Framed Vertex Operator Algebras, Codes and the Moonshine Module

Transcription

1 Commun. Math. Phys. 93, (998) Communications in Mathematical Physics Springer-Verlag 998 Framed Vertex Operator Algebras, Codes and the Moonshine Module Chongying Dong,, Robert L. Griess Jr.,, Gerald Höhn 3, Department of Mathematics, University of California, Santa Cruz, CA 95064, USA Department of Mathematics, University of Michigan, Ann Arbor, MI , USA 3 Mathematisches Institut, Universität Freiburg, Eckerstr., D-7904 Freiburg, Germany Received: 4 July 997 / Accepted: 8 September 997 Abstract: For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras.. Introduction Vertex operator algebras (VOAs) have been studied by mathematicians for more than a decade, but still very little is known about the general structure of VOAs. Most of the examples so far come from an auxiliary mathematical structure like affine Kac-Moody algebras, Virasoro algebras, integral lattices or are modifications of these (like orbifolds and simple current extensions). We use the definition of VOA as in [FLM], Sect In this paper we develop a general structure theory for a class of VOAs containing a subvoa of the same rank and relatively simple form, namely a tensor product of simple Virasoro VOAs of central charge. We call this the class of framed VOAs, abbreviated FVOAs. It contains important examples of VOAs. We show how VOAs constructed from certain integral lattices can be described as framed VOAs. In the case that the lattice itself comes from a binary code, this can be done even more explicitly. As an application of the general structure theory we describe VOAs of small central charge as FVOAs, especially the moonshine VOA V of central charge 4. The first author is supported by NSF grants DMS , DMS and a research grant from the Committee on Research, UC Santa Cruz. The second author is supported by NSF grant DMS and the University of Michigan Faculty Recognition Grant (993 96). The third author is supported by a research fellowship of the DFG, grant Ho 84/-.

2 408 C. Dong, R. L. Griess Jr., G. Höhn The modules of a VOA together with the intertwining operators can be put together into a larger structure which is a called an intertwining algebra [Hu, Hu]. In the case where the fusion algebra of the VOA is the group algebra of an abelian group G, like for lattice VOAs, this specializes to an abelian intertwining algebra [DL]; also see [Mo]. The description of VOAs containing a fixed VOA with abelian intertwining algebra is relatively simple: They correspond to the subgroups H G such that all the conformal weights of the VOA-modules indexed by H are integral [H3]. The Virasoro VOA of rank gives one of the easiest examples of non abelian intertwining algebras. Sect. can be considered as a study of the extension problem for tensor products of this Virasoro VOAs. It is our hope that the ideas used in this work can be extended to structure theories for VOAs based on other classes of rational subvoas with nonabelian intertwining algebras, like the VOAs belonging to the discrete series representations of the Virasoro algebra [W]. We continue with a more detailed description of the results in this paper. The Virasoro algebra of central charge has just three irreducible unitary highest weight representations, with highest weights h =0,, 6, and the one with h = 0 carries the structure of a simple VOA whose irreducible modules are exactly these irreducible unitary highest weight representations. The relevant fusion rules here (Theorem.3) are relatively simple-looking. A tensor product of r such VOAs, denoted T r, has irreducible representations in bijection with r-tuples (h,...,h r ) such that each h i {0,, 6 }. We are interested in the case of a VOA V containing a subvoa isomorphic to T r. Such a subvoa arises from a Virasoro frame, a set of elements ω,..., ω r such that for each i, the vertex operator components of ω i along with the vacuum element span a copy of the simple Virasoro VOA of central charge and such that these subvoas are mutually commutative and ω + +ω r is the Virasoro element of V. We abbreviate VF for Virasoro frame. Such elements may be characterized internally up to a factor as the unique indecomposable idempotents in the weight subalgebra of T r with respect to the algebra product u v induced from the VOA structure on T r. It was shown in [DMZ] that the moonshine VOA V is a FVOA with r = 48. Partial results on decompositions of V into a direct sum of irreducible T 48 -modules were obtained in [DMZ] and [H]. These results were fundamental in proving that V is holomorphic [D3]. In fact, the desire to understand V was one of the original motivations for us to study FVOAs. In Sect., we describe how the set of r-tuples which occur lead to two linear codes C, D F r where D is contained in the annihilator code C. For self-dual (also called holomorphic) FVOAs we give a proof that they are equal: C = D. Associated to these codes are normal -subgroups G D G C of the subgroup G of the automorphism group Aut(V )ofv which stabilizes the VF (as a set). The group G is finite. We get an accounting of all subvoas of V which contain V 0, the subvoa of G D -invariants. We obtain a general result (Theorem.) that FVOAs are rational, establishing the existence of a new broad class of rational VOAs. The rationality of FVOAs is a very important aspect of their representation theory. In particular, a FVOA has only finitely many irreducible modules. In Sect. 3, we describe the Virasoro decompositions of the lattice VOAs V D d, and closely related VOAs, with respect to a natural subvoa T d. In Sect. 4, we study the familiar situation of the twisted or untwisted lattice associated to a binary doubly-even code of length d 8Z and the twisted and untwisted VOA associated to a lattice. A marking of the code is a partition of its coordinates into

3 Framed Vertex Operator Algebras, Codes and Moonshine Module 409 -sets. A marking determines a D d sublattice in the associated lattices and a VF in the associated VOAs. We give an explicit description of the coset decomposition of the lattices under the D d sublattice, a Z 4-code, and the decomposition of the VOA as a module for the subvoa generated by the VF. As a corollary, we give information about various multiplicities of the decompositions under this subvoa using the symmetrized marked weight enumerator of the marked code or the symmetrized weight enumerator of the Z -code. Finally, Sect. 5 is devoted to applications. Two examples are discussed in detail. Example I is about the Hamming code of length 8, the root lattice E 8 and the VOA V E8. Here, r = 6, and we find at least 5 different VFs. Example II is about the Golay code, the Leech lattice and the moonshine module, V, where r = 48. For every VF inside V, the code C has dimension at most 4. There is a special marking of the Golay code for which this bound of 4 is achieved, and for this marking the complete decomposition polynomial is explicitly given. The D -frames inside the Leech lattice which arise from a marking of the Golay code are characterized by properties of the corresponding Z 4 -codes. Appendix A contains a few special results about orbits on markings of the length 8 Hamming code, Appendix B the stabilizer in M 4 of the above special marking for the Golay code, and Appendix C the structure of the automorphism group of the above code of dimension 4. Appendix D shows that all automorphisms of a lattice VOA which correspond to on the lattice are conjugate. In [M M3], there is a new treatment of the moonshine VOA and there is some overlap with results of this article. In particular, the vertex operator subalgebra similar to our V 0 (see Sect. ) and its representation theory have been independently investigated in [M3]. Notation and terminology. a n b The value of the endomorphism a n on b (see Y (v, z)) The vacuum element of a VOA Aut(V ) The automorphism group of the VOA V binary composition on V see: n th binary compostion B V The conformal block on the torus of the VOA V B n The FVOA (M(0, 0) M(, )) n with binary code C(B n)={(0, 0), (, )}n of length n (B n) 0 The subvoa of B n belonging to the subcode of C(Bn ) consisting of codewords of weights divisible by 4 c An element of F n C A linear binary code, often self-annihilating and doubly-even C The annihilator code of C C = C(V ) The binary code determined by the T r -module structure of V 0. C[L] The complex group algebra of the group L C{L} The twisted complex group algebra of the lattice L; it is the group algebra C[ ˆL] modulo the ideal generated by κ + Co 0 The Conway group which is Aut( ), a finite group of order ; its quotient by the center {±} is a finite simple group d The length of a binary code C, usually divisible by 8 d n 4 The marked binary code {(0, 0, 0, 0), (,,, )} n of length 4n (d n 4 ) 0 The subcode of d n 4 consisting of codewords of weights divisible by 8

4 40 C. Dong, R. L. Griess Jr., G. Höhn D n The index sublattice of Z n consisting of vectors whose coordinate sum is even (the checkerboard lattice") D = D(V ) The binary code of the I {,...,r}with V I 0 δ n The marked Kleinian or F 4 -code {(0, 0), (, )} n of length n (δ n) 0 δ(c) The subcode of δ n consisting of codewords of weights divisible by 4 The number of k with c(k) =(c k,c k ) {(0, ), (, 0)} (L) The Z 4 -code associated to a lattice L with fixed D -frame E 8 The root lattice of the Lie group E 8 (C) ɛ A vector with components + or FVOA Abbreviation for framed vertex operator algebra F The set {M(0),M( ),M( 6 )} G The subgroup of Aut(V ) fixing a VF of V. G C The normal subgroup of G acting trivially on T r G D The normal subgroup of G acting trivially on V 0 G = G 4 The Golay code of length 4 γ γɛ a k An element of Z n 4 A map F Z 4 Ɣ a ɛ A map F n Z n 4 h, h i Weights of elements or modules of a VOA, usually h i {0,, 6 } H 8 The Hamming code of length 8 H = H 6 The hexacode of length 6, a code over F 4 = {0,,ω, ω}or over the Kleinian fourgroup Z Z = {0,a,b,c} I A subset of {,...,r} I +J The symmetric difference, for subsets of {,...,r} κ A central element of order in the group ˆL L An integral lattice, often self-dual and even ˆL A central extension of L by a central subgroup κ L The dual lattice of L L C The even lattice constructed from a doubly-even code C L C The twisted even lattice constructed from a doubly-even code C L(n), L i (n) The generator of a Virasoro algebra given by the expansion Y (ω, z) = n Z L(n)z n, resp. Y (ω i,z)= n Z Li (n)z n. The Leech lattice IM The Monster simple group M 4 The simple Mathieu group of order = 44, 83, 040 M(0), M( ), M( 6 ) The irreducible modules for the Virasoro algebra with central charge M() The canonical irreducible module for Heisenberg algebras M(h,...,h r ) The irreducible T r -module of highest weight (h,...,h r ) m h (V)=m h,...,h r The multiplicity of the T r -module M(h,...,h r ) in the FVOA V ; We think of this as a function of (h,...,h r ) {0,, 6 }r. M A marking of a binary code n A natural number n th binary composition The map V V V which takes the pair (a, b)toa n b Nµ ab k ɛ k A map F C[F 4 ] Nµ,ɛ ab A map F n C[F 4n ] µ A vector with components + or. P V (a, b, c) The decomposition polynomial of a FVOA V r The number of elements in a VF

5 Framed Vertex Operator Algebras, Codes and Moonshine Module 4 Rµ a k A map Z 4 C[F ] Rµ a A map Z n 4 C[F n ] smwe C (x, y, z) The symmetrized marked weight enumerator of a binary code C with marking M swe (A, B, C) The symmetrized weight enumerator of a Z 4 -code Sym r, Sym The symmetric group on a set of r objects, usually the index set {,...,r}, resp. the symmetric group on the set n The Z 4 -code {(0, 0), (, )} n of length n ( n ) 0 The subcode of n consisting of codewords of weights divisible by 4 T A faithful module of dimension m for an extraspecial group of order +m, for some m, or for a finite quotient of some ˆL. T r = M(0) r The tensor product of r simple Virasoro VOAs of rank V An arbitrary VOA, often holomorphic = self-dual V (c) The submodule of the FVOA V isomorphic to M( c ) V L The VOA constructed from an even lattice L VL T The Z -twisted module of the lattice VOA V L Ṽ L The twisted VOA constructed from an even lattice L VF Abbreviation for Virasoro frame VOA Abbreviation for vertex operator algebra V I The sum of irreducible T r -submodules of V isomorphic to M(h,...,h r ) with h i = 6 if and only if i I V 0 = V This is V I, for I =0 V The moonshine VOA, or moonshine module W (R) The Weyl group of type R, a root system. Y (v, z) = n Z v nz n The vertex operator associated to a vector v Ξ, Ξ 3 Two D8 /D 8-codes of length and 3 ω, ω i Virasoro elements of rank r,, respectively The all ones vector (,,...,) in F n.. Framed Vertex Operator Algebras Recall that the Virasoro algebra of central charge has three irreducible unitary representations M(h) of highest weights h =0,, 6 (cf. [FQS, GKO, KR]). Moreover, M(0) can be made into a simple vertex operator algebra with central charge (cf. [FZ]). In [DMZ], a class of simple vertex operator algebras (V,Y,,ω) containing an even number of commuting Virasoro algebras of rank were defined. Definition.. Let r be any natural number. A simple vertex operator algebra V is called a framed vertex operator algebra (FVOA) if the following conditions are satisfied: There exist ω i V for i =,..., r such that (a) each ω i generates a copy of the simple Virasoro vertex operator algebra of central charge and the component operators L i (n) of Y (ω i,z) = n Z Li (n)z n satisfy [L i (m),l i (n)] = (m n)l i (m+n)+ m3 m 4 δ m, n, (b) the r Virasoro algebras are mutually commutative, and (c) ω = ω + +ω r. The set {ω,...,ω r }is called a Virasoro frame (VF). In this paper we assume that V is a FVOA. It follows that V is a unitary representation for each of the r Virasoro algebras of central charge. In [DMZ] it is also assumed that V 0 is one-dimensional. This assumption is now a consequence of the simplicity of V :

6 4 C. Dong, R. L. Griess Jr., G. Höhn Lemma.. A FVOA is truncated below from zero: V dimensional: V 0 = C. = n 0 V n and V 0 is one Proof. Let Y (ω i,z)= n Z Li (n)z n. Since V is a unitary representation for the Virasoro algebra generated by the components for Y (ω, z) = n Z L(n)z n as L(n) = r i= Li (n) all weights of V are nonnegative that is, V = n 0 V n. Then each nonzero vector v V 0 is a highest weight vector for the r Virasoro algebras with highest weight (0,...,0). The highest weight module for the i th Virasoro algebra generated by v is necessarily isomorphic to M(0). From the construction of M(0) we see immediately that L i (0)v =0.SoL( )v = i Li ( )v = 0, i.e. v is a vacuum-like vector (see [L]). It is proved in [L] that a simple vertex operator algebra has at most one vacuum-like vector up to a scalar. Since is a vacuum like vector, we conclude that V 0 = C. The following theorem can be found in [DMZ]: Theorem.3. () The VOA M(0) has exactly three irreducible M(0)-modules, M(h), with h =0,, 6, and any module is completely reducible. () The nontrivial fusion rules among these modules are given by: M( ) M( )= M(0), M( ) M( 6 )=M( 6 ) and M( 6 ) M( 6 )=M(0) + M( ). (3) Any module for the tensor product vertex operator algebra T r = M(0) r, where r is a positive integer, is a direct sum of irreducible modules M(h,...,h r ):= M(h ) M(h r )with h i {0,, 6 }. (4) As T r -modules, V = m h,...,h r M(h,...,h r ), h i {0,, 6 } where the nonnegative integer m h,...,h r is the multiplicity of M(h,...,h r )in V. In particular, all the multiplicities are finite and m h,...,h r is at most if all h i are different from 6. Let I be a subset of {,...,r}. Define V I as the sum of all irreducible submodules isomorphic to M(h,...,h r ) such that h i = 6 if and only if i I. Then V = V I. I {,...,r} Here and elsewhere we identify a subset of {,,..., r}with its characteristic function, an integer vector of zeros and ones. We further identify such vectors with their image under the reduction modulo, i.e. we consider them as binary codewords in F r. Interpretation should be clear from the context, e.g. we think of the codeword c as an r-tuple of integers in the expression c. For each c F r let V (c) be the sum of the irreducible submodules isomorphic to M( c,..., c r). Then V 0 = c F V (c). Recall the important fact mentioned in r Theorem.3 (4) that for c Cthe T r -module M( c,..., c r) has multiplicity in V. So, V (c) = 0 or is isomorphic to M( c,..., c r). We can now define two important binary codes C = C(V ) and D = D(V ). Definition.4. For every FVOA V, let C = C(V )={c F r V(c) 0}, and D = D(V )={I F r V I 0}. (.)

7 Framed Vertex Operator Algebras, Codes and Moonshine Module 43 The vector of all multiplicities m h,...,h r will be denoted by m h (V ). Note that the codes C and D are completely determined by m h (V ). The following proposition generalizes Proposition 5. of [DMZ] and Theorem 4.. of [H]. In particular it shows C and D are linear binary codes. As usual we use u n for the component operators of Y (u, z) = n Z u nz n. Proposition.5. () V 0 = V is a simple vertex operator algebra and the V I are irreducible V 0 -modules. Moreover V I and V J are inequivalent if I J. () For any I and J and 0 v V J, span{u n v u V I } = V I+J, where I + J is the symmetric difference of I and J. Moreover, D is an abelian group under the symmetric difference. (3) There is one to one correspondence between the subgroups D 0 of D and the vertex operator subalgebras which contain V 0 via D 0 V D0, where we define V S := I S V I for any subset S of D 0. Moreover V I+D0 is an irreducible V D0 -module for I Dand V I+D0 and V J+D0 are nonisomorphic if the two cosets are different. (4) Let I {,...,r} be given and suppose that (h,...,h r ) and (h,...,h r) are r-tuples with h i, h i {0,, 6 } such that h i = 6 (resp. h i = 6 ) if and only if i I. If both m h,...,h r and m h,...,h are nonzero then m r h,...,h r = m h,...,h. That r is, all irreducible modules inside V I for T r have the same multiplicities. (5) The binary code C is linear and span{u n v u V (c)} = V (c + d) for any c, d C and 0 v V (d). (6) Moreover, there is a one to one correspondence between vertex operator subalgebras of V 0 which contain T r and the subgroups of C, and V is completely reducible for such vertex operator subalgebras whose irreducible modules in V 0 are indexed by the corresponding cosets in C. Proof. Let v V J be nonzero. It follows from Proposition.4 of [DM] or Lemma 6.. of [L] and the simplicity of V that V = span{u n v u V,n Z}. From the fusion rules given in Theorem.3 () and Proposition.0 of [DMZ] we see that u n v V I+J exactly for u V I. In particular, span{u n v u V 0,n Z}=V J. So, V J can be generated by any nonzero vector and V J is a irreducible V 0 -module. Since V I and V J are inequivalent T r -modules if I J they are certainly inequivalent V 0 -modules. By Proposition.9 of [DL], we know that Y (u, z)v 0ifuand v are not 0. Thus V I+J 0 if neither V I or V J are 0. This shows that D is a group. So, we finish the proof of () and (). For (3), we first observe that for a subgroup D 0 of D, () implies that V D0 is a subvoa which contains V 0. On the other hand, since V = V D, V is a completely reducible V 0 - module. Also V I and V J are inequivalent V 0 -modules if I and J are different. Let U be any vertex operator subalgebra of V which contains V 0. Then U is a direct sum of certain V I. Let D 0 be the set of I Dsuch that V I U. Then 0 D 0. Also from () if I, J D 0 then I +J D 0. Thus D 0 is a subgroup of D. In order to see the simplicity of U, we take a vector v V I for some I D 0. Then span{u n v u V J,n Z}=V I+J for any J D 0. It is obvious that {I + J J D 0 }=D 0. Thus U is simple. The proof of the irreducibility of V I+D0 is similar to that of simplicity of V D0. Inequivalence of V I+D0 and V J+D0 is clear as they are inequivalent T r -modules. The proofs of (5) and (6) are similar to that of () and (3). For (4) we set p = m h,...,h r and q = m h,...,h. Let W r,...,w p be submodules of V isomorphic to M(h,...,h r ) such that p i= W i is a direct sum. Let d =(d,...,d r ) C such that V (d) M(h,...,h r )=M(h,...,h r). Set W i = span{u n W i u

8 44 C. Dong, R. L. Griess Jr., G. Höhn V (d), n Z}for i =,...,p. Then W i is isomorphic to M(h,...,h r) for all i. Note that (cf. Proposition 4. of [DM]). Thus p that p q. Similarly, p q. span{u n W i u V (d),n Z} = span{u n v m W i u, v V (d), m,n Z} = span{u n W i u T r,n Z}=W i i= W i must be a direct sum in V. This shows Remark.6. We can also define framed vertex operator superalgebras. The analogue of Proposition.5 still holds. In particular we have the binary codes C and D. Definition.7. Let G be the subgroup of Aut(V ) consisting of automorphisms which stabilize the Virasoro frame {ω i }. Namely, G = {g Aut(V ) g{ω,...,ω r }={ω,...,ω r }}. (.) The two subgroups G C and G D are defined by: G C = {g G g Tr =}, G D ={g G g V 0 =}. Finally, we define the automorphism group Aut(m h (V )) as the subgroup of the group Sym r of permutations of {,...,r} which fixes the multiplicity function m h (V ), i.e. which consists of the permutations σ Sym r such that m h,...,h r = m hσ(),...,h σ(r). It is easy to see that both G D and and G C are normal subgroups of G and G D is a subgroup of G C. Following Miyamoto [M], we define for i =,...,ran involution τ i on V which acts on V I as ifi Iand as otherwise. The group generated by all τ i is a subgroup of the group of all automorphisms of V and is isomorphic to the dual group ˆD of D. We define another group F C which is a subgroup of Aut(V 0 ) and is generated by σ i which acts on M(h,...,h r )by ifh i = and otherwise. The group F C is isomorphic to the dual group Ĉ of C. Theorem.8. () The subgroup G D is isomorphic to the dual group ˆD of D. () G C /G D is isomorphic to a subgroup of the dual group Ĉ of C. (3) G/G C is isomorphic to a subgroup of Aut(m h (V )) Sym r. In particular, G is a finite group. (4) For any g G and a T r -submodule W of V isomorphic to M(h,...,h r )then gw is isomorphic to M(h µ g (),...,h µ g (r) ), where µ g Sym r such that gω i = ω µg(i) for all i. (5) If the eigenvalues of g G C on V I are i and i, then i and i have the same multiplicity. Proof. () Let g G such that g V 0=. Recall from Proposition.5 that V = I D V I. Since each V I is an irreducible V 0 -module we have V I = span{v n u v V 0,n Z} for any nonzero vector u V I. Note that g preserves each homogeneous subspace V I n, which is finite-dimensional. Take u V I to be an eigenvector of g with eigenvalue x I and let v V 0. Then g(v n u)=v n gu = x I v n u. Thus g acts on V I as the constant x I. For any 0 u V I and 0 v V J we have

9 Framed Vertex Operator Algebras, Codes and Moonshine Module 45 0 Y (u, z)v V I+J [[z,z ]]. Since x I+J Y (u, z)v = gy (u, z)v = x I x J Y (u, z)v, we see that x I x J = x I+J. In particular x ˆD and x takes values in {±}. Clearly, each g ˆD acts on V 0 trivially since ˆD is generated by the τ i. This proves (). For () we take g G C. A similar argument as in the first paragraph shows that g V (c) is a constant y c = ± and y c+d = y c y d. In other words we have defined an element y of Ĉ which maps c Cto y c. One can easily see that this gives a group homomorphism from G C to Ĉ with kernel G D. For (3) let g G. Then there exists a unique µ g Sym r such that gω i = ω µg(i). Clearly we have µ gg = µ g µ g for g, g G. It is obvious that the kernel of the map g µ g is G C. In order to prove (4), we take a highest weight vector v of W. Then L i (0)v = h i v for i =,..., r.sol i (0)gv = gl µ g (i) (0)v = h µ v and gv is a highest g weight vector with highest weight (h µ g (),...,h µ g (r)). That is, gw is isomorphic to M(h µ g (),...,h µ g (r) ). Finally, we turn to (5). We first mention how a general g G C acts on V I for I D. Note that g =onv 0 by the proof of (), that is, g G D.Sog =± on each V I. This implies that g is diagonalizable on V I whose eigenvalues are ± ifg =onv I and are ±i if g = onv I. In the second case, let V I = W W p M M q, where all W j, M k are irreducible T r -modules and g = i on each W j and g = i on each M k.onv 0,gis not, since otherwise g is in G D and g would have only ± for eigenvalues, by (). Take an irreducible T r -submodule U of V 0 so that g U =. Set W j = { u nw j u U, n Z}. Then g = i on each W j. Claim. p j= W j is a direct sum. Using associativity, we see that span{u n W j u U, n Z} = span{u m v n W j u, v U, m, n Z} = span{v n W j v T r,n Z}=W j. This proves the claim. Thus p q. Similarly, q p. So they must be equal. This finishes the proof. The results in Proposition.5 () and (3) resp. (5) and (6) can be interpreted by the quantum Galois theory developed in [DM] and [DLM]. For example, Proposition.5 () and (3) is now a special case of Theorems and 3 of [DM] applied for the group G D : Remark.9. Note that V 0 is the space of G D -invariants. There is a one to one correspondence between the subgroups of G D and vertex operator subalgebras of V containing V 0 via H V H. In fact, V H = I H V I, where H = {I D H V I =}. Under the identification of G D with ˆD, the subcode H of D corresponds to the common kernel of the functionals in H. Next we prove that a FVOA is always rational. Recall the definition of rationality and regularity as defined in [DLM]. A vertex operator algebra is called rational if any admissible module is a direct sum of irreducible admissible modules and a rational vertex operator algebra is regular if any weak module is a direct sum of ordinary irreducible modules. (The reader is referred to [DLM3] for the definitions of weak module, admissible module, and ordinary module.)

10 46 C. Dong, R. L. Griess Jr., G. Höhn It is proved in [DLM3] that if V is a rational vertex operator algebra then V has only finitely many irreducible admissible modules and each is an ordinary irreducible module. We need two lemmas. Lemma.0. Let V be a FVOA such that D(V )=0. Then any nonzero weak V -module W contains an ordinary irreducible module. Proof. Since D(V ) = 0 we have the decomposition V = c C V (c). Since T r is regular (see Proposition 3.3 of [DLM]), W is a direct sum of ordinary irreducible T r -modules. Let M be an irreducible T r -submodule of W. Then N := span{u n M u V, n Z} is an ordinary V -module as each span{u n M u V (c), n Z}is an ordinary irreducible T r -module and C is a finite set. For an ordinary V -module X we define m(x) to be the sum of the multiplicities m h,...,h r of all modules M(h,...,h r )in X, i.e., the T r -composition length. Let K be a V -submodule of N such that m(k) is the smallest among all nonzero V -submodules of N. Then K is an irreducible ordinary V -submodule of N and of W. Lemma.. Any FVOA V with D(V )=0is rational. Proof. We must show that any admissible V -module is a direct sum of irreducible ones. Let W be an admissible V -module and M the sum of all irreducible V -submodules. We prove that W = M. Otherwise by Lemma.0 the quotient module W/M has an irreducible submodule W /M, where W is a submodule of W which contains M. Let U be an irreducible T r submodule of W such that U M = 0 and set X := span{v n U v V, n Z}. Then X is a submodule of W and W = M + X. Note that U[c] := span{v n U v V (c), n Z}for each c Cis an irreducible T r -module. Then either U[c] M =0orU[c] M=U[c]. If the latter happens, then Y (v, z)(u + M/M) = 0 in the quotient module W/M, which is impossible by Proposition.9 of [DL]. Thus U[c] M = 0 for all c Cand W = M X. By Lemma.0, X has an irreducible V -submodule Y and certainly M Y strictly contains M. This is a contradiction. Theorem.. Any FVOA V is rational. Proof. Let W be an admissible V -module. Then W is a direct sum of irreducible V 0 - modules by Lemma.. Let M be an irreducible V 0 -module. It is enough to show that M is contained in an irreducible V -submodule of W. First note that there exists a subset I of {,...,r}such that for every irreducible T r -module M(h,...,h r ) inside M we have h i = 6 if and only if i I. Let X be the V -submodule generated by M. Then X = J D X[J] W, where X[J] = span{u nm u V J,n Z}is a V 0 -module. We will show that X is an irreducible V -module. By the fusion rules, we know that for every irreducible T r -submodule of V which is isomorphic to M(h,...,h r ) has h k = 6 if and only if k I + J. The X[J] for J Dare nonisomorphic V 0 -modules as they are nonisomorphic T r -modules. Thus X = J D X[J]. Let Y be a nonzero V -submodule of X. Then Y = J D Y [J], where Y [J] = Y X[J]isaV 0 -module. If Y [J] 0 then span{v n Y J v V J,n Z} 0. Otherwise use the associativity of vertex operators to obtain

11 Framed Vertex Operator Algebras, Codes and Moonshine Module 47 0=span{u m v n Y [J] u, v V J,m,n Z}=span{v n Y [J] v V 0,n Z}=Y[J]. By associativity again we see that span{v n Y [J] v V J,n Z}is a nonzero V 0 - submodule of M. Since M is irreducible it follows immediately that span{v n Y [J] v V J,n Z}=M.SoMis a subspace of Y. Since X is generated by M as a V -module we immediately have X = Y. This shows that X is indeed an irreducible V -module. It should be pointed out that each X[J] in fact is an irreducible V 0 -module. Let 0 u X[J]. Since X = span{v n u v V, n Z} we see that span{v n u v V 0,n Z}=X[J]. Corollary.3. Let V be a FVOA. Then () V has only finitely many irreducible admissible modules and every irreducible admissible V -module is an ordinary irreducible V -module. () V is regular, that is, any weak V -module is a direct sum of ordinary irreducible V -modules. Proof. We have already mentioned that () is true for all rational vertex operator algebra (see [DLM3]). So, () is an immediate consequence of Theorem.. In [DLM] we proved that () is true for any rational vertex operator algebra which has a regular vertex operator subalgebra with the same Virasoro element. Note that T r is such a vertex operator subalgebra of V. Theorem. is very useful. We will see in the later sections that the FVOAs V + and V are rational vertex operator algebras. Theorem. simplifies the original proofs of the rationality of V + in [D3] and V in [DLM]. Most important, we do not use the self-dual property of V (i.e., V is the only irreducible module for itself) as proved in [D3]. It is a interesting problem to find suitable invariants for a FVOA V. Two invariants of V are the binary codes C and D of length r as defined before. They cannot be arbitrary but must satisfy the following conditions: Proposition.4. () The code C is even, i.e. the weight wt(c) = r i= c i Z + of every codeword c Cis divisible by. () The weights of all codewords d Dare divisible by 8. (3) The binary code D is a subcode of the annihilator code C = {d =(d i ) F r (d, c) = i d ic i =0for all c =(c i ) C}. Proof. Let W be a T r -submodule isomorphic to M(h,...,h r ). Then the weight of a highest weight vector of W is h + h + +h r which is necessarily an integer as V is Z-graded. The parts () and () now follow immediately. To see (3), note that for c Cand M V I isomorphic to M(g,...,g r ) one has from the fusion rules given in Theorem.3 () that M = span{u n M(g,...,g r ) u V(c),n Z} V I is isomorphic to M(h,...,h r ) with h i = g i = 6 if i I and h i = 0 (resp. h i = ) if c i +g i =0inF (resp. c i +g i = ). Since the conformal weights g + +g r and h + +h r of M(g,...,g r ) and M(h,...,h r ) are both integral we see that #({i {,...,r} c i =}\I) is an even integer. Thus #{i I c i =}is also even as wt(c) is even. This implies that (d, c) = 0, as required, where d Dis the codeword belonging to I {,...,r}.

12 48 C. Dong, R. L. Griess Jr., G. Höhn Here are a few remarks on the action of Aut(V )onv, which is an action preserving the algebra product a b coming from the VOA structure. Remark.5. () If V is a VOA and is generated as a VOA by V, then Aut(V ) acts faithfully on V. This happens in the case V = V L +, where L is a lattice spanned by its vectors x such that (x, x) =4. () If V is a FVOA, the kernel of the action of Aut(V )onv is contained in the intersection of the groups G C, as we vary over all frames. Hence, this kernel is a finite -group, of nilpotence class at most two and order dividing r, where r = rank(v ). The framed vertex operator algebras with D = 0 can be completely understood in an easy way. Proposition.6. For every even linear code C F r there is up to isomorphism exactly one FVOA V C such that the associated binary codes are C = C and D =0. Proof. Let V Fermi = M(0) M( ) be the super vertex operator algebra as described in [KW]. The (graded) tensor product V r Fermi is a super vertex operator algebra whose code C is the complete code F r (see Remark.6). It has the property, that the even vertex operator subalgebra is the vertex operator algebra associated to the level irreducible highest weight representation for the affine Kac-Moody algebra D r/ if r is even and B (r )/ if r odd (see [H], chapter ). The code C for this vertex operator algebra is the even subcode of F r. Proposition.5 (6) gives, for every even code C Fr,aFVOAV such that C(V )=Cand D(V ) = 0. The uniqueness of the FVOA with code C(V )=C up to isomorphism follows from a general result on the uniqueness of simple current extensions of vertex operator algebras [H3]. This proposition is also proved in a different way by Miyamoto in [M, M3]. Recall that a holomorphic (or self-dual) VOA is a VOA V whose only irreducible module is V itself. In the case of holomorphic FVOAs, we can show that the subcode D C is in fact equal to C. We need some basic facts from [Z] and [DLM4] about the conformal block on the torus B V [Z] of a VOA V. To apply Zhu s modular invariance theorems one has to assume that V is rational and satisfies the C condition. It was proved in [DLM4] that the moonshine VOA satisfies the C condition. The same proof in fact works for any FVOA. We also know from Theorem. that a FVOA is rational. Applying Zhu s result to a FVOA V yields that B V is a finite dimensional complex vector space with a canonical base T Mi indexed by the inequivalent irreducible V - modules M i and that B V carries a natural SL (Z)-module structure ρ V : SL (Z) GL(B V ). Let V and W be two rational VOAs satisfying the C condition. The following two properties of the conformal block follow directly from the definition: (B) B V W = B V B W as SL (Z)-modules and T Mi M j = T Mi T Mj. (B) If W is a subvoa of V with the same Virasoro element then there is a natural SL (Z)-module map ι : B V B W. We also need the following well-known result: The condition that V be a direct sum of highest weight representations for the Virasoro algebra was also required in [Z], but was removed in [DLM4].

13 Framed Vertex Operator Algebras, Codes and Moonshine Module 49 ( ) (B3) For the vertex operator algebra M(0), the action of S = 0 0 SL (Z) on B M(0) in the canonical basis {T M(0),T M( ),T M( 6 )} is given by the matrix / / / / / / / / (.3) 0 Here is a result about binary codes used in the proof of Theorem.9 below: Lemma.7. Let µ n be the n-fold tensor product of the matrix µ =. ( ) considered as a linear endomorphism of the vector space C[F n ] = C[F ] n on the canonical base {e v v F n }. For a subset X Fn denote by χ X = v X e v the characteristic function of X. Then the following relation between a linear code C and its annihilator C holds: χ C = C µ n (χ C ). Remark.8. µ n is a Hadamard matrix of size n and the corresponding linear map is called the Hadamard transform. Proof. For every Z-module R and function f : F n R the following relation holds (cf. Ch. 5, after Lemma of [MaS]) C f(v) = ( ) (u,v) f(v). (.4) v C u C Now let R be the abelian group C[F n ] and define f by f(v) =e vfor all v F n. The left hand side of (.4) is C χ C. Expansion of the right side gives: n µ n (e u )=µ n (χ C ). u C v,...,v n F v F n i= ( ) uivi e v e vn = u C Theorem.9. For a holomorphic FVOA the binary codes C and D satisfy D = C. Proof. The vector of multiplicities m h (V ) can be regarded as an element in the vector space C[F r ] = C[F] r, where F = {M(0),M( ),M( 6 )}. Define two linear maps π, θ : C[F] C[F ]=Ce 0 Ce by π(m(0)) = e 0, π(m( )) = e 0, π(m( 6 )) = e, and θ(m(0)) = e 0, θ(m( )) = e, θ(m( 6 ))=0. Finally let σ : C[F] C[F ] the linear map given by the matrix (.3) relative to the basis F. Now one has π σ = µ θ and thus the following diagram commutes: C[F r ] θ r σ r C[F r ] π r (.5) C[F r ] µ r C[F r ].

14 40 C. Dong, R. L. Griess Jr., G. Höhn By definition, the support of π r (m h (V )) is D F r. From Lemma.7 µ r θ r (m h (V )) = C χ C C[F r ]. Note that the support of χ C is C. These facts together with (.5) imply the theorem if we can show that σ r (m h (V )) = m h (V ). We identify C[F r ] with the conformal block on the torus of the VOA T r by identifying the canonical bases: M = T M. Using (B) and (B3) we observe that σ r = ρ Tr (S), where ρ Tr is the representation ρ Tr : SL (Z) GL(B Tr ) of degree 3 r. Define the shifted graded character ch V (τ) := q c/4 n 0 (dim V n)q n, where q = e πiτ and c is the central charge of V. Since V is holomorphic, the conformal block B V is one dimensional. Then ρ V (S) = (the case ρ V (S) = isimpossible since ch V (i) > 0, where i is the square root of in upper half plane; cf. [H], proof of Cor...3). Now we use (B). The generator T V of B V is mapped by ι to m h,...h r T M(h,...,h r) = m h (V ). Since ι is SL (Z)-equivariant we get σ r (m h (V )) = ρ Tr (S)(m h (V )) = ι (ρ V (S)(T V )) = m h (V ). The same kind of argument was used in the proof of Theorem 4..5 in [H]. 3. Vertex Operator Algebras V D d Let D n = {(x,...,x n ) Z n n i=i x i even} R n,n, be the root lattice of type D n, the checkerboard lattice. In this section, we describe the Virasoro decomposition of modules and twisted modules for the vertex operator algebra V D d. We work in the setting of [FLM] and [DMZ]. In particular L is an even lattice with nondegenerate symmetric Z-bilinear form, ; h = L Z C; ĥ Z is the corresponding Heisenberg algebra; M() is the associated irreducible induced module for ĥ Z such that the canonical central element of ĥ Z acts as ; ( ˆL, ) is the central extension of L by κ κ =, a group of order, with commutator map c 0 (α, β) = α, β +Z;c(, )is the alternating bilinear form given by c(α, β) =( ) c0(α,β) for α, β L; χ is a faithful linear character of κ such that χ(κ) = ; C{L} = Ind ˆL κ C χ ( C[L], linearly), where C χ is the one-dimensional κ -module defined by χ; ι(a) =a C{L}for a ˆL; V L = M() C{L}; = ι(); ω = d r= β r( ), where {β,...,β d }is an orthonormal basis of h; it was proved in [B] and [FLM] that there is a linear map V L (End V L )[[z,z ]], v Y (v, z) = n Zv n z n (v n End V L ) such that V L =(V L,Y,,ω) is a simple vertex operator algebra. Let L = {x h x, L Z}be the dual lattice of L. Then the irreducible modules of V L are the V L+γ (which are defined in [D]) indexed by the elements of the quotient group L /L (see [D]). In fact, V L is a rational vertex operator algebra (see [DLM]). Let θ be the automorphism of ˆL such that θ(a) =a κ ā,ā /. Then θ is a lift of the automorphism of L. We have an automorphism of V L, denoted again by θ, such that θ(u ι(a)) = θ(u) ι(θa) for u M() and a ˆL. (See Appendix D for a fuller discussion.) Here the action of θ on M()isgivenbyθ(α (n ) α k (n k )) = ( ) k α (n ) α k (n k ). The θ-invariants V L + of V L form a simple vertex operator subalgebra and the -eigenspace V L is an irreducible V L + -module (see Theorem of [DM]). Clearly V L = V L + V L.

15 Framed Vertex Operator Algebras, Codes and Moonshine Module 4 Now we take for L the lattice D d = d Zα i, α i,α j =4δ i,j. i= Then L is an even lattice and the central extension ˆL is a direct product of D d with κ and C{L} is simply the group algebra C[L] with basis e α for α L. It is clear that θ(e α )=e α for α D d. We extend the action of θ from V D to V d (D ) = d M() C[L ] such that θ(u e α )=(θu) e α for u M() and α L. One can easily verify that θ has order and θy (u, z)θ = Y (θu, z) for u V D d, where Y (v, z) (v V D d) are the vertex operators on V (D ) d. For any θ-invariant subspace V of V L we use V ± to denote the ±-eigenspaces. First we turn our attention to the case that d =. Then L = Zα = Z = D, where α, α = 4. Note that the dual lattice D is 4 D and {0,,, } is a system of coset representatives of D /D. Set Then ω i V + D. ω = 6 α( ) + 4 (eα + e α ), ω = 6 α( ) 4 (eα +e α ). (3.) Lemma 3.. For D = L = Zα, α, α =4, we have: () V D is a FVOA with r =. () We have the following Virasoro decompositions of V + D and V D : V + D = M(0, 0), V D = M(, ) with highest weight vectors and α( ), respectively. (3) The decompositions for V ± D + are: V + D + = M(, 0), V D + = M(0, ) with highest weight vectors (e α e α ) and (e α + e α ), respectively. (4) For V D+ V D we get, in both cases, (V D+ V D ) ± = M( 6, 6 ) are irre- with highest weight vectors e 4 α ± e 4 α. In fact, both V D+ ducible V D + -modules. and V D

16 4 C. Dong, R. L. Griess Jr., G. Höhn Proof. It was proved in [DMZ] (see Theorem 6.3 there) that Y (ω,z ) = n Z L (n)z n and Y (ω,z )= n Z L (n)z n give two commuting Virasoro algebras with central charge. We first show that the highest weight of α( ) is (, ). Since α( ) V D has the smallest weight in V D it is immediate to see that L i (n)α( )=0 if n>0. It is a straightforward computation by using the definition of vertex operators to show that L (0)α( ) = L (0)α( ) = α( ). Clearly, (V D ) + is a highest weight vector for the Virasoro algebras with highest weight (0, 0). So V D contains two highest weight modules for the two Virasoro algebras with highest weights (0, 0) and (, ). Since M(0, 0) M(, ) and V D have the same graded dimension we conclude that V D = M(0, 0) M(, ) and V D + = M(0, 0), V D = M(, ). This proves () and shows also (): V D is a FVOA with r =. Additionally we see that V D is a unitary representation of the two Virasoro algebras. By Theorem.3 (3) we know that V D+λ, for λ = 0, ±,, is a direct sum of irreducible modules M(h,h ) with h i {0,, 6 }. It is easy to find all highest weight vectors in V D+λ. Part (3) and (4) follow immediately then. We return to the lattice L = d i= Zα i, α i,α j =4δ i,j, L = D d =(Z)d.We sometimes identify L with (Z) d. The component Zα i gives two Virasoro elements ω i and ω i, as in (3.), above. Definition 3.. The VF associated to the FVOAs derived from the D d -lattice is the set {ω,...,ω d }. Corollary 3.3. () The decomposition of V ± into irreducible modules for T D d d is given by = M(h,...,h d ). V ± D d (h i,h i ) {(0, 0), (, )} ( ) #{i h i=0} = ± In particular, V ± is a direct sum of d irreducible modules for T D d d. () Let γ =(γ i ) (D )d such that γ i {0,}. Then we get the decomposition (V D d +γ) ± = M(h,...,h d ). { (h i,h i ) {(0, 0), (, )} if γ i =0, {(0, ), (, 0)} if γ i = ( ) #{i h i =0} =± (3) Let γ =(γ i ) (D )d, such that γ D d, i.e. there is at least one i such that γ i = ±. Then (V D d+γ V D d γ) ±, V D d ±γ have the same decomposition: M(h,...,h d ). { {(0, 0), (, )} if γ i =0, (h i,h i) {(,0), (0, )} if γ i =, {( 6, 6 )} if γ i = ± Proof. Note that V D d is isomorphic to the tensor product vertex operator algebra V D V D (dfactors) and that V D d +γ is isomorphic to the tensor product module V Zα+γ V Zαd +γ d. Thus

17 Framed Vertex Operator Algebras, Codes and Moonshine Module 43 (V D d +γ V D d γ) ± = µ {+, } d µi=± V µ D +γ V µ d D +γ d. The results () and () now follow from Lemma 3. immediately. For (3) it is clear that the decompositions for V D d ±γ hold by Lemma 3.. It remains to show that V D d ±γ and (V D d +γ V D d γ) ± are all isomorphic T d -modules. Note from Lemma 3. that V D+h and V D h are isomorphic T -modules for any h {0,,± }. Thus V D d +γ and V D d γ are isomorphic T d -modules. In fact, θ : V D d +γ V D d γ is such an isomorphism. Thus, (V D d +γ V D d γ) ± = {v±θv v V D d +γ} are isomorphic to V D d +γ as T d -modules. Next we discuss the twisted modules of V L for an arbitrary d-dimensional positive definite even lattice L. Recall from [FLM] the definition of the twisted sectors associated to an even lattice L. Let K = {θ(a)a a ˆL}. Then K =L(bar is the quotient map ˆL L). Also set R := {α L α, L Z}; then R L. Then the inverse image ˆR of R in ˆL is the center of ˆL and K is a subgroup of ˆR. It was proved in [FLM] (Proposition 7.4.8) there are exactly R/L central characters χ : ˆR/K C of ˆL/K such that χ(κk) =. For each such χ, there is a unique (up to equivalence) irreducible ˆL/Kmodule T χ with central character χ and every irreducible ˆL/K-module on which κk acts as is equivalent to one of these. In particular, viewing T χ as an ˆL-module, θa and a agree as operators on T χ for a ˆL. Let ĥ[ ] be the twisted Heisenberg algebra. As in Sect..7 of [FLM] we also denote by M() the unique irreducible ĥ[ ]-module with the canonical central element acting by. Define the twisted space V Tχ L = M() T χ. It was shown in [FLM] and [DL] that there is a linear map such that V Tχ L V L (End V Tχ L )[[z/,z / ]], v Y (v, z) = n Z v n z n is an irreducible θ-twisted module for V L. Moreover, every irreducible for some χ. such that θ-twisted V L -module is isomorphic to V Tχ L Define a linear operator ˆθ d on V Tχ L ˆθ(α ( n ) α k ( n k ) t)=( ) k e dπi/8 α ( n ) α k ( n k ) t for α i h, n i + Z and t T. Then ˆθ d Y (u, z)( ˆθ d ) = Y (θu, z) for u V L (cf. [FLM]). We have the decomposition V Tχ L =(VTχ L )+ (V Tχ L ), where (V Tχ L )+ and (V Tχ L ) are the ˆθ d -eigenspaces with eigenvalues e dπi/8 and e dπi/8 respectively. Then both (V Tχ L )+ and (V Tχ L ) are irreducible V L + -modules (cf. Theorem 5.5 of [DLi]). As before, we now take L = Zα = D with α, α = 4. Then K =L,R=Land R/K = Z. Let χ be the trivial character of R/K and χ the nontrivial character. Then both T χ and T χ are one-dimensional L modules and α acts on T χ± as ±. Lemma 3.4. We have the Virasoro decompositions: () (V Tχ D ) + = M( 6, Tχ ), (VD ) = M( 6, 0).

18 44 C. Dong, R. L. Griess Jr., G. Höhn Proof. Recall from [DL] that () (V Tχ D ) + = M(, 6 ), (V Tχ D ) = M(0, 6 ). V Tχ L = (V Tχ L ) 6 +n n Z,n 0 (see Proposition 6.3 and formula (6.8) of [DL]). Note that (V Tχ L )+ = (V Tχ L ) 6 + +n n Z, n 0 and that (V Tχ L ) = (V Tχ L ) +n. 6 n Z, n 0 Since both (V Tχ L )+ and (V Tχ L ) are irreducible V L + -modules we only need to calculate highest weights for nonzero highest weight vectors in these spaces. Note that T χ is a space of highest weight vectors of (V Tχ L ). One can easily verify that L (0) = 6 and L (0)=0onT χ. Thus (V Tχ L ) = M( 6, 0). Also observe that α( /) T χ is a space of highest weight vectors of (V Tχ L )+. From Lemma 3. we know that α( ) V L = M(, ). Now use the fusion rule given in Theorem.3 to conclude that (V Tχ L )+ = M( 6, ). Part () is proved similarly. As we did in the untwisted case, we now consider the twisted modules for the lattice L = d i= Zα i = D d, α i,α j =4δ i,j, where d is now a positive integer divisible by 8. Then K =L,R=Land R/L = Z d. Thus, there are d irreducible characters for R/L which are denoted by χ J, (where J is a subset of {,...,d}) sending α j to if j J and to otherwise. Then we have χ J = j χ x j, where χ xj is a character of Zα j /Zα j and x j = χ J (a j ). Moreover, T χj = Tχx T χxd. In particular, each T χj is one dimensional. Corollary 3.5. We have the Virasoro decompositions: (V Tχ J ) ± = D d { {( (h i i,h i ) 6, 0), ( 6, )} if i J {(0, 6 ), (, 6 )} if i J ( ) #{j h j = } =±( ) d/8 M(h,...,h d ). Proof. Recall from the proof of Corollary 3.3 that V D d is isomorphic to the tensor product vertex operator algebra V D V D. Note that V Tχ J is isomorphic to the D d tensor product V Tχx D V Tχx d D and ˆθ d is also a tensor product ˆθ ˆθ d.by Lemma 3.4, = M(h,...,h d ). V Tχ J D d { {( (h i i,h i ) 6, 0), ( 6, )} if i J {(0, 6 ), (, 6 )} if i J

19 Framed Vertex Operator Algebras, Codes and Moonshine Module 45 Since ˆθ d =( ) #{j hj= } ( ) d/8 =( ) #{j hj=0} ( ) d/8 on M(h,...,h d )wesee that M(h,...,h d ) embeds in (V Tχ J L ) ± if and only if ( ) #{j hj= } ( ) d/8 = ±. The proof is complete. Remark 3.6. Note that V Tχ J D d is Z graded if d is divisible by 8 (cf. [DL]). In fact (V Tχ J ) + is then the subspace of V Tχ J consisting of vectors of integral weights while D d D d (V Tχ J ) is the subspace of V Tχ J consisting of vectors of non-integral weights. D d D d 4. Vertex Operator Algebras Associated to Binary Codes Let C be a doubly-even linear binary code of length d 8Z containing the all ones vector =(,...,). As mentioned in Sect., we can regard a vector of F d as an element in Z d in an obvious way. One can associate (cf. [CS]) to such a code the two even lattices L C = { (c + x) c C, x (Z) d } and L C = { (c + y) c C, y (Z) d, 4 y i } { (c+y+(,..., )) c C, y (Z)d, 4 ( ( ) d/8 + y i )} and for every self-dual even lattice there are two vertex operator algebras V L and ṼL = V L + (V L T )+ (see [FLM, DGM]). Definition 4.. A marking for the code C is a partition M = {(i,i ),...,(i d,i d )} of the positions,,...,dinto d pairs. A marking M = {(i,i ),...,(i d,i d )} determines the D d sublattice d l= Zα l inside L C and L C, where α k = (e ik + e ik ) and α k = (e ik e ik ) for k =,..., d using {e i} as the standard base of L C Q = Q. Let us simplify notation and arrange for the marking to be M = {(, ), (3, 4),...,(d,d)}. From Definition 3., we see that every such marking defines a system of d commuting Virasoro algebras inside the vertex operator algebras V LC, V LC = Ṽ LC and Ṽ L.As C the main theorem we describe explicitly the decomposition into irreducible T d -modules in terms of the marked code. The triality symmetry of Ṽ L given in [FLM] and [DGM] C is directly visible in this decomposition. (See also [G].) In order to give the Virasoro decompositions in a readable way, we need the next lemma which describes L C and L C in the coordinate system spanned by the α i.we use the following notation. Let γ+, 0 γ, 0 γ+ and γ be the maps F (D /D ) = {0,,, } defined by the table (0, 0) (, ) (, 0) (0, ) γ+ 0 (0, 0) (, 0) (, ) (, ) γ 0 (, ) (0, ) (, )(, ) γ+ (,0) (, 0) (, ) (, ) γ (,) (, ) (0, ) (0, )