ON THE CHIRAL RINGS IN N=2 AND N=4 SUPERCONFORMAL ALGEBRAS

We study the chiral primary rings of N = 2 and N = 4 superconformal algebras (SCA's) constructed over triple systems. The chiral primary states of N = 2 SCA's realized over Hermitian Jordan triple systems are given. Their coset spaces G/H are Hermitian-symmetric and can be compact or noncompact. In the noncompact case under the requirement of unitarity of the representations of G, we find an infinite discrete set of chiral primary states associated with the holomorphic discrete series representations of G and their analytic continuation. A further requirement that the corresponding N = 2 module be unitary truncates this infinite set to a finite subset. There are no chiral primary states associated with the other unitary representations of noncompact groups. Remarkably, the only noncompact groups G that admit holomorphic discrete series unitary representations are such that their quotients G/H with their maximal compact subgroups H are Hermitian-symmetric. The chiral primary states of N = 2 SCA's constructed over the 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) x SU(2) x U(1). We then generalize the concept of chiral rings to these maximal N = 4 superconformal algebras. We find four different rings associated with each sector (left- or right-moving). Inclusion of both sectors gives 16 different rings. We also show that our analysis yields all the possible rings of N = 4 SCA's.