Counting chiral primaries in N=1N=1, d=4d=4 superconformal field theories

I derive a procedure to count chiral primary states in N=1N=1 superconformal field theories in four dimensions. The chiral primaries are counted by putting the N=1N=1 field theory on S3×RS3×R. I also define an index that counts semi-short multiplets of the superconformal theory. I construct N=1N=1 supersymmetric Lagrangians on S3×RS3×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, whereas for gauge theories I use chiral ring arguments. I count chiral primaries for SU(2)SU(2) SYM with three flavors and its Seiberg dual. Those two results agree provided a new chiral ring relation holds.