Geometry and representations of affine W-algebras – GRAW

Affine W-algebras are vertex algebras that extend the Virasoro algebra. They were introduced by Zamolodchikov in physics and developed further in mathematics by Kac, Wakimoto, Arakawa, etc. They are significant for the representation theory of affine Lie algebras or affine quantum groups. Recently new constructions of affine W-algebras in quantum field theories have garnered the attention of many mathematicians. The project GRAW aims to strengthen collaborations, in particular between mathematicians and physicists, and create new ones between reserachers with complementary skills around the representation theory of affine W-algebras. We’ll concentrate on the following objectives.
Objective 1 : Cohomological realization of affine W-algebras and Coulomb branches. We’ll work on realization of W-algebras and their representations in terms of various cohomological models : Coulomb branches, cohomology of affine Springer fibers, cohomology of instanton moduli spaces, AGT correspondence.
Objective 2 : Representation theory of affine W-algebras and symplectic singularities. We’ll work on representations of W-algebras and on their associated varieties, and the relations with symplectic singularities.
Objective 3 : Deformed affine W-algebras and representations of quantum loop groups. We’ll work on the deformed W-algebras introduced by Frenkel-Reshetikhin and their relations with affine quantum groups, the Nakajima quiver varieties and the deformed Coulomb branches.