Zhu Algebra Research Articles

A generalization of intertwining operators for vertex operator algebras

For a vertex operator algebra V, we generalize the notion of an intertwining operator among an arbitrary triple of V-modules to an arbitrary triple of N-graded weak V-modules and study their properties. We show a formula for the dimensions of the spaces of these intertwining operators in terms of modules over the Zhu algebras under some conditions on N-graded weak modules.

Superconformal minimal models and admissible Jack polynomials

We give new proofs of the rationality of the N=1 superconformal minimal model vertex operator superalgebras and of the classification of their modules in both the Neveu–Schwarz and Ramond sectors. For this, we combine the standard free field realisation with the theory of Jack symmetric functions. A key role is played by Jack symmetric polynomials with a certain negative parameter that are labelled by admissible partitions. These polynomials are shown to describe free fermion correlators, suitably dressed by a symmetrising factor. The classification proofs concentrate on explicitly identifying Zhu's algebra and its twisted analogue. Interestingly, these identifications do not use an explicit expression for the non-trivial vacuum singular vector. While the latter is known to be expressible in terms of an Uglov symmetric polynomial or a linear combination of Jack superpolynomials, it turns out that standard Jack polynomials (and functions) suffice to prove the classification.

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.

Structure of classical (finite and affine) $\mathcal W$-algebras

First, we derive an explicit formula for the Poisson bracket of the classical finite \mathcal W -algebra \mathcal W^{\mathrm {fin}}(\mathfrak g,f) , the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra \mathfrak g and its nilpotent element f . On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine \mathcal W -algebra \mathcal W(\mathfrak g,f) . As an immediate consequence, we obtain a Poisson algebra isomorphism between \mathcal W^{\mathrm {fin}}(\mathfrak g,f) and the Zhu algebra of \mathcal W(\mathfrak g,f) . We also study the generalized Miura map for classical \mathcal W -algebras.

Classical affine W-algebras associated to Lie superalgebras

In this paper, we prove classical affine W-algebras associated to Lie superalgebras (W-superalgebras), which can be constructed in two different ways: via affine classical Hamiltonian reductions and via taking quasi-classical limits of quantum affine W-superalgebras. Also, we show that a classical finite W-superalgebra can be obtained by a Zhu algebra of a classical affine W-superalgebra. Using the definition by Hamiltonian reductions, we find free generators of a classical W-superalgebra associated to a minimal nilpotent. Moreover, we compute generators of the classical W-algebra associated to spo(2|3) and its principal nilpotent. In the last part of this paper, we introduce a generalization of classical affine W-superalgebras called classical affine fractional W-superalgebras. We show these have Poisson vertex algebra structures and find generators of a fractional W-superalgebra associated to a minimal nilpotent.

Rationality of W-algebras: principal nilpotent cases

We prove the rationality of all the minimal series principal W-algebras discovered by Frenkel, Kac and Wakimoto in 1992, thereby giving a new family of rational and C_2-cofinite vertex operator algebras. A key ingredient in our proof is the study of Zhu's algebra of simple W-algebras via the quantized Drinfeld-Sokolov reduction. We show that the functor of taking Zhu's algebra commutes with the reduction functor. Using this general fact we determine the maximal spectrums of the associated graded of Zhu's algebra of all the admissible affine vertex algebras as well.

Relaxed singular vectors, Jack symmetric functions and fractional level [formula omitted] models

The fractional level models are (logarithmic) conformal field theories associated with affine Kac–Moody (super)algebras at certain levels k∈Q. They are particularly noteworthy because of several longstanding difficulties that have only recently been resolved. Here, Wakimoto's free field realisation is combined with the theory of Jack symmetric functions to analyse the fractional level slˆ(2) models. The first main results are explicit formulae for the singular vectors of minimal grade in relaxed Wakimoto modules. These are closely related to the minimal grade singular vectors in relaxed (parabolic) Verma modules. Further results include an explicit presentation of Zhu's algebra and an elegant new proof of the classification of simple relaxed highest weight modules over the corresponding vertex operator algebra. These results suggest that generalisations to higher rank fractional level models are now within reach.

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 $\mathcal{W}(p)$.

ADESubalgebras of the Triplet Vertex Algebra 𝒲(p):E6,E7

It is the forth paper in a series devoted to the investigation of the |$ADE$| subalgebras of the triplet vertex algebra |$\mathcal {W}(p)$|⁠. In this paper, we prove |$C_{2}$|-cofiniteness of the invariant subalgebras |$\mathcal {W}(p)^{\Gamma }$| and study their representations when |$\Gamma$| is the tetrahedral group |$T$| (of |$E_6$|-type), or the octahedral group |$O$| (of |$E_7$|-type). We give a list of |$21p$| irreducible |$\mathcal {W}(p)^{T}$|-modules and a list of |$28p$| irreducible |$\mathcal {W}(p)^{O}$|-modules. The completeness of these two lists can be reduced to some conjectures about constant term identities which have been verified in the case |$p=1$|⁠. For the proof of |$C_{2}$|-cofiniteness of |$\mathcal {W}(p)^{\Gamma }$|⁠, and the calculations of the Zhu's algebras |$A(\mathcal {W}(p)^{\Gamma })$|⁠, we develop several quite universal formulas (Theorem 4.3 and its corollaries). In Appendix, we further apply them to derive a useful tool of structure coefficients for the triplet vertex algebra |$\mathcal {W}(p)$| and its irreducible modules, which unifies almost all the calculations in this paper; as an unexpected byproduct, we give a new proof of |$\mathfrak {sl}_{2}$|-action on |$\mathcal {W}(p)$|⁠.

Zhu's algebra, [formula omitted]-algebra and [formula omitted]-cofiniteness of parafermion vertex operator algebras

We study Zhu's algebra, C2-algebra and C2-cofiniteness of parafermion vertex operator algebras. We first give a detailed study of Zhu's algebra and C2-algebra of parafermion vertex operator algebras associated with the affine Kac–Moody Lie algebra slˆ2. We show that they have the same dimension and Zhu's algebra is semisimple. The classification of irreducible modules is also established. Finally, we prove that the parafermion vertex operator algebras for any finite dimensional simple Lie algebras are C2-cofinite.

On the ExtendedW-Algebra of Type 𝔰𝔩2at Positive Rational Level

The extended W-algebra of type sl_2 at positive rational level, denoted by M_{p_+,p_-}, is a vertex operator algebra that was originally proposed in [1]. This vertex operator algebra is an extension of the minimal model vertex operator algebra and plays the role of symmetry algebra for certain logarithmic conformal field theories. We give a construction of M_{p_+,p_-} in terms of screening operators and use this construction to prove that M_{p_+,p_-} satisfies Zhu's c_2-cofiniteness condition, calculate the structure of the zero mode algebra (also known as Zhu's algebra) and classify all simple M_{p_+,p_-}-modules.

On functors between module categories for associative algebras and for [formula omitted]-graded vertex algebras

We prove that the weak associativity for modules for vertex algebras is equivalent to a residue formula for iterates of vertex operators, obtained using the weak associativity and the lower truncation property of vertex operators, together with a known formula expressing products of components of vertex operators as linear combinations of iterates of components of vertex operators. By requiring that these two formulas instead of the commutator formula hold, we construct a functor S from the category of modules for Zhu's algebra of a vertex operator algebra V to the category of N-gradable weak V-modules. We prove that S has a universal property and the functor T of taking top levels of N-gradable weak V-modules is a left inverse of S. In particular, S is equal to a functor implicitly given by Zhu and explicitly constructed by Dong, Li and Mason and we obtain a new construction without using relations corresponding to the commutator formula. The hard part of this new construction is a technical theorem stating roughly that in a module for Zhu's algebra, the relation corresponding to the residue formula mentioned above can in fact be obtained from the relations corresponding to the action of Zhu's algebra.

${\mathbbm {C}}$ C -Graded vertex algebras and conformal flow

We consider $\mathbbm {C}$C-graded vertex algebras, which are vertex algebras V with a $\mathbbm {C}$C-grading such that V is an admissible V-module generated by “lowest weight vectors.” We show that such vertex algebras have a “good” representation theory in the sense that there is a Zhu algebra A(V) and a bijection between simple admissible V-modules and simple A(V)-modules. We also consider pseudo vertex operator algebras (PVOAs), which are $\mathbbm {C}$C-graded vertex algebras with a conformal vector such that the homogeneous subspaces of V are generalized eigenspaces for L(0); essentially, these are VOAs that lack any semisimplicity or integrality assumptions on L(0). As a motivating example, we show that deformation of the conformal structure (conformal flow) of a strongly regular VOA (e.g., a lattice theory, or Wess-Zumino-Witten model) is a path in a space whose points are PVOAs.

ADE SUBALGEBRAS OF THE TRIPLET VERTEX ALGEBRA 𝒲(p): D-SERIES

We are continuing our study of ADE-orbifold subalgebras of the triplet vertex algebra 𝒲(p) initiated in D. Adamović, X. Lin and A. Milas, ADE subalgebras of the triplet vertex algebra 𝒲(p): Am-series, Commun. Contemp. Math.15 (2013), Article ID: 1350028, 1–30. This part deals with the dihedral series. First, subject to a certain constant term identity, we classify all irreducible modules for the vertex algebra [Formula: see text], the ℤ2-orbifold of the singlet vertex algebra [Formula: see text]. Then, we classify irreducible modules and determine Zhu's and C2-algebra for the vertex algebra 𝒲(p)D2. A general method for construction of twisted 𝒲(p)-modules is also introduced. We also discuss classification of twisted [Formula: see text]-modules including the twisted Zhu's algebra [Formula: see text], which is of independent interest. The category of admissible Ψ-twisted [Formula: see text]-modules is expected to be semisimple. We also prove C2-cofiniteness of 𝒲(p)Dm for all m, and give a conjectural list of irreducible 𝒲(p)Dm-modules. Finally, we compute characters of the relevant irreducible modules and describe their modular closure.

ADE SUBALGEBRAS OF THE TRIPLET VERTEX ALGEBRA $\mathcal{W}(p)$: A-SERIES

Motivated by [On the triplet vertex algebra [Formula: see text], Adv. Math.217 (2008) 2664–2699], for every finite subgroup Γ ⊂ PSL(2, ℂ) we investigate the fixed point subalgebra [Formula: see text] of the triplet vertex [Formula: see text], of central charge [Formula: see text], p ≥ 2. This part deals with the A-series in the ADE classification of finite subgroups of PSL(2, ℂ). First, we prove the C2-cofiniteness of the Am-fixed subalgebra [Formula: see text]. Then we construct a family of [Formula: see text]-modules, which are expected to form a complete set of irreducible representations. As a strong support to our conjecture, we prove modular invariance of (generalized) characters of the relevant (logarithmic) modules. Further evidence is provided by calculations in Zhu's algebra for m = 2. We also present a rigorous proof of the fact that the full automorphism group of [Formula: see text] is PSL(2, ℂ).

SOME REMARKS ON PSEUDO-TRACE FUNCTIONS FOR ORBIFOLD MODELS ASSOCIATED WITH SYMPLECTIC FERMIONS

We propose a method to give pseudo-trace functions for vertex operator algebras satisfying Zhu's finiteness condition and apply our method to the ℤ2-orbifold model associated with d-pairs of symplectic fermions. Pseudo-trace functions are defined by using symmetric linear functions on (higher) Zhu's algebras. But our approach does not use such algebras. For d = 1, we determine the dimension of the space of one-point functions, that is, the one of the triplet [Formula: see text]-algebra. For d > 1, we construct 22d-1 + 3 linearly independent one-point functions and study their values at the vacuum vector.

A Commutant Realization of 𝒲n(2) at Critical Level

For n\geq 2, there is a free field realization of the affine vertex superalgebra A associated to psl(n|n) at critical level inside the bc\beta\gamma system W of rank n^2. We show that the commutant C=Com(A,W) is purely bosonic and is freely generated by n+1 fields. We identify the Zhu algebra of C with the ring of invariant differential operators on the space of n\times n matrices under SL_n \times SL_n, and we classify the irreducible, admissible C-modules with finite dimensional graded pieces. For n\leq 4, C is isomorphic to the W_n^{(2)}-algebra at critical level, and we conjecture that this holds for all n.

On [formula omitted]-algebra extensions of [formula omitted] minimal models: [formula omitted

This is a continuation of Adamović and Milas (2010) [5], where, among other things, we classified irreducible representations of the triplet vertex algebra W 2 , 3 . In this part we extend the classification to W 2 , p , for all odd p > 3 . We also determine the structure of the center of the Zhu algebra A ( W 2 , p ) which implies the existence of a family of logarithmic modules having L ( 0 ) -nilpotent ranks 2 and 3. A logarithmic version of Macdonald–Morris constant term identity plays a key role in the paper.

Higher level twisted Zhu algebras

The study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory. In this paper, we consider the general setup of a vertex algebra V, graded by \documentclass[12pt]{minimal}\begin{document}$\Gamma /\mathbb {Z}$\end{document}Γ/Z for some subgroup Γ of \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}$\end{document}R containing \documentclass[12pt]{minimal}\begin{document}$\mathbb {Z}$\end{document}Z, and with a Hamiltonian operator H having real (but not necessarily integer) eigenvalues. We construct the directed system of twisted level p Zhu algebras \documentclass[12pt]{minimal}\begin{document}$\operatorname{Zhu}_{p, \Gamma }(V)$\end{document}Zhup,Γ(V), and we prove the following theorems: For each p, there is a bijection between the irreducible \documentclass[12pt]{minimal}\begin{document}$\operatorname{Zhu}_{p, \Gamma }(V)$\end{document}Zhup,Γ(V)-modules and the irreducible Γ-twisted positive energy V-modules, and V is (Γ, H)-rational if and only if all its Zhu algebras \documentclass[12pt]{minimal}\begin{document}$\operatorname{Zhu}_{p, \Gamma }(V)$\end{document}Zhup,Γ(V) are finite dimensional and semisimple. The main novelty is the removal of the assumption of integer eigenvalues for H. We provide an explicit description of the level p Zhu algebras of a universal enveloping vertex algebra, in particular of the Virasoro vertex algebra \documentclass[12pt]{minimal}\begin{document}$\operatorname{Vir}^c$\end{document}Virc and the universal affine Kac-Moody vertex algebra \documentclass[12pt]{minimal}\begin{document}$V^k(\mathfrak {g})$\end{document}Vk(g) at non-critical level. We also compute the inverse limits of these directed systems of algebras.

Zhu's algebras, [formula omitted]-algebras and abelian radicals

This paper consists of three parts. In the first part we prove that Zhu's and C 2 -algebras in type A have the same dimensions. In the second part we compute the graded decomposition of the C 2 -algebras in type A, thus proving the Gaberdiel–Gannon conjecture. Our main tool is the theory of abelian radicals, which we develop in the third part.