The Hilbert Series of Adjoint SQCD

TL;DR

The paper develops a comprehensive framework for counting gauge-invariant operators in ${ m N}=1$ adjoint SQCD for gauge groups ${ m SU}(N_c)$, ${ m Sp}(N_c)$, ${ m SO}(N_c)$, and ${ m G}_2$ by combining the Plethystic Exponential with the Molien–Weyl formula. It extracts the chiral-ring generators and relations from the plethystic logarithm, and provides detailed, explicit Hilbert series across many cases, including the SU$(N_c)$ cases with various $N_f$, as well as Sp, SO, and G$_2$ examples. A central finding is that the moduli space is an irreducible affine Calabi–Yau cone, with palindromic numerators signaling Calabi–Yau structure and complete-intersection properties emerging in several low-flavor scenarios. The work also develops combinatorial tools (adjoint baryon partitions) and asymptotic formulas for counting adjoint baryons, linking algebraic geometry with gauge-invariant operator counting and offering a rich geometric interpretation of adjoint SQCD vacua. Overall, the results illuminate the structure of the chiral rings, reveal large-scale combinatorics (e.g., adjoint baryons), and establish geometric properties of moduli spaces relevant to dualities and effective field theories in supersymmetric gauge theories.

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.

Figures (1)

• Figure 1: The graph of $\xi(a_0)$ against $a_0$.