Searcharxiv⌕ Search

arXiv subjects

Ching Hung Lam

Publications and source records attributed to Ching Hung Lam.

At least 19 recordsLinked to original sources
arXiv2026

Lattice vertex algebras over a field of positive characteristic: degenerate cases

A vertex operator algebra $V_L$ associated with a positive definite even lattice $L$ has a standard integral form, which we denote it by $V_{L,\mathbb{Z}} $. If $F$ is a field of characteristic $p>0$, it is known that $V_{L,F}:= F\otimes_\mathbb{Z} V_{L,\mathbb{Z}}$, a vertex algebra over $F$, is simple if and only if $(p, \det(L))=1$. In this article, we study $V_{L,F}$ when the characteristic of $F$ divides $\det(L)$. We determine the radical $\mathrm{Rad}$ of the invariant bilinear form on $V_{L,F}$, show that it is the unique maximal ideal and study the quotient vertex algebra $V_{L,F}/\mathrm{Rad}$.

math.QA↗
arXiv2026

Permutation orbifolds and simple current extensions

In this article, we study permutation orbifolds and simple current extensions in the framework of vertex operator (super)algebras. We extend the construction of permutation-twisted modules for tensor products of vertex operator algebras to vertex operator superalgebras with $\frac12\mathbb Z$-grading, including the effect of the canonical involution. Using tensor category methods and simple current extensions, we build an induction theory for permutation-twisted modules associated with solvable automorphism groups, arising from semidirect products of simple current automorphisms and cyclic permutations. In particular, we describe the structure and classification of irreducible twisted modules in terms of stabilizer subgroups and associated projective representations, and determine their multiplicities explicitly. As applications, we illustrate the theory with explicit examples from code vertex operator algebras, lattice-type simple current extensions, and the Moonshine vertex operator algebra.

math.QA↗
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.

math.QA↗
arXiv2024

On irreducibility of modules of Whittaker type: twisted modules and nonabelian orbifolds

In arXiv:1811.04649, we extended the Dong-Mason theorem on irreducibility of modules for cyclic orbifold vertex algebras to the entire category weak modules and applied this result to Whittaker modules. In this paper we present further generalizations of these results for nonabelian orbifolds of vertex operator superalgebras. Let $V$ be a vertex superalgebra with a countable dimension and let $G$ be a finite subgroup of $\mathrm{Aut}(V)$. Assume that $h\in Z(G)$ where $Z(G)$ is the center of the group $G$. For any irreducible $h$-twisted (weak) $V$-module $M$, we prove that if $M\not\cong g\circ M$ for all $g\in G$ then $M$ is also irreducible as $V^G$-module. We also apply this result to examples and give irreducibility of modules of Whittaker type for orbifolds of Neveu-Schwarz vertex superalgebras, Heisenberg vertex algebras, Virasoro vertex operator algebra and Heisenberg-Virasoro vertex algebra.

math.QA↗
arXiv2024

Completely fixed point free isometry and cyclic orbifold of lattice vertex operator algebras

We continue our study of cyclic orbifolds of lattice vertex operator algebras and their full automorphism groups. We consider some special isometry $g\in O(L)$ such that $g^i$ is fixed point free on $L$ for any $1\leq i\leq |g|-1$. We show that when $L_2=\emptyset$ and $g^i$ is fixed point free on $L$ for any $1\leq i\leq |g|-1$, $V_L^{\hat{g}}$ has extra automorphisms implies either (1) the order of $g$ is a prime or (2) $L$ is isometric to the Leech lattice or some coinvariant sublattices of the Leech lattice.

math.QA↗
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.

math.QA↗
arXiv2022

Unitary forms for holomorphic vertex operator algebras of central charge $24$

We prove that all holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one subspaces are unitary. The main method is to use the orbifold construction of a holomorphic VOA $V$ of central charge $24$ directly from a Niemeier lattice VOA $V_N$. We show that it is possible to extend the unitary form for the lattice VOA $V_N$ to the holomorphic VOA $V$ by using the orbifold construction and some information of the automorphism group $\mathrm{Aut}(V)$.

math.QA↗
arXiv2022

A lattice theoretical interpretation of generalized deep holes of the Leech lattice vertex operator algebra

We give a lattice theoretical interpretation of generalized deep holes of the Leech lattice VOA $V_Λ$. We show that a generalized deep hole defines a "true" automorphism invariant deep hole of the Leech lattice. We also show that there is a correspondence between the set of isomorphism classes of holomorphic VOA $V$ of central charge $24$ having non-abelian $V_1$ and the set of equivalence classes of pairs $(τ, \tildeβ)$ satisfying certain conditions, where $τ\in Co_0$ and $\tildeβ$ is a $τ$-invariant deep hole of squared length $2$. It provides a new combinatorial approach towards the classification of holomorphic VOAs of central charge $24$. In particular, we give an explanation for an observation of G. Höhn, which relates the weight one Lie algebras of holomorphic VOAs of central charge $24$ to certain codewords associated with the glue codes of Niemeier lattices.

math.QA↗
arXiv2021

Orbifold construction and Lorentzian construction of Leech lattice vertex operator algebra

We generalize Conway-Sloane's constructions of the Leech lattice from Niemeier lattices using Lorentzian lattice to holomorphic vertex operator algebras (VOA) of central charge 24. It provides a tool for analyzing the structures and relations among holomorphic VOAs related by orbifold construction. In particular, we are able to get some useful information about certain lattice subVOAs associated with the Cartan subalgebra of the weight one Lie algebra. We also obtain a relatively elementary proof that any strongly regular holomorphic VOA of central charge $24$ with a non-trivial weight one subspace can be constructed directly by a single orbifold construction from the Leech lattice VOA without using a dimension formula.

math.QA↗
arXiv2021

The Conway-Miyamoto correspondences for the Fischer 3-transposition groups

In this paper, we present a general construction of 3-transposition groups as automorphism groups of vertex operator algebras. Applying to the moonshine vertex operator algebra, we establish the Conway-Miyamoto correspondences between Fischer 3-transposition groups $\mathrm{Fi}_{23}$ and $\mathrm{Fi}_{22}$ and $c=25/28$ and $c=11/12$ Virasoro vectors of subalgebras of the moonshine vertex operator algebra.

math.QA↗
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 $24$.

math.QA↗
arXiv2021

Cliffold algebras, modular Virasoro vertex operator algebras and Z[1/2]-forms

This paper consists of two parts: (1) Using a Z[1/2]-form of Virasoro vertex operator algebra L(1/2,0) with central charge 1/2, we obtain a modular vertex operator algebra over any field F of finite characteristic different from 2. We determine the generators and classify the irreducible modules for this vertex operator algebra. (2) We investigate modular framed vertex operator algebras. In particular, the rationality of modular framed vertex operator algebras is established. For a modular code vertex operator algebra, the irreducible modules are constructed and classified. Moreover, a Z[1/2]-form for any framed vertex operator algebra over complex field C is constructed. As a result, one can obtain a modular framed vertex operator algebra from any framed vertex operator algebra over C.

math.QA↗
arXiv2021

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 $p$.

math.QA↗
arXiv2021

Fourvolutions and automorphism groups of orbifold lattice vertex operator algebras

Let $L$ be an even positive definite lattice with no roots, i.e., $L(2)=\{x\in L\mid (x|x)=2\}=\emptyset$. Let $g\in O(L)$ be an isometry of order $4$ such that $g^2=-1$ on $L$. In this article, we determine the full automorphism group of the orbifold vertex operator algebra $V_L^{\hat{g}}$. As our main result, we show that $Aut(V_L^{\hat{g}})$ is isomorphic to $N_{Aut(V_L)}(\langle \hat{g}\rangle)/ \langle\hat{g}\rangle $ unless $L\cong \sqrt{2}E_8$ or $BW_{16}$.

math.QA↗
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↗
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.

math.QA↗
arXiv2020

Classification of extremal vertex operator algebras with two simple modules

In recent work, Wang and the third author defined a class of 'extremal' vertex operator algebras (VOAs), consisting of those with at least two simple modules and conformal dimensions as large as possible for the central charge. In this article we show that there are exactly 15 character vectors of extremal VOAs with two simple modules. All but one of the 15 character vectors is realized by a previously known VOA. The last character vector is realized by a new VOA with central charge 33.

math-ph↗