SQCD: A Geometric Apercu

TL;DR

This work reframes SQCD in a geometric language, treating the vacuum moduli space as an affine algebraic variety and using the Hilbert series, the Plethystic Programme, and Molien–Weyl integrals to systematically count gauge-invariant operators. It demonstrates that the classical moduli spaces for ${\cal N}=1$ SQCD are affine Calabi–Yau cones over weighted projective varieties, with precise descriptions of their irreducible structure via primary decomposition and their generation by mesons plus baryons in the $N_f\ge N_c$ regime. The paper provides explicit analytic Hilbert series and plethystic logarithms across a wide range of $(N_f,N_c)$, including the special cases ${N_f<N_c}$, ${N_f=N_c}$, and ${N_f>N_c}$, and shows, in particular, that the numerators of these Hilbert series are palindromic, signaling Calabi–Yau geometry. It also develops refined character expansions that encode the global $SU(N_f)_L\times SU(N_f)_R$ symmetry, offering compact, representation-theoretic expressions for the counting of GIOs and illuminating Seiberg-duality-related structures through a geometric lens.

Abstract

We take new algebraic and geometric perspectives on the old subject of SQCD. We count chiral gauge invariant operators using generating functions, or Hilbert series, derived from the plethystic programme and the Molien-Weyl formula. Using the character expansion technique, we also see how the global symmetries are encoded in the generating functions. Equipped with these methods and techniques of algorithmic algebraic geometry, we obtain the character expansions for theories with arbitrary numbers of colours and flavours. Moreover, computational algebraic geometry allows us to systematically study the classical vacuum moduli space of SQCD and investigate such structures as its irreducible components, degree and syzygies. We find the vacuum manifolds of SQCD to be affine Calabi-Yau cones over weighted projective varieties.