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Hiromichi Yamada

Publications and source records attributed to Hiromichi Yamada.

16 recordsLinked to original sources
arXiv2020

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.

math.QA↗
arXiv2019

$\mathbb{Z}_{2k}$-code vertex operator algebras

We study a simple, self-dual, rational, and $C_2$-cofinite vertex operator algebra of CFT-type whose simple current modules are graded by $\mathbb{Z}_{2k}$. Based on those simple current modules, a vertex operator algebra associated with a $\mathbb{Z}_{2k}$-code is constructed. The classification of irreducible modules for such a vertex operator algebra is established. Furthermore, all the irreducible modules are realized in a module for a certain lattice vertex operator algebra.

math.RT↗
arXiv2019

Simple current extensions of tensor products of vertex operator algebras

We study simple current extensions of tensor products of two vertex operator algebras satisfying certain conditions. We establish the relationship between the fusion rule for the simple current extension and the fusion rule for a tensor factor. In a special case, we construct a chain of simple current extensions. We discuss certain irreducible twisted modules for the simple current extension as well.

math.RT↗
arXiv2019

$\mathbb{Z}_k$-code vertex operator algebras

We introduce a simple, self-dual, rational, and $C_2$-cofinite vertex operator algebra of CFT-type associated with a $\mathbb{Z}_k$-code for $k \ge 2$ based on the $\mathbb{Z}_k$-symmetry among the simple current modules for the parafermion vertex operator algebra $K(\mathfrak{sl}_2,k)$. We show that it is naturally realized as the commutant of a certain subalgebra in a lattice vertex operator algebra. Furthermore, we construct all the irreducible modules inside a module for the lattice vertex operator algebra.

math.RT↗
arXiv2017

Extensions of tensor products of ${\mathbb Z}_p$-orbifold models of the lattice vertex operator algebra $V_{\sqrt{2}A_{p-1}}$

Let $p$ be an odd prime and let $\widehatσ$ be an order $p$ automorphism of $V_{\sqrt{2}A_{p-1}}$ which is a lift of a $p$-cycle in the Weyl group ${\rm Weyl}(A_{p-1})\cong {\mathfrak S}_p$. We study a certain extension $V$ of a tensor product of finitely many copies of the orbifold model $V_{\sqrt{2}A_{p-1}}^{\langle \widehatσ \rangle}$ and give a criterion for $V$ that every irreducible $V$-module is a simple current.

math.QA↗
arXiv2017

A remark on ${\mathbb Z}_p$-orbifold constructions of the Moonshine vertex operator algebra

For $p = 3,5,7,13$, we consider a ${\mathbb Z}_p$-orbifold construction of the Moonshine vertex operator algebra $V^\natural$. We show that the vertex operator algebra obtained by the ${\mathbb Z}_p$-orbifold construction on the Leech lattice vertex operator algebra $V_Λ$ and a lift of a fixed-point-free isometry of order $p$ is isomorphic to the Moonshine vertex operator algebra $V^\natural$. We also describe the relationship between those ${\mathbb Z}_p$-orbifold constructions and the ${\mathbb Z}_2$-orbifold construction in a uniform manner. In Appendix, we give a characterization of the Moonshine vertex operator algebra $V^\natural$ by two mutually orthogonal Ising vectors.

math.QA↗
arXiv2017

Parafermion vertex operator algebras and W-algebras

We prove the conjectual isomorphism between the level $k$ $\widehat{sl}_2$-parafermion vertex operator algebra and the $(k+1,k+2)$ minimal series $W_k$-algebra for all integers $k \ge 2$. As a consequence, we obtain the conjectural isomorphism between the $(k+1,k+2)$ minimal series $W_k$-algebra and the coset vertex operator algebra $SU(k)_1 \otimes SU(k)_1/SU(k)_2$.

math.RT↗
arXiv2014

Commutant of $\mathcal{L}_{\widehat{\mathfrak{sl}}_2}(4,0)$ in the cyclic permutation orbifold of $\mathcal{L}_{\widehat{\mathfrak{sl}}_2}(1,0)^{\otimes 4}$

We study the commutant of the vertex operator algebra $\mathcal{L}_{\widehat{\mathfrak{sl}}_2}(4,0)$ in the cyclic permutation orbifold model $(\mathcal{L}_{\widehat{\mathfrak{sl}}_2}(1,0)^{\otimes 4})^τ$ with $τ=(1\,2\,3\,4)$. It is shown that the commutant is isomorphic to a ${\mathbb Z}_2\times{\mathbb Z}_2$-orbifold model of a tensor product of two lattice type vertex operator algebras of rank one.

math.QA↗
arXiv2013

Fixed point subalgebras of lattice vertex operator algebras by an automorphism of order three

We study the fixed point subalgebra of a certain class of lattice vertex operator algebras by an automorphism of order 3, which is a lift of a fixed-point-free isometry of the underlying lattice. We classify the irreducible modules for the subalgebra. Moreover, the rationality and the $C_2$-cofiniteness of the subalgebra are established. Our result contains the case of the vertex operator algebra associated with the Leech lattice.

math.QA↗
arXiv2009

The structure of parafermion vertex operator algebras

It is proved that the parafermion vertex operator algebra associated to the irreducible highest weight module for the affine Kac-Moody algebra A_1^{(1)} of level k coincides with a certain W-algebra. In particular, a set of generators for the parafermion vertex operator algebra is determined.

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↗
arXiv1999

Decomposition of the vertex operator algebra V_{\sqrt{2}A_3}

For vertex operator algebra V_{\sqrt{2}A_l} associated to the even lattice \sqrt{2}A_l which is \sqrt{2} times root lattice of type A_l, it was shown by Dong-Li-Maosn-Norton that the Virasoro vector is a sum of l+1 mutually orthogonal conformal vectors with central charges c_i=1-6/(i+2)(i+3) for i=1,...,l and c_{l+1}=2l/(l+3) and the subalgebra T generated by these vectors is a tensor product of Virasoro vertex operator algebras L(c_i,0). In this paper we determine the decomposition of V_{\sqrt{2}A_3} into the sum of irreducible T-modules completely.

math.QA↗