SQCD: A Geometric Apercu

TL;DR

This paper introduces algebraic and geometric methods to analyze SQCD, enabling systematic counting of operators and detailed study of its vacuum moduli space, revealing its structure as affine Calabi-Yau cones.

Contribution

It develops new algebraic and geometric techniques for analyzing SQCD, including character expansions and computational algebraic geometry, to study operator counting and vacuum moduli space structure.

Findings

Character expansions for arbitrary colors and flavors

Vacuum moduli space as affine Calabi-Yau cones

Systematic analysis of irreducible components and syzygies

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.