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I will explain how the characters of various rational and non-rational VOAs of type A are obtained from sl_r invariants of torus links. Specifically, we will consider principal W algebras, (1,p) singlet and (1,p) triplet VOAs.
In this talk I will present the construction of combinatorial bases of standard modules with rectangular highest weights for affine Lie algebras, which relies on the construction of quasi--particle bases of the Feigin--Stoyanovsky principal subspaces. This talk is based on a joint project with S. Kožić and M. Primc.
Given two left modules $W_1, W_2$ for a meromorphic open-string vertex algebra $V$, we will first use Huang's cohomology to describe the equivalence classes of modules $U$ fitting in the exact sequence $0 \to W_2 \to U \to W_1 \to 0$ while satisfying a technical convergence condition. Then, we will explain that the technical convergence condition automatically holds if $V$ contains a nice subalgebra $V_0$, such that $W_1$ and $W_2$ are semisimple $V_0$-modules, and the products of intertwining operators converge. Here $V_0$ is not required to be conformally embedded into $V$. Nor will we need $V$-modules to form a tensor category. The result leads to a new method for computing $\text{Ext}^1(W_1, W_2)$.
The goal of the talk is to construct embeddings of the N=2 superconformal vertex algebra, motivated by mirror symmetry, into the chiral de Rham complex, provided that we have solutions to the Killing spinor equations. Our approach to the chiral de Rham complex is based on the universal construction by Bressler and Heluani, which applies to any Courant algebroid over a smooth manifold. The Killing spinor equations that are considered come from the approach to special holonomy based on Courant algebroids in generalized geometry and are inspired by the physics of heterotic supergravity and string theory. The embeddings are given in two different set-ups. Firstly, for equivariant Courant algebroids over homogeneous manifolds, where the construction reduces to embeddings into the superaffinization of a quadratic Lie algebra, and the Killing spinor equations become purely algebraic conditions that can be checked on explicit examples. As an application, we present the first examples of (0,2) mirror symmetry on compact non-Kähler complex manifolds. These results are included in axiv:2012.01851, recently published in International Mathematical Research Notices. Secondly, for transitive Courant algebroids over complex manifolds, where these equations are equivalent to the Hull-Strominger system, with origins in the heterotic sigma-model studied by physicists. Several examples have been studied where the obtained results are applied. These results are included in arxiv:2305.06836. This talk is based on my PhD thesis, and is a joint work with Luis Álvarez-Cónsul and Mario Garcia-Fernandez.
Given a vertex operator algebra V, one can attach a graded Poisson algebra called the C2-algebra. The associated Poisson variety is an important invariant for V and is known as the associated variety of V. In this talk, we will introduce the cohomological variety of a vertex operator algebra, a notion cohomologically dual to that of the associated variety. First, we will motivate and define this variety, as well as give some of its structural properties. Then we will explain how to extract information on the Yoneda algebra defining this variety. Lastly, we will apply those results to the simple vertex operator algebras constructed from the Virasoro Lie algebra and finite dimensional simple Lie algebras. This is a joint work with Cuipo Jiang (Shanghai Jiao Tong University) and Zongzhu Lin (Kansas State University).
Unimodularity a classical notion shows up in various areas like linear algebra, lattices, Poisson algebras, etc. In this talk, we focus on unimodular Hopf algebras and unimodular tensor categories. We will introduce unimodular module categories and use them to construct Frobenius algebras and unimodular tensor categories. These ideas will be illustrated with examples drawn from Hopf algebras.
A commutative algebra A together with a Lie bracket satisfying the Leibniz rule is called a Poisson algebra, which is named in honor of Siméon Denis Poisson. Poisson structures appear in many contexts, including string theory, classical (quantum) mechanics, and differential geometry. In this talk, we will talk about Poisson structure on weighted polynomial rings and introduce Poisson valuations. Furthermore, we will see that the Poisson valuations play an important role in characterizing the Poisson automorphism groups of certain Poisson algebras.
At its core, this talk will be about a relation between characters of modules of logarithmic vertex algebras in the positive and negative Kazhdan-Lusztig (KL) zones. The main concrete result will be an easy-to-use step-by-step computational algorithm that produces a character of a log-VOA, say, in the positive KL zone from the expression in the negative KL zone, and vice versa. An equally (if not more!) valuable conceptual message of this talk will be an explanation that the same bijective relation plays an important role in very different areas of mathematics (and even physics) under other guises.
In this talk, we look at the classification problem for symmetric fusion categories in positive characteristic. We recall the second Adams operation on the Grothendieck ring and use its properties to obtain some classification results. In particular, we show that the Adams operation is not the identity for any non-trivial symmetric fusion category. We also give lower bounds for the rank of a (non-super-Tannakian) symmetric fusion category in terms of the characteristic of the field. As an application of these results, we classify all symmetric fusion categories of rank 3 and those of rank 4 with exactly two self-dual simple objects.
W-(super)algebras have generated great interest in recent years due to their numerous applications in mathematics and physics. The process of Hamiltonian reduction in stages suggests that W-(super)algebras often arise as extensions of tensor products basic building blocks. In type A, we expect that the building blocks are the Gaiotto-Rapcak Y-algebras which arise as 1-parameter quotients of the universal 2-parameter VOA of type W(2,3,…). For types B, C, and D, the quotients of the universal 2-parameter VOA of type W(2,4,…) provide some, but not all, of the necessary building blocks. In this talk we discuss a new universal 2-parameter VOA of type W(1^3,2,3^3,4,…), whose 1-parameter quotients are expected to account for the missing building blocks for W-(super)algebras of types B, C, and D. There are 8 infinite families of such quotients, which are analogues of the Gaiotto-Rapcak Y-algebras. We will explain the process behind the construction of this universal two-parameter VOA and discuss several applications. This is a join work with Thomas Creutzig and Andrew Linshaw.