Searcharxiv⌕ Search

arXiv subjects

Hiroshi Yamauchi

Publications and source records attributed to Hiroshi Yamauchi.

31 records · Page 2Linked to original sources
arXiv2012

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.

math.QA↗
arXiv2011

"McKay's E7 observation on the Baby Monster" and "McKay's E6 observation on the largest Fischer group"

E7 part: In this paper, we study McKay's E7 observation on the Baby Monster. By investigating so called derived c=7/10 Virasoro vectors, we show that there is a natural correspondence between dihedral subgroups of the Baby Monster and certain subalgebras of the Baby Monster vertex operator algebra which are constructed by the nodes of the affine E7 diagram. This allows us to reinterpret McKay's E7 observation via the theory of vertex operator algebras. For a class of vertex operator algebras including the Moonshine module, we will show that the product of two Miyamoto involutions associated to derived c=7/10 Virasoro vectors in certain commutant vertex operator algebras is an element of order at most 4. For the case of the Moonshine module, we obtain the Baby monster vertex operator algebra as the commutant and we can identify the group generated by these Miyamoto involutions with the Baby Monster and recover the {3,4}-transposition property of the Baby Monster in terms of vertex operator algebras. E6 part: In this paper, we study McKay's E6-observation on the largest Fischer 3-transposition group Fi24. We investigate a vertex operator algebra VF of central charge 23+1/5 on which the Fischer group Fi24 naturally acts. We show that there is a natural correspondence between dihedral subgroups of Fi24 and certain vertex operator subalgebras constructed by the nodes of the affine E6 diagram by investigating so called derived Virasoro vectors of central charge 6/7. This allows us to reinterpret McKay's E6-observation via the theory of vertex operator algebras. It is also shown that the product of two non-commuting Miyamoto involutions of sigma-type associated to derived c=6/7 Virasoro vectors is an element of order 3, under certain general hypotheses on the vertex operator algebra. For the case of VF, we identify these involutions with the 3-transpositions of the Fischer group Fi24.

math.QA↗
arXiv2010

On the structure of framed vertex operator algebras and their pointwise frame stabilizers

In this paper, we study the structure of a general framed vertex operator algebra. 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 V_C. This result would give a prospect on the classification of framed vertex operator algebras. In addition, the pointwise frame stabilizer of V is studied. We completely determine all automorphisms in this 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 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 the moonshine VOA is isomorphic to the moonshine VOA itself.

math.QA↗
arXiv2005

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 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.

math.QA↗
arXiv2005

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 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 VOA. The existence of U inside the Moonshine VOA for the cases of 1A, 2A, 2B and 4A is also established. Moreover, the cases for 3A, 5A and 3C are discussed.

math.QA↗
arXiv2004

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 algebra of U coincides with the algebra described in Conway's paper. There is a canonical automorphism of U of order |E_8/L|. Such an automorphism can be extended to the Leech lattice VOA V_Λand it is in fact a product of two Miyamoto involutions. In the sequel [LYY] to this article we shall develop the representation theory of $U$. It is expected that if U is actually contained in the Moonshine VOA V^\natural, the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group.

math.QA↗
arXiv2004

Modularity on vertex operator algebras arising from semisimple primary vectors

In this article, using an idea of the physics superselection principal, we study a modularity on vertex operator algebras arising from semisimple primary vectors. We generalizes the theta functions on vertex operator algebras and prove that the internal automorphisms do not change the genus one twisted conformal blocks.

math.QA↗
arXiv2004

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.

math.QA↗
arXiv2002

Orbifold Zhu theory associated to intertwining operators

We study intertwining operators among the twisted modules for rational VOAs and show the modular invariance property of the space spanned by the trace functions concerning to some representations of rational orbifold VOAs. Our result generalizes the results obtained by Dong-Li-Mason and Miyamoto, which are basically extensions of Zhu's celebrated works.

math.QA↗
arXiv2002

Vertex operator algebra with two Miyamoto involutions generating $S_3$

In this article we study a VOA with two Miyamoto involutions generating $S_3$. In \cite{M3}, Miyamoto showed that a VOA generated by two conformal vectors whose Miyamoto involutions generate an automorphism group isomorphic to $S_3$ is isomorphic to one of the four candidates he listed. We construct one of them and prove that our VOA is actually the same as $\mathrm{VA}(e,f)$ studied by Miyamoto. We also show that there is an embedding into the moonshine VOA. Using our VOA, we can define the 3A-triality in the Monster.

math.QA↗
arXiv2002

On Z_2-twisted representation of vertex operator superalgebras and the Ising model SVOA

We investigate a general theory of the Z_2-twisted representations of vertex operator superalgebras. Certain one-to-one correspondence theorems are established. We also give an explicit realization of the Ising model SVOA and its Z_2-twisted modules. As an application, we obtain the Gerald Hoehn's Babymonster SVOA VB and its Z_2-twisted module VB_{tw} from the moonshine VOA V^\nat by cutting off the Ising models. It is also shown in this paper that Aut (VB) is finite.

math.QA↗