QM Research Seminar: $W$-algebras and $W_{\infty}$-algebras

Speaker: Arim Song (​Friedrich-Alexander University Erlagen-Nürnberg).

Abstract: 
$W$-algebras are families of vertex algebras associated with Lie algebras and their nilpotent elements. They can be viewed as chiral analogues of the Feigin-Frenkel center of the Lie algebra and exhibit rich structures, including Feigin–Frenkel duality and coset realizations. 

The $W_\infty$-algebra is a two-parameter family of vertex algebras whose quotients recover type A principal $W$-algebras. It provides a powerful framework for classifying vertex algebras with prescribed strong generating type and has led to the discovery of isomorphisms among various families of vertex algebras, including results that partially establish the celebrated triality conjecture of Gaiotto and Rapčák. 

The construction of $W$-algebras extends naturally to Lie superalgebras, giving rise to $W$-superalgebras. However, many questions concerning dualities among these algebras remain open. In recent joint work with Creutzig, Kovalchuk, Linshaw, and Suh, we extended the $W_\infty$-algebra framework to type A Lie superalgebras and established dualities for type A principal $W$-superalgebras. 

In this talk, I will introduce $W$-algebras and $W_\infty$-algebras, and discuss their structural properties. If time permits, I will briefly discuss the recent developments in the super setting.