Theory Of Vertex Operator Algebras Research Articles

Convergence in Conformal Field Theory

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory. The author reviews some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. He also reviews the related analytic extension results, conjectures and problems. He discusses the convergence and analytic extensions of products of intertwining operators (chiral conformal fields) and of q-traces and pseudo-q-traces of products of intertwining operators. He also discusses the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result. He then explains conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras.

Codes, vertex operators and topological modular forms

We describe a new link between the theory of topological modular forms and representations of vertex operator algebras obtained by certain lattices. The construction is motivated by the arithmetic Whitehead tower of the orthogonal groups. The tower discloses the role of codes in representation theory.

Representation theory in chiral conformal field theory: from fields to observables

This article develops new techniques for understanding the relationship between the three different mathematical formulations of two-dimensional chiral conformal field theory: conformal nets (axiomatizing local observables), vertex operator algebras (axiomatizing fields), and Segal CFTs. It builds upon previous work (Tener in Adv Math 349:488–563, 2019), which introduced a geometric interpolation procedure for constructing conformal nets from VOAs via Segal CFT, simultaneously relating all three frameworks. In this article, we extend this construction to study the relationship between the representation theory of conformal nets and the representation theory of vertex operator algebras. We define a correspondence between representations in the two contexts, and show how to construct representations of conformal nets from VOAs. We also show that this correspondence is rich enough to relate the respective ‘fusion product’ theories for conformal nets and VOAs, by constructing local intertwiners (in the sense of conformal nets) from intertwining operators (in the sense of VOAs). We use these techniques to show that all WZW conformal nets can be constructed using our geometric interpolation procedure.

Categories and functors in the representation theory of vertex operator algebras

In this paper, we use the language of categories to describe representation theory of vertex operator algebras. The category of all vertex operator algebras and the category of modules of a vertex operator algebra are discussed. A homomorphism between two vertex operator algebras should preserve the Virasoro vectors, which is equivalent to commuting with the operators $L(n)$ for all $n\in \mathbb{Z}$. We expand the morphisms of this category so that morphisms are semi-conformal in the sense that they commute with those $L(n)$ with $ n\geq 0$. This expansion does not change the classification of problem and makes the category into a tensor category. The coset construction becomes more natural in this category and relations between the module categories of vertex operator algebras can be described in terms of Hom-functors. As an application, we also construct the corresponding Jacquet functors. The semi-conformal vertex operator subalgebras plays the role of the Levi subgroups of a reductive group.

Twisted representations of vertex operator algebras associated to affine Lie algebras

In this paper, we prove that the categories of lower bounded twisted modules of positive integer levels for simple vertex operator algebras associated with affine Lie algebras and general automorphisms are semisimple, using the twisted generalizations of Zhu's algebras for these vertex operator algebras, constructed in [13]. We also show that the category of lower bounded twisted modules for a general automorphism is equivalent to the category of lower bounded twisted modules for the corresponding diagram automorphism.

A note on the zeroth products of Frenkel–Jing operators

Let [Formula: see text] be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable [Formula: see text]-module is the finite-dimensional irreducible [Formula: see text]-module, where the action of the simple Lie algebra [Formula: see text] is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level [Formula: see text] Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra [Formula: see text]. By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator [Formula: see text], we obtain an infinite-dimensional vector space [Formula: see text]. Next, we introduce an associative algebra [Formula: see text], a certain quantum analogue of the universal enveloping algebra [Formula: see text], and construct some infinite-dimensional [Formula: see text]-modules [Formula: see text] corresponding to the finite-dimensional irreducible [Formula: see text]-modules [Formula: see text]. We show that the space [Formula: see text] carries a structure of an [Formula: see text]-module and, furthermore, we prove that the [Formula: see text]-module [Formula: see text] is isomorphic to the [Formula: see text]-module [Formula: see text].

The varieties of Heisenberg vertex operator algebras

For a vertex operator algebra $V$ with conformal vector $\omega$, we consider a class of vertex operator subalgebras and their conformal vectors. They are called semi-conformal vertex operator subalgebras and semi-conformal vectors of $(V,\omega)$, respectively, and were used to study duality theory of vertex operator algebras via coset constructions. Using these objects attached to $(V,\omega)$, we shall understand the structure of the vertex operator algebra $(V,\omega)$. At first, we define the set $\on{Sc}(V,\omega)$ of semi-conformal vectors of $V$, then we prove that $\on{Sc}(V,\omega)$ is an affine algebraic variety with a partial ordering and an involution map. Corresponding to each semi-conformal vector, there is a unique maximal semi-conformal vertex operator subalgebra containing it. The properties of these subalgebras are invariants of vertex operator algebras. As an example, we describe the corresponding varieties of semi-conformal vectors for Heisenberg vertex operator algebras. As an application, we give two characterizations of Heisenberg vertex operator algebras using the properties of these varieties.

On Regularised Quantum Dimensions of the Singlet Vertex Operator Algebra and False Theta Functions

We study a family of non-C2-cofinite vertex operator algebras, called the singlet vertex operator algebras, and connect several important concepts in the theory of vertex operator algebras, quantum modular forms, and modular tensor categories. More precisely, starting from explicit formulae for characters of modules over the singlet vertex operator algebra, which can be expressed in terms of false theta functions and their derivatives, we first deform these characters by using a complex parameter ϵϵ. We then apply modular transformation properties of regularisedpartial theta functions to study asymptotic behaviour of regularised characters of irreducible modules and compute their regularised quantum dimensions. We also give a purely geometric description of the regularisation parameter as a uniformisation parameter of the fusion variety coming from atypical blocks. It turns out that the quantum dimensions behave very differently depending on the sign of the real part of ϵϵ. The map from the space of characters equipped with the Verlinde product to the space of regularised quantum dimensions turns out to be a genuine ring isomorphism for positive real part of ϵϵ, while for sufficiently negative real part of ϵϵ its surjective image gives the fusion ring of a rational vertex operator algebra. The category of modules of this rational vertex operator algebra should be viewed as obtained through the process of a semi-simplification procedure widely used in the theory of quantum groups. Interestingly, the modular tensor category structure constants of this vertex operator algebra can be also detected from vector-valued quantum modular forms formed by distinguished atypical characters.

Principal subspaces of higher-level standard $\widehat{\mathfrak{sl}(n)}$-modules

Using completions of certain universal enveloping algebras, we provide a natural setting for families of defining relations for the principal subspaces of standard modules for untwisted affine Lie algebras. We also use the theory of vertex operator algebras and intertwining operators to construct exact sequences among principal subspaces of certain standard [Formula: see text]-modules, n ≥ 3. As a consequence, we obtain the multigraded dimensions of the principal subspaces W(k1Λ1 + k2Λ2) and W(kn-2Λn-2 + kn-1Λn-1). This generalizes earlier work by Calinescu on principal subspaces of standard [Formula: see text]-modules.

On associative algebras, modules and twisted modules for vertex operator algebras

We give a new construction of functors from the category of modules for the associative algebras An(V) and Ag(V) associated with a vertex operator algebra V, defined by Dong, Li and Mason, to the category of admissible V-modules and admissible twisted V-modules, respectively, using the method developed in the joint work [5] with Y.-Z. Huang. The functors were first constructed by Dong, Li and Mason, but the importance of the new method, as in [5], is that we can apply the method to study objects without the commutator formula in the representation theory of vertex operator algebras.

Jacobi trace functions in the theory of vertex operator algebras

We describe a type of n-point function associated to strongly regular vertex operator algebras V and their irreducible modules. Transformation laws with respect to the Jacobi group are developed for 1-point functions. For certain elements in V , the finitedimensional space spanned by the corresponding 1-point functions for the inequivalent irreducible modules is shown to be a vector-valued weak Jacobi form. A decomposition of 1-point functions for general elements is proved, and shows that such functions are typically quasi-Jacobi forms. Zhu-type recursion formulas are proved; they show how an n-point function can be written as a linear combination of (n − 1)-point functions with coefficients that are quasi-Jacobi forms. MSC(2012): 17B69.

Presentations of the principal subspaces of the higher-level standard [formula omitted]-modules

Using the theory of vertex operator algebras and intertwining operators, we obtain presentations for the principal subspaces of all the standard sl(3)ˆ-modules. Certain of these presentations had been conjectured and used in work of Calinescu to construct exact sequences leading to the graded dimensions of certain principal subspaces. We prove the conjecture in its full generality for all standard sl(3)ˆ-modules. We then provide a conjecture for the case of sl(n+1)ˆ.

Representations of vertex operator algebras over an arbitrary field

An associative algebra A(V) for a vertex operator algebra over an arbitrary algebraically closed field F is constructed such that there is a one to one correspondence between irreducible A(V)-modules and irreducible admissible V-modules. Moreover, A(V) is semisimple if V is rational. An A(V)-bimodule A(M) is also constructed for any admissible V-module M and its connection with the fusion rules is investigated. These results are then used to compute the fusion rules for the rational vertex operator algebra L(12,0)F associated to the irreducible highest weight module for the Virasoro algebra with central charge 12.

Combinatorial bases of principal subspaces for the affine Lie algebra of type [formula omitted

We consider principal subspaces WL(kΛ0) and WN(kΛ0) of standard module L(kΛ0) and generalized Verma module N(kΛ0) at level k≥1 for the affine Lie algebra of type B2(1). By using the theory of vertex operator algebras, we find combinatorial bases of principal subspaces in terms of quasi-particles. From quasi-particle bases, we obtain character formulas for WL(kΛ0) and WN(kΛ0).

Representations of vertex operator algebras and bimodules

For a vertex operator algebra V, a V-module M and a nonnegative integer n, an An(V)-bimodule An(M) is constructed and studied. The connections between An(M) and intertwining operators are discussed. Moreover, the An(V)-bimodule At,s(V) is a quotient of An(V) for all s,t⩽n. In the case that V is rational, An(M) for irreducible V-module M is given explicitly.

Combinatorial bases of modules for affine Lie algebra B 2(1)

Abstract We construct bases of standard (i.e. integrable highest weight) modules L(Λ) for affine Lie algebra of type B 2(1) consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces W(Λ) of L(Λ) by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence W(kΛ0) for B 2(1) and the integrable highest weight module L(kΛ0) for A 1(1) have the same parametrization of combinatorial bases and the same presentation P/I.

A new existence proof of the Monster by VOA theory

We use uniqueness of a VOA (vertex operator algebra) extension of (V + EE8) 3 to a Moonshine type VOA to give a new existence proof of a finite simple group of Monster type. The proof is relatively direct. Our methods depend on VOA representation theory and are free of many special calculations which traditionally occur in theory of the Monster.

McKay’s E 6 Observation on the Largest Fischer Group

In this paper, we study McKay’s E6-observation on the largest Fischer 3-transposition group Fi24. We investigate a vertex operator algebra \({VF^{\natural}}\) of central charge \({23\frac{1}{5}}\) on which the Fischer group Fi24 naturally acts. We show that there is a natural correspondence between dihedral subgroups of Fi24 and certain vertex operator subalgebras constructed by the nodes of the affine E6 diagram by investigating so-called derived Virasoro vectors of central charge 6/7. This allows us to reinterpret McKay’s E6-observation via the theory of vertex operator algebras.

On Majorana Representations of A 6 and A 7

The Majorana representations of groups were introduced in Ivanov (The Monster Group and Majorana Involutions, 2009) by axiomatising some properties of the 2A-axial vectors of the 196 884-dimensional Monster algebra, inspired by the sensational classification of such representations for the dihedral groups achieved by Sakuma (Int Math Res Notes, 2007). This classification took place in the heart of the theory of Vertex Operator Algebras and expanded earlier results by Miyamoto (J Alg 268:653–671, 2003). Every subgroup G of the Monster which is generated by its intersection with the conjugacy class of 2A-involutions possesses the (possibly unfaithful) Majorana representation obtained by restricting to G the action of the Monster on its algebra. This representation of G is said to be based on an embedding of G in the Monster. So far the Majorana representations have been classified for the groups G isomorphic to the symmetric group S 4 of degree 4 (Ivanov et al. in J Alg 324:2432–2463, 2010), the alternating group A 5 of degree 5 (Ivanov AA, Seress A in Majorana Representations of A 5, 2010), and the general linear group GL 3(2) in dimension 3 over the field of two elements (Ivanov AA, Shpectorov S in Majorana Representations of L 3(2), 2010). All these representations are based on embeddings in the Monster of either the group G itself or of its direct product with a cyclic group of order 2. The dimensions and shapes of these representations are given in the following table:

McKay’s E7 Observation on the Baby Monster

In this paper, we study McKay’s E7 observation on the Baby Monster. By investigating the so-called derived c=7/10 Virasoro vectors, we show that there is a natural correspondence between dihedral subgroups of the Baby Monster and certain subalgebras of the Baby Monster vertex operator algebra which are constructed by the nodes of the affine E7 diagram. This allows us to reinterpret McKay’s E7 observation via the theory of vertex operator algebras. For a class of vertex operator algebras including the Moonshine module, we will show that the product of two Miyamoto involutions associated to derived c=7/10 Virasoro vectors in certain commutant vertex operator algebras is an element of order at most 4. For the case of the Moonshine module, we obtain the Baby monster vertex operator algebra as the commutant and we can identify the group generated by these Miyamoto involutions with the Baby Monster and recover the {3,4}-transposition property of the Baby Monster in terms of vertex operator algebras.