6-transposition property of $τ$-involutions of vertex operator algebras

Sigma involutions associated with parafermion vertex operator algebra $K(\mathfrak{sl}_2,k)$

An irreducible module for the parafermion vertex operator algebra $K(\mathfrak{sl}_2,k)$ is said to be of $σ$-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 $K(\mathfrak{sl}_2,k)^{\langle θ\rangle}$ spanned by the irreducible direct summands of $σ$-type irreducible $K(\mathfrak{sl}_2,k)$-modules, where $θ$ is an involution of $K(\mathfrak{sl}_2,k)$. We discuss some examples of such an automorphism as well.

Vertex operator algebras, extended E_8 diagram, and McKay's observation on the Monster simple group

We study McKay's observation on the Monster simple group, which relates the 2A-involutions of the Monster simple group to the extended E_8 diagram, using the theory of vertex operator algebras (VOAs). We first consider the sublattices L of the E_8 lattice obtained by removing one node from the extended E_8 diagram at each time. We then construct a certain coset (or commutant) subalgebra U associated with L in the lattice VOA V_{\sqrt{2}E_8}. There are two natural conformal vectors of central charge 1/2 in U such that their inner product is exactly the value predicted by Conway. The Griess alge…

Fusion rules from root systems I: case ${\rm A}_n$

Axial algebras are commutative algebras generated by idempotents; they generalise associative algebras by allowing the idempotents to have additional eigenvectors, controlled by fusion rules. If the fusion rules are $\mathbb{Z}/2$-graded, axial algebras afford representations of transposition groups. We consider axial representations of Weyl groups of simply-laced root systems, which are examples of regular $3$-transposition groups. We introduce coset axes, a special class of idempotents based on embeddings of transposition groups, and use them to study the propagation of fusion rules in axial…

A nonmeromorphic extension of the moonshine module vertex operator algebra

We describe a natural structure of an abelian intertwining algebra (in the sense of Dong and Lepowsky) on the direct sum of the untwisted vertex operator algebra constructed {}from the Leech lattice and its (unique) irreducible twisted module. When restricting ourselves to the moonshine module, we obtain a new and conceptual proof that the moonshine module has a natural structure of a vertex operator algebra. This abelian intertwining algebra also contains an irreducible twisted module for the moonshine module with respect to the obvious involution. In addition, it contains a vertex operator s…

Norton's Trace Formulae for the Griess Algebra of a Vertex Operator Algebra with Larger Symmetry

Formulae expressing the trace of the composition of several (up to five) adjoint actions of elements of the Griess algebra of a vertex operator algebra are derived under certain assumptions on the action of the automorphism group. They coincide, when applied to the moonshine module $V^\natural$, with the trace formulae obtained in a different way by S.Norton, and the spectrum of idempotents related to 2A, 2B, 3A and 4A element of the Monster is determined by the representation theory of Virasoro algebra at $c=1/2$, $W_3$ algebra at $c=4/5$ or $W_4$ algebra at $c=1$. The generalization to the t…

On the correspondence between mirror-twisted sectors for N=2 supersymmetric vertex operator superalgebras of the form $V \otimes V$ and N=1 Ramond sectors of $V$

Using recent results of the author along with Vander Werf, we present the classification and construction of mirror-twisted modules for N=2 supersymmetric vertex operator superalgebras of the form $V \otimes V$ for the signed transposition mirror map automorphism. In particular, we show that the category of such mirror-twisted sectors for $V\otimes V$ is isomorphic to the category of N=1 Ramond sectors for $V$.

Lattices, Vertex Algebras and Modular Categories

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.

2A-orbifold construction and the baby-monster vertex operator superalgebra

In this article we prove that the full automorphism group of the baby-monster vertex operator superalgebra constructed by Hoehn is isomorphic to 2xB, where B is the baby-monster sporadic finite simple group and determine irreducible modules for the baby-monster vertex operator algebra. Our result has many corollaries. In particular, we can prove that the Z_2-orbifold construction with respect to a 2A-involution of the Monster applied to the moonshine vertex operator algebra yields the moonshine vertex operator algebra itself again.

Ising vectors in the vertex operator algebra $V_Λ^+$ associated with the Leech lattice $Λ$

In this article, we study the Ising vectors in the vertex operator algebra $V_Λ^+$ associated with the Leech lattice $Λ$. The main result is a characterization of the Ising vectors in $V_Λ^+$. We show that for any Ising vector $e$ in $V_Λ^+$, there is a sublattice $E\cong \sqrt{2}E_8$ of $Λ$ such that $e\in V_E^+$. Some properties about their corresponding $τ$-involutions in the moonshine vertex operator algebra $V^\natural$ are also discussed. We show that there is no Ising vector of $σ$-type in $V^\natural$. Moreover, we compute the centralizer $C_{\aut V^\natural}(z, τ_e)$ for any Ising vec…

McKay's observation and vertex operator algebras generated by two conformal vectors of central charge 1/2

This paper is a continuation of our paper math.QA/0403010 at which several coset subalgebras of the lattice VOA $V_{\sqrt{2}E_8}$ were constructed and the relationship between such algebras with the famous McKay observation on the extended E_8 diagram and the Monster simple group were discussed. In this article, we shall provide the technical details. We completely determine the structure of the coset subalgebras constructed and show that they are all generated by two conformal vectors of central charge 1/2. We also study the representation theory of these coset subalgebras and show that the p…

Weyl groups and vertex operator algebras generated by Ising vectors satisfying $(2B,3C)$ condition

In this article, we construct explicitly certain moonshine type vertex operator algebras generated by a set of Ising vectors $I$ such that (1) for any $e\neq f\in I$, the subVOA $\mathrm{VOA}(e,f)$ generated by $e$ and $f$ is isomorphic to either $U_{2B}$ or $U_{3C}$; and (2)the subgroup generated by the corresponding Miyamoto involutions $\{τ_e|\,e\in I\}$ is isomorphic to the Weyl group of a root system of type $A_n$, $D_n$, $E_6$, $E_7$ or $E_8$. The structures of the corresponding vertex operator algebras and their Griess algebras are also studied. In particular, the central charge of thes…

On Majorana representations of the group $3^2{:}2$ of 3C-pure type and the corresponding vertex operator algebras

In this article, we study Griess algebras and vertex operator subalgebras generated by Ising vectors in a moonshine type VOA such that the subgroup generated by the corresponding Miyamoto involutions has the shape $3^2{:}2$ and any two Ising vectors generate a 3C subVOA $U_{3C}$. We show that such a Griess algebra is uniquely determined, up to isomorphisms. The structure of the corresponding vertex operator algebra is also discussed. In addition, we give a construction of such a VOA inside the lattice VOA $V_{E_8^3}$, which gives an explicit example for Majorana representations of the group $3…

Extended Griess algebras and Matsuo-Norton trace formulae

We introduce the Z_2-extended Griess algebra of a vertex operator superalgebra with an involution and derive the Matsuo-Norton trace formulae for the extended Griess algebra based on conformal design structure. We illustrate an application of our formulae by reformulating the one-to-one correspondence between 2A-elements of the Baby-monster simple group and N=1 c=7/10 Virasoro subalgebras inside the Baby-monster vertex operator superalgebra.

Modularity of trace functions in orbifold theory for Z-graded vertex operator superalgebras

We study the trace functions in orbiford theory for Z-graded vertex operator superalgebras and obtain a modular invariance result. More precisely, let V be a C_2-cofinite Z-graded vertex operator superalgebra and G a finite automorphism group of V. Then for any commuting pairs (g,h) in G, the hσ-trace functions associated to the simple g-twisted V-modules are holomorphic in the upper half plane where σis the canonical involution on V coming from the superspace structure of V. If V is further g-rational for every g n G, the trace unctions afford a representation for the full modular group SL(2,…

The varieties of Heisenberg vertex operator algebras

For a vertex operator algebra $V$ with conformal vector $ω$, we consider a class of vertex operator subalgebras and their conformal vectors. They are called semi-conformal vertex operator subalgebras and semi-conformal vectors of $(V,ω)$, respectively, and were used to study duality theory of vertex operator algebras via coset constructions. Using these objects attached to $(V,ω)$, we shall understand the structure of the vertex operator algebra $(V,ω)$. At first, we define the set $\on{Sc}(V,ω)$ of semi-conformal vectors of $V$, then we prove that $\on{Sc}(V,ω)$ is an affine algebraic variety…

$\frac{\mathbb{Z}}{2}$-分次亚纯开弦顶点代数及其 $\mathbb{Z}_2$-扭曲模的费米子构造，II

Fermionic construction of the $\frac{\mathbb{Z}}{2}$-graded meromorphic open-string vertex algebra and its $\mathbb{Z}_2$-twisted module, II

本文定义了被对合扭曲的亚纯开弦顶点代数模并证明其收敛性，通过费米子构造给出了其典范扭曲模的具体实例。

置换轨道和简单电流扩张

Permutation orbifolds and simple current extensions

本文在顶点算子（超）代数框架下研究置换轨道和简单电流扩张，通过张量范畴方法和简单电流扩张建立可解自同构群的诱导理论，并分类不可约扭模。

On permutation-twisted free fermions and two conjectures

We conjecture that the category of permutation-twisted modules for a multi-fold tensor product vertex operator superalgebra and a cyclic permutation of even order is isomorphic to the category of parity-twisted modules for the underlying vertex operator superalgebra. This conjecture is based on our observations of the cyclic permutation-twisted modules for free fermions as we discuss in this work, as well as previous work of the first author constructing and classifying permutation-twisted modules for tensor product vertex operator superalgebras and a permutation of odd order. In addition, we …

顶点代数与TKK代数

Vertex algebras and TKK algebras

本文通过twisted modules将TKK代数与顶点代数联系起来，证明了特定范畴的同构关系，并对不可约twisted modules进行了分类。

Twist-field representations of W-algebras, exact conformal blocks and character identities

We study the twist-field representations of W-algebras and generalize construction of the corresponding vertex operators to D- and B-series. It is shown, how the computation of characters of these representations leads to nontrivial identities involving lattice theta-functions. We also propose a way to calculate their exact conformal blocks, expressing them for D-series in terms of geometric data of the corresponding Prym variety for covering curve with involution.

计算顶点算子代数的 G-交叉扩张与轨形

Computing $G$-Crossed Extensions and Orbifolds of Vertex Operator Algebras

本文发展计算辫子张量范畴 G-交叉扩张的工具，利用唯一性结果和与交换代数凝聚的交换性，确定格顶点算子代数轨形的模张量范畴，并推广 Tambara-Yamagami 范畴。

Quasifinite representations of classical Lie subalgebras of W_{1+infty}

We show that there are precisely two, up to conjugation, anti-involutions sigma_{\pm} of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension \hat{D}^{\pm} of the Lie subalgebra of this algebra fixed by - sigma_{\pm}, and find the unitary ones. We realize them in terms of highest weight representations of the central extension of the Lie algebra of infinite matrices with finitely many non-zero diagonals over the truncated polynomial algebra C[u] / (u^{m+1}) and its…