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Most non principal W-superalgebras allow for modules that neither have finite dimensional conformal weight spaces nor is the conformal weight bounded below. Already the categories of relaxed-highest weight modules of affine vertex algebras at admissible non-integral levels are an example. These properties make it a big challenge to completely understand these categories. On the other hand W-superalgebras enjoy various dualities and the aim of this talk is to convince you that dualities can be rather helpful.
Tensor categories of vertex operator algebras play an important role in the study of vertex operator algebras and conformal field theories. A central problem of tensor category theory of Huang-Lepowsky-Zhang is the existence of the vertex tensor category structure. We develop a few general methods to establish the existence of tensor structure on module categories for vertex operator algebras, especially for non-rational and non-C_2 cofinite vertex operator algebras. As applications, we obtain the tensor structure of affine Lie algebras at various levels, affine Lie superalgebra gl(1|1), the Virasoro algebra at all central charges as well as the singlet algebras. We also study important properties, including constructions of projective covers, fusion rules and the rigidity of these tensor categories. This talk is based on joint work with T. Creutzig, Y.-Z. Huang, F. Orosz Hunziker, C. Jiang, R. McRae and D. Ridout.
We prove a new Fermionic quasiparticle sum expression for the character of the Ising model vertex algebra, related to the Jackson-Slater q-series identity of Rogers-Ramanujan type. We find, as consequences, an explicit monomial basis for the Ising model, and a description of its singular support. We find that the ideal sheaf of the latter, defining it as a subscheme of the arc space of its associated scheme, is finitely generated as a differential ideal. We prove three new q-series identities of the Rogers-Ramanujan-Slater type associated with the three irreducible modules of the Virasoro Lie algebra of central charge 1/2. This is joint work with G. E. Andrews and J. van Ekeren and is based on arxiv.org:2005.10769
(joint work with Lam, Moeller and Shimakura) If V is a holomorphic vertex algebra of central charge 24 then its weight one space V_1 is known to be a reductive Lie algebra which is either trivial, abelian of dimension 24 (in which case V is the Leech lattice vertex algebra) or else one of 69 semisimple Lie algebras first determined by Schellekens in 1993. Until now the only known proof of Schelekens result was a heavily computational one involving case analysis and difficult integer programming problems. Recently Moeller and Scheithauer have established a bound on the dimension of the weight one space of a holomorphic orbifold vertex algebra, using the Deligne bound on the growth of coefficients of weight 2 cusp forms. In this talk I will describe how the dimension bound together with Kac's very strange formula implies that all holomorphic vertex algebras of central charge 24 and nontrivial weight one space are orbifolds of the Leech lattice algebra. Since the automorphism group of the latter algebra is known one can, with a little more work, recover Schellekens result in this way.
Screening operators are useful tools to characterize free field realizations of vertex algebras, and give new perspectives in their structures. We explain screening operators of the beta-gamma system, affine vertex (super)algebras and W-(super)algebras. We also explain the applications to the coset constructions, representations and trialities of W-algebras.
I will discuss the level two Zhu algebra for the Heisenberg vertex operator algebra and techniques used in determining its structure. I will also discuss more general results helpful in determining generators and relations for higher level Zhu algebras, and in particular, will provide an example to clarify the necessity of an extra condition required in the definition of higher level Zhu algebras. (Joint with Katrina Barron.)
The Kazama-Suzuki coset vertex operator superalgebra associated with a simple Lie algebra g and its Cartan subalgebra h is a ``super-analog'' of the parafermion vertex operator algebra associated with g. At positive integer levels, the coset superalgebra turns out to be C_2-cofinite and rational by the general theory of orbifolds (Miyamoto) and Heisenberg cosets (Creutzig-Kanade-Linshaw-Ridout), respectively. On the other hand, at Kac-Wakimoto admissible levels, the coset superalgebra is not C_2-cofinite nor rational. In this talk we discuss a relationship between the category of weight modules for the admissible affine vertex algebra associated with g and that for the corresponding Kazama-Suzuki coset vertex superalgebra. In our discussion the inverse Kazama-Suzuki coset construction, which is originally due to Feigin-Semikhatov-Tipunin in the g=sl_2 case, plays an important role. As an application, for g=sl_2 at level -1/2, we determine all the fusion rules between simple weight modules of the Kazama-Suzuki coset vertex superalgebra and verify the conjectural Verlinde formula in this case (corresponding to Creutzig-Ridout's result in the affine side). The last part is based on the joint work with Shinji Koshida.
I will introduce an associative algebra A^{∞}(V) constructed using infinite matrices with entries in a grading-restricted vertex algebra V. The Zhu algebra and its generalizations by Dong-Li-Mason are very special subalgebras of A^{∞}(V). I will also introduce the new subalgebras A^{N}(V) of A^{∞}(V), which can be viewed as obtained from finite matrices with entries in V. I will then discuss the relations between lower-bounded generalized V-modules and suitable modules for these associative algebras. This talk is based on the paper arXiv:2009.00262.
We prove several families of q-series identities that are motivated by the correspondence between 4d N = 2 superconformal field theories (SCFTs) and vertex operator superalgebras. We also discuss identities coming from certain non-commutative q-series and quivers
To any vertex algebra one can attach in a canonical way a certain Poisson variety, called the associated variety. Nilpotent Slodowy slices appear as associated varieties of admissible (simple) W-algebras. They also appear as Higgs branches of the Argyres-Douglas theories in 4d N=2 SCFT’s. These two facts are linked by the so-called Higgs branch conjecture. In this talk I will explain how to exploit the geometry of nilpotent Slodowy slices to study some properties of W-algebras whose motivation stems from physics. In particular I will be interested in collapsing levels for W-algebras. This is a joint work (still in preparation) with Tomoyuki Arakawa and Jethro van Ekeren.
For a finite dimensional simply-laced simple Lie algebra $g$ and an integer $p\geq 2$, we can attach the logarithmic $W$-algebra $W(p)_Q$. When $g=sl_2$, $W(p)_Q$ is called the triplet $W$-algebra, and studied by many people as one of the most famous examples of $C_2$-cofinite but irrational vertex operator algebra. However, apart from the triplet $W$-algebra, not much is known about the log $W$-algebras $W(p)_Q$. In this talk, after we construct $W(p)_Q$ and their modules $W(p,\lambda)_Q$ geometrically along the preprint of Feigin-Tipunin, first we show the simplicity, $W_k(g)$-module structure, and character formula of $W(p,\lambda)_Q$ when $\sqrt{p}\bar\lambda$ is in the closure of the fundamental alcove. In particular, for $p\geq h-1$, $W(p)_Q$ is simple and decomposed into simple $W_k(g)$-modules. Second we give a purely $W$-algebraic algorithm to calculate nilpotent elements in the Zhu's $C_2$-algebra of $W(p)_Q$ much easier than straightforward way. Using this algorithm to the cases $g=sl_3$ and $p=2,3$, we show that $W(p)_Q$ is $C_2$-cofinite in these cases.
In this talk I will discuss certain properties of sheaves of covacua and conformal blocks attached to modules over vertex operator algebras. After briefly recalling how these objects are constructed from a geometric point of view, I will focus on the conditions required to construct Cohomological Field Theories from these sheaves. If time permits I will also discuss open problems which naturally arise. This is based on joint works with A. Gibney and N. Tarasca.
Recently, dualities among W-superalgebras and their affine cosets conjectured by Gaiotto-Rapcak have been established in many cases by Creutzig-Linshaw and Creutzig-Linshaw-Kanade by using universal objects of such algebras. Independently, Creutzig-Genra and I proved the duality in the case of subregular W-algebras and principal W-superalgebras of type A by using free field realizations of those algebras. This point of view upgrades the duality to a "reconstruction theorem" of one of the algebra from the other one. The simplest example is the Kazama-Suzuki coset construction of N=2 superconformal algebra from the affine sl2 vertex algebra and its inverse by Feigin-Semikhatov-Tipunin. In this talk, I will explain this reconstruction theorem and then its application to the representation theory of principal W-superalgebra side in the rational cases. This talk is based on on-going project with Thomas Creutzig, Naoki Genra and Ryo Sato
We will review recent results appearing in the last five years including the representation theory of affine vertex algebras beyond the category O, semi-simplicity of representations at collapsing levels and some applications to logarithmic vertex algebras