We provide a rigorous mathematical foundation to the study of strongly rational, holomorphic vertex operator algebras V of central charge c = 8, 16 and 24 initiated by Schellekens. If c = 8 or 16 we show that V is isomorphic to a lattice theory corresponding to a rank c even, self-dual lattice. If c = 24 we prove, among other things, that either V is isomorphic to a lattice theory corresponding to a Niemeier lattice or the Leech lattice, or else the Lie algebra on the weight one subspace V_1 is semisimple (possibly 0) of Lie rank less than 24.
This article is a continuation of our work on the classification of holomorphic framed vertex operator algebras of central charge 24. We show that a holomorphic framed VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one subspace. As a consequence, we completely classify all holomorphic framed vertex operator algebras of central charge 24 and show that there exist exactly 56 such vertex operator algebras, up to isomorphism.
In this article, we describe a construction of a holomorphic vertex operator algebras of central charge $24$ whose weight one Lie algebra has type $A_{6,7}$.
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 article, we construct three new holomorphic vertex operator algebras of central charge $24$ using the $\mathbb{Z}_2$-orbifold construction associated to inner automorphisms. Their weight one subspaces has the Lie algebra structures $D_{7,3}A_{3,1}G_{2,1}$, $E_{7,3}A_{5,1}$, and $A_{8,3}A_{2,1}^2$. In addition, we discuss the constructions of holomorphic vertex operator algebras with Lie algebras $A_{5,6}C_{2,3}A_{1,2}$ and $D_{6,5}A_{1,1}^2$ from holomorphic vertex operator algebras with Lie algebras $C_{5,3}G_{2,2}A_{1,1}$ and $A_{4,5}^2$, respectively.
In this article, we develop a general technique for proving the uniqueness of holomorphic vertex operator algebras based on the orbifold construction and its "reverse" process. 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 $E_{6,3}G_{2,1}^3$, $A_{2,3}^6$ or $A_{5,3}D_{4,3}A_{1,1}^3$.
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 ver…
We continue our program on classification of holomorphic vertex operator algebras of central charge $24$. In this article, we show that there exists a unique strongly regular holomorphic VOA of central charge $24$, up to isomorphism, if its weight one Lie algebra has the type $C_{4,10}$, $D_{7,3}A_{3,1}G_{2,1}$, $A_{5,6}C_{2,3}A_{1,2}$, $A_{3,1}C_{7,2}$, $D_{5,4}C_{3,2}A_{1,1}^2$, or $E_{6,4}C_{2,1}A_{2,1}$. As a consequence, we have verified that the isomorphism class of a strongly regular holomorphic vertex operator algebra of central charge $24$ is determined by its weight one Lie algebra s…
The vertex operator algebra structure of a strongly regular holomorphic vertex operator algebra $V$ of central charge $24$ is proved to be uniquely determined by the Lie algebra structure of its weight one space $V_1$ if $V_1$ is a Lie algebra of the type $A_{1,4}^{12}$, $B_{2,2}^6$, $B_{3,2}^4$, $B_{4,2}^3$, $B_{6,2}^2$, $B_{12,2}$, $D_{4,2}^2B_{2,1}^4$, $D_{8,2}B_{4,1}^2$, $A_{3,2}^4A_{1,1}^4$, $D_{5,2}^2A_{3,1}^2$, $D_{9,2}A_{7,1}$, $C_{4,1}^4$ or $D_{6,2}B_{3,1}^2C_{4,1}$.
In this article, we determine the automorphism groups of $14$ holomorphic vertex operator algebras of central charge $24$ obtained by applying the $\mathbb{Z}_2$-orbifold construction to the Niemeier lattice vertex operator algebras and lifts of the $-1$-isometries.
We describe the automorphism groups of all holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one Lie algebras by using their constructions as simple current extensions. We also confirm a conjecture of G. Höhn on the numbers of holomorphic vertex operator algebras of central charge $24$ obtained as inequivalent simple current extensions of certain vertex operator algebras, which gives another proof of the uniqueness of holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one Lie algebras.
In this note we study holomorphic integer graded vertex superalgebras. We prove that all such vertex superalgebras of central charge 8 and 16 are purely even. For the case of central charge 24 we prove that the weight-one Lie superalgebra is either zero, of superdimension 24, or else is one of an explicit list of 1332 semisimple Lie superalgebras.
In this thesis we develop an orbifold theory for a finite, cyclic group $G$ acting on a suitably regular, holomorphic vertex operator algebra $V$. To this end we describe the fusion algebra of the fixed-point vertex operator subalgebra $V^G$ and show that $V^G$ has group-like fusion. Then we solve the extension problem for vertex operator algebras with group-like fusion. We use these results to construct five new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds, contributing to the classification of the $V_1$-structures of suitably regular, holomorphic vertex oper…
In 1993, Schellekens obtained a list of possible 71 Lie algebras of holomorphic vertex operator algebras with central charge 24. However, not all cases are known to exist. The aim of this article is to construct new holomorphic vertex operator algebras using the theory of framed vertex operator algebras and to determine the Lie algebra structures of their weight one subspaces. In particular, we study holomorphic framed vertex operator algebras associated to subcodes of the triply even codes $\RM(1,4)^3$ and $\RM(1,4)\oplus \EuD(d_{16}^+)$ of length 48. These vertex operator algebras correspond…
We establish that the Lie algebra of weight one states in a (strongly) rational vertex operator algebra is reductive, and that its Lie rank is bounded above by the effective central charge. We show that lattice vertex operator algebras may be characterized by the equalities of the effective central charge, the Lie rank and the central charge, and in particular holomorphic lattice theories may be characterized among all holomorphic vertex operator algebras by the equality of the Lie rank and the central charge.
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 sub…
We give a lattice theoretical interpretation of generalized deep holes of the Leech lattice VOA $V_Λ$. 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 $(τ, \tildeβ)$ satisfying certain conditions, where $τ\in Co_0$ and $\tildeβ$ is a $τ$-invariant deep hole of squared length $2$. It provides a new combinatorial approach towards the classif…
Let $X$ be a complex manifold, $π: E \rightarrow X$ a locally trivial holomorphic fibration with fiber $F$, and $\mathfrak{g}$ a Lie algebra with an invariant symmetric form. We associate to this data a holomorphic prefactorization algebra $\mathcal{F}_{\mathfrak{g}, π}$ on $X$ in the formalism of Costello-Gwilliam. When $X=\mathbb{C}$, $\mathfrak{g}$ is simple, and $F$ is a smooth affine variety, we extract from $\mathcal{F}_{\mathfrak{g}, π}$ a vertex algebra which is a vacuum module for the universal central extension of the Lie algebra $\mathfrak{g} \otimes H^{0}(F, \mathcal{O})[z,z^{-1}]$…
In this article, we show that a framed vertex operator algebra V satisfying the conditions: (1) V is holomorphic (i.e., V is the only irreducible V-module); (2) V is of rank 24; and (3) V_1=0; is isomorphic to the moonshine vertex operator algebra constructed by Frenkel-Lepowsky-Meurman.
We determine the automorphism groups of the cyclic orbifold vertex operator algebras associated with coinvariant lattices of isometries of the Leech lattice in the conjugacy classes $4C,6E,6G,8E$ and $10F$. As a consequence, we have determined the automorphism groups of all the $10$ vertex operator algebras in [Hö], which are useful to analyze holomorphic vertex operator algebras of central charge $24$.
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, …
Let V be the even part of the vertex operator super-algebra of r pairs of symplectic fermions. Up to two conjectures, we show that V admits a unique holomorphic extension if r is a multiple of 8, and no holomorphic extension otherwise. This is implied by two results obtained in this paper: 1) If r is a multiple of 8, one possible holomorphic extension is given by the lattice vertex operator algebra for the even self dual lattice $D_r^+$ with shifted stress tensor. 2) We classify Lagrangian algebras in SF(h), a ribbon category associated to symplectic fermions. The classification of holomorphic…
The extensions of the affine vertex operator algebras $L_{sl_2}(k,0)$ and the preunitary vertex operator algebras with central charges $c<1$ are classified. In particular, the unitary vertex operator algebras with central charges $c<1$ are classified.
We construct orbifolds of holomorphic lattice Vertex Operator Algebras for non-Abelian finite automorphism groups $G$. To this end, we construct twisted modules for automorphisms $g$ together with the projective representation of the centralizer of $g$ on the twisted module. This allows us to extract the irreducible modules of the fixed point VOA $V^G$, and to compute their characters and modular transformation properties. We then construct holomorphic VOAs by adjoining such modules to $V^G$. Applying these methods to extremal lattices in $d=48$ and $d=72$, we construct more than fifty new hol…
By applying Miyamoto's $\mathbb{Z}_{3}$-orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices and their automorphisms of order 3, we construct holomorphic vertex operator algebras of central charge 24 whose Lie algebras of the weight one spaces are of types $A_{2,3}^6$, $E_{6,3}G_{2,1}^{3}$, and $A_{5,3}D_{4,3}A_{1,1}^{3}$, which correspond to No.6, No.17, and No.32 on Schellekens' list, respectively.
It is one of the remarkable results of vertex operator algebras (VOAs) that the graded traces (one-point correlation functions) of holomorphic VOAs are modular functions. This paper explores the question of which modular functions arise as the graded traces of holomorphic VOAs. For VOAs of small central charge, i.e., $c\le 24$, and a non-zero weight-one subspace, we find that the only conditions imposed on the modular fuctions are those that arise easily out of our condition that the VOAs be of CFT type, that is that they have no negative-weight subspaces and their zero-weight subspace is gene…
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 isom…
We study graded traces of vectors in free bosonic vertex operator algebras and lattice vertex operator algebras. We show in particular that trace functions in these two theories always have the shape f(q)/η(q)^d where f(q) is quasi-modular in the case of d free bosons, and modular (i.e., a sum of holomorphic modular forms of various weights) in the case of theories based on a lattice L of rank d. We also show how spherical harmonic polynomials with respect to L are related to primary fields in lattice theories.
A large class of supersymmetric quantum field theories, including all theories with $\mathcal{N} = 2$ supersymmetry in three dimensions and theories with $\mathcal{N} = 2$ supersymmetry in four dimensions, possess topological-holomorphic sectors. We formulate Poisson vertex algebras in such topological-holomorphic sectors and discuss some examples. For a four-dimensional $\mathcal{N} = 2$ superconformal field theory, the associated Poisson vertex algebra is the classical limit of a vertex algebra generated by a subset of local operators of the theory.
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…
Let $\D$ be the Lie algebra of regular differentialoperators on ${\C} \setminus \{0\}$, and ${\hD}= {\D} + {\C} C$ be the central extension of ${\D}$. Let $W_{1+\infty,-N}$ be the vertex algebra associated to the irreducible vacuum $\hD$-module with the central charge $c=-N$. We show that $W_{1+\infty,-N}$ is a subalgebra of the Heisenberg vertex algebra M(1) with $2 N$ generators, and construct 2N-dimensional family of irreducible $W_{1+\infty,-N}$-modules. Considering these modules as $\hD$-modules, we identify the corresponding highest weights.
We consider correlation functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We describe a generalisation of genus one Zhu recursion where we express an arbitrary genus two $n$--point correlation function in terms of $(n-1)$--point functions. We consider several applications including the correlation functions for the Heisenberg vertex operator algebra and its modules, Virasoro correlation functions and genus two Ward identities. We derive novel differential equations in terms of a differential operator on the genus two Siegel upper half pl…
We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vaccum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naru…
In hep-th/0010293 Kapustin and Orlov introduce the notion of an OPE-algebra and propose that it formalizes conformal field theories in the same way as vertex algebras formalize chiral algebras, i.e. the subalgebras of holomorphic fields of conformal field theories. In this thesis we study the question which concepts and results of the general theory of vertex algebras can be extended to OPE-algebras.
Let $L$ be an even (positive definite) lattice and $g\in O(L)$. In this article, we prove that the orbifold vertex operator algebra $V_{L}^{\hat{g}}$ has group-like fusion if and only if $g$ acts trivially on the discriminant group $\mathcal{D}(L)=L^*/L$ (or equivalently $(1-g)L^*<L$). We also determine their fusion rings and the corresponding quadratic space structures when $g$ is fixed point free on $L$. By applying our method to some coinvariant sublattices of the Leech lattice $Λ$, we prove a conjecture proposed by G. Höhn. In addition, we also discuss a construction of certain holomorphic…
Rational chiral conformal field theories are organized according to their genus, which consists of a modular tensor category $\mathcal{C}$ and a central charge $c$. A long-term goal is to classify unitary rational conformal field theories based on a classification of unitary modular tensor categories. We conjecture that for any unitary modular tensor category $\mathcal{C}$, there exists a unitary chiral conformal field theory $V$ so that its modular tensor category $\mathcal{C}_V$ is $\mathcal{C}$. In this paper, we initiate a mathematical program in and around this conjecture. We define a cla…
We will partially classify spaces of characters of vertex operator algebras $V$ with central charges 8 and 16, such that the spaces of characters is 3-dimensional and the characters forms a basis of the solution space of a third order monic modular linear differential equation with rational indicial roots. Assuming a mild arithmetic condition, we show that the space of characters of $V$ coincides with the space of characters of lattice vertex operators associated with integral lattices $\sqrt{2}E_8$ or the affine vertex operator algebra of type $D_{20}^{(1)}$ for $c=8$, and the Barnes--Wall la…
A characterization of vertex operator algebra $V_L^+$ for any rank one positive definite even lattice $L$ is given in terms of dimensions of homogeneous subspaces of small weights. This result reduces the classification of rational vertex operator algebras of central charge 1 to the characterization of three vertex operator algebras in the $E$-series of central charge one.
We define and compute the continuous orbifold partition function and a generating function for all $n$-point correlation functions for the rank two free fermion vertex operator superalgebra on a genus two Riemann surface formed by self-sewing a torus. The partition function is proportional to an infinite dimensional determinant with entries arising from torus Szego kernel and the generating function is proportional to a finite determinant of genus two Szego kernels. These results follow from an explicit analysis of all torus $n$-point correlation functions for intertwiners of the irreducible m…
In this note we give an account of recent progress on the construction of holomorphic vertex algebras as cyclic orbifolds as well as related topics in lattices and modular categories. We present a novel computation of the Schur indicator of a lattice involution orbifold using finite Heisenberg groups and discriminant forms.
This paper is about the orbifold theory for vertex operator superalgebras. Given a vertex operator superalgebra V and a finite automorphism group G of V, we show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in G under the C_2-cofinite condition. We also establish that these functions afford a representation of the full modular group if V is C_2-cofinite and g-rational for any g in G.
Motivated by the notion of extremal vertex operator algebras, we investigate cyclic orbifolds of vertex operator algebras coming from extremal even self-dual lattices in $d=48$ and $d=72$. In this way we construct about one hundred new examples of holomorphic VOAs with a small number of low weight states.
We introduce the notion of a genus and its mass for vertex algebras. For lattice vertex algebras, their genera are the same as those of lattices, which plays an important role in the classification of lattices. We derive a formula relating the mass for vertex algebras to that for lattices, and then give a new characterization of some holomorphic vertex operator algebras.
Using techniques from the theory of mock modular forms and harmonic Maass forms, especially Weierstrass mock modular forms, we establish several dimension formulas for certain strongly rational, holomorphic vertex operator algebras, complementing previous work by van Ekeren, Möller, and Scheithauer.
The rational and C_2-cofinite simple vertex operator algebras whose effective central charges and the central charges c are equal and less than 1 are classified. Such a vertex operator algebra is zero if c<0 and C if c=0. If c>0, it is an extension of discrete Virasoro vertex operator algebra L(c_{p,q},0) by its irreducible modules. It is also proved that for any rational and C_2-cofinite simple vertex operator algebra whose effective central charge and central charge are equal, the vertex operator subalgebra generated by the Virasoro vector is simple.
We construct holomorphic local conformal framed nets extended from a tensor power of the Virasoro net with c=1/2 with a pair of binary codes (C,D) satisfying the conditions given by Lam and Yamauchi for holomorphic framed vertex operator algebras. Our result is an operator algebraic counterpart of theirs, but our proof is entirely different. We apply the alpha-induction in order to identify the representation theory of "code local conformal net" and this gives rise to the existence of the desired local conformal net.
We introduce the notion of N=1 supergeometric vertex operator superalgebra motivated by the worldsheet geometry underlying genus-zero, two-dimensional, holomorphic N=1 superconformal field theory. We then show, assuming the convergence of certain projective factors, that the category of such objects is isomorphic to the category of N=1 Neveu-Schwarz vertex operator superalgebras.
