Operators and vacua of N=1 field theories

Abstract

We review the idea of Hilbert Series as a tool to study the moduli space and the BPS operators of four dimensional N=1 supersymmetric field theories. We concentrate on the particular case of N=1 superconformal field theories living on N D3 branes at toric Calabi-Yau singularities. The main claim is: it is possible to write down explicit partition functions counting all the local BPS operators for generic N number of branes, and obtain important informations about the BPS operators, the moduli space and the dual geometry.