On the Chiral Rings in N=2 and N=4 Superconformal Algebras

TL;DR

This paper explores the structure of chiral rings in N=2 and N=4 superconformal algebras, revealing new classifications and properties of chiral primary states through algebraic and geometric methods.

Contribution

It introduces a comprehensive analysis of chiral rings in N=2 and N=4 superconformal algebras, including classifications over hermitian and Freudenthal triple systems, and extends the concept of chiral rings to N=4.

Findings

Classification of chiral primary states over hermitian symmetric spaces

Identification of infinite and finite sets of chiral states under unitarity constraints

Derivation of all possible chiral rings in N=4 superconformal algebras

Abstract

We study the chiral rings in N=2 and N=4 superconformal algebras. The chiral primary states of N=2 superconformal algebras realized over hermitian triple systems are given. Their coset spaces G/H are hermitian symmetric which can be compact or non-compact. In the non-compact case, under the requirement of unitarity of the representations of G we find an infinite set of chiral primary states associated with the holomorphic discrete series representations of G. Further requirement of the unitarity of the corresponding N=2 module truncates this infinite set to a finite subset. The chiral primary states of the N=2 superconformal algebras realized over Freudenthal triple systems are also studied. These algebras have the special property that they admit an extension to N=4 superconformal algebras with the gauge group SU(2)XSU(2)XU(1). We generalize the concept of the chiral rings to N=4 superconformal algebras. We find four different rings associated with each sector (left or right moving). We also show that our analysis yields all the possible rings of N=4 superconformal algebras.