The Hilbert Series of Adjoint SQCD

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.