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Principal characters of standard (i.e., highest weight, integrable) modules for affine Lie algebras have been a rich source of q-series and partition identities. The algebra A_1^{(1)} (or, sl_2^) was "understood" in this sense a few decades ago. On q-series side, this leads to identities of Gordon-Andrews and Andrews-Bressoud. In this talk, I'll present q-series identities related to the next "simplest" affine Lie algebra, namely, A_2^{(2)}. Here, we get six families of q-series identities confirming a conjecture of McLaughlin and Sills. The main machinery used is that of Bailey pairs and Bailey lattices. This is a joint work with Matthew C. Russell. (N.B.: These q-series include Vir(3,p) minimal model characters.)
We prove that the universal affine vertex algebra associated with a simple Lie algebra g is simple if and only if the associated
variety of its unique simple quotient is equal to g*. We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra. This is a joint work with T. Arakawa and A. Moreau.
Algebras in tensor categories appear in several interesting research areas, like e.g. VOA extensions or spin topological field theories, but they are usually tricky to find. In this talk, we will explain how to generalize a result by Ostrik and Natale on algebra objects in categories related to lattice VOAs to the case of so-called group-theoretical fusion categories. The algebra objects we find for these also have very good properties that we will describe in detail. We will assume little knowledge of categories. Joint work with the WINART2 team Y. Morales, M. Mueller, J. Plavnik, A. Tabiri and C. Walton
Physicists are interested in holographic families of VOAs. These are families of VOAs that on the one hand have dim V_n `small' for `small' n, and on the other hand have some kind of large central charge limit. I will discuss the motivation behind these requirements and the connection to extremal VOAs. I will then discuss some attempts at constructing such families, namely permutation orbifold VOAs and lattice orbifold VOAs. This talk is based on joint work with Thomas Gemuenden.
The 4D/2D duality discovered by Beem et al in physics gives a remarkable connection between 4D N=2 SCFTs and VOAs. It gives not only many new interesting examples of VOAs but also new perspectives to known VOAs, such as Frenkel-Styrkas’s modified regular representation of the Virasoro algebra and Adamovic’s realization of N=4 small superconformal algebra. In this talk I will discuss the 4D/2D duality from the VOA perspective, starting from these examples.
Given a strongly rational unitary VOA $V$, a Hermitian form on the space of its intertwining operators was introduced recently to understand the unitarity of the representation modular tensor category $Rep(V)$. It was actually shown that, along with some natural assumptions, if this Hermitian form (which is necessarily non-degenerate) is positive, namely, if it is an inner product, then $Rep(V)$ is unitary. The crucial step of this story is to prove the positivity of the Hermitian form. In this talk, I give a geometric interpretation of this positivity problem using the idea (self)conjugate Riemann surfaces and (self)conjugate conformal blocks.
I will discuss work in progress related to proving semisimplicity of the module category for a suitable positive-energy, self-contragredient, C_2-cofinite vertex operator algebra V. The goal is to show that the category of V-modules is semisimple if the Zhu algebra of V is a semisimple algebra. The idea for proving this is to show that the braided tensor category of V-modules is rigid with a non-degenerate braiding, using tensor-categorical methods combined with the modular invariance methods used by Huang to prove the Verlinde conjecture for rational vertex operator algebras.
Vertex operator algebras exhibit a feature much like Lie algebras in that they admit too many modules for the category of all their modules to exhibit nice structure. However, good choices of module category can lead to categories with very rich structure. For example the categories of admissible modules over rational vertex operator algebras are modular tensor categories, as proved by Huang. I will present some recent work on making the study of vertex operator algebra module categories more tractable by replacing them by Hopf algebras, an arguably simpler algebraic structure. The guiding example will be the free boson.
Skew representations (corresponding to skew Young diagrams) of Yangian and quantum affine algebra of type A were introduced by Cherednik and extensively studied by Nazarov and Tarasov. In this talk, we will discuss some known results about skew representations of super Yangian of type A such as Jacobi-Trudi identities, Drinfeld functor, irreducibility conditions of tensor products, and extended T-systems. We also discuss some open problems related to tame modules of super Yangian. Some essential differences comparing to the even case will be discussed as well.
Motivated by the generalized AGT conjecture in this talk I will construct surjective homomorphisms from the affine super Yangians to the universal enveloping algebras of rectangular W-superalgebras. This result is a super affine analogue of a result of Ragoucy and Sorba, which gave surjective homomorphisms from finite Yangians of type A to rectangular finite W-algebras of type A.
I shall summarise recent results (and ongoing work) regarding the classification of strongly rational, holomorphic VOAs (or CFTs) of central charge 24 (together with Jethro van Ekeren, Gerald Höhn, Ching Hung Lam, Nils Scheithauer and Hiroki Shimakura). First, we show that there is an abstract bijection (without classifying either side) between these VOAs and the generalised deep holes of the Leech lattice VOA. The proof uses a dimension formula obtained by pairing the VOA character with a vector-valued Eisenstein series and an averaged version of Kac's Lie theoretic "very strange formula". This is a quantum analogue of the beautiful result by Conway, Parker and Sloane (and Borcherds) that the deep holes of the Leech lattice are in natural bijection with the Niemeier lattices. Then, we explain how this can be used to classify the (exactly 70) strongly rational, holomorphic VOAs of central charge 24 with non-zero weight-one space. (The case of zero weight-one space, which includes the Moonshine module, is more difficult and still open.)
We generalize the theory of linked partition ideals due to Andrews using finite automata in formal language theory and apply it to prove three Rogers-Ramanujan type identities of modulo 14 that were posed by Nandi through vertex operator theoretic construction of the level 4 standard modules of the affine Lie algebra A^{(2)}_{2}. This is a joint work with Motoki Takigiku.
One of the simplest examples of $\mathcal{W}$-algebras is the Bershadsky-Polyakov vertex algebra W^k(g f_{min}), associated to g = sl(3) and the minimal nilpotent element f_{min}. We study the simple Bershadsky-Polyakov algebra \mathcal W_k at positive integer levels and obtain a classification of their irreducible modules. In the case k=1, we show that this vertex algebra has a Kazama-Suzuki-type dual isomorphic to the simple affine vertex superalgebra L_{k'} (osp(1 \vert 2))for k'=-5/4. This is joint work with D. Adamovic.
Meromorphic open-string vertex algebras (abbre. MOSVAs) is a noncommutative generalization of the usual vertex algebra defined by Yi-Zhi Huang in 2012. Vertex operators still satisfy the associativity but do not necessarily satisfy commutativity. In this talk I will illustrate nontrivial examples of MOSVAs and modules we know so far, including the universal bosonic construction, the universal fermionic construction, and the example from the geometry over constant curvature manifolds.
In this talk, I will discuss an impact of Leibniz algebras on the algebraic structure of $\mathbb{N}$-graded vertex algebras. Along the way, I will provide easy ways to characterize several types of N-graded vertex algebras.
Minimal models are simple vertex operator algebras (VOAs) for which the structure of the associated universal VOA is somehow maximally degenerate. Some minimal models are rational and C_2-cofinite, eg those for Virasoro or N=1, and some are not. I will look at some examples which are not, specifically the admissible-level affine minimal models associated with sl_3. The novelty here is the fact that the rank of the associated algebra is not 1.
I will discuss joint work with Terry Gannon in which we provide a ribbon tensor equivalence between the representation category of small quantum SL(2), at parameter q=exp(pi i/p), and the representation category of the triplet vertex operator algebra at integral parameter p>1. We provide similar quantum group equivalences for representation categories associated to the Virasoro, and singlet vertex operator algebras at central charge c=1-6(p-1)^2/p.
While regular W-algebras have enjoyed many years of study and attention, recent developments in physics have the less popular subregular W-algebras playing an important role. Moreover, these subregular W-algebras appear at levels where the corresponding conformal field theory is likely non-rational. This necessitates a deeper understanding of the representation theory of such vertex operator algebras at non-rational levels. In type A_n, only the n=1 (sl_2) and n=2 (Bershadsky-Polyakov algebra) cases are particularly well-understood. In both cases an 'inverse reduction-by-stages' approach, first described for sl_2 in vertex operator algebra language by D. Adamovic, relates much of the representation theory to that of the corresponding regular W-algebra. In this talk, I will describe how to generalise this approach to all type A_n subregular W-algebras using screening operators developed by N. Genra.