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

TL;DR

This paper develops a method to count chiral primary states in four-dimensional N=1 superconformal field theories by placing them on S^3 x R and introduces an index for semi-short multiplets, with applications to specific gauge theories.

Contribution

It presents a novel procedure for counting chiral primaries in 4D N=1 SCFTs using a new index and supersymmetric Lagrangians on S^3 x R, including applications to SU(2) SYM and Seiberg duality.

Findings

Counted chiral primaries in SU(2) SYM with three flavors.

Confirmed agreement of chiral ring relations in dual theories.

Established a new method for analyzing superconformal states.

Abstract

I derive a procedure to count chiral primary states in N=1 superconformal field theories in four dimensions. The chiral primaries are counted by putting the N=1 field theory on S^3 X R. I also define an index that counts semi-short multiplets of the superconformal theory. 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, whereas for gauge theories I use chiral ring arguments. I count chiral primaries for SU(2) SYM with three flavors and its Seiberg dual. Those two results agree provided a new chiral ring relation holds.