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In this talk, I will explore the algebraic properties of an $\mathbb{N}$-graded vertex algebra associated with Gorenstein rings. Through this framework, I will demonstrate how these vertex algebras offer a natural bridge to the shifted theory of vertex algebras of the CFT type.
Thanks to the advances over the last decade, we now have a reasonable understanding of when a vertex operator algebra (VOA) admits a vertex tensor category that is braided monoidal. However, rigidity of such categories is often quite difficult to establish and is proven by ad hoc methods.
I will present recent joint work (https://arxiv.org/abs/2409.14618) with Thomas Creutzig, Robert McRae and Kenichi Shimizu where we develop techniques for proving rigidity of vertex tensor categories. Namely, given an extension of VOAs V ⊂ W, we establish results that allow us to prove rigidity to Rep(V) given rigidity of Rep(W) and vice versa.
We ascertain properties of the algebraic structures in towers of codes, lattices, and vertex operator algebras (VOAs) by studying the associated subobjects fixed by lifts of code automorphisms. In the case of sublattices fixed by subgroups of code automorphisms, we identify replicable functions that occur as quotients of the associated theta functions by suitable eta products. We show that these lattice theta quotients can produce replicable functions not associated to any individual automorphisms. Moreover, we show that the structure of the fixed subcode can induce certain replicable lattice theta quotients and we provide a general code theoretic characterization of order doubling for lifts of code automorphisms to the lattice-VOA. Finally, we prove results on the decompositions of characters of fixed subVOAs. This talk is based on joint work with Jennifer Berg, Eva Goedhart, Hussain M. Kadhem, Allechar Serrano López, and Stephanie Treneer.
W-algebras are vertex algebras obtained form quantum Hamiltonian reductions of an affine vertex algebra. These reductions are naturally upgraded to functors from the category of modules over the affine vertex algebras to the category of the modules over the corresponding W-algebras. However, they are difficult to control in general. Recently, two approaches have been developed to improve our understanding of the functors. One consists in spitting the functor into small pieces that are easier to deal with (partial reductions), the other aims to reverse the quantum Hamiltonian reduction procedure (inverse Hamiltonian reductions). In this talk, I will discuss about recent advances in these complementary approaches. The talk is based on recent papers with T. Creutzig, A. Linshaw and S. Nakatsuka and with Z. Fehily, E. Fursman and S. Nakatsuka.
Spaces of coinvariants have classically been constructed by assigning representations of affine Lie algebras, and more generally, vertex operator algebras, to pointed algebraic curves. Removing curves out of the picture, I will construct spaces of coinvariants at abelian varieties with respect to the action of an infinite-dimensional Lie algebra. I will show how these spaces globalize to twisted D-modules on moduli of abelian varieties, extending the classical picture from moduli of curves. This is based on the preprint arXiv:2301.13227.
Affine Lie algebras are universal central extensions of algebras of matrices over Laurent polynomials. In the case of sl_2, the ring of Laurent polynomials can be replaced with any unital Jordan algebra. This gives a very large family of Lie algebras. Motivated by this connection between Lie and Jordan theory, we describe a category of Lie algebra weight modules, whose homological properties are related to the long-standing open problem of computing graded dimensions of free Jordan algebras. No prior knowledge of Jordan algebras will be assumed. This talk is based on joint work with Olivier Mathieu.