Ising Vectors Research Articles

Vertex operator algebras generated by two Ising vectors

In this paper we mainly study the vertex operator algebra CVA(e,f) generated by two Ising vectors e and f with 〈e,f〉=5210. We prove CVA(e,f) is isomorphic to the 6A-algebra U6A constructed in [20]. We also discuss the cases 〈e,f〉=14,125, 0 and the case 〈e,f〉=128 with (τeτf)4=1.

The VOAs generated by two Ising vectors e and f with [formula omitted], |τeτf| = 3 or [formula omitted

In this paper we study two VOAs: first we study the VOA generated by two Ising vectors whose inner product is 128 with |τeτf|=3, then we study the VOA generated by two Ising vectors whose inner product is 329. We also determine they both have unique VOA structures.

6A-Algebra and its representations

In this paper, we study the structure and representation of a 6A-algebra which is a vertex operator algebra generated by two Ising vectors e,f with inner product 〈e,f〉=5210. In particular, we prove the uniqueness of the vertex operator algebra structure of this 6A-algebra, classify the irreducible modules, and determine the fusion rules.

Vertex operator algebras generated by Ising vectors of $$\sigma $$ σ -type

We prove the uniqueness of the simple vertex operator algebra of OZ-type generated by Ising vectors of $$\sigma $$ -type. We also prove that the simplicity can be omitted if the Griess algebra is isomorphic to the Matsuo algebra associated with the root system of type $$A_n$$ .

Griess algebras generated by the Griess algebras of two $3A$-algebras with a common axis

In this article, we study Griess algebras generated by two pairs of Ising vectors $(a_0, a_1)$ and $(b_0,b_1)$ such that each pair generates a $3A$-algebra $U_{3A}$ and their intersection contains the $W_3$-algebra $\mathcal{W}(4/5)\cong L(4/5,0)\oplus L(4/5,3)$. We show that there are only 3 possibilities, up to isomorphisms and they are isomorphic to the Griess algebras of the VOAs $V_{F(1A)}$, $V_{F(2A)}$ and $V_{F(3A)}$ constructed by Hohn–Lam–Yamauchi.

An explicit Majorana representation of the group 32:2 of 3C-pure type

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 $3^2{:}2$ of 3C-pure type.

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

In this paper, we construct explicitly certain moonshine type vertex operator algebras generated by a set of Ising vectors I such that(1) for any e ≠ f ∈ I, the subVOA VOA (e, f) generated by e and f is isomorphic to either U2Bor U3C; and(2) the subgroup generated by the corresponding Miyamoto involutions {τe| e ∈ I} is isomorphic to the Weyl group of a root system of type An, Dn, E6, E7or E8.The structures of the corresponding vertex operator algebras and their Griess algebras are also studied. In particular, the central charge of these vertex operator algebras are determined.

Griess algebras generated by 3 Ising vectors of central 2A-type

We study the Griess algebra generated by three Ising vectors e,f, and g in a CFT type vertex operator algebra V with V1=0 such that 〈e,f〉=〈e,g〉=132. We call such a configuration of central 2A-type. Under this assumption, we show that there are only 5 possible structures of Griess algebras and they correspond exactly to the Griess algebras GVB(nX) of the five VOA VB(nX), nX∈{1A,2B,3A,4B,2C}, constructed by Höhn–Lam–Yamauchi.

Classification of Ising vectors in the vertex operator algebraVL+

Let $L$ be an even lattice without roots. In this article, we classify all Ising vectors in the vertex operator algebra $V_L^+$ associated with $L$.

Ising vectors and automorphism groups of commutant subalgebras related to root systems

In this article we study and obtain a classification of Ising vectors in vertex operator algebras (VOAs) associated to binary codes and \(\sqrt{2}\) times root lattices, where an Ising vector is a conformal vector with central charge 1/2 generating a simple Virasoro sub-VOA. Then we apply our results to study certain commutant subalgebras related to root systems. We completely classify all Ising vectors in such commutant subalgebras and determine their full automorphism groups.