An index to count chiral primaries in N=1 d=4 superconformal field theories

Abstract

I derive an index that counts chiral primary states in N=1 superconformal field theories in four dimensions. The index is calculated by putting the N=1 field theory on S^3 X R and allows to count chiral primaries of a given spin. The spin statistics theorem then ensures that there are no cancellations between bosons and fermions in the index. In order to calculate the index I construct N=1 supersymmetric Lagrangians on S^3 X R for theories which are believed to flow to a conformal fixed point in the IR. For ungauged theories I reduce the field theory to a supersymmetric quantum mechanics. For gauge theories I use chiral ring arguments to calculate the index. I calculate the index for SU(2) SYM with three flavors and its Seiberg dual. Those two indices agree provided a new chiral ring relation holds.