The Hilbert Series of Adjoint SQCD

TL;DR

This paper computes the Hilbert series for gauge invariant operators in various N=1 supersymmetric gauge theories with adjoint and fundamental matter, revealing the structure of their moduli spaces.

Contribution

It introduces a method to compute the Hilbert series for adjoint SQCD using plethystic exponential and Molien-Weyl formula, analyzing the chiral ring structure.

Findings

Hilbert series computed for multiple gauge groups

Chiral ring structure analyzed via plethystic logarithm

Classical moduli space identified as an affine Calabi-Yau cone

Abstract

We use the plethystic exponential and the Molien-Weyl formula to compute the Hilbert series (generating funtions), which count gauge invariant operators in N=1 supersymmetric SU(N_c), Sp(N_c), SO(N_c) and G_2 gauge theories with 1 adjoint chiral superfield, fundamental chiral superfields, and zero classical superpotential. The structure of the chiral ring through the generators and relations between them is examined using the plethystic logarithm and the character expansion technique. The palindromic numerator in the Hilbert series implies that the classical moduli space of adjoint SQCD is an affine Calabi-Yau cone over a weighted projective variety.