One-point restricted conformal blocks and the fusion tensor product

Regularity of Rational Vertex Operator Algebras

Rational vertex operator algebras are finitely generated

A bstract . We discuss the foundations of vertex operator algebras and their representations, concentrating on rational vertex operator algebras. We use the Witt-Grothendieck group of conformal field theories as a vehicle to describe results, in particular the connections with modular forms. INTRODUCTION Vertex operator algebras suggest themselves as objects that could play a role in a geometric description of elliptic cohomology and related topics. As linear spaces with lots of symmetry they can participate in K-theoretic type constructions, and many (but not all) vertex operator algebras are endowed with a natural modular form as part of their structure. The incorporation of vertex operator algebras into topology is well under way (e.g. [Bo], [MS], [MSV], [T]), but it seems true to say that the underlying algebraic theory is not well understood by many potential users of the subject. What follows is an attempt to convey some of the basic ideas about vertex operator algebras, in particular the rapidly advancing theory of rational vertex operator algebras. These are the algebras most naturally associated to modular forms, and in the guise of RCFT (rational conformal field theory) they constitute an active area of research in physics ([FMS]) as well as mathematics. For more information and background the reader may refer to the following: [DM4], [FLM], [G], [K1], [KR], [QFS]. Additional references will be mentioned below. In my talk at the Newton Institute I emphasized questions about group actions (orbifold theory) and in particular how one can get information about finite group cohomology (e.g., for the Monster group) by looking at maps of equivariant Witt-Grothendieck groups of vertex operator algebras into group cohomology.

The level one Zhu algebra for the Heisenberg vertex operator algebra is calculated, and implications for the use of Zhu algebras of higher level for vertex operator algebras are discussed. In particular, we show the Heisenberg vertex operator algebra gives an example of when the level one Zhu algebra, and in fact all its higher level Zhu algebras, do not provide new indecomposable non simple modules for the vertex operator algebra beyond those detected by the level zero Zhu algebra.

Yoneda algebras of the triplet vertex operator algebra

The two pillars of rational conformal field theory and rational vertex operator algebras are modularity of characters, and the interpretation of its category of modules as a modular tensor category. Overarching these pillars is the Verlinde formula. In this paper we consider the more general class of logarithmic conformal field theories and C2-cofinite vertex operator algebras. We suggest logarithmic variants of those pillars and of Verlinde’s formula. We illustrate our ideas with the -triplet algebras and the symplectic fermions.

In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the λ-bracket. In Section 2 we construct, in the most general framework, the Zhu algebra ZhuΓV, an associative algebra which “controls” Γ-twisted representations of the vertex algebra V with a given Hamiltonian operator H. An important special case of this construction is the H-twisted Zhu algebra Zhu H V. In Section 3 we review the theory of non-linear Lie conformal algebras (respectively non-linear Lie algebras). Their universal enveloping vertex algebras (resp. universal enveloping algebras) form an important class of freely generated vertex algebras (resp. PBW generated associative algebras). We also introduce the H-twisted Zhu non-linear Lie algebra Zhu H R of a non-linear Lie conformal algebra R and we show that its universal enveloping algebra is isomorphic to the H-twisted Zhu algebra of the universal enveloping vertex algebra of R. After a discussion of the necessary cohomological material in Section 4, we review in Section 5 the construction and basic properties of affine and finite W-algebras, obtained by the method of quantum Hamiltonian reduction. Those are some of the most intensively studied examples of freely generated vertex algebras and PBW generated associative algebras. Applying the machinery developed in Sections 3 and 4, we then show that the H-twisted Zhu algebra of an affine W-algebra is isomorphic to the finite W-algebra, attached to the same data. In Section 6 we define the Zhu algebra of a Poisson vertex algebra, and we discuss quasiclassical limits. In the Appendix, the equivalence of three definitions of a finite W-algebra is established.

Conformal field theory (CFT) has proven to be one of the richest and deepest subjects of modern theoretical and mathematical physics research, especially as regards statistical mechanics and string theory. It has also stimulated an enormous amount of activity in mathematics, shaping and building bridges between seemingly disparate fields through the study of vertex operator algebras, a (partial) axiomatisation of a chiral CFT. One can add to this that the successes of CFT, particularly when applied to statistical lattice models, have also served as an inspiration for mathematicians to develop entirely new fields: Schramm-Loewner evolution and Smirnov’s discrete complex analysis being notable examples. When the energy operator fails to be diagonalisable on the quantum state space, the CFT is said to be logarithmic. Consequently, a logarithmic CFT is one whose quantum space of states is constructed from a collection of representations which includes reducible but indecomposable ones. This qualifier arises because of the consequence that certain correlation functions will possess logarithmic singularities, something that contrasts with the familiar case of power law singularities. While such logarithmic singularities and reducible representations were noted by Rozansky and Saleur in their study of the U(1|1) Wess-Zumino-Witten model in 1992, the link between the non-diagonalisability of the energy operator and logarithmic singularities in correlators is usually ascribed to Gurarie’s 1993 article (his paper also contains the first usage of the term “logarithmic conformal field theory”). The class of CFTs that were under control at this time was quite small. In particular, an enormous amount of work from the statistical mechanics and string theory communities had produced a fairly detailed understanding of the (so-called) rational CFTs. However, physicists from both camps were well aware that applications from many diverse fields required significantly more complicated non-rational theories. Examples include critical percolation, supersymmetric string backgrounds, disordered electronic systems, sandpile models describing avalanche processes, and so on. In each case, the non-rationality and non-unitarity of the CFT suggested that a more general theoretical framework was needed. Driven by the desire to better understand these applications, the mid-nineties saw significant theoretical advances aiming to generalise the constructs of rational CFT to a more general class. In 1994, Nahm introduced an algorithm for computing the fusion product of representations which was significantly generalised two years later by Gaberdiel and Kausch who applied it to explicitly construct (chiral) representations upon which the energy operator acts non-diagonalisably. Their work made it clear that underlying the physically relevant correlation functions are classes of reducible but indecomposable representations that can be investigated mathematically to the benefit of applications. In another direction, Flohr had meanwhile initiated the study of modular properties of the characters of logarithmic CFTs, a topic which had already evoked much mathematical interest in the rational case. Since these seminal theoretical papers appeared, the field has undergone rapid development, both theoretically and with regard to applications. Logarithmic CFTs are now known to describe non-local observables in the scaling limit of critical lattice models, for example percolation and polymers, and are an integral part of our understanding of quantum strings propagating on supermanifolds. They are also believed to arise as duals of three-dimensional chiral gravity models, fill out hidden sectors in non-rational theories with non-compact target spaces, and describe certain transitions in various incarnations of the quantum Hall effect. Other physical

Let [Formula: see text] be a commutative algebra in a braided monoidal category [Formula: see text]. For example, [Formula: see text] could be a vertex operator algebra (VOA) extension of a VOA [Formula: see text] in a category [Formula: see text] of [Formula: see text]-modules. We first find conditions for the category [Formula: see text] of [Formula: see text]-modules in [Formula: see text] and its subcategory [Formula: see text] of local modules to inherit rigidity from [Formula: see text]. Second and more importantly, we prove a converse result, finding conditions under which [Formula: see text] and [Formula: see text] inherit rigidity from [Formula: see text]. For our first results, we assume that [Formula: see text] is a braided finite tensor category and identify mild conditions under which [Formula: see text] and [Formula: see text] are also rigid. These conditions are based on criteria due to Etingof and Ostrik for [Formula: see text] to be an exact algebra in [Formula: see text]. As an application, we show that if [Formula: see text] is a simple [Formula: see text]-graded VOA containing a strongly rational vertex operator subalgebra [Formula: see text], then [Formula: see text] is also strongly rational, without requiring the dimension of [Formula: see text] in the modular tensor category of [Formula: see text]-modules to be non-zero. We also identify conditions under which the category of [Formula: see text]-modules inherits rigidity from the module category of a [Formula: see text]-cofinite non-rational subalgebra [Formula: see text]. For our converse result, we assume that [Formula: see text] is a Grothendieck–Verdier category, which means that [Formula: see text] admits a weaker duality structure than rigidity. We first show that [Formula: see text] is also a Grothendieck–Verdier category. Using this, we then prove that if [Formula: see text] is rigid, then so is [Formula: see text] under conditions that include a mild non-degeneracy assumption on [Formula: see text], as well as assumptions that every simple object of [Formula: see text] is local and that induction [Formula: see text] commutes with duality. These conditions are motivated by free field-like VOA extensions [Formula: see text] where [Formula: see text] is often an indecomposable [Formula: see text]-module, and thus our result will make it more feasible to prove rigidity for many vertex algebraic braided monoidal categories. In a follow-up work, our results are used to prove rigidity of the category of weight modules for the simple affine VOA of [Formula: see text] at any admissible level, which embeds by Adamović’s inverse quantum Hamiltonian reduction into a rational Virasoro VOA tensored with a half-lattice VOA.

On generators and relations for higher level Zhu algebras and applications

Cohomological varieties associated to vertex operator algebras

In several examples it has been observed that a module category of a vertex operator algebra (VOA) is equivalent to a category of representations of some quantum group. The present article is concerned with developing such a duality in the case of the Virasoro VOA at generic central charge; arguably the most rudimentary of all VOAs, yet structurally complicated. We do not address the category of all modules of the generic Virasoro VOA, but we consider the infinitely many modules from the first row of the Kac table. Building on an explicit quantum group method of Coulomb gas integrals, we give a new proof of the fusion rules, we prove the analyticity of compositions of intertwining operators, and we show that the conformal blocks are fully determined by the quantum group method. Crucially, we prove the associativity of the intertwining operators among the first-row modules, and find that the associativity is governed by the 6j-symbols of the quantum group. Our results constitute a concrete duality between a VOA and a quantum group, and they will serve as the key tools to establish the equivalence of the first-row subcategory of modules of the generic Virasoro VOA and the category of (type-1) finite-dimensional representations of {mathcal {U}}_q (mathfrak {sl}_2).

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.

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.

The main result is that the category of ordinary modules of an affine vertex operator algebra of a simply laced Lie algebra at admissible level is rigid and thus a braided fusion category. If the level satisfies a certain coprime property then it is even a modular tensor category. In all cases open Hopf links coincide with the corresponding normalized S-matrix entries of torus one-point functions. This is interpreted as a Verlinde formula beyond rational vertex operator algebras. A preparatory Theorem is a convenient formula for the fusion rules of rational principal W-algebras of any type.