Search

It is known from Zhu's results that under modular transformations, correlators of rational $C_2$-cofinite vertex operator algebras transform like Jacobi forms. We investigate the modular transformation properties of VOA correlators that have zero modes inserted. We derive recursion relations for such correlators and use them to establish modular transformation properties. We find that correlators with only zero modes transform like quasi-modular forms, and mixed correlators with both zero modes and vertex operators transform like quasi-Jacobi forms. As an application of our results, we introduce algebras of higher weight fields whose zero mode correlators mimic the properties of those of weight 1 fields. We also give a simplified proof of the weight 1 transformation properties originally proven by Miyamoto.

The representation theory of the Nappi-Witten VOA was initiated in arXiv:1104.3921 and arXiv:2011.14453. In this paper we use the technique of inverse quantum hamiltonian reduction to investigate the representation theory of the Nappi-Witten VOA $ V^1(\mathfrak h_4)$. We first prove that the quantum hamiltonian reduction of $ V^1(\mathfrak h_4)$ is the Heisenberg-Virasoro VOA $L^{HVir}$ of level zero investigated in arXiv:math/0201314 and arXiv:1405.1707. We invert the quantum hamiltonian reduction in this case and prove that $ V^1(\mathfrak h_4)$ is realized as a vertex subalgebra of $L^{HVir} \otimes \Pi$, where $\Pi$ is a certain lattice-like vertex algebra. Using such an approach we shall realize all relaxed highest weight modules which were classified in arXiv:2011.14453. We show that every relaxed highest weight module, whose top components is neither highest nor lowest weight $\mathfrak h_4$-module, has the form $M_1 \otimes \Pi_{1} (\lambda)$ where $M_1$ is an irreducible, highest weight $L^{HVir}$-module and $\Pi_{1} (\lambda)$ is an irreducible weight $\Pi$-module. Using the fusion rules for $L^{HVir}$-modules and the previously developed methods of constructing logarithmic modules we are able to construct a family of logarithmic $V^1(\mathfrak h_4)$-modules. The Loewy diagrams of these logarithmic modules are completely analogous to the Loewy diagrams of projective modules of weight $L_k(\mathfrak{sl}(2))$-modules, so we expect that our logarithmic modules are also projective in a certain category of weight $ V^1(\mathfrak h_4)$-modules., Comment: 22 pages

Frenkel, Lepowsky, and Meurman constructed a vertex operator algebra (VOA) associated to any even, integral, Euclidean lattice. In the language of physics, these are examples of chiral conformal field theories (CFT). In this paper, we define non-chiral vertex operator algebra and some associated notions. We then give a construction of a non-chiral VOA associated to an even, integral, Lorentzian lattice and construct their irreducible modules. We obtain the moduli space of such modular invariant non-chiral CFTs based on even, self-dual Lorentzian lattices of signature $(m,n)$ assuming the validity of a technical result about automorphisms of the lattice. We finally show that Narain conformal field theories in physics are examples of non-chiral VOA. Our formalism helps us to identify the chiral algebra of Narain CFTs in terms of a particular sublattice and give us the decomposition of its partition function into sum of characters., Comment: 100 Pages, Comments are welcome! v2:minor corrections and clarifications

This paper is about the orbifold theory of parafermion vertex operator algebras $K(osp(1|2),k)$ associated to the affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(k,0)$ with any positive integer $k$. Among the main results, we classify the irreducible modules for the orbifold of parafermion vertex operator algebra $K(osp(1|2),k)$., Comment: 22 pages. arXiv admin note: substantial text overlap with arXiv:2306.03334

In this paper, we introduce and study new classes of subVOAs of the lattice vertex operator algebras, which we call the conic, Borel, and parabolic-type subVOAs. These CFT-type VOAs, which are not necessarily strongly finitely generated, satisfy similar properties with the usual Borel and parabolic subalgebras of a Lie algebra. For the lowest-rank nontrivial example of Borel-type subVOA $V_{B}$ of $V_{\Z\al}$, we explicitly determine its Zhu's algebra $A(V_B)$ in terms of generators and relations.

We define the notion of strongly interlocked for indecomposable generalized modules for a vertex operator algebra, and show that the notion of graded pseudo-trace is well defined for modules which satisfy this property. We prove that the graded pseudo-trace is a symmetric linear operator that satisfies the logarithmic derivative property. As an application, we prove that all the indecomposable reducible generalized modules for the rank one Heisenberg (one free boson) vertex operator algebras are strongly interlocked, independent of the choice of conformal vector and thus have well-defined graded pseudo-traces. We also completely characterize which indecomposable reducible generalized modules for the universal Virasoro vertex operator algebras induced from the level zero Zhu algebra are strongly interlocked. In particular, we prove that the universal Virasoro vertex operator algebra with central charge c has modules induced from the level zero Zhu algebra with conformal weight h that are strongly interlocked if and only if either (c,h) is outside the extended Kac table, or the central charge is either c = 1 or 25, the conformal weight satisfies a certain property, and the level zero Zhu algebra module being induced is determined by a Jordan block of size less than a certain specified parameter. We give several examples of graded pseudo-traces for Heisenberg and Virasoro strongly interlocked modules., Comment: The title of the paper was changed to better reflect the central role that the notion of strongly interlocked module plays in the theory of graded pseudo-traces. A proof that graded pseudo-traces for any strongly interlocked modules are symmetric, linear, and satisfy the logarithmic derivative property was added. We rewrote and reorganized parts of the exposition and fixed minor typos

A unitary and strongly rational vertex operator algebra (VOA) $V$ is called strongly unitary if all irreducible $V$-modules are unitarizable. A strongly unitary VOA $V$ is called completely unitary if for each unitary $V$-modules $W_1$, $W_2$ the canonical nondegenerate Hermitian form on the fusion product $W_1\boxtimes W_2$ is positive. It is known that if $V$ is completely unitary, then the modular category of unitary $V$-modules is unitary [Gui19b], and all simple VOA extensions of V are automatically unitary and moreover completely unitary [Gui22, CGGH23]. In this paper, we give a geometric characterization of the positivity of the Hermitian product on $W_1$ and $W_2$, which helps us prove that the positivity is always true when the fusion product $W_1\boxtimes W_2$ is an irreducible and unitarizable $V$-module. We give several applications: (1) We show that if $V$ is a unitary (strongly rational) holomorphic VOA with a finite cyclic unitary automorphism group $G$, and if $V^G$ is strongly unitary, then $V^G$ is completely unitary. This result applies to the cyclic permutation orbifolds of unitary holomophic VOAs. (2) We show that if $V$ is unitary and strongly rational, and if $U$ is a simple current extension which is unitarizable as a $V$-module, then $U$ is a unitary VOA., Comment: 69 pages. Final revision. Published in Comm. Math. Phys

Let V be a simple unitary vertex operator algebra and U be a (polynomially) energy-bounded unitary subalgebra containing the conformal vector of V. We give two sufficient conditions implying that V is energy-bounded. The first condition is that U is a compact orbifold for some compact group G of unitary automorphisms of V. The second condition is that V is exponentially energy-bounded and it is a finite direct sum of simple U-modules. As consequence of the second condition, we prove that if U is a regular energy-bounded unitary subalgebra of a simple unitary vertex operator V, then $V$ is energy-bounded. In particular, every simple unitary extension (with the same conformal vector) of a simple unitary affine vertex operator algebra associated with a semisimple Lie algebra is energy-bounded., Comment: 19 pages, revised version

Conformal blocks, physical quantities of chiral 2d conformal field theory, are sheaves on the configuration spaces of the complex plane, which are mathematically formulated in terms of a vertex operator algebra, its modules and associated D-modules. We show that the operad of fundamental groupoids of the configuration spaces, the parenthesized braid operad, acts on the conformal blocks by the monodromy representation. More precisely, let $V$ be a vertex operator algebra with $V=\bigoplus_{n\geq 0} V_n$, $\dim V_n <\infty$, $V_0=\mathbb{C}\bf{1}$ and $V\text{-mod}_{\mathrm{f.g}}$ the category of $V$-modules $M$ such that $M$ is $C_1$-cofinite and the dual module $M^\vee$ is a finitely generated $V$-module. We show that the parenthesized braid operad weakly 2-categorically acts on $V\text{-mod}_{\mathrm{f.g}}$, and consequently $V\text{-mod}_{\mathrm{f.g}}$ has a structure of the (unital) pseudo-braided category. Moreover, if $V$ is rational and $C_2$-cofinite, then $V\text{-mod}_{\mathrm{f.g}}$ is a balanced braided tensor category, which gives an alternative proof of a result of Huang and Lepowsky., Comment: 121 pages. In v2, add Appendix B, relax the assupmtion of Main theorem, revise Introduction

For a foliation $\F$ defined on a smooth complex manifold $M$ we introduce the category of vertex operator algebra $V$ bundles with sections provided by vectors of elements of the space of algebraically extended $V$-module $W$-valued differentials. An intrinsic coordinate-independent formulation for such bundles is given. Finally, we identify the cohomology of the spaces of sections for a vertex operator algebra $V$ bundle with vertex operator algebra cohomology of the holonomy groupoid $Hol(M, \F)$.

We determine the level two Zhu algebra for the Heisenberg vertex operator algebra $V$ for any choice of conformal element. We do this using only the following information for $V$: the internal structure of $V$; the level one Zhu algebra of $V$ already determined by the second author, along with Vander Werf and Yang; and the information the lower level Zhu algebras give regarding irreducible modules. We are able to carry out this calculation of the level two Zhu algebra for $V$ with this minimal information by employing the general results and techniques for determining generators and relations for higher level Zhu algebras for a vertex operator algebra, as developed previously by the authors in "On generators and relations for higher level Zhu algebras and applications", by Addabbo and Barron, J. Algebra, 2023. In particular, we show that the level $n$ Zhu algebras for the Heisenberg vertex operator algebra become noncommutative at level $n=2$. We also give a conjecture for the structure of the level $n$ Zhu algebra for the Heisenberg vertex operator algebra, for any $n >2$., Comment: 65 pages; minor typos corrected; some calculations shortened

Given a vertex operator algebra $V$, one can construct two associative algebras, the Zhu algebra $A(V)$ and the $C_2$-algebra $R(V)$. This gives rise to two abelian categories $A(V)-\text{Mod}$ and $R(V)-\text{Mod}$, in addition to the category of admissible modules of $V$. In case $V$ is rational and $C_2$-cofinite, the category of admissible $V$-modules and the category of all $A(V)$-modules are equivalent. However, when $V$ is not rational, the connection between these two categories is unclear. The goal of this paper is to study the triplet vertex operator algebra $\mathcal{W}(p)$, as an example to compare these three categories, in terms of abelian categories. For each of these three abelian categories, we will determine the associated Ext quiver, the Morita equivalent basic algebra, i.e., the algebra $ \text{End} (\oplus_{L\in \text{Irr}} P_L)^{op}$, and the Yoneda algebra $\text{Ext}^{*}(\oplus_{L\in \text{Irr}}L, \oplus_{L\in \text{Irr}}L)$. As a consequence, the category of admissible log-modules for the triplet VOA $ \mathcal W(p)$ has infinite global dimension, as do the Zhu algebra $A(\mathcal W(p))$, and the associated graded algebra $\text{gr} \ A(\mathcal W(p))$ which is isomorphic to $R(\mathcal W(p))$. We also describe the Koszul properties of the module categories of $ \mathcal W(p)$, $A(\mathcal W(p))$ and $\text{gr} \ A(\mathcal W(p))$., Comment: 39 pages. References were added, Sections 3, 6.1 and 6.2 were rewritten, and Koszul categories were introduced. Comments are welcome

The monoidal category of twisted modules of a Vertex Operator Algebra $V$ is defined and reduced to its 2-group of invertible objects $G_\alpha$, which can be described by a 3-cocycle $\alpha$ on its 0-truncation $G$ with values in the group of units $A$ of the field of definition of $V$ serving as its associator. This cocycle also presents the classifying morphism of an $\infty$-group extension of $G$ by the delooping $BA$. Motivated by this, it is proven that the $\infty$-group extension classified by a 3-cocycle $\alpha$ is presented by the skeletal 2-group $G_\alpha$ with associator $\alpha$. The results are discussed in light of current developments in Moonshine and $(\infty,1)$-topos theory., Comment: 22 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 construct the foliation of aspace associated to correlation functions of vertex operator algebras on considered on Riemann surfaces. We prove that the computation of general genus $g$ correlation functions determines a foliation on the space associated to these correlation functions a sewn Riemann surface. Certain further applications of the definition are proposed.

An irreducible module for the parafermion vertex operator algebra $K(\mathfrak{sl}_2,k)$ is said to be of $\sigma$-type if an automorphism of the fusion algebra of $K(\mathfrak{sl}_2,k)$ of order $k$ is trivial on it. For any integer $k \ge 3$, we show that there exists an automorphism of order $2$ of the subalgebra of the fusion algebra of $K(\mathfrak{sl}_2,k)^{\langle \theta \rangle}$ spanned by the irreducible direct summands of $\sigma$-type irreducible $K(\mathfrak{sl}_2,k)$-modules, where $\theta$ is an involution of $K(\mathfrak{sl}_2,k)$. We discuss some examples of such an automorphism as well., Comment: 37 pages

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

The Nappi-Witten model is a Wess-Zumino-Witten model in which the target space is the nonreductive Heisenberg group $H_4$. We consider the representation theory underlying this conformal field theory. Specifically, we study the category of weight modules, with finite-dimensional weight spaces, over the associated affine vertex operator algebra $\mathsf{H}_4$. In particular, we classify the irreducible $\mathsf{H}_4$-modules in this category and compute their characters. We moreover observe that this category is nonsemisimple, suggesting that the Nappi-Witten model is a logarithmic conformal field theory., Comment: 21 pages; introduction expanded, references added, minor notation changes

We first determine the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$.Then, for any integer $t>1$, we introduce a new Lie algebra $\mathcal{L}_{t}$, and show that $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$($\ell_{2}=0$)-modules are in one-to-one correspondence with restricted $\mathcal{L}_{t}$-modules of level $\ell_{13}$, where $\sigma_{t}$ is an order $t$ automorphism of $V_{\mathcal{L}}(\ell_{123},0)$. At the end, we give a complete list of irreducible $\sigma_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$($\ell_{2}=0$)-modules., Comment: 17 pages

We compute the partition and correlation generating functions for the Heisenberg intertwiner generalized vertex operator algebra on a genus $g$ Riemann surface in the Schottky uniformization. These are expressed in terms of differential forms of the first, second and third kind, the prime form and the period matrix and are computed by combinatorial methods using a generalization of the MacMahon Master Theorem., Comment: 21 pages. To appear in Proceedings of the Dubrovnik conference Representation Theory XVI, published in the AMS Contemporary Mathematics book series

We prove a dimension formula for the weight-1 subspace of a vertex operator algebra $V^{\operatorname{orb}(g)}$ obtained by orbifolding a strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with a finite-order automorphism $g$. Based on an upper bound derived from this formula we introduce the notion of a generalised deep hole in $\operatorname{Aut}(V)$. Then we show that the orbifold construction defines a bijection between the generalised deep holes of the Leech lattice vertex operator algebra $V_\Lambda$ with non-trivial fixed-point Lie subalgebra and the strongly rational, holomorphic vertex operator algebras of central charge 24 with non-vanishing weight-1 space. This provides the first uniform construction of these vertex operator algebras and naturally generalises the correspondence between the deep holes of the Leech lattice $\Lambda$ and the 23 Niemeier lattices with non-vanishing root system found by Conway, Parker and Sloane., Comment: 50 pages, LaTeX; some changes to the exposition, numbering in Section 6 changed; to appear in Ann. of Math

For a positive-definite, even, integral lattice $L$, the lattice vertex operator algebra $V_L$ is known to be rational and $C_2$-cofinite, and thus the fusion products of its modules always exist. The fusion product of two untwisted irreducible $V_L$-modules is well-known, namely $V_{L+\lambda} \boxtimes_{V_L} V_{L+\mu} = V_{L + \lambda + \mu}$. In this paper, we determine the other two fusion products: $V_{L+\lambda} \boxtimes_{V_L} V_L^{T_{\chi}}$ and $V_L^{T_{\chi_1}} \boxtimes_{V_L} V_L^{T_{\chi_2}}$.

Herein we study conformal vectors of a Z-graded vertex algebra of (strong) CFT type. We prove that the full vertex algebra automorphism group transitively acts on the set of the conformal vectors of strong CFT type if the vertex algebra is simple. The statement is equivalent to the uniqueness of self-dual vertex operator algebra structures of a simple vertex algebra. As an application, we show that the full vertex algebra automorphism group of a simple vertex operator algebra of strong CFT type uniquely decomposes into the product of certain two subgroups and the vertex operator algebra automorphism group. Furthermore, we prove that the full vertex algebra automorphism group of the moonshine module over the field of real numbers is the Monster., Comment: 13 pages

We give a new factorisable ribbon quasi-Hopf algebra U, whose underlying algebra is that of the restricted quantum group for sl(2) at a 2p'th root of unity. The representation category of U is conjecturally ribbon-equivalent to that of the triplet vertex operator algebra W(p). We obtain U via a simple current extension from the unrolled restricted quantum group at the same root of unity. The representation category of the unrolled quantum group is conjecturally equivalent to that of the singlet vertex operator algebra M(p), and our construction is parallel to extending M(p) to W(p). We illustrate the procedure in the simpler example of passing from the Hopf algebra for the group algebra CZ to a quasi-Hopf algebra for CZ_{2p}, which corresponds to passing from the Heisenberg vertex operator algebra to a lattice extension., Comment: 71pp

In this paper, the irreducible modules for the $\mathbb{Z}_{2}$-orbifold vertex operator subalgebra of the parafermion vertex operator algebra associated to the irreducible highest weight modules for the affine Kac-Moody algebra $A_1^{(1)}$ of level $k$ are classified and constructed., Comment: 21 pages

We derive a vertex operator based expression for the kinematic numerators of Yang-Mills amplitudes by applying the momentum kernel formalism to open string amplitudes. The expression involves an $\alpha'$-weighted commutator induced by the monodromy relations between the colour ordered Yang-Mills amplitudes, which mirrors the $\alpha'$ deformed colour structure observed in open string and semi-abelian $Z$-theory. The kinematic algebra given by this construction contains the Lie algebra of diffeomorphism as an obvious sub-algebra., Comment: 25 pages. v2: minor corrections version to be published

For $p = 3,5,7,13$, we consider a ${\mathbb Z}_p$-orbifold construction of the Moonshine vertex operator algebra $V^\natural$. We show that the vertex operator algebra obtained by the ${\mathbb Z}_p$-orbifold construction on the Leech lattice vertex operator algebra $V_\Lambda$ and a lift of a fixed-point-free isometry of order $p$ is isomorphic to the Moonshine vertex operator algebra $V^\natural$. We also describe the relationship between those ${\mathbb Z}_p$-orbifold constructions and the ${\mathbb Z}_2$-orbifold construction in a uniform manner. In Appendix, we give a characterization of the Moonshine vertex operator algebra $V^\natural$ by two mutually orthogonal Ising vectors., Comment: 13 pages, no figure. An error is fixed

In this article, we study orbifold constructions associated with the Leech lattice vertex operator algebra. As an application, we prove that the structure of a strongly regular holomorphic vertex operator algebra of central charge $24$ is uniquely determined by its weight one Lie algebra if the Lie algebra has the type $A_{3,4}^3A_{1,2}$, $A_{4,5}^2$, $D_{4,12}A_{2,6}$, $A_{6,7}$, $A_{7,4}A_{1,1}^3$, $D_{5,8}A_{1,2}$ or $D_{6,5}A_{1,1}^2$ by using the reverse orbifold construction. Our result also provides alternative constructions of these vertex operator algebras (except for the case $A_{6,7}$) from the Leech lattice vertex operator algebra., Comment: 29 pages

In this paper, a holomorphic vertex operator algebra $U$ of central charge 24 with the weight one Lie algebra $A_{8,3}A_{2,1}^2$ is proved to be unique. Moreover, a holomorphic vertex operator algebra of central charge 24 with weight one Lie algebra $F_{4,6}A_{2,2}$ is obtained by applying a $\mathbb{Z}_2$-orbifold construction to $U$. The uniqueness of such a vertex operator algebra is also established. By a similar method, we also established the uniqueness of a holomorphic vertex operator algebra of central charge 24 with the weight one Lie algebra $E_{7,3}A_{5,1}$. As a consequence, we verify that all $71$ Lie algebras in Schellekens' list can be realized as the weight one Lie algebras of some holomorphic vertex operator algebras of central charge $24$. In addition, we establish the uniqueness of three holomorphic vertex operator algebras of central charge $24$ whose weight one Lie algebras have the type $A_{8,3}A_{2,1}^2$, $F_{4,6}A_{2,2}$, and $E_{7,3}A_{5,1}$., Comment: 49 pages, added Subsection 4.4 and Section 7

In this paper, we study a new kind of vertex operator algebra related to the twisted Heisenberg-Virasoro algebra, which we call the twisted Heisenberg-Virasoro vertex operator algebra, and its modules. Specifically, we present some results concerning the relationship between the restricted module categories of twisted Heisenberg-Virasoro algebras of rank one and rank two and several different kinds of module categories of their corresponding vertex algebras. We also study fully the structures of the twisted Heisenberg-Virasoro vertex operator algebra, give a characterization of it as a tensor product of two well-known vertex operator algebras, and solve the commutant problem., Comment: 39 pages

Let $p$ be an odd prime and let $\widehat{\sigma}$ be an order $p$ automorphism of $V_{\sqrt{2}A_{p-1}}$ which is a lift of a $p$-cycle in the Weyl group ${\rm Weyl}(A_{p-1})\cong {\mathfrak S}_p$. We study a certain extension $V$ of a tensor product of finitely many copies of the orbifold model $V_{\sqrt{2}A_{p-1}}^{\langle \widehat{\sigma} \rangle}$ and give a criterion for $V$ that every irreducible $V$-module is a simple current., Comment: 23 pages, no figure

In this article, we describe some maximal $3$-local subgroups of the Monster simple group using vertex operator algebras (VOA). We first study the holomorphic vertex operator algebra obtained by applying the orbifold construction to the Leech lattice vertex operator algebra and a lift of a fixed-point free isometry of order $3$ of the Leech lattice. We also consider some of its special subVOAs and study their stabilizer subgroups using the symmetries of the subVOAs. It turns out that these stabilizer subgroups are $3$-local subgroups of its full automorphism group. As one of our main results, we show that its full automorphism group is isomorphic to the Monster simple group by using a $3$-local characterization and that the holomorphic VOA is isomorphic to the Moonshine VOA. This approach allows us to obtain relatively explicit descriptions of two maximal $3$-local subgroups of the shape $3^{1+12}.2.\Suz{:}2$ and $3^8.\Omega^-(8,3).2$ in the Monster simple group., Comment: change title, rewrite abstract and introduction

Irreducible modules of the 3-permutation orbifold of a rank one lattice vertex operator algebra are listed explicitly. Fusion rules are determined by using the quantum dimensions. The $S$-matrix is also given., Comment: 21 pages

Let $M(1)$ be the vertex operator algebra with the Virasoro element $\omega$ associated to the Heisenberg algebra of rank $1$ and let $M(1)^{+}$ be the subalgebra of $M(1)$ consisting of the fixed points of an automorphism of $M(1)$ of order $2$. We classify the simple weak $M(1)^{+}$-modules with a non-zero element $w$ such that for some integer $s\geq 2$, $\omega_i w\in{\mathbb C}w$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$), $\omega_{s}w\in{\mathbb C}^{\times}w$, and $\omega_i w=0$ for all $i>s$. The result says that any such simple weak $M(1)^{+}$-module is isomorphic to some simple weak $M(1)$-module or to some $\theta$-twisted simple weak $M(1)$-module., Comment: 20 pages; typos corrected