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Let $V$ be a simple, non-negatively-graded, rational, $C_2$-cofinite, and self dual vertex operator algebra, $g_1, g_2, g_3$ be three commuting finitely ordered automorphisms of $V$ such that $g_1g_2=g_3$ and $g_i^T=1$ for $i=1, 2, 3$ and $T\in \N$. Suppose $M^1$ is a $g_1$-twisted module. For any $n, m\in \frac{1}{T}\N$, we construct an $A_{g_3, n}(V)$-$A_{g_2, m}(V)$-bimodule $\mathcal{A}_{g_3, g_2, n, m}(M^1)$ associated to the quadruple $(M^1, g_1, g_2, g_3)$. Given an $A_{g_2, m}(V)$-module $U$, an admissible $g_3$-twisted module $\mathcal{M}(M^1, U)$ is constructed. For the quadruple $(V, 1, g, g)$ for some $g\in \text{Aut}(V)$, $\mathcal{A}_{g, g, n, m}(V)$ coincides with the $A_{g, n}(V)$-$A_{g, m}(V)$-bimodules $A_{g, n, m}(V)$ constructed by Dong-Jiang, and $\mathcal{M}(V, U)$ is the generalized Verma type admissible $g$-twisted module generated by $U$. For an irreducible $g_1$-twisted module $M^1$ and an irreducible $g_2$-twisted module $M^2$, we give a construction of tensor product of $M^1$ and $M^2$ using the bimodule theory developed in this paper. As an application, a twisted version of the fusion rules theorem is established., Comment: 49 pages

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

We study the Lie algebra of physical states associated with certain vertex operator algebras of central charge 24. By applying the no-ghost theorem from string theory we express the corresponding Lie brackets in terms of vertex algebra operations. In the special case of the Moonshine module this result answers a question of Borcherds, posed in his paper on the Monstrous moonshine conjecture., Comment: 18 pages

In this first of a series of two papers, we investigate two different equivalence relations obtained by generalizing the notion of genus of even lattices to the setting of vertex operator algebras (or two-dimensional chiral algebras). The bulk genus equivalence relation was defined in arXiv:math/0209333 and groups (suitably regular) vertex operator algebras according to their modular tensor category and central charge. Hyperbolic genus arXiv:2004.01441 tests isomorphy after tensoring with a hyperbolic plane vertex algebra. Physically, two rational chiral algebras are said to belong to the same bulk genus if they live on the boundary of the same 2+1d topological quantum field theory; they belong to the same hyperbolic genus if they can be related by current-current exactly marginal deformations after tensoring a non-chiral compact boson. As one main result, we prove the conjecture that the hyperbolic genus defines a finer equivalence relation than the bulk genus. This is based on a new, equivalent characterization of the hyperbolic genus that uses the maximal lattice inside a vertex operator algebra and its commutant (or coset). We discuss the implications of these constructions for the classification of rational conformal field theory. In particular, we propose a program for (partially) classifying $c=32$, holomorphic vertex operator algebras (or chiral conformal field theories), and obtain novel lower bounds, via a generalization of the Smith-Minkowski-Siegel mass formula, on the number of vertex operator algebras at higher central charges. Finally, we conjecture a Siegel-Weil identity which computes the "average" torus partition function of an ensemble of chiral conformal field theories defined by any hyperbolic genus, and interpret this formula physically in terms of disorder-averaged holography., Comment: 66 pages, LaTeX

Let $Q$ be a non-degenerated even lattice, let $V_Q$ be the lattice vertex algebra associated to $Q$, and let $V_Q^\eta$ be a quantum lattice vertex algebra. In this paper, we prove the equivalence between the category $V_Q$-modules and the category of $V_Q^\eta$-modules. As a consequence, we show that every $V_Q^\eta$-module is completely reducible, and the set of simple $V_Q^\eta$-modules are in one-to-one correspondence with the set of cosets of $Q$ in its dual lattice., Comment: arXiv admin note: substantial text overlap with arXiv:2310.03571

Let $\mathfrak g$ be a symmetrizable Kac-Moody Lie algebra, and let $V_{\hat{\mathfrak g},\hbar}^\ell$, $L_{\hat{\mathfrak g},\hbar}^\ell$ be the quantum affine vertex algebras constructed in [11]. For any complex numbers $\ell$ and $\ell'$, we present an $\hbar$-adic quantum vertex algebra homomorphism $\Delta$ from $V_{\hat{\mathfrak g},\hbar}^{\ell+\ell'}$ to the twisted tensor product $\hbar$-adic quantum vertex algebra $V_{\hat{\mathfrak g},\hbar}^\ell\widehat\otimes V_{\hat{\mathfrak g},\hbar}^{\ell'}$. In addition, if both $\ell$ and $\ell'$ are positive integers, we show that $\Delta$ induces an $\hbar$-adic quantum vertex algebra homomorphism from $L_{\hat{\mathfrak g},\hbar}^{\ell+\ell'}$ to the twisted tensor product $\hbar$-adic quantum vertex algebra $L_{\hat{\mathfrak g},\hbar}^\ell\widehat\otimes L_{\hat{\mathfrak g},\hbar}^{\ell'}$. Moreover, we prove the coassociativity of $\Delta$., Comment: arXiv admin note: substantial text overlap with arXiv:2310.03571

Categorical coset constructions are investigated and Kac-Wakimoto Hypothesis associated with pseudo unitary modular tensor categories is proved. In particular, the field identifications are obtained. These results are applied to the coset constructions in the theory of vertex operator algebra., Comment: 23 pages

We continue our study of cyclic orbifolds of lattice vertex operator algebras and their full automorphism groups. We consider some special isometry $g\in O(L)$ such that $g^i$ is fixed point free on $L$ for any $1\leq i\leq |g|-1$. We show that when $L_2=\emptyset$ and $g^i$ is fixed point free on $L$ for any $1\leq i\leq |g|-1$, $V_L^{\hat{g}}$ has extra automorphisms implies either (1) the order of $g$ is a prime or (2) $L$ is isometric to the Leech lattice or some coinvariant sublattices of the Leech lattice., Comment: arXiv admin note: text overlap with arXiv:2103.07035

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

Let $\mathfrak g$ be a finite dimensional simple Lie algebra over $\mathbb C$, and let $\ell$ be a positive integer. In this paper, we construct the quantization $K_{\hat{\mathfrak g},\hbar}^\ell$ of the parafermion vertex algebra $K_{\hat{\mathfrak g}}^\ell$ as an $\hbar$-adic quantum vertex subalgebra inside the simple quantum affine vertex algebra $L_{\hat{\mathfrak g},\hbar}^\ell$. We show that $L_{\hat{\mathfrak g},\hbar}^\ell$ contains an $\hbar$-adic quantum vertex subalgebra isomorphic to the quantum lattice vertex algebra $V_{\sqrt\ell Q_L}^{\eta_\ell}$, where $Q_L$ is the lattice generated by the long roots of ${\mathfrak g}$. Moreover, we prove the double commutant property of $K_{\hat{\mathfrak g},\hbar}^\ell$ and $V_{\sqrt\ell Q_L}^{\eta_\ell}$ in $L_{\hat{\mathfrak g},\hbar}^\ell$.

Let $V$ be a vertex operator superalgebra and $g=\left(1\ 2\ \cdots k\right)$ be a $k$-cycle which is viewed as an automorphism of the tensor product vertex operator superalgebra $V^{\otimes k}$. In this paper, we construct an explicit isomorphism from $A_{g}\left(V^{\otimes k}\right)$ to $A\left(V\right)$ if $k$ is odd and to $A_{\sigma}\left(V\right)$ if $k$ is even where $\sigma$ is the canonical automorphism of $V$ of order 2 determined by the superspace structure of $V.$ These recover previous results by Barron and Barron-Werf that there is a one-to-one correspondence between irreducible $g$-twisted $V^{\otimes k}$-modules and irreducible $V$-modules (resp. irreducible $\sigma$-twisted $V$-modules) when $k$ is odd (resp. even). This explicit isomorphism is expected to be useful in our further study on the Zhu algebra of fixed point subalgebra., Comment: 21 pages. The article was accepted for publication in SCIENCE CHINA Mathematics

We introduce the notion of a $\hbar$-vertex algebra, which is an algebraic structure similar to a vertex algebra, with a deformed translation covariance axiom. We study the structure theory of $\hbar$-vertex algebras, highlighting similarities with the structure theory of vertex algebras. We prove results analogous to Goddard's Uniqueness Theorem, Reconstruction Theorem, Borcherds Identity and OPE Expansion Formula. We also introduce the related notions of a $\hbar$-Lie conformal algebra and of a $\hbar$-Poisson vertex algebra. The main example of $\hbar$-vertex algebras are vertex algebras with deformed vertex operators $Y_\hbar(a,z)=(1+\hbar z)^{\Delta_a}Y(a,z)$, where $\Delta_a$ is the eigenvalue of a Hamiltonian operator $H$. We simplify the construction of the Zhu algebra using the formalism of $\hbar$-vertex algebras. As a further application of the theory developed in the first part of the paper, we discuss the chiralization of classical star-products. We compute explicit formulae for the chiralization of a class of star-products, including the Moyal-Weyl and the Gutt star-products. Putting $\hbar=0$, these formulae yield explicit formulae for deformation quantizations of a class of Poisson vertex algebras., Comment: Typos corrected, Section 3.5 added

In this paper, we study the construction of the supersymmetric extensions of vertex algebras. In particular, for $N = n \in \mathbb{Z}_{+}$, we show the universal enveloping $N = n$ supersymmetric (SUSY) vertex algebra of an $N = n$ SUSY Lie conformal algebra can be extended to an $N = n' > n$ SUSY vertex algebra., Comment: 29 pages; added references and a new section

We classify the self-dual (or holomorphic) vertex operator superalgebras of central charge 24, or in physics parlance the purely left-moving, fermionic 2-dimensional conformal field theories with just one primary field. There are exactly 969 such vertex operator superalgebras under suitable regularity assumptions (essentially strong rationality) and the assumption that the shorter moonshine module $V\!B^\natural$ is the unique self-dual vertex operator superalgebra of central charge 23.5 whose weight-1/2 and weight-1 spaces vanish. Additionally, there might be self-dual vertex operator superalgebras arising as fake copies of $V\!B^\natural$ tensored with a free fermion $F$. We construct and classify the self-dual vertex operator superalgebras by determining the 2-neighbourhood graph of the self-dual vertex operator algebras of central charge 24 and also by realising them as simple-current extensions of a dual pair containing a certain maximal lattice vertex operator algebra. We show that all vertex operator superalgebras besides $V\!B^\natural\otimes F$ and potential fake copies thereof stem from elements of the Conway group $\mathrm{Co}_0$, the automorphism group of the Leech lattice $\Lambda$. By splitting off free fermions $F$, if possible, we obtain the classification for all central charges less than or equal to 24., Comment: 80 pages, LaTeX; improvements to the exposition

Let $V$ be a vertex algebra of countable dimension, $G$ a subgroup of ${\rm Aut} V$ of finite order, $V^{G}$ the fixed point subalgebra of $V$ under the action of $G$, and ${\mathscr S}$ a finite $G$-stable set of inequivalent irreducible twisted weak $V$-modules associated with possibly different automorphisms in $G$. We show a Schur--Weyl type duality for the actions of ${\mathscr A}_{\alpha}(G,{\mathscr S})$ and $V^G$ on the direct sum of twisted weak $V$-modules in ${\mathscr S}$ where ${\mathscr A}_{\alpha}(G,{\mathscr S})$ is a finite dimensional semisimple associative algebra associated with $G,{\mathscr S}$, and a $2$-cocycle $\alpha$ naturally determined by the $G$-action on ${\mathscr S}$. It follows as a natural consequence of the result that for any $g\in G$ every irreducible $g$-twisted weak $V$-module is a completely reducible weak $V^G$-module., Comment: 13 pages

Let $V_{L}$ be the vertex algebra associated to a non-degenerate even lattice $L$, $\theta$ the automorphism of $V_{L}$ induced from the $-1$ symmetry of $L$, and $V_{L}^{+}$ the fixed point subalgebra of $V_{L}$ under the action of $\theta$. In this series of papers, we classify the irreducible weak $V_{L}^{+}$-modules and show that any irreducible weak $V_{L}^{+}$-module is isomorphic to a weak submodule of some irreducible weak $V_{L}$-module or to a submodule of some irreducible $\theta$-twisted $V_{L}$-module. Let $M(1)^{+}$ be the fixed point subalgebra of the Heisenberg vertex operator algebra $M(1)$ under the action of $\theta$. In this paper (Part $2$), we show that there exists an irreducible $M(1)^{+}$-submodule in any non-zero weak $V_{L}^{+}$-module and we compute extension groups for $M(1)^{+}$., Comment: 42 pages. To appear in Journal of the Mathematical Society of Japan. We divide the article arXiv:1910.07126 into 3 parts for publication. This is the 2nd part

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

Let $V$ be a simple vertex algebra of countable dimension, $G$ be a finite automorphism group of $V$ and $\sigma$ be a central element of $G$. Assume that ${\cal S}$ is a finite set of inequivalent irreducible $\sigma$-twisted $V$-modules such that ${\cal S}$ is invariant under the action of $G$. Then there is a finite dimensional semisimple associative algebra ${\cal A}_{\alpha}(G,{\cal S})$ for a suitable $2$-cocycle $\alpha$ naturally determined by the $G$-action on ${\cal S}$ such that $({\cal A}_{\alpha}(G,{\cal S}),V^G)$ form a dual pair on the sum $\cal M$ of $\sigma$-twisted $V$-modules in ${\cal S}$ in the sense that (1) the actions of ${\cal A}_{\alpha}(G,{\cal S})$ and $V^G$ on $\cal M$ commute, (2) each irreducible ${\cal A}_{\alpha}(G,{\cal S})$-module appears in $\cal M,$ (3) the multiplicity space of each irreducible ${\cal A}_{\alpha}(G,{\cal S})$-module is an irreducible $V^G$-module, (4) the multiplicitiy spaces of different irreducible ${\cal A}_{\alpha}(G,{\cal S})$-modules are inequivalent $V^G$-modules. As applications, every irreducible $\sigma$-twisted $V$-module is a direct sum of finitely many irreducible $V^G$-modules and irreducible $V^G$-modules appearing in different $G$-orbits are inequivalent. This result generalizes many previous ones. We also establish a bijection between subgroups of $G$ and subalgebras of $V$ containing $V^G.$, Comment: 24 pages

In this paper, we define the shifted superconformal vector of supersymmetric charged free fermion vertex algebras, which is a 1-parameter deformation of the superconformal vector of the SUSY $bc$-$\beta\gamma$ system. Moreover, we find the corresponding shifted $N=2$ superconformal symmetry of SUSY charged free fermion vertex algebras, by using the $N_{K}=1$ SUSY vertex algebra formalism. Finally, in order to describe the shifted $N=2$ superconformal symmetry of the SUSY charged free fermion vertex algebra by $N=2$ superfields, we construct an $N_{K}=2$ SUSY version of the $bc$-$\beta\gamma$ system., Comment: MSc Thesis, 28 pages

The main goals for this paper is i) to study of an algebraic structure of $\mathbb{N}$-graded vertex algebras $V_B$ associated to vertex $A$-algebroids $B$ when $B$ are cyclic non-Lie left Leibniz algebras, and ii) to explore relations between the vertex algebras $V_B$ and the rank one Heisenberg vertex operator algebra. To achieve these goals, we first classify vertex $A$-algebroids $B$ associated to given cyclic non-Lie left Leibniz algebras $B$. Next, we use the constructed vertex $A$-algebroids $B$ to create a family of indecomposable non-simple vertex algebras $V_B$. Finally, we use the algebraic structure of the unital commutative associative algebras $A$ that we found to study relations between a certain type of the vertex algebras $V_B$ and the vertex operator algebra associated to a rank one Heisenberg algebra., Comment: 26 pages. arXiv admin note: text overlap with arXiv:1908.10446

We prove that $KL_k(\mathfrak{sl}_m)$ is a semi-simple, rigid braided tensor category for all even $m\ge 4$, and $k= -\frac{m+1}{2}$ which generalizes result from arXiv:2103.02985 obtained for $m=4$. Moreover, all modules in $KL_k(\mathfrak{sl}_m)$ are simple-currents and they appear in the decomposition of conformal embeddings $\mathfrak{gl}_m \hookrightarrow \mathfrak{sl}_{m+1} $ at level $ k= - \frac{m+1}{2}$ from arXiv:1509.06512. For this we inductively identify minimal affine $W$-algebra $ W_{k-1} (\mathfrak{sl}_{m+2}, \theta)$ as simple current extension of $L_{k}(\mathfrak{sl}_m) \otimes \mathcal H \otimes \mathcal M$, where $\mathcal H$ is the rank one Heisenberg vertex algebra, and $\mathcal M$ the singlet vertex algebra for $c=-2$. The proof uses previously obtained results for the tensor categories of singlet algebra from arXiv:2202.05496. We also classify all irreducible ordinary modules for $ W_{k-1} (\mathfrak{sl}_{m+2}, \theta)$. The semi-simple part of the category of $ W_{k-1} (\mathfrak{sl}_{m+2}, \theta)$-modules comes from $KL_{k-1}(\mathfrak{sl}_{m+2})$, using quantum Hamiltonian reduction, but this $W$-algebra also contains indecomposable ordinary modules., Comment: 20 pages

We prove that all holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one subspaces are unitary. The main method is to use the orbifold construction of a holomorphic VOA $V$ of central charge $24$ directly from a Niemeier lattice VOA $V_N$. We show that it is possible to extend the unitary form for the lattice VOA $V_N$ to the holomorphic VOA $V$ by using the orbifold construction and some information of the automorphism group $\mathrm{Aut}(V)$.

In this paper, a continuation of \cite{MPS}, we investigate the $S_3$-orbifold subalgebra of $(\mathcal{V}_c)^{\otimes 3}$, that is, we consider the $S_3$-fixed point vertex subalgebra of the tensor product of three copies of the universal Virasoro vertex operator algebras $\mathcal{V}_c$. Our main result is construction of a minimal, strong set of generators of this subalgebra for any generic values of $c$. More precisely, we show that this vertex algebra is of type $(2,4,6^2,8^2,9,10^2,11,12^3)$. We also investigate two prominent examples of simple $S_3$-orbifold algebras corresponding to central charges $c=\frac12$ (Ising model) and $c=-\frac{22}{5}$ (i.e. $(2,5)$-minimal model). We prove that the former is a new unitary $W$-algebra of type $(2,4,6,8)$ and the latter is isomorphic to the affine simple $W$-algebra of type $\frak{g}_2$ at non-admissible level $-\frac{19}{6}$. We also provide another version of this isomorphism using the affine $W$-algebra of type $\frak{g}_2$ coming from a subregular nilpotent element., Comment: 25 pages

Using the Zhu algebra for a certain category of $\mathbb{C}$-graded vertex algebras $V$, we prove that if $V$ is finitely $\Omega$-generated and satisfies suitable grading conditions, then $V$ is rational, i.e. has semi-simple representation theory, with one dimensional level zero Zhu algebra. Here $\Omega$ denotes the vectors in $V$ that are annihilated by lowering the real part of the grading. We apply our result to the family of rank one Weyl vertex algebras with conformal element $\omega_\mu$ parameterized by $\mu \in \mathbb{C}$, and prove that for certain non-integer values of $\mu$, these vertex algebras, which are non-integer graded, are rational, with one dimensional level zero Zhu algebra. In addition, we generalize this result to appropriate $\mathbb{C}$-graded Weyl vertex algebras of arbitrary ranks., Comment: Final version. Typos corrected and bibliography updated. We thank the referee for their comments and suggestions. To appear in Journal of Mathematical Physics

This paper is to study what we call twisted regular representations for vertex operator algebras. Let $V$ be a vertex operator algebra, let $\sigma_1,\sigma_2$ be commuting finite-order automorphisms of $V$ and let $\sigma=(\sigma_1\sigma_2)^{-1}$. Among the main results, for any $\sigma$-twisted $V$-module $W$ and any nonzero complex number $z$, we construct a weak $\sigma_1\otimes \sigma_2$-twisted $V\otimes V$-module $\mathfrak{D}_{\sigma_1,\sigma_2}^{(z)}(W)$ inside $W^{*}$. Let $W_1,W_2$ be $\sigma_1$-twisted, $\sigma_2$-twisted $V$-modules, respectively. We show that $P(z)$-intertwining maps from $W_1\otimes W_2$ to $W^{*}$ are the same as homomorphisms of weak $\sigma_1\otimes \sigma_2$-twisted $V\otimes V$-modules from $W_1\otimes W_2$ into $\mathfrak{D}_{\sigma_1,\sigma_2}^{(z)}(W)$. We also show that a $P(z)$-intertwining map from $W_1\otimes W_2$ to $W^{*}$ is equivalent to an intertwining operator of type $\binom{W'}{W_1\; W_2}$, which is a twisted version of a result of Huang and Lepowsky. Finally, we show that for each $\tau$-twisted $V$-module $M$ with $\tau$ any finite-order automorphism of $V$, the coefficients of the $q$-graded trace function lie in $\mathfrak{D}_{\tau,\tau^{-1}}^{(-1)}(V)$, which generate a $\tau\otimes \tau^{-1}$-twisted $V\otimes V$-submodule isomorphic to $M\otimes M'$., Comment: 36 pages

This paper is to establish a natural connection between regular representations for a vertex operator algebra $V$ and $A_{n}(V)$-$A_{m}(V)$ bimodules of Dong and Jiang. Let $W$ be a weak $V$-module and let $(m,n)$ be a pair of nonnegative integers. We study two quotient spaces $A_{n,m}^{\dagger}(W)$ and $A^{\diamond}_{n,m}(W)$ of $W$. It is proved that the dual space $A^{\dagger}_{n,m}(W)^{*}$ viewed as a subspace of $W^*$ coincides with the level-$(m,n)$ vacuum subspace of the regular representation module $\mathfrak{D}_{(-1)}(W)$. By making use of this connection, we obtain an $A_{n}(V)$-$A_m(V)$ bimodule structure on both $A_{n,m}^{\dagger}(W)$ and $A^{\diamond}_{n,m}(W)$. Furthermore, we obtain an $\N$-graded weak $V$-module structure together with a commuting right $A_m(V)$-module structure on $A^{\diamond}_{\Box,m}(W):=\oplus_{n\in \N}A^{\diamond}_{n,m}(W)$. Consequently, we recover the corresponding results and roughly confirm a conjecture of Dong and Jiang., Comment: 28 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

Let $V$ be a vertex operator algebra, $T\in \mathbb{N}$ and $(M^k, Y_{M^k})$ for $k=1, 2, 3$ be a $g_k$-twisted module, where $g_k$ are commuting automorphisms of $V$ such that $g_k^T=1$ for $k=1, 2, 3$ and $g_3=g_1g_2$. Suppose $I(\cdot, z)$ is an intertwining operator of type $({array}{c} M^{3} M^{1} M^{2} {array}) $. We construct an $A_{g_1g_2}(V)$-$A_{g_2}(V)$-bimodule $A_{g_1g_2, g_2}(M^1)$ which determines the action of $M^1$ from the bottom level of $M^2$ to the bottom level of $M^3$ and explored its connections with fusion rules., Comment: 26 pages

For a fixed positive integer $k$, any element $g$ of the permutation group $S_{k}$ acts on the tensor product vertex operator algebra $V^{\otimes k}$ in the obvious way. In this paper, we determine the $S$-matrix of $\left(V^{\otimes k}\right)^{G}$ if $G=\left\langle g\right\rangle $ is the cyclic group generated by $g=\left(1,\ 2,\cdots,k\right).$, Comment: 20 pages

We consider the Whittaker modules $M_{1}(\lambda,\mu)$ for the Weyl vertex algebra $M$, constructed in arXiv:1811.04649, where it was proved that these modules are irreducible for each finite cyclic orbifold $M^{\Bbb Z_n}$. In this paper, we consider the modules $M_{1}(\lambda,\mu)$ as modules for the ${\Bbb Z}$-orbifold of $M$, denoted by $M^0$. $M^0$ is isomorphic to the vertex algebra $\mathcal W_{1+\infty, c=-1} = \mathcal M(2) \otimes M_1(1)$ which is the tensor product of the Heisenberg vertex algebra $M_1(1)$ and the singlet algebra $\mathcal M(2)$. Furthermore, these modules are also modules of the Lie algebra $\widehat{\mathfrak gl}$ with central charge $c=-1$. We prove they are reducible as $\widehat{\mathfrak gl}$-modules (and therefore also as $M^0$-modules), and we completely describe their irreducible quotients $L(d,\lambda,\mu)$. We show that $L(d,\lambda,\mu)$ in most cases are not tensor product modules for the vertex algebra $ \mathcal M(2) \otimes M_1(1)$. Moreover, we show that all constructed modules are typical in the sense that they are irreducible for the Heisenberg-Virasoro vertex subalgebra of $\mathcal W_{1+\infty, c=-1}$., Comment: 25 pages

This paper is to study vertex operator superalgebras which are strongly generated by their weight-$2$ and weight-$\frac{3}{2}$ homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra $V$ is simple, then $V_{(2)}$ has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and $V_{(\frac{3}{2})}$ is naturally a $V_{(2)}$-module equipped with a $V_{(2)}$-valued symmetric bilinear form and a non-degenerate ($\mathbb{C}$-valued) symmetric bilinear form, satisfying a set of conditions. On the other hand, assume that $A$ is any commutative associative algebra equipped with a non-degenerate symmetric associative bilinear form and assume that $U$ is an $A$-module equipped with a symmetric $A$-valued bilinear form and a non-degenerate ($\mathbb{C}$-valued) symmetric bilinear form, satisfying the corresponding conditions. Then we construct a Lie superalgebra $\mathcal{L}(A,U)$ and a simple vertex operator superalgebra $L_{\mathcal{L}(A,U)}(\ell,0)$ for every nonzero number $\ell$ such that $L_{\mathcal{L}(A,U)}(\ell,0)_{(2)}=A$ and $L_{\mathcal{L}(A,U)}(\ell,0)_{(\frac{3}{2})}=U$., Comment: 25 pages

We introduce a subalgebra $\overline F$ of the Clifford vertex superalgebra ($bc$ system) which is completely reducible as a $L^{Vir} (-2,0)$-module, $C_2$-cofinite, but it is not conformal and it is not isomorphic to the symplectic fermion algebra $\mathcal{SF}(1)$. We show that $\mathcal{SF}(1)$ and $\overline{F}$ are in an interesting duality, since $\overline{F}$ can be equipped with the structure of a $\mathcal{SF}(1)$-module and vice versa. Using the decomposition of $\overline F$ and a free-field realization from arXiv:1711.11342, we decompose $L_k(\mathfrak{osp}(1\vert 2))$ at the critical level $k=-3/2$ as a module for $L_k(\mathfrak{sl}(2))$. The decomposition of $L_k(\mathfrak{osp}(1\vert 2))$ is exactly the same as of the $N=4$ superconformal vertex algebra with central charge $c=-9$, denoted by $\mathcal V^{(2)}$. Using the duality between $\overline{F}$ and $\mathcal{SF}(1)$, we prove that $L_k(\mathfrak{osp}(1\vert 2))$ and $\mathcal V^{(2)}$ are in the duality of the same type. As an application, we construct and classify all irreducible $L_k(\mathfrak{osp}(1\vert 2))$-modules in the category $\mathcal O$ and the category $\mathcal R$ which includes relaxed highest weight modules. We also describe the structure of the parafermion algebra $N_{-3/2}(\mathfrak{osp}(1\vert 2))$ as a $N_{-3/2}(\mathfrak{sl}(2))$-module. We extend this example, and for each $p \ge 2$, we introduce a non-conformal vertex algebra $\mathcal A^{(p)}_{new}$ and show that $\mathcal A^{(p)}_{new} $ is isomorphic to the doublet vertex algebra as a module for the Virasoro algebra. We also construct the vertex algebra $ \mathcal V^{(p)} _{new}$ which is isomorphic to the logarithmic vertex algebra $\mathcal V^{(p)}$ as a module for $\widehat{\mathfrak{sl}}(2)$., Comment: 22 pages

Let $V$ be a simple, rational, $C_2$-cofinite vertex operator algebra and $G$ a finite group acting faithfully on $V$ as automorphisms, which is simply called a rational vertex operator algebra with a $G$-action. It is shown that the category ${\cal E}_{V^G}$ generated by the $V^G$-submodules of $V$ is a symmetric fusion category braided equivalent to the $G$-module category ${\cal E}={\rm Rep}(G)$. If $V$ is holomorphic, then the $V^G$-module category ${\cal C}_{V^G}$ is a minimal modular extension of ${\cal E},$ and is equivalent to the Drinfeld center ${\cal Z}({\rm Vec}_G^{\alpha})$ as modular tensor categories for some $\alpha\in H^3(G,S^1)$ with a canonical embedding of ${\cal E}$. Moreover, the collection ${\cal M}_v({\cal E})$ of equivalence classes of the minimal modular extensions ${\cal C}_{V^G}$ of ${\cal E}$ for holomorphic vertex operator algebras $V$ with a $G$-action form a group, which is isomorphic to a subgroup of $H^3(G,S^1).$ Furthermore, any pointed modular category ${\cal Z}({\rm Vec}_G^{\alpha})$ is equivalent to ${\cal C}_{V_L^G}$ for some positive definite even unimodular lattice $L.$ In general, for any rational vertex operator algebra $U$ with a $G$-action, ${\cal C}_{U^G}$ is a minimal modular extension of the braided fusion subcategory ${\cal F}$ generated by the $U^G$-submodules of $U$-modules. Furthermore, the group ${\cal M}_v({\cal E})$ acts freely on the set of equivalence classes ${\cal M}_v({\cal F})$ of the minimal modular extensions ${\cal C}_{W^G}$ of ${\cal F}$ for any rational vertex operators algebra $W$ with a $G$-action., Comment: 43 pages, correct typos and add more details to the proof of Theorem 7.1