Associated varieties and Higgs branches (a survey)

TL;DR

This survey explores the concept of associated varieties in vertex algebras, their connection to Higgs branches in 4D N=2 SCFTs, and implications for modular invariance of Schur indices.

Contribution

It provides a comprehensive overview of the relationship between vertex algebra associated varieties and Higgs branches in superconformal field theories.

Findings

Associated varieties reflect key properties of vertex algebras.

They are linked to Higgs branches of 4D N=2 SCFTs.

Modular invariance of Schur indices can be deduced from vertex algebra theory.

Abstract

Associated varieties of vertex algebras are analogue of the associated varieties of primitive ideals of the universal enveloping algebras of semisimple Lie algebras. They not only capture some of the important properties of vertex algebras but also have interesting relationship with the Higgs branches of four-dimensional $N=2$ superconformal field theories (SCFTs). As a consequence, one can deduce the modular invariance of Schur indices of 4d $N=2$ SCFTs from the theory of vertex algebras.