Emmy-Noether-Seminar: Introduction to Supersymmetries in W-algebras

Introduction to Supersymmetries in W-algebras

Arim Song

Abstract: W-algebras are vertex algebras constructed via Hamiltonian reduction from Lie (super)algebras. When associated with Lie superalgebras, they are expected to exhibit rich structures. One notable conjecture is that certain families of these W-algebras possess $N=n$ supersymmetries.

These supersymmetries naturally pair fields within the W-algebras, providing a useful framework for understanding their structure.

In particular, for type A Lie superalgebras, the associated principal W-algebra was conjectured to have $N=2$ supersymmetry. In recent joint works, we have proven this conjecture and further explored the properties of the type A principal W-algebra.

In this talk, I will introduce the notion of supersymmetries inside vertex algebras. Then, I will present examples of vertex algebras that exhibit such supersymmetries. Finally, I will briefly introduce W-algebras and outline the $N=2$ supersymmetry of type A principal W-algebras. This talk is based on two joint works: one with Linshaw and Suh, and the other with Creutzig, Kovalchuck, Linshaw and Suh.