Lattice Vertex Research Articles

Application of a ℤ₃-orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices

By applying Miyamoto’s Z 3 \mathbb {Z}_{3} -orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices and their automorphisms of order 3 3 , we construct holomorphic vertex operator algebras of central charge 24 24 whose Lie algebras of the weight one spaces are of types A 2 , 3 6 A_{2,3}^6 , E 6 , 3 G 2 , 1 3 E_{6,3}G_{2,1}^{3} , and A 5 , 3 D 4 , 3 A 1 , 1 3 A_{5,3}D_{4,3}A_{1,1}^{3} , which correspond to No. 6, No. 17, and No. 32 on Schellekens’ list, respectively.

Vertex Algebras W(p)Amand W(p)Dmand Constant Term Identities

We consider $AD$-type orbifolds of the triplet vertex algebras $\mathcal{W}(p)$ extending the well-known $c=1$ orbifolds of lattice vertex algebras. We study the structure of Zhu's algebras $A(\mathcal{W}(p)^{A_m})$ and $A(\mathcal{W}(p)^{D_m})$, where $A_m$ and $D_m$ are cyclic and dihedral groups, respectively. A combinatorial algorithm for classification of irreducible $\mathcal{W}(p)^\Gamma$-modules is developed, which relies on a family of constant term identities and properties of certain polynomials based on constant terms. All these properties can be checked for small values of $m$ and $p$ with a computer software. As a result, we argue that if certain constant term properties hold, the irreducible modules constructed in [Commun. Contemp. Math. 15 (2013), 1350028, 30 pages, arXiv:1212.5453; Internat. J. Math. 25 (2014), 1450001, 34 pages, arXiv:1304.5711] provide a complete list of irreducible $\mathcal{W}(p)^{A_m}$ and $\mathcal{W}(p)^{D_m}$-modules. This paper is a continuation of our previous work on the $ADE$ subalgebras of the triplet vertex algebra $\mathcal{W}(p)$.

Bosonizations of sl<sub>2</sub> and Integrable Hierarchies

We construct embeddings of $\widehat{\mathfrak{sl}}_2$ in lattice vertex algebras by composing the Wakimoto realization with the Friedan-Martinec-Shenker bosonization. The Kac-Wakimoto hierarchy then gives rise to two new hierarchies of integrable, non-autonomous, non-linear partial differential equations. A new feature of our construction is that it works for any value of the central element of $\widehat{\mathfrak{sl}}_2$; that is, the level becomes a parameter in the equations.

Asymptotic Laplacian-energy-like invariant of lattices

Let μ1⩾μ2⩾⋯⩾μn denote the Laplacian eigenvalues of a graph G with n vertices. The Laplacian-energy-like invariant, denoted by LEL(G)=∑i=1n-1μi, is a novel topological index. In this paper, we show that the Laplacian-energy-like per vertex of various lattices is independent of the toroidal, cylindrical, and free boundary conditions. Simultaneously, the explicit asymptotic values of the Laplacian-energy-like in these lattices are obtained. Moreover, our approach implies that in general the Laplacian-energy-like per vertex of other lattices is independent of the boundary conditions.

Asymptotic incidence energy of lattices

The energy of a graph G arising in chemical physics, denoted by E(G), is defined as the sum of the absolute values of the eigenvalues of G. As an analogue to E(G), the incidence energy IE(G), defined as the sum of the singular values of the incidence matrix of G, is a much studied quantity with well known applications in chemical physics. In this paper, based on the results by Yan and Zhang (2009), we propose the incidence energy per vertex problem for lattice systems, and present the closed-form formulae expressing the incidence energy of the hexagonal lattice, triangular lattice, and 33.42 lattice, respectively. Moreover, we show that the incidence energy per vertex of lattices is independent of the toroidal, cylindrical, and free boundary conditions. In particular, the explicit asymptotic values of the incidence energy in these lattices are obtained by utilizing the applications of analysis approach with the help of calculational software.

Regular Resistor Lattice Networks in Two Dimensions (Archimedean Lattices)

In this paper, we investigate the two-vertex resistance on four Archimedean lattices. This technique is based on the Green's function corresponding to the Laplacian matrix of the lattice. The Laplacian matrix of the resistor network is determined by applying basic principles (Kirchhoff's and Ohm's laws) in electrical circuit analysis, and Fourier transforms of the electric potential and current. We present some numerical results regarding the resistance between nearby lattice vertices.

Lattice vertex algebras over fields of prime characteristic

Based on a work of Dong and Griess, we study a class of vertex algebras over fields of prime characteristic obtained from integral forms in lattice vertex operator algebras over complex field. In particular, the generating set is given, a bilinear form is obtained, and the simplicity is studied. Finally, the vertex operator algebra structure is characterized.

A generalized Kac–Moody algebra of rank 14

We construct a vertex algebra of central charge 26 from a lattice orbifold vertex operator algebra of central charge 12. The BRST-cohomology group of this vertex algebra is a new generalized Kac–Moody algebra of rank 14. We determine its root space multiplicities and a set of simple roots.

Coset Constructions of Logarithmic (1, p) Models

One of the best understood families of logarithmic onformal field theories consists of the (1, p) models (p = 2, 3, . . .) of central charge c 1, p =1 − 6(p − 1)2/p. This family includes the theories corresponding to the singlet algebras $${\mathcal{M}(p)}$$ and the triplet algebras $${\mathcal{W}(p)}$$ , as well as the ubiquitous symplectic fermions theory. In this work, these algebras are realised through a coset construction. The $${W^{(2)}_n}$$ algebra of level k was introduced by Feigin and Semikhatov as a (conjectured) quantum hamiltonian reduction of $${\widehat{\mathfrak{sl}}(n)_k}$$ , generalising the Bershadsky–Polyakov algebra $${W^{(2)}_3}$$ . Inspired by work of Adamović for p = 3, vertex algebras $${\mathcal{B}_p}$$ are constructed as subalgebras of the kernel of certain screening charges acting on a rank 2 lattice vertex algebra of indefinite signature. It is shown that for p≤5, the algebra $${\mathcal{B}_p}$$ is a quotient of $${W^{(2)}_{p-1}}$$ at level −(p − 1)2/p and that the known part of the operator product algebra of the latter algebra is consistent with this holding for p> 5 as well. The triplet algebra $${\mathcal{W}(p)}$$ is then realised as a coset inside the full kernel of the screening operator, while the singlet algebra $${\mathcal{M}(p)}$$ is similarly realised inside $${\mathcal{B}_p}$$ . As an application, and to illustrate these results, the coset character decompositions are explicitly worked out for p = 2 and 3.

The automorphism group of the -orbifold of the Barnes–Wall lattice vertex operator algebra of central charge 32

AbstractIn this paper, we prove that the full automorphism group of the${\mathbb Z}_2$-orbifold of the Barnes–Wall lattice vertex operator algebra of central charge 32 has the shape 227.E6(2). In order to identify the group structure, we introduce a graph structure on the Griess algebra and show that it is a rank 3 graph associated toE6(2).

Yang-Mills lattice on CUDA

Abstract The Yang-Mills fields have an important role in the non- Abelian gauge field theory which describes the properties of the quarkgluon plasma. The real time evolution of the classical fields is given by the equations of motion which are derived from the Hamiltonians to contain the term of the SU(2) gauge field tensor. The dynamics of the classical lattice Yang-Mills equations are studied on a 3 dimensional regular lattice. During the solution of this system we keep the total energy on constant values and it satisfies the Gauss law. The physical quantities are desired to be calculated in the thermodynamic limit. The broadly available computers can handle only a small amount of values, while the GPUs provide enough performance to reach out for higher volumes of lattice vertices which approximates the aforementioned limit.

On permutation-twisted free fermions and two conjectures

We conjecture that the category of permutation-twisted modules for a multi-fold tensor product vertex operator superalgebra and a cyclic permutation of even order is isomorphic to the category of parity-twisted modules for the underlying vertex operator superalgebra. This conjecture is based on our observations of the cyclic permutation-twisted modules for free fermions as we discuss in this work, as well as previous work of the first author constructing and classifying permutation-twisted modules for tensor product vertex operator superalgebras and a permutation of odd order. In addition, we observe that the transposition isomorphism for two free fermions corresponds to a lift of the −1 isometry of the integral lattice vertex operator superalgebra corresponding to two free fermions under boson-fermion correspondence. We conjecture that all even order cyclic permutation automorphisms of free fermions can be realized as lifts of lattice isometries under boson-fermion correspondence. We discuss the role of parity stability in the construction of these twisted modules and prove that in general, parity-unstable weak twisted modules for a vertex operator superalgebras come in pairs that form orthogonal invariant subspaces of parity-stable weak twisted modules, clarifying their role in many other settings.

Lattice Vertex Superalgebras, I: Presentation of the Principal Subalgebra

This is the first in a series of papers in which we study the structure of certain distinguished subspaces of lattice vertex operator superalgebras and their modules, modeled on Feigin-Stoyanovsky principal subspaces [18]. In this part we completely describe their structure in the (super)commutative case.

The Intermediate Vertex Subalgebras of the Lattice Vertex Operator Algebras

A notion of intermediate vertex subalgebras of lattice vertex operator algebras is introduced, as a generalization of the notion of principal subspaces. Bases and the graded dimensions of such subalgebras are given.As an application, it is shown that the characters of some modules of an intermediate vertex subalgebra between $E_7$ and $E_8$ lattice vertex operator algebras satisfy some modular differential equations.This result is an analogue of the result concerning the "hole" of the Deligne dimension formulas and the intermediate Lie algebra between the simple Lie algebras $E_7$ and $E_8$.

Quadratic Differential Operators, Bicharacters and • Products

For a commutative cocommutative Hopf algebra, we study the relationship between a certain linear map defined via a bicharacter, an exponential of a quadratic differential operator, and a • product obtained via twisting by a bicharacter. This new relationship between • products and exponentials of quadratic differential operators was inspired by studying the exponential of a quadratic differential operator introduced in [7] and used in the theory of twisted modules of lattice vertex algebras.

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.

Uniqueness Theorem of $${\mathcal{W}}$$ W -Constraints for Simple Singularities

In a recent paper [3], Bakalov and Milanov proved that the total descendant potential of a simple singularity satisfies the W-constraints, which come from the W-algebra of the lattice vertex algebra associated to the root lattice of this singularity and a twisted module of this vertex algebra. In the present paper, we prove that the solution of these W-constraints is unique up to a constant factor, as conjectured by Bakalov and Milanov in their paper.

Half-metallicity of a kagome spin lattice: the case of a manganese bis-dithiolene monolayer

The spin ordering in kagome lattices has long been studied in the search for real materials with a spin-liquid ground state. The synthesis of a nickel bis-dichiolene complex (Ni3C12S12) nanosheet (T. Kambe et al., J. Am. Chem. Soc., 2013, 135, 2462) paved a way for realizing real two-dimensional kagome lattices. Using first-principles calculations, we predicted that a ferromagnetic kagome spin lattice with S = 3/2 on lattice vertices can be achieved in an Mn3C12S12 monolayer formed by substituting Ni with Mn atoms in nonmagnetic Ni3C12S12. Monte Carlo simulations on the basis of the Ising model suggest that it has a Curie temperature of about 212 K. A ferromagnetic Mn3C12S12 monolayer is half metallic with high carrier mobility in one spin channel and a band gap of 1.54 eV in another spin channel, which is quite promising for spintronic device applications. Additionally, a small band gap opens up at the Dirac point of the kagome bands due to the spin-orbital coupling effects, which may be implementable for achieving a quantum anomalous Hall effect.

C 2-Cofiniteness of 2-Cyclic Permutation Orbifold Models

In this article, we consider permutation orbifold models of $C_2$-cofinite vertex operator algebras of CFT type. We show the $C_2$-cofiniteness of the 2-cyclic permutation orbifold model $(V\otimes V)^{S_2}$ for an arbitrary $C_2$-cofinite simple vertex operator algebra $V$ of CFT type. We also give a proof of the $C_2$-cofiniteness of a $\Z_2$-orbifold model $V_L^+$ of the lattice vertex operator algebra $V_L$ associated with a rank one positive definite even lattice $L$ by using our result and the $C_2$-cofiniteness of $V_L$.

Rationality of vertex operator algebra V L + : higher rank

The lattice vertex operator V_L associated to a positive definite even lattice L has an automorphism of order 2 lifted from -1 isometry of L. It is established that the fixed point vertex operator algebra V_L^+ is rational.