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Counting Gauge Invariant Operators in SQCD with Classical Gauge Groups

Amihay Hanany, Noppadol Mekareeya

TL;DR

This work develops a comprehensive framework for counting gauge-invariant operators in SQCD with classical gauge groups by combining the plethystic exponential, Molien–Weyl integration, and representation-theoretic character expansions. It provides full character expansions for SO and Sp theories across arbitrary numbers of colours and flavours, studies orientifold projections from SU to SO/Sp, and demonstrates that the classical moduli space is an irreducible affine Calabi–Yau cone over a weighted projective variety. The results yield explicit Hilbert series, generator/constraint structures via plethystic logarithms, and robust geometric interpretations, enriching the understanding of chiral rings in these theories. The methods and findings have potential applications in dualities, brane constructions, and algebraic geometry descriptions of SQCD moduli spaces. Overall, the paper delivers a detailed, calculable map between gauge invariants, global symmetries, and the geometry of vacua for SU, SO, and Sp SQCD.

Abstract

We use the plethystic programme and the Molien-Weyl fomula to compute generating functions, or Hilbert Series, which count gauge invariant operators in SQCD with the SO and Sp gauge groups. The character expansion technique indicates how the global symmetries are encoded in the generating functions. We obtain the full character expansion for each theory with arbitrary numbers of colours and flavours. We study the orientifold action on SQCD with the SU gauge group and examine how it gives rise to SQCD with the SO and Sp gauge groups. We establish that the classical moduli space of SQCD is not only irreducible, but is also an affine Calabi-Yau cone over a weighted projective variety.

Counting Gauge Invariant Operators in SQCD with Classical Gauge Groups

TL;DR

This work develops a comprehensive framework for counting gauge-invariant operators in SQCD with classical gauge groups by combining the plethystic exponential, Molien–Weyl integration, and representation-theoretic character expansions. It provides full character expansions for SO and Sp theories across arbitrary numbers of colours and flavours, studies orientifold projections from SU to SO/Sp, and demonstrates that the classical moduli space is an irreducible affine Calabi–Yau cone over a weighted projective variety. The results yield explicit Hilbert series, generator/constraint structures via plethystic logarithms, and robust geometric interpretations, enriching the understanding of chiral rings in these theories. The methods and findings have potential applications in dualities, brane constructions, and algebraic geometry descriptions of SQCD moduli spaces. Overall, the paper delivers a detailed, calculable map between gauge invariants, global symmetries, and the geometry of vacua for SU, SO, and Sp SQCD.

Abstract

We use the plethystic programme and the Molien-Weyl fomula to compute generating functions, or Hilbert Series, which count gauge invariant operators in SQCD with the SO and Sp gauge groups. The character expansion technique indicates how the global symmetries are encoded in the generating functions. We obtain the full character expansion for each theory with arbitrary numbers of colours and flavours. We study the orientifold action on SQCD with the SU gauge group and examine how it gives rise to SQCD with the SO and Sp gauge groups. We establish that the classical moduli space of SQCD is not only irreducible, but is also an affine Calabi-Yau cone over a weighted projective variety.

Paper Structure

This paper contains 38 sections, 68 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction and Summary
  2. Outline and Key Points:
  3. Notation for irreducible representations.
  4. $SO(N_c)$ SQCD with $N_f$ flavours
  5. The Case of $N_f < N_c$
  6. The Case of $N_f \geq N_c$
  7. The Case of $N_f=N_c$
  8. Counting Gauge Invariants: the Plethystic Exponential and Molien--Weyl formula
  9. The case of $N_c =3$.
  10. The case of $N_c=4$.
  11. The case of $N_c=5$.
  12. Character Expansions and Global Symmetries
  13. Terminology.
  14. The $N_f< N_c$ theories.
  15. Character expansion of an arbitrary $(N_f, SO(N_c))$ theory.
  16. ...and 23 more sections

Figures (2)

  • Figure 1: The quiver diagram of $SU(N_c)$ SQCD with $N_f$ flavours. The red node represents the $SU(N_c)$ gauge symmetry while the two blue nodes denote the global $U(N_f)$ symmetries. Each node gives rise to a $U(1)$ global symmetry, one of which is redundant.
  • Figure 2: Left diagram: $Sp(N_c)$ SQCD with $N_f$ flavours as a quiver theory. The red node represents the $Sp(N_c)$ gauge symmetry, while the blue node denotes the global $U(2N_f)$ symmetries. Right diagram:$SO(N_c)$ SQCD with $N_f$ flavours as a quiver theory. The red node represents the $SO(N_c)$ gauge symmetry, while the blue node denotes the global $U(N_f)$ symmetries. In each of these cases: On the contrary to the $SU(N_c)$ SQCD, although the blue node give rise to a $U(1)$ factor, the red node does not due to the orientifold projection.