SQCD: A Geometric Apercu

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.

49 pages, 1 figure

References in corpus (31)

• Electric-Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories
• Exact Results on the Space of Vacua of Four Dimensional SUSY Gauge Theories
• Lectures on supersymmetric gauge theories and electric-magnetic duality
• Counting chiral primaries in N=1 d=4 superconformal field theories
• The Hagedorn Transition, Deconfinement and N=4 SYM Theory
• Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories
• Counting Gauge Invariants: the Plethystic Program
• Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics
• Sasaki-Einstein Manifolds and Volume Minimisation
• Lectures on Supersymmetry Breaking
• Phases of N=1 supersymmetric gauge theories in four dimensions
• Diagonal multi-matrix correlators and BPS operators in N=4 SYM
• The Master Space of N=1 Gauge Theories
• Calculating the Superconformal Index and Seiberg Duality
• Counting Chiral Operators in Quiver Gauge Theories
• Counting BPS operators in N=4 SYM
• Chiral Rings and Phases of Supersymmetric Gauge Theories
• STRINGVACUA: A Mathematica Package for Studying Vacuum Configurations in String Phenomenology
• Counting BPS Baryonic Operators in CFTs with Sasaki-Einstein duals
• Exploring the Vacuum Geometry of N=1 Gauge Theories
• Reverse geometric engineering of singularities
• A New Method for Finding Vacua in String Phenomenology
• Baryonic Generating Functions
• Algorithmic Algebraic Geometry and Flux Vacua
• Molien Function for Duality
• Vacuum Geometry and the Search for New Physics
• Counting BPS Operators in the Chiral Ring of N=2 Supersymmetric Gauge Theories or N=2 Braine Surgery
• BPS Partition Functions for Quiver Gauge Theories: Counting Fermionic Operators
• Typicality, Black Hole Microstates and Superconformal Field Theories
• Plethystics and instantons on ALE spaces
• Poincare series of some pure and mixed trace algebras of two generic matrices

Cited by in corpus (85)

• Gauge Theories and Macdonald Polynomials
• Low-derivative operators of the Standard Model effective field theory via Hilbert series methods
• Hilbert Series for Constructing Lagrangians: expanding the phenomenologist's toolbox
• Elliptic hypergeometry of supersymmetric dualities
• The Hilbert Series of the One Instanton Moduli Space
• Hilbert series and operator bases with derivatives in effective field theories
• On the impact of dimension-eight SMEFT operators on Higgs measurements
• Exceptional Indices
• Hilbert Series for Flavor Invariants of the Standard Model
• The supersymmetric index in four dimensions
• The Hilbert Series of Adjoint SQCD
• Hilbert Series for Moduli Spaces of Two Instantons
• Highest Weight Generating Functions for Hilbert Series
• Enumerating the operators of an effective field theory
• Quiver Theories for Moduli Spaces of Classical Group Nilpotent Orbits
• Phases of M2-brane Theories
• Residues, modularity, and the Cardy limit of the 4d $\mathcal{N}=4$ superconformal index
• Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories
• Quivers as Calculators: Counting, Correlators and Riemann Surfaces
• 2, 12, 117, 1959, 45171, 1170086, ...: A Hilbert series for the QCD chiral Lagrangian
• Counting Gauge Invariant Operators in SQCD with Classical Gauge Groups
• Brane Webs and Magnetic Quivers for SQCD
• Complete Intersection Moduli Spaces in N=4 Gauge Theories in Three Dimensions
• Quiver Theories and Formulae for Nilpotent Orbits of Exceptional Algebras
• 2, 84, 30, 993, 560, 15456, 11962, 261485, ...: Higher dimension operators in the SM EFT
• Systematic construction of basis invariants in the 2HDM
• Flavor Invariants and Renormalization-group Equations in the Leptonic Sector with Massive Majorana Neutrinos
• Characters and Group Invariant Polynomials of (Super)fields: Road to "Lagrangian"
• SO and USp Kahler and Hyper-Kahler Quotients and Lumps
• The moduli spaces of $3d$ $\mathcal{N}\ge 2$ Chern-Simons gauge theories and their Hilbert series
• Higgsing M2-brane Theories
• On the 5d instanton index as a Hilbert series
• Discrete Gauging in Coulomb branches of Three Dimensional $\mathcal{N}=4$ Supersymmetric Gauge Theories
• Hilbert Series and Plethystics: Paving the path towards 2HDM- and MLRSM-EFT
• Factorised 3d $\mathcal{N}=4$ orthosymplectic quivers
• Magnetic quivers and line defects -- On a duality between 3d N=4 unitary and orthosymplectic quivers
• EFT Asymptotics: the Growth of Operator Degeneracy
• Hilbert Series for Theories with Aharony Duals
• Quiver Theories and Hilbert Series of Classical Slodowy Intersections
• Toric geometry and local Calabi-Yau varieties: An introduction to toric geometry (for physicists)
• Hilbert Series, Machine Learning, and Applications to Physics
• Numerical Analyses on Moduli Space of Vacua
• A Simple Introduction to Grobner Basis Methods in String Phenomenology
• The Hilbert series of U/SU SQCD and Toeplitz Determinants
• The Hilbert series of N=1 SO(N_c) and Sp(N_c) SQCD, Painlevé VI and Integrable Systems
• Exploring the orthosymplectic zoo
• 3d $\mathcal{N}=4$ mirror symmetry with 1-form symmetry
• Non-Connected Gauge Groups and the Plethystic Program
• The Coulomb and Higgs Branches of $\mathcal{N}=1$ Theories of Class ${\mathcal{S}}_k$
• New 3d $\mathcal{N}=2$ Dualities from Quadratic Monopoles
• Master Space and Hilbert Series for N=1 Field Theories
• Hilbert series and higher-order Lagrangians for the $O(N)$ model
• Hilbert series and mixed branches of $T[SU(N)]$ theory
• Minimally Unbalanced Quivers
• From Hagedorn to Lee-Yang: Partition functions of $\mathcal{N}=4$ SYM theory at finite $N$
• Free quantum fields in 4D and Calabi-Yau spaces
• Refined Counting of Necklaces in One-loop $\mathcal{N}=4$ SYM
• Veronese Geometry and the Electroweak Vacuum Moduli Space
• Constructing Operator Basis in Supersymmetry: A Hilbert Series Approach
• New results for the SQCD Hilbert series
• Moduli space of logarithmic states in critical massive gravities
• Moduli space of supersymmetric QCD in the Veneziano limit
• On the Geometry of Super Yang-Mills Theories: Phases and Irreducible Polynomials
• Counting Operators in $N=1$ Supersymmetric Gauge Theories
• A fully basis invariant Symmetry Map of the 2HDM
• Standard Model Plethystics
• From Spinning Primaries to Permutation Orbifolds
• Highest Weight Generating functions for hyperKahler T*(G/H) spaces
• Refined Partition Functions for Open Superstrings with 4, 8 and 16 Supercharges
• On Fields over Fields
• Probing the Space of Toric Quiver Theories
• Testing R-parity with Geometry
• Futaki Invariants and Reflexive Polygons
• Holomorphic primary fields in free CFT4 and Calabi-Yau orbifolds
• The Hilbert series of 3d N=2 Yang-Mills theories with vectorlike matter
• Chiral Rings, Futaki Invariants, Plethystics, and Groebner Bases
• Vacua, Symmetries, and Higgsing of Chern-Simons Matter Theories
• Universal fine grained asymptotics of free and weakly coupled Quantum Field Theory
• The Statistics of Vacuum Geometry
• Nonlinear (chiral) p-form electrodynamics
• On the Chiral Ring and Vacua of $\mathcal{N}=1$ Adjoint SQCD
• BV Formalism and Partition Functions
• Singlet-Assisted Supersymmetry Breaking For $Sp(2N)$ Theories
• Exploring the Potential Energy Landscape Over a Large Parameter-Space
• Holographic spectral functions for Sasaki-Einstein 5-manifolds