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

TL;DR

The paper develops a symmetry-based framework to count chiral primaries in four-dimensional N=1 superconformal field theories by placing the theory on S^3×R and constructing a twisted index that probes semi-short multiplets. It builds N=1 SUSY Lagrangians on this curved background, reduces ungauged theories to quantum mechanics, and employs chiral-ring arguments for gauge theories, applying the method to SU(2) SYM with three flavors and its Seiberg dual. A key insight is that counting chiral primaries is tied to cohomological constructions in a twisted, curved-space setting, and the results agree across dual descriptions provided a new nonperturbative chiral-ring relation is invoked. The approach offers a concrete route to compare BPS spectra and chiral-ring structures across dualities within curved-space QFT and has potential implications for AdS/CFT via BPS sectors.

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.