Ching Hung Lam

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

This paper is a continuation of (33) at which several coset subalgebras of the lattice VOA Vp 2E8 were constructed and the relationship between such algebras with the famous McKay observation on the extended E8 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 represen- tation theory of these coset subalgebras and show that the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group if a coset subal- gebra U is actually contained in the Moonshine VOA V ♮ . The existence of U inside the Moonshine VOA V ♮ for the cases of 1A,2A,2B and 4A is also established. Moreover, the cases for 3A, 5A and 3C are discussed. and the Monster simple group were discussed. In this article, we shall provide the technical details. We shall determine the structure of the coset subalgebras 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 product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group if a coset subalgebra U is actually contained in the Moonshine

Vertex operator algebras, extended

We study McKays 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 Eg diagram at each time. We then construct a certain coset (or commutant) subalgebra U associated with L in the lattice VOA V √2E8 . 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 (1985). The Griess algebra of U coincides with the algebra described in his Table 3. There is a canonical automorphism of U of order |E 8 /L|. Such an automorphism can be extended to the Leech lattice VOA V A , and it is in fact a product of two Miyamoto involutions. In the sequel (2005) to this article, the properties of U will be discussed in detail. It is expected that if U is actually contained in the Moonshine VOA V 1 , the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group.

E_8

In this paper, we study the structure of a general framed vertex operator algebra (VOA). We show that the structure codes (C,D) of a framed VOA V satisfy certain duality conditions. As a consequence, we prove that every framed VOA is a simple current extension of the associated binary code VOA VC. This result suggests the feasibility of classifying framed vertex operator algebras, at least if the central charge is small. In addition, the pointwise frame stabilizer of V is studied. We completely determine all automorphisms in the pointwise stabilizer, which are of order 1, 2 or 4. The 4A-twisted sector and the 4A-twisted orbifold theory of the famous moonshine VOA

On the Structure of Framed Vertex Operator Algebras and Their Pointwise Frame Stabilizers

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Classification of holomorphic framed vertex operator algebras of central charge 24

are also constructed explicitly. We verify that the top module of this twisted sector is of dimension 1 and of weight 3/4 and the VOA obtained by 4A-twisted orbifold construction of

Tricritical 3-State Potts Model and Vertex Operator Algebras Constructed from Ternary Codes

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