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Hiroki Shimakura

Publications and source records attributed to Hiroki Shimakura.

At least 19 recordsLinked to original sources
arXiv2024

Extra automorphisms of cyclic orbifolds of lattice vertex operator algebras

In this article, we study the automorphism group of the cyclic orbifold of a vertex operator algebra associated with a rootless even lattice for a lift of a fixed-point free isometry of odd prime order $p$. We prove that such a cyclic orbifold contains extra automorphisms, not induced from automorphisms of the lattice vertex operator algebra, if and only if the rootless even lattice can be constructed by Construction B from a code over $\mathbb{Z}_p$ or is isometric to the coinvariant lattice of the Leech lattice associated with a certain isometry of order $p$.

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arXiv2024

Automorphism groups of parafermion vertex operator algebras: general case

We complete the program for determining the full automorphism groups of all parafermion vertex operator algebras associated with simple Lie algebras and positive integral levels. We show that the full automorphism group of the parafermion vertex operator algebra is isomorphic to the automorphism group of the associated root system for the remaining cases: (i) the level is at least $3$; (ii) the level is $2$ and the simple Lie algebra is non simply laced.

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arXiv2024

Generalised Checkerboard Lattices

A series of integral lattices parametrised by integers $k,m,n$ are introduced and investigated, where $n$ is the rank of the lattice, including the root lattices described in a uniform way and unimodular lattices such as the Niemeier lattices of type $A_{24}$ and $D_{24}$. The lattices are characterised by means of a sublattice isomorphic to the root lattice of type $A_{n-1}$. A sufficient condition for existence of an orthogonal $k$-frame of such a lattice is given in terms of symmetric $2$-designs.

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arXiv2023

Automorphism groups and uniqueness of holomorphic vertex operator algebras of central charge $24$

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.

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arXiv2021

Automorphism groups of cyclic orbifold vertex operator algebras associated with the Leech lattice and some non-prime isometries

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 $24$.

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arXiv2020

Schellekens' List and the Very Strange Formula

In 1993 Schellekens proved that the weight-one space $V_1$ of a strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 must be one of 71 Lie algebras. During the following three decades, in a combined effort by many authors, it was proved that each of these Lie algebras is realised by such a vertex operator algebra and that, except for $V_1=\{0\}$, this vertex operator algebra is uniquely determined by $V_1$. In this paper we give a fundamentally different, simpler proof of Schellekens' list of 71 Lie algebras. Using the dimension formula in arXiv:1910.04947 and Kac's "very strange formula" we show that every strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with $V_1\neq\{0\}$ can be obtained by an orbifold construction from the Leech lattice vertex operator algebra $V_\Lambda$. This suffices to restrict the possible Lie algebras that can occur as weight-one space of $V$ to the 71 of Schellekens. Moreover, the fact that each strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 comes from the Leech lattice $\Lambda$ can be used to classify these vertex operator algebras by studying properties of the Leech lattice. We demonstrate this for 43 of the 70 non-zero Lie algebras on Schellekens' list, omitting those cases that are too computationally expensive.

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arXiv2019

Reverse orbifold construction and uniqueness of holomorphic vertex operator algebras

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 $A_{5,3}D_{4,3}A_{1,1}^3$.

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arXiv2018

Inertia groups and uniqueness of holomorphic vertex operator algebras

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 structure if the weight one subspace is nonzero.

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arXiv2017

On orbifold constructions associated with the Leech lattice vertex operator algebra

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 $A_{6,7}$) from the Leech lattice vertex operator algebra.

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arXiv2017

$\Z_3$-orbifold construction of the Moonshine vertex operator algebra and some maximal $3$-local subgroups of the Monster

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, we show that its full automorphism group is isomorphic to the Monster simple group by using a $3$-local characterization and that the holomorphic VOA is isomorphic to the Moonshine VOA. This approach allows us to obtain relatively explicit descriptions of two maximal $3$-local subgroups of the shape $3^{1+12}.2.\Suz{:}2$ and $3^8.Ω^-(8,3).2$ in the Monster simple group.

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arXiv2015

Classification of vertex operator algebras of class $\mathcal{S}^4$ with minimal conformal weight one

In this article, we describe the trace formulae of composition of several (up to four) adjoint actions of elements of the Lie algebra of a vertex operator algebra by using the Casimir elements. As an application, we give constraints on the central charge and the dimension of the Lie algebra for vertex operator algebras of class $\mathcal{S}^4$. In addition, we classify vertex operator algebras of class $\mathcal{S}^4$ with minimal conformal weight one under some assumptions.

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arXiv2015

Orbifold construction of holomorphic vertex operator algebras associated to inner automorphisms

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.

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arXiv2014

Automorphisms of Niemeier lattices for Miyamoto's $\mathbb{Z}_3$-orbifold construction

We classify, up to conjugation, all automorphisms of Niemeier lattices to which we can apply Miyamoto's orbifold construction. Using this classification, we prove that the VOAs obtained in [M] and [SS] are all of holomorphic non-lattice VOAs which we can obtain by applying the $\mathbb{Z}_3$-orbifold construction to a Niemeier lattice and its automorphism.

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arXiv2013

Application of a $\mathbb{Z}_{3}$-orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices

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.

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arXiv2012

Classification of holomorphic framed vertex operator algebras of central charge 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.

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