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We precisely determined an $\bN$-graded structure of Zhu's poisson algebra $V/C_2(V)$ of vertex operator algebras $V$ of moonshine type. Namely, if $V$ is a vertex operator algebra of moonshine type with a central charge $24$, then $C_2(V)=\sum_{n=5}^{\infty}V_n+L(-1)V$., Comment: 12 pages

We prove that if V is a unitary simple holomorphic vertex operator algebra of CFT type, then V is rational, that is, all N-gradable V-modules are direct sums of copies of V., Comment: 6 pages

We give a lattice theoretical interpretation of generalized deep holes of the Leech lattice VOA $V_\Lambda$. We show that a generalized deep hole defines a "true" automorphism invariant deep hole of the Leech lattice. We also show that there is a correspondence between the set of isomorphism classes of holomorphic VOA $V$ of central charge $24$ having non-abelian $V_1$ and the set of equivalence classes of pairs $(\tau, \tilde{\beta})$ satisfying certain conditions, where $\tau\in Co_0$ and $\tilde{\beta}$ is a $\tau$-invariant deep hole of squared length $2$. It provides a new combinatorial approach towards the classification of holomorphic VOAs of central charge $24$. In particular, we give an explanation for an observation of G. H\"ohn, which relates the weight one Lie algebras of holomorphic VOAs of central charge $24$ to certain codewords associated with the glue codes of Niemeier lattices., Comment: 39 pages

We prove an associative law of the fusion products $\boxtimes$ of $C_1$-cofinite ${\mathbb N}$-gradable modules for a vertex operator algebra $V$. To be more precise, for $C_1$-cofinite ${\mathbb N}$-gradable $V$-modules $A,B,C$ and their fusion products $(A\!\boxtimes\! B, {\cal Y}^{AB})$, $((A\!\boxtimes\! B)\!\boxtimes\! C, {\cal Y}^{(AB)C})$, $(B\!\boxtimes\! C, {\cal Y}^{BC})$, $(A\!\boxtimes\! (B\!\boxtimes\! C),{\cal Y}^{A(BC)})$ with logarithmic intertwining operators ${\cal Y}^{AB},\ldots,{\cal Y}^{A(BC)}$ satisfying the universal properties for ${\mathbb N}$-gradable modules, we prove that four-point correlation functions $\langle \theta, {\cal Y}^{A(BC)}(v,x){\cal Y}^{BC}(u,y)w\rangle$ and $\langle \theta', {\cal Y}^{(AB)C}({\cal Y}^{AB}(v,x-y)u,y)w\rangle$ are locally normally convergent over $\{(x,y)\in {\mathbb C}^2 \mid 0\!<\!|x\!-\!y|\!<\!|y|\!<\!|x|\}$. We then take their respective principal branches $\tilde{F}(\langle \theta,{\cal Y}^{A(BC)}(v,x){\cal Y}^{BC}(u,y)w\rangle)$ and $\tilde{F}(\langle \theta,{\cal Y}^{(AB)C}({\cal Y}^{AB)}(v,x-y)u,y)w\rangle)$ on ${\cal D}^2\!=\!\{(x,y)\in {\mathbb C}^2 \mid 0\!<\!|x\!-\!y|\!<\!|y|\!<\!|x|, \mbox{ and } x,y,x\!-\!y\not\in {\mathbb R}^{\leq 0}\}$ and then show that there is an isomorphism $\phi_{[AB]C}:(A\boxtimes B)\boxtimes C \to A\boxtimes (B\boxtimes C)$ such that $$ \widetilde{F}(\langle \theta, {\cal Y}^{A(BC)}(v,x){\cal Y}^{BC}(u,y)w\rangle) =\tilde{F}(\langle \phi_{[AB]C}^{\ast}(\theta), {\cal Y}^{(AB)C}({\cal Y}^{AB}(v,x-y)u,y)w)\rangle $$ on ${\cal D}^2$ for $\theta\in (A\boxtimes (B\boxtimes C))^{\vee}$, $v\in A$, $u\in B$, and $w\in C$, where $W^{\vee}$ denotes the contragredient module of $W$ and $\phi_{[AB]C}^{\ast}$ denotes the dual of $\phi_{[AB]C}$. We also prove the pentagon identity., Comment: 26 pages, I corrected typos and simplified the arguments

We generalize Conway-Sloane's constructions of the Leech lattice from Niemeier lattices using Lorentzian lattice to holomorphic vertex operator algebras (VOA) of central charge 24. It provides a tool for analyzing the structures and relations among holomorphic VOAs related by orbifold construction. In particular, we are able to get some useful information about certain lattice subVOAs associated with the Cartan subalgebra of the weight one Lie algebra. We also obtain a relatively elementary proof that any strongly regular holomorphic VOA of central charge $24$ with a non-trivial weight one subspace can be constructed directly by a single orbifold construction from the Leech lattice VOA without using a dimension formula., Comment: Minor changes; A mistake in Sec. 6.1.5 is fixed; to appear in J. Algebra

We prove that if $V$ is a $C_2$-cofinite simple vertex operator algebra of CFT-type with a nonsingular invariant bilinear form and its an automorphism group $G$ is finite, then an orbifold model $V^G$ is also $C_2$-cofinite., Comment: There is a serious gap in this paper

We show that if $T$ is a simple non-negatively graded regular vertex operator algebra with a nonsingular invariant bilinear form and $\sigma$ is a finite order automorphism of $T$, then the fixed-point vertex operator subalgebra $T^\sigma$ is also regular. This yields regularity for fixed point vertex operator subalgebras under the action of any finite solvable group. As an application, we obtain an $SL_2(\mathbb{Z})$-compatibility between twisted twining characters for commuting finite order automorphisms of holomorphic vertex operator algebras. This resolves one of the principal claims in the Generalized Moonshine conjecture., Comment: (v4) additional explanations; 41 pages

We prove an $\text{SL}_2 (\mathbb{Z})$-invariance property of multivariable trace functions on modules for a regular VOA. Applying this result, we provide a proof of the inversion transformation formula for Siegel theta series. As another application, we show that if $V$ is a regular VOA containing a regular subVOA $U$ whose commutant $U^c$ is regular and satisfies $(U^c)^c =U$, then all simple $U$-modules appear in some simple $V$-module., Comment: 18 pages, additional references added, typos corrected, and some additional information included

We prove an orbifold conjecture for a solvable automorphism group. Namely, we show that if V is a C_2-cofinite simple vertex operator algebra and G is a finite solvable automorphism group of V, then the fixed point vertex operator subalgebra V^G is also C_2-cofinite. This offers a mathematically rigorous background to orbifold theories of finite type with solvable automorphism groups., Comment: 10 pages

Let V be a vertex operator algebra. We prove that if U and W are C_1-cofinite {\mathbb N}-gradable V-modules, then a fusion product U\boxtimes W is well-defined and also a C_1-cofinite {\mathbb N}-gradable V-module, where the fusion product is defined by (logarithmic) intertwining operators. This is also true for C_2-cofinite {\mathbb N}-gradable modules., Comment: This is a proceeding of "Conformal field theories and tensor categories" held in 2011, (editor Y.-Z Huang), Springer Proceedings in Mathematics & Statistics (2013)

In vertex operator algebra theories, most of the general theorems are proved under the assumptions of rationality and C_2-cofiniteness. In this paper, we obtain several general theorems without the assumption of rationality so that we can use them for proving rationality of given C_2-cofinite vertex operator algebras. For example, we apply them to orbifold models and show that if g is a finite automorphism of a rational C_2-cofinite vertex operator algebra T of CFT-type with T'\cong T and a fixed point subVOA T^g is C_2-cofinite, then T^g is also rational., Comment: 29pages, we reorganized arXiv0906.1407, 0909.3665, math/0309350 into one with short proofs

Let $V$ be a simple VOA of CFT-type satisfying $V'\cong V$ and $\sigma$ a finite automorphism of $V$. We prove that if all $V$-modules are completely reducible and a fixed point subVOA $V^\sigma$ is $C_2$-cofinite, then all $V^\sigma$-modules are completely reducible and every simple $V^{\sigma}$-module appears in some twisted or ordinary $V$-modules as a $V^{\sigma}$-submodule. We also prove that $V_L^{\sigma}$ is $C_2$-cofinite for any lattice VOA $V_L$ and $\sigma\in \Aut(V_L)$ lifted from any triality automorphism of $L$. Using these results, we present two $Z_3$-orbifold constructions as examples. One is the moonshine VOA $V^{\natural}$ and the other is a new CFT No.32 in Schellekens' list., Comment: 24 pages, we have revised some part of the proof

Let V be a simple C_2-cofinite VOA of CFT-type and we assume that there is a simple module U such that \Hom_V(U\boxtimes V',V)\not=0 where V' is a restricted dual of V. As the author has shown, an S-transformation S(\Psi_V) of a trace function \Psi_V on V corresponding \begin{pmatrix}0&-1\cr 1&0\end{pmatrix} may contain pseudo-trace functions. Our assumption in this paper is that no pseudo-trace functions appear in S(\Psi_V). Under these assumptions, we prove that every V-module W is semirigid and \Psi_W appears in S(\Psi_V). As a corollary of our main theorem, a fixed point subVOA of a rational VOA of CFT-type by an automorphism of finite order becomes rational if fixed point subVOA is C_2-cofinite., Comment: 14 pages, we avoid the calculations of fusion matrices, we need one assumption to avoid pseudo-trace function, and we also add a proof of the semisimplicity of orbifold models of rational VOAs

We study properties of a C_2-cofinite vertex operator algebra of CFT type. If it is also rational and V'\cong V, then the rigidity of the tensor category of modules has been proved by Huang. When we treat an irrational C_2-cofinite VOA, the rigidity is too strong, because it is almost equivalent to be rational as we see. We introduce a natural weaker condition "semi-rigidity". Under this condition, we prove the following results. For a projective cover P of a V-module V and a finitely generated V-module M, the projective cover of M is a direct summand of the tensor product P\boxtimes M defined by logarithmic intertwining operators. Using this result, we prove the flatness property of finitely generated modules for the tensor products, that is, if 0\to A\to B\to C\to 0 is exact then so is 0\to D\boxtimes A\to D\boxtimes B\to D\boxtimes C\to 0 for any finitely generated V-modules A, B, C and D. As a corollary, we have that if a semi-rigid C_2-cofinite V contains a rational subVOA with the same Virasoro element, then V is rational., Comment: 17 pages, we weaken slightly the definition of semi-rigidity

If a vertex operator algebra $V=\oplus_{n=0}^{\infty}V_n$ satisfies $\dim V_0=1, V_1=0$, then $V_2$ has a commutative (nonassociative) algebra structure called Griess algebra. One of the typical examples of commutative (nonassociative) algebras is a Jordan algebra. For example, the set $Sym_d(\C)$ of symmetric matrices of degree $d$ becomes a Jordan algebra. On the other hand, in the theory of vertex operator algebras, central charges influence the properties of vertex operator algebras. In this paper, we construct vertex operator algebras with central charge $c$ and its Griess algebra is isomorphic to $Sym_d(\C)$ for any complex number $c$., Comment: 8 pages

We reformed the tensor product theory of vertex operator algebras developed by Huang and Lepowsky so that we could apply it to all vertex operator algebras satisfying C_2-cofiniteness. We also showed that the tensor product theory develops naturally if we include not only ordinary modules, but also weak modules with a composition series of finite length (we call it an Artin module). In particular, we don't assume the semisimplicity of the weight operator L(0). Actually, without the assumption of rationality, a C_2-cofiniteness on V is enough to obtain the existence of a tensor product of two Artin modules and natural associativity of tensor products. Namely, the category of Artin modules becomes a braided tensor category. As an application of the tensor product theory under C_2-cofiniteness, we proved the rationality of some orbifold models. For example, if a vertex operator algebra V has a finite automorphism group and the fixed point vertex operator subalgebra V^G is C_2-cofinite, then for any irreducible V^{}-module W, there is an element h\in such that W is contained in some h-twisted V-module. Furthermore, if V^G is rational, then V^{} is also rational for any g\in G., Comment: 27 pages. Latex

We show that C_2-cofiniteness is enough to prove a modular invariance property of vertex operator algebras without assuming the semisimplicity of Zhu algebra. For example, if a VOA V=\oplus_{m=0}^{\infty}V_m is C_2-cofinite, then the space spanned by generalized characters of V-modules is invariant under the action of SL_2(\Z). In this case, the central charge and conformal weights are all rational numbers. Namely, a VOA satisfying C_2-cofiniteness is a rational conformal field theory in a sense. We also show that C_2-cofiniteness is equivalent to the condition that every weak module is an \N-graded weak module which is a direct sum of generalized eigenspaces of L(0)., Comment: 34 pages. We get rid of the condition of being of``CFT type''. Latex

Let V be a vertex operator algebra and G a finite automorphism group of V. For each g\in G and nonnegative rational number n\in {\mathbb Z}/|g|, a g-twisted Zhu algebra A_{g,n}(V) plays an important role in the theory of vertex operator algebras, but the given product in A_{g,n}(V) depends on the eigenspaces of g. We show that there is a uniform definition of products on V and we introduce a G-twisted Zhu algebra A_{G,n}(V) which covers all g-twisted Zhu algebras. Assume that V is simple and let {\cal S} be a finite set of inequivalent irreducible twisted V-modules which is closed under the action of G. There is a finite dimensional semisimple associative algebra {\cal A}_{\alpha}(G,{\cal S}) for a suitable 2-cocycle naturally determined by the G-action on {\cal S}. We show that a duality theorem of Schur-Weyl type holds for the actions of {\cal A}_{\alpha}(G,{\cal S}) and V^G on the direct sum of twisted V-modules in {\cal S} as an application of the theory of A_{G,n}(V). It follows as a natural consequence of the result that for any g\in G every irreducible g-twisted V-module is a completely reducible V^G-module., Comment: 16 pages, Latex We added a duality theorem

We determined the inner products of two conformal vectors with central charge 1/2 whose \tau-involutions generates S_3 if none of \tau-involutions are trivial. We also see that a subVA generated by such conformal vectors is a VOA with central charge 1/2+20/21 or $4/5+6/7 and has a Griess algebra of dimension three or four, respectively., Comment: 19 pages, Latex, corrected some comment

We extend the modular invariance property of the trace functions of vertex operator algebra on the set of irreducible modules (Zhu's theory) to the case of trace functions of intertwining operators., Comment: LaTex-file 25 pages

We introduce theta-functions of VOA-modules and show that the space spanned by them has a modular invariance property., Comment: 15 pages, Latex

We define an automorphism of VOA of order 3 by using a sub VOA isomorphic to a direct sum of 3-state Potts models $L(\ff,0)$ and an its module $L(\ff,3)$. This automorphism is a 3A element of the monster simple group if $V$ is the moonshine VOA $V^{\natural}$., Comment: 16 pages, Latex

We prove the Galois correspondence between the subgroups of a finite automorphism group G of a simple vertex operator algebra V and the vertex operator subalgebras of V containing the set V^G of G-invariants., Comment: revised version (with a longer proof), 6 pages, Latex

We give a new construction of the moonshine VOA V^{\natural} over the real number field. We proved that V^{\natural} has a positive definite invariant bilinear form and its full automorphism group is the Monster simple group. We also construct an infinite series of meromorphic VOAs whose full automorphism groups are finite. We calculate the trace form on V^{\natural} for some element of the Monster., Comment: revised version (We add several things) 66 pages, Latex

We study a vertex operator algebra containing a tensor product of Ising models. It is a direct sum of code vertex operator algebra and its irreducible modules. Therefore, we classify all irreducible modules of code vertex operator algebras and show a new construction of vertex operator algebras., Comment: 47 pages, Latex