Lattice Vertex Algebras Research Articles

Representations of quantum lattice vertex algebras

Let Q be a non-degenerate even lattice, let VQ be the lattice vertex algebra associated to Q, and let VQη be a quantum lattice vertex algebra ([10]). In this paper, we prove that every VQη-module is completely reducible, and the set of simple VQη-modules are in one-to-one correspondence with the set of cosets of Q in its dual lattice.

Modified bosonic integrable hierarchy

In this paper, we construct a new integrable hierarchy by using charged free boson and Friedan–Martinec–Shenker (FMS) bosonization, called modified bosonic integrable hierarchy, which is an anologue of the classical (l−l′)–th modified KP hierarchy. Firstly, the representation of glˆ∞ is given by using charged free bosons. Then the modified bosonic integrable hierarchy is proposed. Next after discussing FMS bosonizations in the language of lattice vertex algebra, bilinear equations of modified bosonic integrable hierarchy are constructed by FMS bosonization, which are rewritten into Hirota bilinear equations by using Hirota derivatives. At last, we consider solutions of this modified bosonic integrable hierarchy.

On the commutant of the principal subalgebra in the $$A_1$$ lattice vertex algebra

On the commutant of the principal subalgebra in the $$A_1$$ lattice vertex algebra

Orbifolds of lattice vertex algebras

Orbifolds of lattice vertex algebras

Subregular W-algebras of type A

Subregular W-algebras are an interesting and increasingly important class of quantum hamiltonian reductions of affine vertex algebras. Here, we show that the [Formula: see text] subregular W-algebra can be realized in terms of the [Formula: see text] regular W-algebra and the half lattice vertex algebra [Formula: see text]. This generalizes the realizations found for [Formula: see text] and [Formula: see text] in [D. Adamović, Realizations of simple affine vertex algebras and their modules: The cases [Formula: see text] and [Formula: see text], Comm. Math. Phys. 366 (2019) 1025–1067, arXiv:1711.11342 [math.QA]; D. Adamović, K. Kawasetsu and D. Ridout, A realization of the Bershadsky–Polyakov algebras and their relaxed modules, Lett. Math. Phys., 111 (2021) 1–30, arXiv:2007.00396 [math.QA]] and can be interpreted as an inverse quantum hamiltonian reduction in the sense of Adamović. We use this realization to explore the representation theory of [Formula: see text] subregular W-algebras. Much of the structure encountered for [Formula: see text] and [Formula: see text] is also present for [Formula: see text]. Particularly, the simple [Formula: see text] subregular W-algebra at nondegenerate admissible levels can be realized purely in terms of the [Formula: see text] minimal model vertex algebra and [Formula: see text].

Smash product construction of modular lattice vertex algebras

<abstract><p>Motivated by a work of Li, we study nonlocal vertex algebras and their smash products over fields of positive characteristic. Through smash products, modular vertex algebras associated with positive definite even lattices are reconstructed. This gives a different construction of the modular vertex algebras obtained from integral forms introduced by Dong and Griess in lattice vertex operator algebras over a field of characteristic zero.</p></abstract>

Lattice structure of modular vertex algebras

In this paper we study the integral form of the lattice vertex algebra VL. We show that divided powers of general vertex operators preserve the integral lattice spanned by Schur functions indexed by partition-valued functions. We also show that the Garland operators, counterparts of divided powers of Heisenberg elements in affine Lie algebras, also preserve the integral form. These construe analogs of the Kostant Z-forms for the enveloping algebras of simple Lie algebras and the algebraic affine Lie groups in the situation of the lattice vertex algebras.

Geometric free field realization for the genus-two class S theory of type a1

We present a free field realisation for the vertex operator algebra associated to the genus-two, class $\mathcal{S}$ superconformal field theory of type $\mathfrak{a}_1$. The free field realisation is in the style of recent work by the authors, and is formulated in terms of a one-dimensional isotropic lattice vertex algebra along with two pairs of symplectic fermions. Our realisation makes manifest an enhanced ${\rm USp}(4)$ outer automorphism group of the VOA that is inherited from the symplectic fermion system. This extends an ${\rm SU(2)}$ outer automorphism that has been observed in recent work of Kiyoshige and Nishinaka and significantly simplifies the structure of the algebra. Along the way, we also produce a realisation of the generic subregular Drinfel'd-Sokolov $\mathcal{W}$ algebra of type $\mathcal{c}_2$ in terms of the generic principle $\mathcal{W}$ algebra of type $\mathfrak{c}_2$ and a one-dimensional isotropic lattice vertex algebra.

Genus of vertex algebras and mass formula

We introduce the notion of a genus and its mass for vertex algebras. For lattice vertex algebras, their genera are the same as those of lattices, which plays an important role in the classification of lattices. We derive a formula relating the mass for vertex algebras to that for lattices, and then give a new characterization of some holomorphic vertex operator algebras.

A realisation of the Bershadsky–Polyakov algebras and their relaxed modules

We present a realisation of the universal/simple Bershadsky--Polyakov vertex algebras as subalgebras of the tensor product of the universal/simple Zamolodchikov vertex algebras and an isotropic lattice vertex algebra. This generalises the realisation of the universal/simple affine vertex algebras associated to $\mathfrak{sl}_2$ and $\mathfrak{osp}(1|2)$ given in arXiv:1711.11342. Relaxed highest-weight modules are likewise constructed, conditions for their irreducibility are established, and their characters are explicitly computed, generalising the character formulae of arXiv:1803.01989.

Representations of twisted toroidal Lie algebras from twisted modules over vertex algebras

Given a simple finite-dimensional Lie algebra and an automorphism of finite order, one defines the notion of a twisted toroidal Lie algebra. In this paper, we construct representations of twisted toroidal Lie algebras from twisted modules over affine and lattice vertex algebras.

(G,χϕ)-equivariant ϕ-coordinated quasi modules for nonlocal vertex algebras

In this paper, we study (G,χϕ)-equivariant ϕ-coordinated quasi modules for nonlocal vertex algebras. Among the main results, we establish several conceptual results, including a generalized commutator formula and a general construction of weak quantum vertex algebras and their (G,χϕ)-equivariant ϕ-coordinated quasi modules. As an application, we also construct (equivariant) ϕ-coordinated quasi modules for lattice vertex algebras by using Lepowsky's work on twisted vertex operators.

ADMISSIBLE LEVEL mathfrak{osp}left(1left|2right.right) MINIMAL MODELS AND THEIR RELAXED HIGHEST WEIGHT MODULES

The minimal model mathfrak{osp}left(1|2right) vertex operator superalgebras are the simple quotients of affine vertex operator superalgebras constructed from the affine Lie super algebra hat{mathfrak{osp}}left(1left|2right.right) at certain rational values of the level k. We classify all isomorphism classes of ℤ2-graded simple relaxed highest weight modules over the minimal model mathfrak{osp}left(1|2right) vertex operator superalgebras in both the Neveu–Schwarz and Ramond sectors. To this end, we combine free field realisations, screening operators and the theory of symmetric functions in the Jack basis to compute explicit presentations for the Zhu algebras in both the Neveu–Schwarz and Ramond sectors. Two different free field realisations are used depending on the level. For k < −1, the free field realisation resembles the Wakimoto free field realisation of affine mathfrak{sl}(2) and is originally due to Bershadsky and Ooguri. It involves 1 free boson (or rank 1 Heisenberg vertex algebra), one βγ bosonic ghost system and one bc fermionic ghost system. For k > −1, the argument presented here requires the bosonisation of the βγ system by embedding it into an indefinite rank 2 lattice vertex algebra.

Twisted logarithmic modules of lattice vertex algebras

Twisted modules over vertex algebras formalize the relations among twisted vertex operators and have applications to conformal field theory and representation theory. A recent generalization, called twisted logarithmic module, involves the logarithm of the formal variable and is related to logarithmic conformal field theory. We investigate twisted logarithmic modules of lattice vertex algebras, reducing their classification to the classification of modules over a certain group. This group is a semidirect product of a discrete Heisenberg group and a central extension of the additive group of the lattice.

Boson-fermion correspondence from factorization spaces

Abstract We give a proof of the boson-fermion correspondence, an isomorphism of lattice and fermion vertex algebras, in terms of a natural isomorphism of corresponding factorization spaces.

Cosets of Bershadsky–Polyakov algebras and rational $${\mathcal W}$$ W -algebras of type A

The Bershadsky-Polyakov algebra is the $\mathcal{W}$-algebra associated to $\mathfrak{s}\mathfrak{l}_3$ with its minimal nilpotent element $f_{\theta}$. For notational convenience we define $\mathcal{W}^{\ell} = \mathcal{W}^{\ell - 3/2} (\mathfrak{s}\mathfrak{l}_3, f_{\theta})$. The simple quotient of $\mathcal{W}^{\ell}$ is denoted by $\mathcal{W}_{\ell}$, and for $\ell$ a positive integer, $\mathcal{W}_{\ell}$ is known to be $C_2$-cofinite and rational. We prove that for all positive integers $\ell$, $\mathcal{W}_{\ell}$ contains a rank one lattice vertex algebra $V_L$, and that the coset $\mathcal{C}_{\ell} = \text{Com}(V_L, \mathcal{W}_{\ell})$ is isomorphic to the principal, rational $\mathcal{W}(\mathfrak{s}\mathfrak{l}_{2\ell})$-algebra at level $(2\ell +3)/(2\ell +1) -2\ell$. This was conjectured in the physics literature over 20 years ago. As a byproduct, we construct a new family of rational, $C_2$-cofinite vertex superalgebras from $\mathcal{W}_{\ell}$

Conformal embeddings of affine vertex algebras in minimal W-algebras I: Structural results

We find all values of k∈C, for which the embedding of the maximal affine vertex algebra in a simple minimal W-algebra Wk(g,θ) is conformal, where g is a basic simple Lie superalgebra and −θ its minimal root. In particular, it turns out that if Wk(g,θ) does not collapse to its affine part, then the possible values of these k are either −23h∨ or −h∨−12, where h∨ is the dual Coxeter number of g for the normalization (θ,θ)=2. As an application of our results, we present a realization of simple affine vertex algebra V−n+12(sl(n+1)) inside the tensor product of the vertex algebra Wn−12(sl(2|n),θ) (also called the Bershadsky–Knizhnik algebra) with a lattice vertex algebra.

Quantum dimensions and fusion products for irreducible VQσ-modules with σ2 = 1

ABSTRACTEvery isometry σ of a positive-definite even lattice Q can be lifted to an automorphism of the lattice vertex algebra VQ. An important problem in vertex algebra theory and conformal field theory is to classify the representations of the σ-invariant subalgebra of VQ, known as an orbifold. In the case when σ is an isometry of Q of order two, we have classified the irreducible modules of the orbifold vertex algebra and identified them as submodules of twisted or untwisted VQ-modules in [6]. Here we calculate their quantum dimensions and fusion products.The examples where Q is the orthogonal direct sum of two copies of the A2 root lattice and σ is the 2-cycle permutation as well as where Q is the An root lattice and σ is a Dynkin diagram automorphism are presented in detail.

Orbifolds of lattice vertex algebras under an isometry of order two

Every isometry σ of a positive-definite even lattice Q can be lifted to an automorphism of the lattice vertex algebra VQ. An important problem in vertex algebra theory and conformal field theory is to classify the representations of the σ-invariant subalgebra VQσ of VQ, known as an orbifold. In the case when σ is an isometry of Q of order two, we classify the irreducible modules of the orbifold vertex algebra VQσ and identify them as submodules of twisted or untwisted VQ-modules. The examples where Q is a root lattice and σ is a Dynkin diagram automorphism are presented in detail.

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 $\mathcal{W}(p)$.