Seminars & Colloquia Calendar

Abstract Arc spaces were originally introduced in algebraic geometry to study singularities. More recently they show in connections to vertex algebras. There is a closed embedding from the singular support of a vertex algebra V into the arc space of associated scheme of V. We call a vertex algebra "classically free" if this embedding is an isomorphism. In this introductory survey talk, we will first introduce arc spaces and some of its backgrounds. Then we will provide several examples of classically free vertex algebras including Feigin-Stoyanovsky principal subspaces, and explain their applications in differential algebras, \(q\)-series identities, etc. In particular, we will show the classically freeness of principal subspaces of type A at level 1 by using a method of filtrations and identities from quantum dilogarithm or quiver representations. As a result, we obtain new presentations and graded dimensions of the principal subspaces of type A at level 1, which can be thought of as the continuation of previous works by Calinescu, Lepowsky and Milas. The classically freeness of some principal subspaces which possess free fields realisation will also be discussed. Most of the talk is based on the joint work with A. Milas.