Tomoyuki Arakawa (RIMS): Symplectic singularities and vertex algebras

Symplectic singularities introduced by Beauville appear in various aspects of representation theory and are referred to as “21st-century Lie theory” by Okounkov. On theother hand, symplectic singularities also arise in the context of quantum field theory in physics, particularly in the Higgs and Coulomb branches of three-dimensional theories, as well as in the Higgs branches of four-dimensional theories.

Additionally, in vertex algebra theory, certain Poisson varieties called associated varieties are defined as geometric invariants, and they often turn out to be symplectic singularities. In such cases, vertex algebras can be regarded as chiral noncommutative deformations of symplectic singularities.

In particular, the 4D/2D duality proposed by Beem et al. in theoretical physics determines vertex algebras as invariants for superconformal four-dimensional theories. It is claimed that the Higgs branch of four-dimensional theories can be reconstructed as the associated variety of vertex algebras. Therefore, all vertex algebras arising from four-dimensional theories are supposed to be chiral noncommutative deformations of singular symplectic varieties.

In this lecture, we will discuss such vertex algebras and their representation theory.