Heisenberg Vertex Operator Algebra Research Articles

On [formula omitted]-graded vertex algebras associated with cyclic Leibniz algebras with small dimensions

The main goals for this paper are i) to study of the algebraic structure of N-graded vertex algebras VB associated with vertex A-algebroids B when B are cyclic non-Lie left Leibniz algebras, and ii) to explore relations between the vertex algebras VB and the rank one Heisenberg vertex operator algebra. To achieve these goals, we first classify vertex A-algebroids B associated with given cyclic non-Lie left Leibniz algebras B. Next, we use the constructed vertex A-algebroids B to create a family of indecomposable non-simple vertex algebras VB. Finally, we use the algebraic structure of the unital commutative associative algebras A that we found to study relations between certain types of the vertex algebras VB and the vertex operator algebra associated to the rank one Heisenberg algebra.

The level two Zhu algebra for the Heisenberg vertex operator algebra

We determine the level two Zhu algebra for the Heisenberg vertex operator algebra V for any choice of conformal element. We do this using only the following information for V: the internal structure of V; the level one Zhu algebra of V already determined by the second author, along with Vander Werf and Yang; and the information the lower level Zhu algebras give regarding irreducible modules. We are able to carry out this calculation of the level two Zhu algebra for V with this minimal information by employing the general results and techniques for determining generators and relations for higher level Zhu algebras for a vertex operator algebra, as developed previously by the authors in On generators and relations for higher level Zhu algebras and applications, by Addabbo and Barron, J. Algebra, 2023. In particular, we show that the level n Zhu algebras for the Heisenberg vertex operator algebra become noncommutative at level n = 2. We also give a conjecture for the structure of the level n Zhu algebra for the Heisenberg vertex operator algebra, for any n > 2.

On generators and relations for higher level Zhu algebras and applications

We give some general results about the generators and relations for the higher level Zhu algebras for a vertex operator algebra. In particular, for any element u in a vertex operator algebra V, such that u has weight greater than or equal to −n for n∈N, we prove a recursion relation in the nth level Zhu algebra An(V) and give a closed formula for this relation. We use this and other properties of An(V) to reduce the modes of u that appear in the generators for An(V) as long as u∈V has certain properties (properties that apply, for instance, to the conformal vector for any vertex operator algebra or if u generates a Heisenberg vertex subalgebra), and we then prove further relations in An(V) involving such an element u. We present general techniques that can be applied once a set of reasonable generators is determined for An(V) to aid in determining the relations of those generators, such as using the relations of those generators in the lower level Zhu algebras and the zero mode actions on V-modules induced from those lower level Zhu algebras. We prove that the condition that (L(−1)+L(0))v acts as zero in An(V) for n∈Z+ and for all v in V is a necessary added condition in the definition of the Zhu algebra at level higher than zero. We discuss how these results on generators and relations apply to the level n Zhu algebras for the Heisenberg vertex operator algebra and the Virasoro vertex operator algebras at any level n∈N.

The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order 2 (Part 1)

Let VL be the vertex algebra associated to a non-degenerate even lattice L, θ the automorphism of VL induced from the −1-isometry of L, and VL+ the fixed point subalgebra of VL under the action of θ. In this series of papers we classify the irreducible weak VL+-modules and show that any irreducible weak VL+-module is isomorphic to a weak submodule of some irreducible weak VL-module or to a submodule of some irreducible θ-twisted VL-module.In this paper (Part 1), we show that when the rank of L is 1, every non-zero weak VL+-module contains a non-zero M(1)+-module, where M(1)+ is the fixed point subalgebra of the Heisenberg vertex operator algebra M(1) under the action of θ.

On irreducibility of modules of Whittaker type for cyclic orbifold vertex algebras

We extend the Dong-Mason theorem on irreducibility of modules for orbifold vertex algebras (cf. [18]) to the category of weak modules. Let V be a vertex operator algebra, g an automorphism of order p. Let W be an irreducible weak V–module such that W,W∘g,…,W∘gp−1 are inequivalent irreducible modules. We prove that W is an irreducible weak V〈g〉–module. This result can be applied on irreducible modules of certain Lie algebra L such that W,W∘g,…,W∘gp−1 are Whittaker modules having different Whittaker functions. We present certain applications in the cases of the Heisenberg and Weyl vertex operator algebras.

Simple Whittaker modules over free bosonic orbifold vertex operator algebras

We construct weak (i.e. nongraded) modules over the vertex operator algebra M ( 1 ) + M(1)^+ , which is the fixed-point subalgebra of the higher rank free bosonic (Heisenberg) vertex operator algebra with respect to the − 1 -1 automorphism. These weak modules are constructed from Whittaker modules for the higher rank Heisenberg algebra. We prove that the modules are simple as weak modules over M ( 1 ) + M(1)^+ and calculate their Whittaker type when regarded as modules for the Virasoro Lie algebra. Lastly, we show that any Whittaker module for the Virasoro Lie algebra occurs in this way. These results are a higher rank generalization of some results by Tanabe [Proc. Amer. Math. Soc. 145 (2017), no. 10, pp. 4127–4140].

Simple weak modules for the fixed point subalgebra of the Heisenberg vertex operator algebra of rank $1$ by an automorphism of order $2$ and Whittaker vectors

Let $M(1)$ be the vertex operator algebra with the Virasoro element $\omega$ associated to the Heisenberg algebra of rank $1$ and let $M(1)^{+}$ be the subalgebra of $M(1)$ consisting of the fixed points of an automorphism of $M(1)$ of order $2$. We classify the simple weak $M(1)^{+}$-modules with a non-zero element $w$ such that for some integer $s\geq 2$, $\omega_i w\in{\mathbb C}w$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$), $\omega_{s}w\in{\mathbb C}^{\times}w$, and $\omega_i w=0$ for all $i>s$. The result says that any such simple weak $M(1)^{+}$-module is isomorphic to some simple weak $M(1)$-module or to some $\theta$-twisted simple weak $M(1)$-module.

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.

𝒲-Algebra constraints and topological recursion for AN-singularity (with an Appendix by Danilo Lewanski)

We derive a Bouchard–Eynard type topological recursion for the total descendant potential of [Formula: see text]-singularity. Our argument relies on a certain twisted representation of a Heisenberg Vertex Operator Algebra (VOA) constructed via the periods of [Formula: see text]-singularity. In particular, our approach allows us to prove that the topological recursion for the total descendant potential is equivalent to a certain generating set of [Formula: see text]-algebra constraints.

Ward identities from recursion formulas for correlation functions in conformal field theory

A conformal block formulation for the Zhu recursion procedure in conformal field theory which allows to find $n$-point functions in terms of the lower correlations functions is introduced. Then the Zhu reduction operators acting on a tensor product of VOA modules are defined. By means of these operators we show that the Zhu reduction procedure generates explicit forms of Ward identities for conformal blocks of vertex operator algebras. Explicit examples of Ward identities for the Heisenberg and free fermionic vertex operator algebras are supplied.

RIEMANN SURFACES AND VERTEX OPERATOR ALGEBRAS

We show that for any fixed point P0 on a Riemann surface Σ the distinct realizations of cocycles in [Formula: see text] correspond to the natural appearances of the standard Heisenberg vertex operator algebra Π(P0) and to the commutative Heisenberg vertex operator algebra Π0(P0), respectively.

A generalized vertex operator algebra for Heisenberg intertwiners

We consider the extension of the Heisenberg vertex operator algebra by all its irreducible modules. We give an elementary construction for the intertwining vertex operators and show that they satisfy a complex parameterized generalized vertex operator algebra. We illustrate some of our results with the example of integral lattice vertex operator superalgebras.

Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I

We define the partition and $n$-point functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and for any pair of simple Heisenberg modules. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties for the Heisenberg and lattice vertex operator algebras and a continuous orbifolding of the rank two fermion vertex operator super algebra. We compute the genus two Heisenberg vector $n$-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity for these theories.

Logarithmic Intertwining Operators and Vertex Operators

This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg vertex operator algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra $M(1)_a$, of central charge $1-12a^2$. We classify these operators in terms of {\em depth} and provide explicit constructions in all cases. Furthermore, for $a=0$ we focus on the vertex operator subalgebra L(1,0) of $M(1)_0$ and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of {\em hidden} logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1,0)-module.

TWISTED VERTEX OPERATORS AND BERNOULLI POLYNOMIALS

Using general principles in the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an arbitrary twisting automorphism. The construction involves the Bernoulli polynomials in a fundamental way. We develop new identities and principles in the theory of vertex operator algebras and their twisted modules, and explain the construction by applying general results, including an identity that we call modified weak associativity, to the Heisenberg vertex operator algebra. This paper gives proofs and further explanations of results announced earlier. It is a generalization to twisted vertex operators of work announced by the second author some time ago, and includes as a special case the proof of the main results of that work.