Zhu Algebra Research Articles

Differential graded vertex operator algebras and their Poisson algebras

In this paper, we define differential graded vertex operator algebras and the algebraic structures on the associated Zhu algebras and C2-algebras. We also introduce the corresponding notions of modules, and investigate the relations between the different module categories.

Yoneda algebras of the triplet vertex operator algebra

Given a vertex operator algebra V, one can construct two associative algebras, the Zhu algebra A(V) and the C2-algebra R(V). This gives rise to two abelian categories A(V)-Mod and R(V)-Mod, in addition to the category of admissible modules of V. In case V is rational and C2-cofinite, the category of admissible V-modules and the category of all A(V)-modules are equivalent. However, when V is not rational, the connection between these two categories is unclear. The goal of this paper is to study the triplet vertex operator algebra W(p), as an example to compare these three categories, in terms of abelian categories. For each of these three abelian categories, we will determine the associated Ext quiver, the Morita equivalent basic algebra, i.e., the algebra End(⊕L∈IrrPL)op, and the Yoneda algebra Ext⁎(⊕L∈IrrL,⊕L∈IrrL). As a consequence, the category of admissible log-modules for the triplet VOA W(p) has infinite global dimension, as do the Zhu algebra A(W(p)), and the associated graded algebra grA(W(p)) which is isomorphic to R(W(p)). We also describe the Koszul properties of the module categories of W(p), A(W(p)) and grA(W(p)).

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.

一类酉Virasoro顶点算子代数扩张的Zhu代数

一类酉Virasoro顶点算子代数扩张的Zhu代数

On rationality of C-graded vertex algebras and applications to Weyl vertex algebras under conformal flow

Using the Zhu algebra for a certain category of C-graded vertex algebras V, we prove that if V is finitely Ω-generated and satisfies suitable grading conditions, then V is rational, i.e., it has semi-simple representation theory, with a one-dimensional level zero Zhu algebra. Here, Ω denotes the vectors in V that are annihilated by lowering the real part of the grading. We apply our result to the family of rank one Weyl vertex algebras with conformal element ωμ parameterized by μ∈C and prove that for certain non-integer values of μ, these vertex algebras, which are non-integer graded, are rational, with a one-dimensional level zero Zhu algebra. In addition, we generalize this result to appropriate C-graded Weyl vertex algebras of arbitrary ranks.

Formula omitted] at level [formula omitted

Let Ll=L(sl2l+1,−l−12) be the simple vertex operator algebra based on the affine Lie algebra slˆ2l+1 at boundary admissible level −l−12.We consider a lift ν of the Dynkin diagram involution of A2l=sl2l+1 to an involution of Ll. The ν-twisted Ll-modules are A2l(2)-modules of level −l−12 with an anti-homogeneous realization. We classify simple ν-twisted highest-weight (weak) Ll-modules using twisted Zhu algebras and singular vectors for slˆ2l+1 at level −l−12 obtained by Perše.We find that there are finitely many such modules up to isomorphism, and the ν-twisted (weak) Ll-modules that are in category O for A2l(2) are semi-simple.

First cohomologies of affine, Virasoro, and lattice vertex operator algebras

In this paper we study the first cohomologies for the following three examples of vertex operator algebras: (i) the simple affine VOA associated to a simple Lie algebra with positive integral level; (ii) the Virasoro VOA corresponding to minimal models; (iii) the lattice VOA associated to a positive definite even lattice. We prove that in all these cases, the first cohomology $H^1(V, W)$ are given by the zero-mode derivations when $W$ is any $V$-module with an $\N$-grading (not necessarily by the operator $L(0)$). This agrees with the conjecture made by Yi-Zhi Huang and the author in 2018. For negative energy representations of Virasoro VOA, the same conclusion holds when $W$ is $L(0)$-graded with lowest weight greater or equal to $-3$. Relationship between the first cohomology of the VOA and that of the associated Zhu's algebra is also discussed.

G-twisted associative algebras of vertex operator superalgebras

Let V be a vertex operator superalgebra and G a finite automorphism group of V. Let σ be the order 2 automorphism of V associated with the superstructure of V. A sequence of associative algebras AG,n(V) are constructed to study the twisted representations of V for nonnegative n∈1T′Z where T′ is the order of group generated by G and σ. This result which generalizes many previous results on Zhu's algebras is then used to investigate the super orbifold theory. If V is a simple vertex operator superalgebra and S is a finite set of inequivalent irreducible twisted V-modules which is closed under the action of G, a duality theorem of Schur-Weyl type is obtained for the actions of certain finite dimensional semisimple associated algebra Aα(G,S) and VG on the direct sum of twisted V-modules in S. In particular, for any g∈G every irreducible g-twisted V-module is completely reducible VG-module.

Vertex superalgebras over fields of prime characteristic

This paper is to study vertex superalgebras over a field of prime characteristic. For a general 12Z-graded vertex superalgebra V, Zhu's A(V) theory is examined with some suitable modifications. Among the main results, Zhu's algebras of certain affine vertex superalgebras and Neveu-Schwarz vertex superalgebras, and irreducible modules for certain quotient vertex superalgebras associated to p-center, are determined. Irreducible modules for Clifford vertex superalgebras are also determined and a complete reducibility theorem is obtained.

From VOAs to Short Star Products in SCFT

We build a bridge between two algebraic structures in superconformal field theories (SCFT): a vertex operator algebra (VOA) in the Schur sector of 4d $$\mathcal {N}=2$$ theories and an associative algebra in the Higgs sector of 3d $$\mathcal {N}=4$$ . The natural setting is a 4d $$\mathcal {N}=2$$ SCFT placed on $$S^3\times S^1$$ : by sending the radius of $$S^1$$ to zero, we recover the 3d $$\mathcal {N}=4$$ theory, and the corresponding VOA on the torus degenerates to the associative algebra on the circle. We prove that: (1) the Higgs branch operators remain in the cohomology; (2) all the Schur operators of the non-Higgs type are lifted by line operators wrapped on the $$S^1$$ ; (3) no new cohomology classes are added. We show that the algebra in 3d is given by the quotient $$\mathcal {A}_H = \mathrm{Zhu}_{s}(V)/N$$ , where $$\mathrm{Zhu}_{s}(V)$$ is the non-commutative Zhu algebra of the VOA V (for $${s}\in \mathrm{Aut}(V)$$ ), and N is a certain ideal. This ideal is the null space of the (s-twisted) trace map $$T_{s}: \mathrm{Zhu}_{s}(V) \rightarrow \mathbb {C}$$ determined by the torus 1-point function in the high temperature (or small complex structure) limit. It therefore equips $$\mathcal {A}_H$$ with a non-degenerate (twisted) trace, leading to a short star-product according to the recent results of Etingof and Stryker. The map $$T_{s}$$ is easy to determine for unitary VOAs, but has a much subtler structure for non-unitary and non- $$C_2$$ -cofinite VOAs of our interest. We comment on relation to the Beem-Rastelli conjecture on the Higgs branch and the associated variety. A companion paper will explore further details, examples, and some applications of these ideas.

Chiral Homology of Elliptic Curves and the Zhu Algebra

We study the chiral homology of elliptic curves with coefficients in a quasiconformal vertex algebra. Our main result expresses the nodal curve limit of the first chiral homology group in terms of the Hochschild homology of the Zhu algebra of V. A technical result of independent interest regarding the equivalence between the associated graded with respect to Li's filtration and the arc space of the C_2 algebra is presented.

On Zhu's algebra and C2–algebra for symplectic fermion vertex algebra SF(d)+

In this paper, we study the family of vertex operator algebras SF(d)+, known as symplectic fermions. This family is of a particular interest because these VOAs are irrational and C2-cofinite. We determine Zhu's algebra A(SF(d)+) and show that the equality of dimensions of A(SF(d)+) and the C2–algebra P(SF(d)+) holds for d≥2 (the case of d=1 was treated by T. Abe in [1]). We use these results to prove a conjecture by Y. Arike and K. Nagatomo ([8]) on the dimension of the space of one-point functions on SF(d)+.

Classification of irreducible modules for Bershadsky–Polyakov algebra at certain levels

We study the representation theory of the Bershadsky–Polyakov algebra [Formula: see text]. In particular, Zhu algebra of [Formula: see text] is isomorphic to a certain quotient of the Smith algebra, after changing the Virasoro vector. We classify all modules in the category [Formula: see text] for the Bershadsky–Polyakov algebra [Formula: see text] for [Formula: see text]. In the case [Formula: see text], we show that the Zhu algebra [Formula: see text] has two-dimensional indecomposable modules.

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.

Vertex Algebras Associated with Hypertoric Varieties

Abstract We construct a family of vertex algebras associated with a family of symplectic singularity/resolution, called hypertoric varieties. While the hypertoric varieties are constructed by a certain Hamiltonian reduction associated with a torus action, our vertex algebras are constructed by (semi-infinite) BRST reduction. The construction works algebro-geometrically, and we construct sheaves of $\hbar $-adic vertex algebras over hypertoric varieties, which localize the vertex algebras. We determine when it is a vertex operator algebra by giving an explicit conformal vector. We also discuss the Zhu algebra of the vertex algebra and its relation with a quantization of the hypertoric variety. In certain cases, we obtain the affine ${\mathcal{W}}$-algebra associated with the subregular nilpotent orbit in $\mathfrak{s}\mathfrak{l}_N$ at level $N-1$ and simple affine vertex operator algebra for $\mathfrak{s}\mathfrak{l}_N$ at level $-1$.

A short construction of the Zhu algebra

We investigate associative quotients of vertex algebras. We also give a short construction of the Zhu algebra, and a proof of its associativity using elliptic functions.

Higher level Zhu algebras and modules for vertex operator algebras

Motivated by the study of indecomposable, nonsimple modules for a vertex operator algebra V, we study the relationship between various types of V-modules and modules for the higher level Zhu algebras for V, denoted An(V), for n∈N, first introduced by Dong, Li, and Mason in 1998. We resolve some issues that arise in a few theorems previously presented when these algebras were first introduced, and give examples illustrating the need for certain modifications of the statements of those theorems. We establish that whether or not An−1(V) is isomorphic to a direct summand of An(V) affects the types of indecomposable V-modules which can be constructed by inducing from an An(V)-module, and in particular whether there are V-modules induced from An(V)-modules that were not already induced by A0(V). We give some characterizations of the V-modules that can be constructed from such inducings, in particular as regards their singular vectors. To illustrate these results, we discuss two examples of A1(V): when V is the vertex operator algebra associated to either the Heisenberg algebra or the Virasoro algebra. For these two examples, we show how the structure of A1(V) in relationship to A0(V) determines what types of indecomposable V-modules can be induced from a module for the level zero versus level one Zhu algebras. We construct a family of indecomposable modules for the Virasoro vertex operator algebra that are logarithmic modules and are not highest weight modules.

Modular Virasoro vertex algebras and affine vertex algebras

In this paper, we study Virasoro vertex algebras and affine vertex algebras over a general field of characteristic p>2. More specifically, we study certain quotients of the universal Virasoro and affine vertex algebras by ideals related to the p-centers of the Virasoro algebra and affine Lie algebras. Among the main results, we classify their irreducible N-graded modules by explicitly determining their Zhu algebras and show that these vertex algebras have only finitely many irreducible N-graded modules and they are C2-cofinite.

Higher level Zhu algebras are subquotients of universal enveloping algebras

We prove that higher level Zhu algebras of a vertex operator algebra are isomorphic to subquotients of its universal enveloping algebra.