτ ) = 12 ( Z U (0 , ( τ ) + Z U (0 ,s ) ( τ ) + Z U ( h, ( τ ) + Z U ( h,s ) ( τ )) . Since Z U (0 ,s ) ( τ ) = Z V ( τ ) − Z U (0 , ( τ ), we have Z U (0 ,s ) ( Sτ ) = 12 ( Z U (0 , ( τ ) + Z U (0 ,s ) ( τ ) − Z U ( h, ( τ ) − Z U ( h,s ) ( τ )) . Let S = ( S P,Q ) be the S -matrix indexed by U (0 , , U (0 ,s ) , U ( h, , U ( h,s ) . Note that S issymmetric and S is the permutation matrix which sends an irreducible U (0 , -module toits contragredient module. By the S -transformations above , we have S = 12 − − − a − a − − a a , where a = 1 or −
1. In particular, S is the identity matrix, which shows that anyirreducible U (0 , -module is self-dual. By the Verlinde formula, we have the fusion rules N QP,P = X i,j ∈{ , } S P U ( ih,js ) S U ( ih,js ) Q S U (0 , U ( ih,js ) = X i,j ∈{ , } S U ( ih,js ) Q . ence N QP,P = 0 if and only if Q = U (0 , , and N U (0 , P,P = 1 for P = U ( ih,js ) , i, j ∈ { , } .Thus we have the following proposition: Proposition 5.2.
For i, j ∈ { , } , U ( ih,is ) is a self-dual simple current U (0 , -module. Z -orbifold construction associated to inner automorphisms. In this subsec-tion, we prove the following proposition by using the results in [DLM96, Section 3].
Proposition 5.3.
The V σ h -module ˜ V = V σ h ⊕ ( V ( h ) ) Z has a VOA structure as a simplecurrent extension of V σ h graded by Z .Proof. By [DLM96, Theorem 3.21] ¯ U = U (0 , ⊕ U ( h, ⊕ U (0 ,s ) ⊕ U ( h,s ) is an abelianintertwining algebra. For the notations U ( ih,js ) and ( V ( h ) ) Z , see (5.1) and (5.2). Since thesubspace ˜ V of ¯ U is a V σ h -module, it satisfies the axiom of a VOA except for the Jacobiidentity on ( V ( h ) ) Z . Note that the generalized Jacobi identity in [DLM96, (3.88)] on ¯ U is z − δ (cid:18) z − z z (cid:19) (cid:18) z − z z (cid:19) η (( λ i ,α ) , ( λ j ,β )) ¯ Y ( u, z ) ¯ Y ( v, z ) w − ¯ C (( λ i , α ) , ( λ j , β )) z − δ (cid:18) z − z − z (cid:19) (cid:18) z − z z (cid:19) η (( λ i ,α ) , ( λ j ,β )) ¯ Y ( v, z ) ¯ Y ( u, z ) w = z − δ (cid:18) z − z z (cid:19) (cid:18) z + z z (cid:19) η (( λ i ,α ) , ( λ k ,γ )) h ( i, j, k ) ¯ Y ( ¯ Y ( u, z ) v, z ) w, (5.3)where u ∈ U ( λ i ,α ) , v ∈ U ( λ j ,β ) , w ∈ U ( λ k ,γ ) . We will check that the maps η ( · , · ), ¯ C ( · , · ), h ( · , · , · ) are trivial on ( V ( h ) ) Z , equivalently, (5.3) is the usual Jacobi identity of a VOA on( V ( h ) ) Z .First, we consider the map η : ( L/ L × P/ P ) × ( L/ L × P/ P ) → ( Z / / Z definedin [DLM96, (3.19)], where η (( λ i , α ) , ( λ j , β )) = −h λ i | λ j i − h λ i | β i − h λ j | α i + 2 Z . It is obvious that η ≡ { ( h, } (resp. { ( h, s ) } ) if h h | h i ∈ Z (resp. h h | h i ∈ Z + 1).Next we consider the map ¯ C : ( L × P ) × ( L × P ) → C ∗ defined in [DLM96, (3.20) and(3.87)], where ¯ C (( λ i , α ) , ( λ j , β )) = e ( h λ i | β i−h λ j | α i ) π √− C ( α, β ) . Notice that by [DLM96, Remark 3.16], C ≡ P since the rank of P is one. Hence¯ C ≡ { ( h, } and { ( h, s ) } .Finally, we consider the map h : ¯ A × ¯ A × ¯ A → C ∗ defined in [DLM96, (3.83)], where h (( λ i , α ) , ( λ j , α ) , ( λ k , α )) = e − ( λ i + λ j − λ i + j ,λ k ) π √− C ( λ i + λ j − λ i + j , λ k ) , ¯ A = ( L × P ) / { ( α, − α ) | α ∈ L } and λ i + j ∈ { , h } so that λ i + j ≡ λ i + λ j (mod 2 L ). No-tice that ¯ A is identified with { (0 , α ) , ( h, α ) | α ∈ P } and ¯ U is isomorphic to L ( λ,α ) ∈ ¯ A V ( λ,α ) s a vector space. In our case, λ i + λ j − λ i + j must be 0 or 2 h , which shows that e − ( λ i + λ j − λ i + j ,λ k ) π √− ≡
1. As we mentioned above, C ≡ P . Hence h ( · , · , · ) ≡ A × ¯ A × ¯ A . Thus (5.3) is the usual Jacobi identity on ( V ( h ) ) Z .By Proposition 5.2, ˜ V is a simple current extension of V σ h graded by Z . (cid:3) Properties of the VOAs obtained by Z -orbifold construction. In this sub-section, under some assumptions, we show that the VOA ˜ V constructed in Proposition5.3 is holomorphic and strongly regular and the Lie algebras V and ˜ V share a commonCartan subalgebra. In particular, the Lie ranks of V and ˜ V are the same.Recall that h is a semisimple element in V , σ h ∈ Aut V is of order 2 on V and h h | h i ∈ Z . Theorem 5.4.
Assume that V is semisimple. Let V = L ti =1 g i be the decomposition of V into the direct sum of t simple ideals g i . Let k i be the level of the affine representationof g i on V . Let H be a Cartan subalgebra of V such that h ∈ H . (1) Assume the following: (a) the conformal vectors of V and the subVOA generated by V are the same; (b) ( h | α ) ≥ − for all roots α of V ; (c) for some i , − h i is not a fundamental weight for H on V , where h i is the image of h under the canonical projection from H to H ∩ g i .Then the VOA ˜ V is of CFT-type. (2) If ˜ V is of CFT-type, then it is strongly regular and holomorphic. (3) Assume that ˜ V is strongly regular and holomorphic and that (d) − P ti =1 k i h i is not a weight of V for H ,then H is a Cartan subalgebra of ˜ V . In particular, the Lie ranks of V and ˜ V are thesame.Proof. (1) By Proposition 3.8, along with the assumptions (a), (b) and (c), the lowest L (0)-weight of ( V ( h ) ) Z is greater than 0. Since V σ h is of CFT-type, so is ˜ V = V σ h ⊕ ( V ( h ) ) Z .(2) Since V σ h is rational and C -cofinite ([Mi, Mi15]), so is ˜ V [Li97, Theorem 5.6]. Hence˜ V is strongly regular. Moreover, by Proposition 5.1 and the classification of irreducible˜ V -modules ([Li97, Theorem 5.6]), ˜ V is holomorphic.(3) By Proposition 2.2, ˜ V is reductive. Since σ h acts trivially on H , ( V σ h ) contains H .By Proposition 2.3, any element of H is semisimple in ˜ V . Hence H is a toral subalgebraof ˜ V . Let us show that the centralizer z of H in ˜ V is H itself. Clearly, H ⊂ z . Let v = v + v ∈ z , where v ∈ ( V σ h ) and v ∈ ( V ( h ) ) . Since H ⊂ ( V σ h ) and ( V ( h ) ) isa ( V σ h ) -module, both v and v belong to z . If v = 0, then, by Lemma 3.7, v is alsoa weight vector in V for H and its weight is − P ti =1 k i h i , which contradicts (d). Hence v = 0. Since H is a Cartan subalgebra of V and v = v ∈ ( V σ h ) ⊂ V , we have v ∈ H ,and hence z ⊂ H . Thus z = H , and H is a Cartan subalgebra of ˜ V . (cid:3) he next proposition will be used to identify the Lie algebra structure of ˜ V . Proposition 5.5.
Let V be a strongly regular, simple VOA. Let H be a Cartan subalgebraof the reductive Lie algebra V . Let s be a simple Lie subalgebra of V of type X n,k . Assumethat s is spanned by weight vectors for H . (1) There exists a unique simple ideal a of V such that s ⊂ a . (2) Let Y m,k ′ be the type of a in (1). If a long root of s is also a long root of a , then X n iscontained in Y m as a root system, and k = k ′ . Otherwise, X n is contained in the rootsystem consisting of short roots of Y m . In particular, X = A, D , and k ′ = k/ (resp. k ′ = k/ ) if Y = B, C, F (resp. Y = G ).Proof. By the assumption, s is spanned by root vectors of V . Let a be the ideal generatedby s . Then a is a simple ideal by the simplicity of s . The uniqueness of a follows from theuniqueness of the decomposition of V into the direct sum of simple ideals and the center.Let ( ·|· ) s and ( ·|· ) a be the normalized Killing forms on s and on a so that the norm ofany long root is 2, respectively. By the simplicity of s , there exists non-zero ξ ∈ C suchthat(5.4) ξ ( ·|· ) s = ( ·|· ) a on s . Let α be a root of s . Let E α and E − α be root vectors of s associated to α and − α ,respectively. By the assumption, both vectors are also root vectors of a . Then(5.5) ( E α ) (0) ( E − α ) = ( E α | E − α ) s α = ( E α | E − α ) a α ′ , where α ′ is the root of a corresponding to E α . Combining (5.4) and (5.5), we obtain α = ξα ′ .Assume that α is a long root of both s and a . Then ξ = 1 by ( α | α ) s = ( α | α ) a = 2.Thus any long root of s is also a long root of a . Clearly, the restriction of the normalizedinvariant form h·|·i on V to a subVOA is also the normalized invariant form. Hence, k = k ′ since k ( ·|· ) s = h·|·i = k ′ ( ·|· ) a on s (see Lemma 2.1).Assume that α is a long root of s and is not a long root of a . Then ξ = 1, and α ′ = α/ξ is a root of a . It follows from ( α/ξ | α/ξ ) a = (1 /ξ )( α | α ) s = 2 /ξ = 2 that α/ξ is a short rootof a . Since a is simple, it contains at most two different norms of roots. Hence the normsof roots of s are the same, and X is contained in the root system consisting of short rootsof Y . Note that X = A, D and that ξ = 2 (resp. ξ = 3) if Y = B, C, F (resp. Y = G ).By k ( α | α ) s = h α | α i = k ′ ( α | α ) a = k ′ ξ ( α | α ) s , we have k ′ = k/ξ . (cid:3) . Holomorphic VOA of central charge with Lie algebra D , A , G , In this section, applying the Z -orbifold construction to a holomorphic VOA of cen-tral charge 24 with Lie algebra E , G , and certain inner automorphism, we obtain aholomorphic VOA of central charge 24 with Lie algebra D , A , G , .6.1. Simple affine VOA of type E , . Let α , . . . , α be simple roots of type E suchthat ( α i | α j ) = − δ j − i, + 2 δ i,j , ( α p | α ) = − δ p, + 2 δ p, , ( α p | α ) = − δ p, + 2 δ p, for 3 ≤ i ≤ j ≤
6, 1 ≤ p ≤
6. Let { Λ i | ≤ i ≤ } be the set of the fundamental weights with respectto { α i | ≤ i ≤ } . Let L g (3 ,
0) be the simple affine VOA associated with the simpleLie algebra g of type E at level 3. There exist exactly 20 (non-isomorphic) irreducible L g (3 , L g (3 , λ ) with highest weight λ , which are summarized in Table 2. Table 2.
Irreducible L g (3 , E Highest weight 0 Λ , Λ Λ Λ , Λ Λ lowest L (0)-weight 0 26 /
45 4 / / / + Λ , Λ + Λ , Λ + Λ , Λ + Λ + Λ Λ + Λ Λ + Λ Λ + Λ Λ + 2Λ lowest L (0)-weight 13 / / / / / ,2Λ ,3Λ lowest L (0)-weight 56 /
45 2One can easily verify the following lemma, which will be used later.
Lemma 6.1.
For every α ∈ Π( θ, E ) , i.e. root α of E , we have ((Λ − Λ ) / | α ) ≥ − / . Simple affine VOAs of type G , and G , . Let α and α be simple roots oftype G such that ( α | α ) = 2 /
3, ( α | α ) = 2 and ( α | α ) = −
1. Let Λ and Λ bethe fundamental weights with respect to α and α , respectively. Let L g ( k,
0) be thesimple affine VOA associated with the simple Lie algebra g of type G at level k . Thereexist exactly two (resp. four) (non-isomorphic) irreducible L g (1 , L g (1 , λ ) (resp. L g (2 , L g (2 , λ )) with highest weight λ , which are summarized in Tables 3 (resp.Table 4). Table 3.
Irreducible L g (1 , G Highest weight 0 Λ lowest L (0)-weight 0 2 / Lemma 6.2.
Let
Λ = Λ / . able 4. Irreducible L g (2 , G Highest weight 0 Λ Λ lowest L (0)-weight 0 1 / / / For every α ∈ Π( θ, G ) , i.e. root α of G , we have (Λ | α ) ≥ − . (2) For every α ∈ Π(Λ , G ) , we have (Λ | α ) ≥ − / . (3) For every α ∈ Π(2Λ , G ) , we have (Λ | α ) ≥ − . Inner automorphism of a holomorphic VOA with Lie algebra E , G , . Let V be a strongly regular, holomorphic VOA of central charge 24 with Lie algebra E , G , .Note that such a VOA was constructed in [Mi13, SS]. Let V = L i =1 g i be the decom-position into the direct sum of 4 simple ideals, where the type of g is E , , and thetypes of g , g and g are G , . Let H be a Cartan subalgebra of V . Then H ∩ g i is aCartan subalgebra of g i . Let U be the subVOA generated by V . By Proposition 2.2, U ∼ = L g (3 , ⊗ L g (1 , ⊗ L g (1 , ⊗ L g (1 , L ( λ , λ , λ , λ ) denote the irreducible U -module L g (3 , λ ) ⊗ L g (1 , λ ) ⊗ L g (1 , λ ) ⊗ L g (1 , λ ).Let h = 12 (Λ − Λ , Λ , Λ , ∈ M i =1 h i . Note that Λ = θ in g i ( i = 2 , , h h | h i = 34 (Λ − Λ | Λ − Λ ) | g + 14 (Λ | Λ ) | g + 14 (Λ | Λ ) | g = 2 . Lemma 6.3.
All the highest weights of irreducible U -modules L ( λ , λ , λ , λ ) with in-tegral L (0) -weights are given by Table 5. In particular, for any weight ( λ , λ , λ , λ ) inTable 5, we have ( h | ( λ , λ , λ , λ )) ∈ Z / , that is, the spectrum of h (0) on a highest weightvector in L ( λ , λ , λ , λ ) is half-integral.Proof. An irreducible module L ( λ , λ , λ , λ ) has integral L (0)-weights if and only if thesum of the lowest L (0)-weights of L g (3 , λ ), L g (1 , λ ), L g (1 , λ ) and L g (1 , λ ) is integral.Hence, this lemma is immediate from Tables 2 and 3. (cid:3) Lemma 6.4.
The spectrum of the -th mode h (0) on V is half-integral. In particular, σ h is an automorphism of V of order , and the L (0) -weights of irreducible σ h -twisted V -module V ( h ) are half-integral.Proof. Let M be an irreducible U -submodule of V with highest weight λ . Clearly L (0)-weights of M are integral. Hence we can apply Lemma 6.3 to M , and obtain ( h | λ ) ∈ Z / h , we have ( h | L ) ⊂ Z /
2, where L is the root lattice of V . Hencefor any weight µ of M , we have ( h | µ ) ∈ Z / µ ∈ λ + L . Thus the spectra of h (0) able 5. Irreducible modules with integral L (0)-weights: Case E , G , Highest weight (0 , , ,
0) (3Λ , , ,
0) (Λ , Λ , , , , ,
0) (Λ , , Λ , , , , Λ )lowest L (0)-weight 0 2 2Highest weight (Λ + Λ , Λ , Λ ,
0) (Λ , Λ , Λ , Λ ) (Λ + Λ , Λ , Λ , Λ )(Λ + Λ , Λ , , Λ ) (Λ + Λ , Λ , Λ , Λ )(Λ + Λ , , Λ , Λ )lowest L (0)-weight 2 2 3on M are half-integral, and so is on V . The last assertion follows from Lemma 3.2 and(6.1). (cid:3) Identification of the Lie algebra: Case D , A , G , . In this subsection, weidentify the Lie algebra structure of ˜ V .First, we check the assumptions of Section 5 and Theorem 5.4. Proposition 6.5.
The VOA V and the vector h satisfy the assumptions of Section 5 andTheorem 5.4. In particular, ˜ V = V σ h ⊕ ( V ( h ) ) Z is a strongly regular, holomorphic VOAof central charge . In addition, V and ˜ V share a common Cartan subalgebra and theirLie ranks are the same.Proof. By (6.1), we have h h | h i = 2 ∈ Z . By Lemma 6.4 the order of σ h is 2. By Proposition2.4, V is semisimple. Let us check the remaining assumptions (a)–(d) in Theorem 5.4.The assumption (a) (resp. (b)) follows from Proposition 2.4 (resp. Lemmas 6.2 and 6.1).By the definition of h and the level of g i , the assumptions (c) and (d) hold. (cid:3) Next, let us determine the Lie algebra structure of ( V σ h ) . Proposition 6.6.
The set of weights of ( V σ h ) for H is given as follows: { ( a, , , | a ∈ ( M k =2 Z α k ⊕ Z θ ) ∩ Π( θ, E ) } ∪ {± (0 , Λ , , , ± (0 , , Λ , }∪ {± (0 , α , , , ± (0 , , α , } ∪ { (0 , , , α ) | α ∈ Π( θ, G ) } . Moreover, the Lie algebra structure of ( V σ h ) is D , A , A , G , U (1) and the dimensionof ( V σ h ) is , where U (1) means a -dimensional abelian Lie algebra.Proof. One can calculate all the weights of the Lie subalgebra ( V σ h ) with respect to H directly, which determines the Lie algebra structure of ( V σ h ) .The weights {± (0 , α , , } , {± (0 , , α , } are short roots of g ⊕ g and they forma root system of type A , up to scalar. Hence, by Proposition 5.5 (2), the type of the orresponding ideal in ( V σ h ) is A , . Since the other components contain long roots of V , one can determine the levels of the corresponding Lie subalgebras by Proposition 5.5(2). (cid:3) Now, we describe some weights of ( V ( h ) ) for H and find a root subsystem of ˜ V . Proposition 6.7. (1)
The L (0) -weights of the following four vectors in V ( h ) are : (6.2) , E (0 , − Λ , , − , E (0 , , − Λ , − , E (0 , − Λ , , − E (0 , , − Λ , − . Moreover, they are root vectors in ˜ V for H and their roots are given by
12 (3Λ − , Λ , Λ , ,
12 (3Λ − , − Λ , Λ , ,
12 (3Λ − , Λ , − Λ , ,
12 (3Λ − , − Λ , − Λ , , (6.3) respectively. (2) There exist root vectors in ˜ V for H whose roots are the negatives of the roots givenin (6.3) .Proof. It follows from h h | h i = 2, h (0) = 0 and (3.2) that ∈ ( V ( h ) ) . Note that( h | α ) = − α is one of roots (0 , − Λ , , , (0 , , − Λ , L (0)-weights of the four vectors in (6.2) are 1 in ( V ( h ) ) . The explicit construction of L g ( k, H with desired weights, which shows (1).Recall from Theorem 5.4 that ˜ V is a semisimple Lie algebra and that H is a Cartansubalgebra of ˜ V . Hence the vectors in (6.2) are root vectors of ˜ V . Obviously (2) holdssince the negative of a root is also a root in any root system. (cid:3) Proposition 6.8.
The set
Ψ = (cid:26) ± (0 , Λ , , , ± (0 , , Λ , , ±
12 (3Λ − , ( − δ Λ , ( − ε Λ , (cid:12)(cid:12)(cid:12)(cid:12) δ, ε ∈ { , } (cid:27) . consists of roots of ˜ V for H and it forms a root system of type A . Moreover, the level ofthe Lie subalgebra of ˜ V corresponding to Ψ is .Proof. By Propositions 6.6 and 6.7, any element of Ψ is a root of ˜ V . One can see thatthe rank of Ψ is 3, | Ψ | = 12 and { (0 , Λ , , , (0 , , Λ , ,
12 (3Λ − , − Λ , − Λ , } is a set of simple roots of type A . Since the level of the Lie subalgebra of type A corresponding to {± (0 , Λ , , } is 1 (see Proposition 6.6), the level of the Lie subalgebracorresponding to Ψ is also 1 (see Proposition 5.5). (cid:3) inally, we identify the Lie algebra structure of ˜ V . Theorem 6.9.
Let V be a strongly regular, holomorphic VOA of central charge . As-sume that the Lie algebra structure of V is E , G , . Let V = L i =1 g i be the decomposi-tion into the direct sum of simple ideals, where the types of g , g , g and g are E , , G , , G , , G , , respectively. Let h be the vector in a Cartan subalgebra H of V given by h = 12 (Λ − Λ , Λ , Λ , ∈ M i =1 ( H ∩ g i ) , where Λ j are the fundamental weights. Then, applying the Z -orbifold construction to V and σ h , we obtain a strongly regular, holomorphic VOA ˜ V of central charge whoseweight subspace ˜ V has a Lie algebra structure D , A , G , .Proof. By Proposition 6.5, ˜ V is a strongly regular, holomorphic VOA of central charge24 and H is a Cartan subalgebra of ˜ V . Note that ˜ V is semisimple (Proposition 2.4) andthat the rank of ˜ V is 12.By Proposition 6.6, the root system of ˜ V contains a root system of type G . Bythe classification of root systems, it must be an indecomposable component. Hence ˜ V contains a simple ideal of the type G , . It follows from Proposition 2.4 that dim ˜ V = 120;hence we obtain the ratio h ∨ /k = 4 by Proposition 2.4.Recall from Propositions 6.6 and 6.8 that ˜ V contains simple Lie subalgebras of type D , and A , which are spanned by weight vectors for H . By Proposition 5.5 (1), thereexists the simple ideal a (resp. b ) of ˜ V at level k a (resp. k b ) containing the Lie subalgebraof type A , (resp. D , ). By Proposition 5.5 (2), k a (resp. k b ) must be 1 (resp. 3), and bythe ratio h ∨ /k = 4, the dual Coxeter number of a (resp. b ) is 4 (resp. 12). In addition,the root system of a (resp. b ) contains A (resp. D ) as in Proposition 5.5 (2). Hencethe only possible type of a is A , , and possible types of b are E , and D , . If the typeof b is E , , then the remaining Lie rank is 1, and hence the Lie algebra structure of ˜ V is E A G A but its dimension is 110, which contradicts dim ˜ V = 120. Hence the type of b is D , , and we obtain an ideal of type D , A , G , . Comparing the dimensions of thisideal and ˜ V , we complete the proof. (cid:3) Holomorphic VOA of central charge 24 with Lie algebra E , A , In this section, applying the Z -orbifold construction to a holomorphic VOA of centralcharge 24 with Lie algebra D , A , G , and certain inner automorphism, we obtain aholomorphic VOA of central charge 24 with Lie algebra E , A , .7.1. Simple affine VOA of type D , . Let α , . . . , α be simple roots of type D suchthat ( α i | α j ) = − δ | i − j | , + 2 δ i,j , ( α i | α ) = − δ i, and ( α | α ) = 2 for 1 ≤ i, j ≤
6. Let Λ i | ≤ i ≤ } be the set of the fundamental weights with respect to { α i | ≤ i ≤ } .Let L g (3 ,
0) be the simple affine VOA associated with the simple Lie algebra g of type D at level 3. There exist exactly 36 (non-isomorphic) irreducible L g (3 , L g (3 , λ )with highest weight λ , which are summarized in Table 6. Table 6.
Irreducible L g (3 , D ( j ∈ { , } )Highest weight 0 Λ Λ Λ Λ Λ lowest L (0)-weight 0 13 /
30 4 / /
10 4 / / j Λ + Λ Λ + Λ Λ + Λ lowest L (0)-weight 91 /
120 14 /
15 3 / /
10 8 / / + Λ Λ + Λ j + Λ j Λ + 2Λ j Λ + Λ + Λ Λ + Λ j lowest L (0)-weight 2 49 /
40 211 /
120 32 /
15 21 /
10 13 / + Λ j Λ + Λ j Λ + Λ j j j Λ + Λ lowest L (0)-weight 47 /
24 89 /
40 97 /
40 49 /
30 21 / / + Λ Λ + 2Λ lowest L (0)-weight 307 /
120 307 / Lemma 7.1.
Set
Λ = (1 / − Λ )(= (1 / α − α )) . Let j ∈ { , } . (1) For every α ∈ Π( θ, D ) , i.e., root α of D , we have (Λ | α ) ≥ − / . (2) Let λ ∈ { Λ , Λ , Λ + Λ } . Then for every α ∈ Π( λ, D ) , we have (Λ | α ) ≥ − / . (3) Let λ ∈ { Λ + Λ , Λ + Λ , Λ + Λ + Λ } . Then for every α ∈ Π( λ, D ) , we have (Λ | α ) ≥ − . (4) For every α ∈ Π(3Λ , D ) , we have (Λ | α ) ≥ − / . (5) Let λ ∈ { Λ + Λ j , Λ + Λ j , j , Λ + Λ j } . Then for every α ∈ Π( λ, D ) , we have (Λ | α ) ≥ − / .Proof. One can verify this lemma by using explicit descriptions of the weights of irreducible g -modules (cf. [Ka90]). (cid:3) Simple affine VOA of type A , . Let α , α , α be simple roots of type A suchthat ( α i | α j ) = − δ | i − j | , + 2 δ i,j for 1 ≤ i, j ≤
3. Let { Λ i | ≤ i ≤ } be the set of thefundamental weights with respect to { α i | ≤ i ≤ } . Let L g (1 ,
0) be the simple affineVOA associated with the simple Lie algebra g of type A at level 1. There exist exactly4 (non-isomorphic) irreducible L g (1 , L g (1 , λ ) with highest weight λ , which aresummarized in Table 7.One can easily verify the following lemma, which will be used later. able 7. Irreducible L g (1 , A Highest weight 0 Λ Λ Λ lowest L (0)-weight 0 3 / / / Lemma 7.2. (1)
For every α ∈ Π( θ, A ) , i.e., root α of A , we have (Λ | α ) ≥ − . (2) For every α ∈ Π(Λ , A ) , we have (Λ | α ) ≥ − / . (3) For every α ∈ Π(Λ , A ) , we have (Λ | α ) ≥ − / . (4) For every α ∈ Π(Λ , A ) , we have (Λ | α ) ≥ − / . Inner automorphism of a holomorphic VOA with Lie algebra D , A , G , . Let V be a strongly regular, holomorphic VOA of central charge 24 with Lie algebra D , A , G , . Note that such a VOA was constructed in the previous section. Let V = L i =1 g i be the decomposition into the direct sum of 3 simple ideals, where the typesof g , g and g are D , , A , and G , , respectively. Let H be a Cartan subalgebra of V . Then g i ∩ H is a Cartan subalgebra of g i . Let U be the subVOA generated by V .By Proposition 2.2, U ∼ = L g (3 , ⊗ L g (1 , ⊗ L g (1 , L ( λ , λ , λ ) denote theirreducible U -module L g (3 , λ ) ⊗ L g (1 , λ ) ⊗ L g (1 , λ ). Table 8.
Irreducible modules with integral L (0)-weights: Case D , A , G , Highest weight (0 , ,
0) (Λ , Λ , Λ ) (Λ + Λ + Λ , Λ , Λ )(Λ , Λ ,
0) (Λ + Λ j , Λ k , Λ )(Λ + Λ , ,
0) (3Λ j , Λ k , , Λ , + Λ j , Λ k , Λ )(Λ + Λ , , Λ )(Λ + Λ , , Λ )(Λ + Λ j , Λ k , L (0)-weight 0 2 3Let h = 12 (Λ − Λ , , Λ ) ∈ M i =1 ( g i ∩ H ) . Then h h | h i = 34 (Λ − Λ | Λ − Λ ) | g + (Λ | Λ ) | g + 14 (Λ | Λ ) | g = 2 . Lemma 7.3.
All the highest weights of irreducible U -modules L ( λ , λ , λ ) with integral L (0) -weights are given by Table 8, where k ∈ { , } and j ∈ { , } in the table. Inparticular, for any weight ( λ , λ , λ ) in Table 8, we have ( h | ( λ , λ , λ )) ∈ Z / , that is,the spectrum of h (0) on a highest weight vector in L ( λ , λ , λ ) is half-integral. roof. This lemma is immediate from Tables 3, 6 and 7 (cf. the proof of Lemma 6.3). (cid:3)
Lemma 7.4.
The spectrum of the -th mode h (0) on V is half-integral. In particular, σ h is an automorphism of V of order , and the irreducible σ h -twisted V -module V ( h ) hashalf-integral L (0) -weights.Proof. One can prove this lemma by the exactly the same way as in Lemma 6.4 if we useLemma 7.3 instead of Lemma 6.3. (cid:3)
Identification of the Lie algebra: Case E , A , . In this subsection, we identifythe Lie algebra structure of ˜ V .First, we determine the Lie algebra structure of ( V σ h ) . Proposition 7.5.
The set of all the weights of ( V σ h ) for H is given as follows: { ( α, , | α ∈ ( M k =1 Z α k ⊕ Z θ ) ∩ Π( θ, D ) } ∪ { (0 , α, | α ∈ Π( θ, A ) }∪ {± (0 , , α ) , ± (0 , , Λ ) } . Moreover, the Lie algebra structure of ( V σ h ) is D , A , A , A , U (1) and dim( V σ h ) = 88 .Proof. We can find all the weights of ( V σ h ) for H ⊂ ( V σ h ) from the definition of h , andwe know that the Lie algebra structure of ( V σ h ) is D A A U (1).By the exactly the same arguments as in the proof of Proposition 6.6, we can determinethe levels of the simple ideals of ( V σ h ) . (cid:3) Next, we determine the lowest L (0)-weight of V ( h ) . Proposition 7.6.
The lowest L (0) -weight of the irreducible σ h -twisted V -module V ( h ) is . In particular, ( V ( h ) ) / = 0 .Proof. By h h | h i = 2, we have ∈ ( V ( h ) ) by (3.2). Let M ∼ = L ( λ , λ , λ ) be an irre-ducible U -submodule of V . Let ℓ and ℓ ( h ) be the lowest L (0)-weights of M and of M ( h ) ,respectively. It suffices to show that ℓ ( h ) ≥
1. By (3.7) and h h | h i = 2, we have ℓ ( h ) = X i =1 (cid:18) ( λ i + 2 ρ | λ i )2( k i + h ∨ ) (cid:19) + X i =1 (min { ( h i | λ i ) | λ ∈ Π( λ i , X i ) } ) + X i =1 (cid:18) k i ( h i | h i )2 (cid:19) = ℓ + X i =1 (min { ( h i | λ i ) | λ ∈ Π( λ i , X i ) } ) + 1 , where X = D , X = A and X = G . By Lemma 7.3, ( λ , λ , λ ) is one of Table 8,and one can see that ℓ ( h ) ≥ (cid:3) Finally, we identify the Lie algebra structure of ˜ V . heorem 7.7. Let V be a strongly regular, holomorphic VOA of central charge . As-sume that the Lie algebra structure of V is D , A , G , . Let V = L i =1 g i be the decom-position into the direct sum of simple ideals, where the types of g , g and g are D , , A , and G , , respectively. Let h be the vector in a Cartan subalgebra H given by h = 12 (Λ − Λ , , Λ ) ∈ M i =1 ( H ∩ g i ) , where Λ i are the fundamental weights. Then, applying the Z -orbifold construction to V and σ h , we obtain a strongly regular, holomorphic VOA ˜ V of central charge whoseweight subspace ˜ V has a Lie algebra structure E , A , .Proof. By the exactly the same way as in Proposition 6.5, we can apply the Z -orbifoldconstruction to V and σ h , and we obtain a strongly regular, holomorphic VOA ˜ V ofcentral charge 24. Notice that ˜ V is a semisimple Lie algebra of rank 12. By Proposition7.5, we have dim( V σ h ) = 88, and by Proposition 7.6, we have ( V ( h ) ) / = 0. By Theorem4.3 (2), dim ˜ V = 3 × dim( V σ h ) − dim V + 24 = 168;hence we obtain the ratio h ∨ /k = 6 by Proposition 2.4. By Proposition 7.5. ˜ V containssimple Lie subalgebras of type D , and A , which are spanned by weight vectors for H .By Proposition 5.5 (1), there exists a simple ideal a (resp. b ) of ˜ V at level k a (resp. k b )containing the Lie subalgebra of type D , (resp. A , ). By Proposition 5.5 (2), k a (resp. k b ) must be 3 (resp. 1), and by the ratio h ∨ /k = 6, the dual Coxeter number of a (resp. b ) is 18 (resp. 6). In addition, the root system of a (resp. b ) contains D (reps. A ) asin Proposition 5.5 (2). Hence the possible types of a (resp. b ) are E , and D , (resp. A , and D , ). Since dim ˜ V = 168 and the dimension of a Lie algebra of type D is 190,the type of a is E , . If the type of a is D , , then the remaining rank is 1, and hencethe type of ˜ V must be A D E but its dimension is 164, which is a contradiction. Thusthe type of b is A , . Therefore we obtain an ideal of ˜ V of type E , A , . Comparing thedimensions of this ideal and ˜ V , we complete the proof. (cid:3) Holomorphic VOA of central charge 24 with Lie algebra A , A , In this section, applying the Z -orbifold construction to a holomorphic VOA of cen-tral charge 24 with Lie algebra E , A , and certain inner automorphism, we obtain aholomorphic VOA of central charge 24 with Lie algebra A , A , .8.1. Simple affine VOA of type E , . Let α , . . . , α be simple roots of type E suchthat ( α i | α j ) = − δ j − i, for 3 ≤ i < j ≤
7, and ( α p | α p ) = 2, ( α p | α ) = − δ p, and ( α p | α ) = − δ p, for 1 ≤ p ≤
7. Let { Λ i | ≤ i ≤ } be the set of the fundamental weights withrespect to { α i | ≤ i ≤ } . Let L g (3 ,
0) be the simple affine VOA associated with he simple Lie algebra g of type E at level 3. There exist exactly 12 (non-isomorphic)irreducible L g (3 , L g (3 , λ ) with highest weight λ , which are summarized inTable 9. Table 9.
Irreducible L g (3 , E Highest weight 0 Λ Λ Λ Λ Λ lowest L (0)-weight 0 6 / / / /
28 4 / Λ + Λ Λ + Λ Λ + Λ lowest L (0)-weight 19 /
28 19 /
12 2 59 /
28 10 / / Lemma 8.1.
For every α ∈ Π( θ, E ) , i.e., root α of E , we have (Λ / | α ) ≥ − . Simple affine VOA of type A , . Let α , α , . . . , α be simple roots of type A suchthat ( α i | α j ) = − δ | i − j | , + 2 δ i,j . Let { Λ i | ≤ i ≤ } be the set of the fundamental weightswith respect to { α i | ≤ i ≤ } . Let L g (1 ,
0) be the simple affine VOA associated withthe simple Lie algebra g of type A at level 1. There exist exactly 6 (non-isomorphic)irreducible L g (1 , L g (1 , λ ) with highest weight λ , which are summarized inTable 10. Table 10.
Irreducible L g (1 , A Highest weight 0 Λ Λ Λ Λ Λ lowest L (0)-weight 0 5 /
12 2 / / / / Lemma 8.2.
For every α ∈ Π( θ, A ) , i.e., root α of A , (Λ / | α ) ≥ − / . Inner automorphism of a holomorphic VOA with Lie algebra E , A , . Let V be a strongly regular, holomorphic VOA of central charge 24 with Lie algebra E , A , .Note that such a VOA was constructed in the previous section. Let V = L i =1 g i bethe decomposition into the direct sum of 2 simple ideals, where the types of g and g are E , and A , , respectively. Let H be a Cartan subalgebra of V . Then g i ∩ H is aCartan subalgebra of g i . Let U be the subVOA generated by V . By Proposition 2.2, U ∼ = L g (3 , ⊗ L g (1 , L ( λ , λ ) denote the irreducible U -module isomorphic to L g (3 , λ ) ⊗ L g (1 , λ ).Let h = 12 (Λ , Λ ) ∈ M i =1 ( g i ∩ H ) . able 11. Irreducible modules with integral L (0)-weights: Case E , A , Highest weight (0 ,
0) (Λ , Λ ) (3Λ , Λ )(Λ , Λ )(Λ , Λ )(Λ + Λ , Λ )(Λ + Λ , Λ )(Λ + Λ , L (0)-weight 0 2 3Then h h | h i = 34 (Λ | Λ ) | g + 14 (Λ | Λ ) | g = 3 . Lemma 8.3.
All the highest weights of irreducible U -modules L ( λ , λ ) with integral L (0) -weights are given by Table 11. In particular, for any weight ( λ , λ ) in Table 11, we have ( h | ( λ , λ )) ∈ Z / , that is, the spectrum of h (0) on a highest weight vector in L ( λ , λ ) ishalf-integral.Proof. This lemma is immediate from Tables 9 and 10 (cf. the proof of Lemma 6.3). (cid:3)
Lemma 8.4.
The spectrum of the -th mode h (0) on V is half-integral. In particular, σ h is an automorphism of V of order , and the irreducible σ h -twisted V -module V ( h ) hashalf-integral L (0) -weights.Proof. One can prove this lemma by the exactly the same way as in Lemma 6.4 if we useLemma 8.3 instead of Lemma 6.3. (cid:3)
Identification of the Lie algebra: Case A , A , . In this subsection, we identifythe Lie algebra structure of ˜ V . Proposition 8.5.
The set of all the weights of ( V σ h ) for H is given as follows: { ( α, | α ∈ Π( θ, E ) , ( α | Λ ) ∈ Z } ∪ { (0 , α ) | α ∈ Π( θ, A ) , ( α | Λ ) ∈ Z } . Moreover, the Lie algebra structure of ( V σ h ) is A , A , U (1) and dim( V σ h ) = 80 .Proof. We can find all the weights of ( V σ h ) for H ⊂ ( V σ h ) from the definition of h , andwe can check that the Lie algebra structure of ( V σ h ) is A A U (1).Since the type of any simple ideal of V is A n , the level of a simple ideal of ( V σ h ) isequal to that of the ideal of V containing it by Proposition 5.5 (2). (cid:3) Theorem 8.6.
Let V be a strongly regular, holomorphic VOA of central charge . As-sume that the Lie algebra structure of V is E , A , . Let V = L i =1 g i be the decomposi-tion into the direct sum of simple ideals, where the types of g and g are E , and A , , espectively. Let h be the vector in a Cartan subalgebra H of V given by h = 12 (Λ , Λ ) ∈ M i =1 ( H ∩ g i ) , where Λ i are the fundamental weights. Then applying the Z -orbifold construction to V and σ h , we obtain a strongly regular, holomorphic VOA ˜ V of central charge whoseweight subspace ˜ V has a Lie algebra structure A , A , .Proof. By the exactly the same way as in Proposition 6.5, we can apply the Z -orbifoldconstruction to V and σ h , and we obtain a strongly regular, holomorphic VOA ˜ V ofcentral charge 24. Notice that ˜ V is a semisimple Lie algebra of rank 12. By Proposition8.5, dim( V σ h ) = 88. By Theorem 4.3 (2), we havedim ˜ V = 3 × dim( V σ h ) − dim V + 24 × (1 − dim( V ( h ) ) / ) = 96 − × dim( V ( h ) ) / . Since dim ˜ V ≥ dim( V σ h ) = 88, we have dim( V ( h ) ) / = 0, and dim ˜ V = 96. Note thatthe ratio h ∨ /k is 3 by Proposition 2.4. By Proposition 8.5, ˜ V contains a simple Liesubalgebra of type A , which is spanned by weight vectors for H .By Proposition 5.5 (1), there exists a simple ideal a of ˜ V at level k containing the Liesubalgebra of type A , . By Proposition 5.5 (2), k must be 3, and by the ratio h ∨ /k = 3,the dual Coxeter number of a is 9. In addition, the root system of a contains A as inProposition 5.5 (2). Hence the only possible type of a is A , . By Proposition 8.5, ˜ V also contains a Lie subalgebra of type A , , which has trivial intersection with a sincethe levels are different and A , is spanned by weight vectors for H . Hence ˜ V contains aLie subalgebra of type A , A , . Comparing the dimensions of this subalgebra and ˜ V , wecomplete this theorem. (cid:3) Holomorphic VOA of central charge 24 with Lie algebra A , C , A , In this section, applying the Z -orbifold construction to a holomorphic VOA of centralcharge 24 with Lie algebra C , G , A , and certain inner automorphism, we obtain aholomorphic VOA of central charge 24 with Lie algebra A , C , A , .9.1. Simple affine VOA of type C , . Let α , α , . . . , α be simple roots of type A such that ( α i | α j ) = − δ | i − j | , / δ i,j , 1 ≤ i, j ≤
4, ( α k | α ) = − δ k, and ( α | α ) = 2. Let { Λ i | ≤ i ≤ } be the set of the fundamental weights with respect to { α i | ≤ i ≤ } .Let L g (3 ,
0) be the simple affine VOA associated with the simple Lie algebra g of type C at level 3. There exist exactly 56 (non-isomorphic) irreducible L g (3 , L g (3 , λ )with highest weight λ , which are summarized in Table 12.One can easily verify the following lemma, which will be used later. Lemma 9.1.
For every α ∈ Π( θ, C ) , i.e., root α of C , (Λ / | α ) ≥ − / . able 12. Irreducible L g (3 , C Highest weight 0 Λ Λ Λ Λ Λ Λ lowest L (0)-weight 0 11 /
36 5 / / / /
36 2 / Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ lowest L (0)-weight 11 /
12 10 / / / / /
12 14 / Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ lowest L (0)-weight 59 /
36 5 / /
36 17 / /
12 20 / Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ lowest L (0)-weight 13 /
12 4 / /
36 5 / / /
36 11 / Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ lowest L (0)-weight 71 /
36 37 /
18 25 /
12 20 / /
36 29 /
12 5 / Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ lowest L (0)-weight 95 /
36 2 79 /
36 7 / /
12 22 / / Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ lowest L (0)-weight 8 / / /
36 3 11 / / / Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ lowest L (0)-weight 37 /
12 19 / /
36 10 / /
12 32 / / Simple affine VOA of type A , . Let α be a simple root of type A such that( α | α ) = 2. Then the fundamental weight is Λ = α /
2. Let L g (1 ,
0) be the simple affineVOA associated with the simple Lie algebra g of type A at level 1. There exist exactly2 (non-isomorphic) irreducible L g (1 , L g (1 , λ ) with highest weight λ , which aresummarized in Table 13. Table 13.
Irreducible L g (1 , A Highest weight 0 Λ lowest L (0)-weight 0 1 / Lemma 9.2.
For every α ∈ Π( θ, A ) , i.e. root α of A , we have (Λ / | α ) ≥ − / . Inner automorphism of a holomorphic VOA with Lie algebra C , G , A , . Let V be a strongly regular, holomorphic VOA of central charge 24 with Lie algebra C , G , A , . In this subsection, we assume the existence of such a VOA, which has notbeen confirmed yet. Let V = L i =1 g i be the decomposition into the direct sum of 3simple ideals, where the types of g , g and g are C , , G , and A , , respectively. Let H be a Cartan subalgebra of V . Then for i = 1 , , g i ∩ H is a Cartan subalgebra of g i . et U be the subVOA generated by V . Note that U ∼ = L g (3 , ⊗ L g (2 , ⊗ L g (1 , L ( λ , λ , λ ) denote the irreducible U -module L g (3 , λ ) ⊗ L g (2 , λ ) ⊗ L g (1 , λ ). Table 14.
Irreducible modules with lowest L (0)-weights in Z ≥ : Case C , G , A , Highest weight (2Λ , ,
0) (Λ + 2Λ , ,
0) (3Λ , Λ , , ,
0) (Λ + Λ + Λ , Λ ,
0) (3Λ , , Λ )(2Λ , Λ ,
0) (2Λ , ,
0) (2Λ + Λ , Λ , Λ )(2Λ + Λ , Λ ,
0) (Λ + Λ + Λ , ,
0) (2Λ + Λ , , Λ )(2Λ , ,
0) (2Λ + Λ , Λ ,
0) (Λ + 2Λ , Λ , Λ )(Λ + Λ , Λ ,
0) (3Λ , , Λ )(2Λ + Λ , Λ ,
0) (Λ + 2Λ , Λ , Λ )(2Λ + Λ , , Λ ) (2Λ + Λ , Λ , Λ )(Λ + Λ , Λ , Λ ) (Λ + Λ + Λ , , Λ )(Λ , , Λ ) (Λ + Λ , Λ , Λ )(3Λ , Λ , Λ ) (Λ + 2Λ , Λ , Λ )lowest L (0)-weight 2 3 4Let h = 12 (Λ , Λ , Λ ) ∈ M i =1 ( g i ∩ H ) . Then h h | h i = 34 (Λ | Λ ) | g + 12 (Λ | Λ ) | g + 14 (Λ | Λ ) | g = 3 . Lemma 9.3.
All the highest weights of irreducible U -modules L ( λ , λ , λ ) whose lowest L (0) -weights belong to Z ≥ are given by Table 14. In particular, for any weight ( λ , λ , λ ) in Table 14, we have ( h | ( λ , λ , λ )) ∈ Z / , that is, the spectrum of h (0) on a highest weightvector in L ( λ , λ , λ ) is half-integral.Proof. This lemma is immediate from Tables 4, 12 and 13 (cf. the proof of Lemma 6.3). (cid:3)
Lemma 9.4.
The spectrum of the -th mode h (0) on V is half-integral. In particular, σ h is an automorphism of V of order , and the irreducible σ h -twisted V -module V ( h ) hashalf-integral L (0) -weights.Proof. It follows from V i = U i ( i = 0 ,
1) that any irreducible U -submodule is isomorphicto L ( λ , λ , λ ) for some ( λ , λ , λ ) in Table 14. One can prove this lemma by the exactlythe same way as in Lemma 6.4 if we use Lemma 9.3 instead of Lemma 6.3. (cid:3) .4. Identification of the Lie algebra: Case A , C , A , . In this subsection, weidentify the Lie algebra structure of ˜ V . Proposition 9.5.
The set of all the weights of ( V σ h ) for H is given as follows: { ( α, , | α ∈ Π( θ, C ) ∩ M i =1 Z α i } ∪ {± (0 , α , , ± (0 , Λ , } . Moreover, the Lie algebra structure of ( V σ h ) is A , A , A , U (1) and dim( V σ h ) = 32 .Proof. We can find all the weights of ( V σ h ) for H ⊂ ( V σ h ) from the definition of h , andwe can check that the Lie algebra structure of ( V σ h ) is A A U (1) .By the same argument as in Proposition 6.6, we see that the level of the Lie subalgebraof g of type A with roots {± (0 , Λ , } (resp. {± (0 , α , } ) is 2 (resp. 6) and that thelevel of the Lie algebra of type A is 6. (cid:3) Theorem 9.6.
Let V be a strongly regular, holomorphic VOA of central charge . As-sume that the Lie algebra structure of V is C , G , A , . Let V = L i =1 g i be the decom-position into the direct sum of simple ideals, where the types of g , g and g are C , , G , and A , , respectively. Let h be the vector in a Cartan subalgebra H given by h = 12 (Λ , Λ , Λ ) ∈ M i =1 ( H ∩ g i ) , where Λ i is the fundamental weight. Then applying the Z -orbifold construction to V and σ h , we obtain a strongly regular, holomorphic VOA ˜ V of central charge whose weight subspace ˜ V has the Lie algebra structure A , C , A , .Proof. By the exactly the same way as in Proposition 6.5, we can apply the Z -orbifoldconstruction to V and σ h , and we obtain a strongly regular, holomorphic VOA ˜ V ofcentral charge 24. Notice that ˜ V is a semisimple Lie algebra of rank 8. By Proposition9.5, dim( V σ h ) = 32. By Theorem 4.3 (2), we havedim ˜ V = 3 × dim( V σ h ) − dim V + 24 × (1 − dim( V ( h ) ) / ) = 48 − × dim( V ( h ) ) / . By dim ˜ V ≥ dim( V σ h ) = 32, we have dim( V ( h ) ) / = 0, and dim ˜ V = 48; hence the ratio h ∨ /k is 1 by Proposition 2.4. By Proposition 9.5, ˜ V contains simple Lie subalgebras oftype A , and A , which are spanned by weight vectors for H .By Propositions 5.5 (1), there exists a simple ideal a (resp. b ) of ˜ V at level k a (resp. k b ) containing the Lie subalgebra of type A , (resp. A , ).By Proposition 5.5 (2), k a must be 3 or 6, and by the ratio h ∨ /k = 1 the dual Coxeternumber of a is equal to k a . There is no indecomposable root system such that it contains A and its dual Coxeter number is 3. Hence k a = 6 and the dual Coxeter number is 6.The only possible type of a is A , . y Proposition 5.5 (2), k b must be 1 or 2, and by the ratio h ∨ /k = 1, the dual Coxeternumber of b is equal to k b . There are no indecomposable root system with dual Coxeternumber 1. Hence k b = 2 and the dual Coxeter number of b is 2. The only possible typeof b is A , .Let c be the ideal of ˜ V such that ˜ V = a ⊕ b ⊕ c . Since the type of a ⊕ b is A , A , ,we have dim c = 10 and the rank of c is 2. By the semisimplicity of c , the only possibletype of c is C (= B ). In addition, by the ratio h ∨ /k = 1, its level is 3. Comparing thedimensions, we have ˜ V = a ⊕ b ⊕ c , and the type of ˜ V is A , C , A , . (cid:3) Holomorphic VOA of central charge with Lie algebra D , A , In this section, we will explain how to obtain a holomorphic VOA of central charge24 with Lie algebra D , A , from a holomorphic VOA constructed by applying the Z -orbifold construction to the Niemeier lattice VOA V N ( A ) .10.1. Holomorphic VOA of central charge with Lie algebra A , . Let N be aNiemeier lattice with root lattice A . Let ( ·|· ) be the positive-definite symmetric bilinearform of Q ⊗ Z N , which will be identified with the normalized Killing form ( ·|· ) on a Cartansubalgebra of the weight 1 space of the lattice VOA V N . So we use the same notation.For explicit calculation, we use the standard model for a root lattice of type A , i.e., A = { ( a , . . . , a ) ∈ Z | a + · · · + a = 0 } . Let { α = (1 , − , , , , α = (0 , , − , , , α = (0 , , , − , , α = (0 , , , , − } bea set of simple roots. We also use the glue code given in [CS99, Chapter 16], which is the Z -code generated by the row vectors of . Let τ be the automorphism of N which acts on A as a 5-cycle on the last 5 copies of A ’s. We denote the induced automorphism on V N by the same symbol τ . For the detailsof the lattice VOAs V N , see [Bo86, FLM88].Let h = C ⊗ Z N . We extend the form ( ·|· ) C -bilinearly to h . We also extend theautomorphism τ C -linearly to h . Let h (0) be the subspace of fixed-points of τ in h . Notethat for r ∈ {± , ± } , h (0) is also the subspace of fixed-points of τ r in h since the orderof τ is 5. Define M = ((1 − P ) h ) ∩ N = { α ∈ N | ( α | x ) = 0 for all x ∈ h (0) } , here P is the orthogonal projection from N to h (0) . Let V N [ τ r ], ( r = ± , ± τ r -twisted V N -module ([DLM00]). Such a module was constructed in[DL96] (see [SS, Section 2.2] for a review) explicitly; as a vector space, V N [ τ r ] ∼ = M (1)[ τ r ] ⊗ C [ P ( N )] ⊗ T r , where M (1)[ τ r ] is the “ τ r -twisted” free bosonic space and T r is the unique irreduciblemodule of ˆ M / \ (1 − τ r ) N with certain condition (see [Le85, Propositions 6.1 and 6.2] and[DL96, Remark 4.2] for details). It follows from (1 − τ r ) N = M that dim T r = 1 for r ∈ {± , ± } . By direct calculation, we have(10.1) P ( N ) = (cid:26)
15 (5 a, b, b, b, b, b ) | a ∈ A ∗ , b ∈ A (cid:27) . For r ∈ { , } , set δ = 15 (2 , , , − , − , δ = 15 ( − , , , − , ∈ Q ⊗ Z A , f r = ( δ r , , , , , ∈ h . Then ( f r | f r ) = 2 / r = 1 ,
2. Note that 5 δ is the sum of all fundamental weights of A and δ is a vector in 2 δ + A with minimum norm. We regard f r as a vector in ( V N ) via the canonical injective map h → h ( − ⊂ ( V N ) . Remark . We regard N as a lattice in ( V N ) via the injective map above. Then thebilinear form ( ·|· ) on Q ⊗ Z N coincides with the restriction of the normalized Killing formof ( V N ) to Q ⊗ Z N . In addition, since the level of any simple ideal of ( V N ) is 1, therestriction of the normalized invariant from h·|·i of V N to ( V N ) coincides with ( ·|· ) (seeLemma 2.1).Set σ f = exp( − π √− f ). It follows from f ∈ N/ σ f is an automorphismof V N of order 5. It also follows from τ ( f ) = f that σ f commutes with τ . Thus, theautomorphism g = τ σ f ∈ Aut ( V N )has order 5. Since δ ∈ δ + A , we have f ∈ f + N . Hence( σ f ) = σ f = exp( − π √− f )on V N . By Proposition 3.1, we obtain the irreducible g ǫr -twisted V N -module V N [ τ ǫr ] ( ǫf r ) for ǫ = ± r = 1 ,
2. For convenience, we fix a non-zero vector t ǫr ∈ T ǫr . Then T ǫr = C t ǫr . By (3.2), we have(10.2) L ( ǫf r ) (0) = L (0) + ǫf r (0) + | ǫf r | id. Note that | εf r | = h f r | f r i = ( f r | f r ) = 2 / or ǫ ∈ {± } and r ∈ { , } , we set S ǫr = (cid:26) a + ǫδ r (cid:12)(cid:12)(cid:12)(cid:12) a ∈ A ∗ , | a + ǫδ r | = 25 (cid:27) . Lemma 10.2.
Set β = (0 , − , − , , , β = (2 , , , − , − , β = ( − , − , , , ,β = (1 , , − , − , , β = ( − , , , , − . Then S = { β i | i = 0 , , , , } , S = { β i + β i +1 | i = 0 , , , , } , S − = { β i + β i +1 + β i +2 | i = 0 , , , , } , S − = { β i + β i +1 + β i +2 + β i +3 | i = 0 , , , , } , (10.3) where i + j is interpreted as an integer modulo . In particular, for ǫ ∈ {± } and r ∈ { , } , we have |S ǫr | = 5 .Proof. Let a ∈ A ∗ . It follows from | ǫδ r | = 2 / | a + ǫδ r | = 2 / a | ǫδ r ) = − | a | . By the Schwarz inequality, we also have | ( a | ǫδ r ) | ≤ r | a | . Thus, ( a | ǫδ r ) = − | a | implies | a | ≤ q or | a | ≤ /
5. Therefore, a is a vector withminimum norm in a coset of A ∗ /A . By direct calculations, it is easy to verify that thereexists a unique a in each coset of A ∗ /A such that | a + ǫδ r | = 2 /
5. Indeed, we obtain allvectors in each S ǫr as in (10.3). Hence we have proved this lemma. (cid:3) Lemma 10.3.
For ǫ ∈ {± } and r ∈ { , } , (cid:26) e ( a, , , , , ⊗ t ǫr (cid:12)(cid:12)(cid:12)(cid:12) a ∈ A ∗ , | a + ǫf r | = 25 (cid:27) is a basis of (cid:0) V N [ τ ǫr ] ( ǫf r ) (cid:1) . Moreover, the dimension of (cid:0) V N [ τ ǫr ] ( ǫf r ) (cid:1) is .Proof. Let w ⊗ e x ⊗ t ǫr ∈ V N [ τ ǫr ] ( ǫf r ) ( w ∈ M (1)[ τ ], x ∈ P ( N )) be a vector whose L (0)-weight is 1. By [DL96, (6.28)], it is straightforward to show that the L (0)-weight of t ǫr ∈ V N [ τ ǫr ] is 4 /
5. Let ℓ be the L (0)-weight of w in M (1)[ τ ], which belongs to Z ≥ .Then by (10.2), the L (0)-weight of w ⊗ e x ⊗ t ǫr in the twisted module V N [ τ ǫr ] ( ǫf r ) is ℓ + | x | ǫ ( f r | x ) + | f r | ℓ + | x + ǫf r | , hich is equal to 1 by the assumption. Hence ℓ = 0, and we may assume that w = 1. Inaddition, we obtain(10.4) | x + ǫf r | = 25 . Let a ∈ A ∗ , b ∈ A such that x = (5 a, b, b, b, b, b ) / ∈ P ( N ) (see (10.1)). Then(10.5) | x + ǫf r | = | a + ǫδ r | + | b | . Let us show that | a + ǫδ r | ≥ for any a ∈ A ∗ . Clearly, | a + ǫδ r | = | a | + | δ r | + 2 ǫ ( a | δ r ) . Since δ r ∈ A and a ∈ A ∗ , we have | a | ∈ Z and ( a | δ r ) ∈ Z . Hence | a + ǫδ r | ∈ Z ≥ as | δ r | = . Moreover, δ r / ∈ A ∗ and hence a + ǫδ r = 0. Thus(10.6) | a + ǫδ r | ≥ . By (10.4), (10.5) and (10.6), we have b = 0 and | a + ǫδ r | = 2 /
5, which proves the formerassertion.The latter assertion follows from a + ǫδ r ∈ S ǫr and Lemma 10.2. (cid:3) Now consider the following V gN -module:˜ V N,g = V gN ⊕ ( V N [ τ ] ( f ) ) Z ⊕ ( V N [ τ ] ( f ) ) Z ⊕ ( V N [ τ − ] ( − f ) ) Z ⊕ ( V N [ τ − ] ( − f ) ) Z . Remark . It is claimed by M¨oller Sven that ˜ V N,g is a strongly regular, holomorphicVOA of central charge 24 and that it is a simple current extension of V gN graded by Z . Proposition 10.5.
Suppose that ˜ V N,g defined as above is a strongly, holomorphic VOA ofcentral charge . Then dim( ˜ V N,g ) = 48 and the Lie algebra structure of ( ˜ V N,g ) is A , .Proof. Recall that h (0) = { ( α, β, β, β, β, β ) | α, β ∈ C ⊗ Z A } . We view h (0) as a subspaceof ( V gN ) . First we note that( V gN ) = h (0) ⊕ Span C ( X r = − τ r (cid:0) e (0 ,α, , , , (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ∈ A , ( α | α ) = 2 ) . The corresponding Lie algebra structure on ( V gN ) is A , U (1) and h (0) is a Cartan sub-algebra of ( V gN ) .Recall from (3.1) that for any x = ( x , x , . . . , x ) ∈ h (0) ⊂ ( V gN ) , we have x ( ǫf r )(0) = x (0) + ( x | ǫf r ) id on V N [ τ ǫr ]( σ ǫrf ). Hence for w ⊗ e ( a, , , , , ⊗ t ǫr ∈ ( V N [ τ ǫr ] ( ǫf r ) ) ,(10.7) x ( ǫf r )(0) ( w ⊗ e ( a, , , , , ⊗ t ǫr ) = ( x | a + ǫδ r ) w ⊗ e ( a, , , , , ⊗ t ǫr . ecall also that x (0) = 0 on M (1)[ τ ] and on T ǫr by the explicit description of vertexoperators in [Le85, DL96] (cf. [SS]).Let β , β , . . . , β be the vectors in Q ⊗ Z A given in Lemma 10.2. Notice that S ǫr inLemma 10.2 is the set of all weights of ( V N [ τ ǫr ] ( ǫf r ) ) for the Cartan subalgebra h of ( V gN ) .It is easy to see that ( β i | β j ) = / i = j ; − / | i − j | = 1;0 otherwise . Then, up to a scaling, { β , β , β , β } is a set of simple roots for a root system of A . Bythe descriptions of S ǫr in Lemma 10.2, we see that { ( α, , , , , | α ∈ C ⊗ Z A } ⊕ M r ∈{ , } ,ǫ ∈{± } Span C { x ⊗ e ( a, , , , , ⊗ t ǫr | a + ǫδ r ∈ S ǫr } , forms a Lie subalgebra of type A . Notice that { (0 , β, β, β, β, β ) | β ∈ C ⊗ Z A } is theorthogonal complement of { ( α, , , , , | α ∈ C ⊗ Z A } in the Cartan subalgebra h (0) and that it acts trivially on this Lie subalgebra (see (10.7)). Hence this Lie subalgebra isan ideal.In order to determine the level of this ideal, we modify the invariant form as ( ·|· ) =( ·|· ) / { ( α, , , , , | α ∈ C ⊗ Z A } of h (0) . Then one can see that { ˜ β i = 5 β i | ≤ i ≤ } is a set of simple roots and that h ˜ β i | ˜ β i i = 5( ˜ β i | ˜ β i ) . Hence byLemma 2.1, the level is 5. Therefore the Lie algebra structure of ( ˜ V N,g ) is A , . (cid:3) Remark . It was already claimed in [EMS] that the Lie algebra structure of ( ˜ V N,g ) is A , by using Schellekens’ list.10.2. Inner automorphism of the holomorphic VOA ˜ V N,g . In this subsection, wedefine an inner automorphism of order 2 on ˜ V N,g .Let Λ = ( α + 2 α + 3 α + 4 α ) = (1 , , , , − ∈ A ∗ and Λ ′ = β + 2 β + 3 β + 4 β =(1 , − , , − , ∈ A . Then (Λ ′ , Λ , Λ , Λ , Λ , Λ) ∈ N since (0 , , , , ,
1) belongs to the gluecode of N . Set h = 12 (Λ ′ , Λ , Λ , Λ , Λ , Λ) ∈ N/ . Then h h | h i = ( h | h ) = 2. Since h is fixed by τ , we have h ∈ h (0) ⊂ ( V gN ) ⊂ ( ˜ V N,g ) . Notethat ( h | ǫf r ) = 0 for ǫ ∈ {± } , r ∈ { , } . Now let σ h = exp( − π √− h (0) ) . Then σ h alsodefines an automorphism in Aut ˜ V N,g . Lemma 10.7. On V N and V N [ τ ǫr ] ( ǫf r ) ( ǫ = ± , r = 1 , ), Spec h (0) ⊂ Z / . In particular,the order of σ h is on ˜ V N,g . roof. Since h ∈ N/ N is unimodular, we have Spec h (0) ⊂ Z / V N .Since ( h | ǫf r ) = 0 and the vectors ǫf r and h belong to M (1) ⊂ V N , we have ǫf r ( n ) h = 0for n ≥
0. Hence ∆( ǫf r , z ) h = h , and for w ⊗ e x ⊗ t ǫr ∈ V N [ τ ǫr ] ( ǫf r ) , h ( ǫf r )(0) ( w ⊗ e x ⊗ t ǫr ) = ( h | x ) w ⊗ e x ⊗ t ǫr , where w ∈ M (1)[ τ ǫr ], x = (1 / a, b, b, b, b, b ) ∈ P ( N ), ( a ∈ A ∗ , b ∈ A ) and t ǫr ∈ T ǫr .Since Λ ′ ∈ A and Λ ∈ A ∗ , we obtain( h | x ) = (cid:18) h (cid:12)(cid:12)(cid:12)(cid:12)
15 (5 a, b, b, b, b, b ) (cid:19) = (Λ ′ | a )2 + (Λ | b )2 ∈ Z / , which completes this lemma. (cid:3) Identification of the Lie algebra: Case D , A , . In this subsection, we identifythe Lie algebra structure of the weight 1 subspace of the holomorphic VOA ˜ V which isobtained by applying the Z -orbifold construction to ˜ V N,g and σ h . Proposition 10.8.
Let ˜ V N,g and σ h be defined as above. Then the Lie algebra structureof (cid:16) ( ˜ V N,g ) σ h (cid:17) is A , U (1) and dim (cid:16) ( ˜ V N,g ) σ h (cid:17) = 32 .Proof. By the definitions of α i , β i , Λ and Λ ′ , it follows immediately that( α i | Λ) = δ i, and ( β i | Λ ′ ) = δ i, for i = 1 , , , . Hence, the Lie algebra structure of (cid:16) ( ˜ V N,g ) σ h (cid:17) is A U (1) . In addition, the level of theLie subalgebra of type A is 5 by Proposition 5.5 (2). (cid:3) Lemma 10.9.
The lowest L (0) -weight of the (unique) irreducible σ h -twisted ˜ V N,g -module ( ˜ V N,g ) ( h ) is .Proof. By h h | h i = 2, (3.2), and Lemma 10.7, we know that the L (0)-weights of ( ˜ V N,g ) ( h ) are half-integral. In addition, we have L ( h ) (0) = . In order to prove this lemma,it suffices to show that the lowest L (0)-weights of both ( V N ) ( h ) and (cid:0) V N [ τ ǫr ] ( ǫf r ) (cid:1) ( h ) aregreater than 1 / (cid:16) ˜ V N,g (cid:17) ( h ) ⊂ ( V N ) ( h ) ⊕ M ǫ ∈{± } ,r ∈{ , } (cid:0) V N [ τ ǫr ] ( ǫf r ) (cid:1) ( h ) . Case ( V N ) ( h ) . For a ( − n ) · · · a i ( − n i ) ⊗ e α ∈ V N , ( n i ∈ Z > , α ∈ N, a i ∈ h ), we have L ( h ) (0) ( a ( − n ) · · · a i ( − n i ) ⊗ e α )= ( n + · · · + n i + 12 | α + h | ) ( a ( − n ) · · · a i ( − n i ) ⊗ e α ) . Hence it suffices to show that | α + h | > h ∈ N/
2, we have α + h ∈ N/
2. Let x i ∈ A ∗ (1 ≤ i ≤
6) defined by2( α + h ) = ( x , x , x , x , x , x ) ∈ N. ince Λ ′ , Λ2 / ∈ A ∗ and α ∈ N ⊂ ( A ∗ ) , none of the x i ’s is zero and hence4 | ( α + h ) | = X i =1 x i ≥ × > A ∗ is 4 /
5. Hence, | α + h | > Case (cid:0) V N [ τ ǫr ] ( ǫf r ) (cid:1) ( h ) . On (cid:0) V N [ τ ǫr ] ( ǫf r ) (cid:1) ( h ) , we have( L ( ǫf r ) ) ( h ) (0) = L ( h ) (0) + ǫf r ( h )(0) + | ǫf r | id = L (0) + ( ǫf r + h ) (0) + | h + ǫf r | id. Let w ⊗ e x ⊗ t ǫr ∈ (cid:0) V N [ τ ǫr ] ( ǫf r ) (cid:1) ( h ) with L (0) w = ℓw ( ℓ ≥ L ( ǫf r ) ) ( h ) (0) ( w ⊗ e x ⊗ t ǫr )= (cid:18) ℓ + ( x | x )2 + 45 + ( ǫf r + h | x ) + | h + ǫf r | (cid:19) ( w ⊗ e x ⊗ t ǫr )= (cid:18) ℓ + 45 + | h + ǫ ¯ f r + x | (cid:19) ( w ⊗ e x ⊗ t ǫr ) . Thus, the lowest L (0)-weight of (cid:0) V N [ τ ǫr ]( σ ǫrf ) (cid:1) ( h ) is greater than or equal to 4 /
5, whichcompletes this case. (cid:3)
Theorem 10.10.
Let ˜ V be the strongly regular, holomorphic VOA of central charge which is obtained by applying the Z -orbifold construction to ˜ V N,g and σ h . Then the Liealgebra structure of ˜ V is D , A , .Proof. By h h | h i = 2 and | σ h | = 2, we can apply the Z -orbifold construction to ˜ V N,g and σ h and obtain a VOA ˜ V of central charge 24 (Proposition 5.3). By Lemma 10.9,˜ V is of CFT-type. Similarly to Theorem 5.4, we can see that ˜ V is strongly regular andholomorphic. By the definition of h , the assumption (d) of Theorem 5.4 holds. Hence byTheorem 5.4 (3), ˜ V is a semisimple Lie algebra of rank 8. By Theorem 4.3 (2), Proposition10.8 and Lemma 10.9,dim ˜ V = 3 × dim (cid:16) ( ˜ V N,g ) σ h (cid:17) − dim( ˜ V N,g ) + 24 × (1 − dim(( ˜ V N,g ) ( h ) ) / ) = 72;hence the ratio h ∨ /k is 2 by Proposition 2.4. By Proposition 10.8, ˜ V contains two simpleLie subalgebras of type A , which are spanned by weight vectors for H .By Proposition 5.5 (1), there exists a simple ideal a of ˜ V at level k a containing (oneof) the Lie subalgebra of type A , . By Proposition 5.5 (2), k a is 5, and by Proposition2.4 the dual Coxeter number of a is 10. Hence the possible types of a are A , and D , .Since dim ˜ V = 72 and the dimension of a simple Lie algebra of type A is 99, the type of a is D , . et b be the ideal of ˜ V such that ˜ V = a ⊕ b . Then dim b = 6 and the rank of b is 2.Since b is semisimple, the only possible type of b is A . By the ratio h ∨ /k = 2, the levelof b is 1. Thus the Lie algebra structure of ˜ V is D , A , . (cid:3) Acknowledgement.
The authors wish to thank the referee for useful comments andvaluable suggestions.
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J. Amer. Math. Soc. (1996), 237–302.(C. H. Lam) Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan and Na-tional Center for Theoretical Sciences of Taiwan.
E-mail address : [email protected] (H. Shimakura) Graduate School of Information Sciences, Tohoku University, Sendai980-8579, Japan
E-mail address : [email protected]@m.tohoku.ac.jp