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A while ago and with David Ridout a Verlinde formula for the affine VOAs of sl(2) at admissible level was conjectured. This is a VOA whose representation category is neither finite nor semisimple. Our idea was to replace the Verlinde S-matrix by an S-kernel and atypical modules by their resolutions of typical modules to get a natural Verlinde formula conjecture. I will explain why this formalism is true.
Supersymmetric(SUSY) W-algebras are vertex algebras constructed through SUSY Hamiltonian reduction based on Lie superalgebras with osp(1|2) embeddings. By construction, they have a supersymmetric structure that naturally couples the generators of the algebras. Madsen and Ragoucy conjectured that SUSY W-algebras are isomorphic to W-algebras up to a tensor product with free field algebras. In recent joint work with Genra and Suh, we proved this conjecture for principal SUSY W-algebras.
In this talk, I will introduce SUSY W-algebras and the concept of SUSY vertex algebras, which provides a mathematical framework for the study. Then, I will discuss key properties of SUSY W-algebras in comparison to W-algebras, a part of which contributes to the proof of the conjecture. This talk is based on my recent paper and the one with Genra and Suh.
In order to investigate the representation theory of a vertex algebra $V$, a fruitful strategy is to look at the properties of its $C_2$-algebra $R(V)$. This Poisson algebra reflects interesting properties of the vertex algebra and is often easier to handle than the vertex algebra itself. In this talk, we are interested in studying vertex algebras in a closed monoidal category, and in providing a description of the dual versions of $V$ and $R(V)$ when those exist. We introduce vertex algebras graded by an abelian group and explain how to "dualize" the definition to obtain a graded vertex coalgebra. This leads to the notion of the $C_2$-coalgebra of a vertex coalgebra. We will describe its properties and show that the duality vertex algebra / vertex coalgebra passes down to a duality $C_2$-algebra / $C_2$-coalgebra. We will also explain how this dualities carry on to the respective modules / comodules.
In this talk we introduce a canonical decreasing filtration on intertwiners of a vertex algebra. We study the associated graded spaces. Then, we define Poisson vertex intertwiners and Poisson intertwiners. We obtain relations between the associated varieties of modules of a vertex algebra.
In this talk, I will present a sewing-factorization theorem for conformal blocks in arbitrary genus associated to a (possibly nonrational) $C_2$-cofinite VOA $V$. This result gives a higher genus analog of Huang-Lepowsky-Zhang's tensor product theory. Moreover, I will explain the relation between our result and pseudotraces, and confirm some of the conjectures by Gainuditnov-Runkel. The relationship between our result and coends will also be discussed. The talk is based on an ongoing project (arXiv: 2305.10180, 2411.07707, 2503.23995) joint with Bin Gui.