{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.supersymmetry-duality.n1-duality.seiberg-map-operator-dictionary",
  "artifact_class": "bidirectional SQCD dictionary and deformation square",
  "title": "Seiberg duality uses a normalized core dictionary with a closing flow square",
  "evidence_date": "2026-08-24",
  "reader_question": "What data define the electric and magnetic SQCD pair, and how does a one-flavor mass test the proposed infrared equivalence?",
  "dominant_point": "The electric and magnetic theory cards are linked by a normalized protected dictionary and holomorphic scale relation; a one-flavor electric mass closes the daughter square only because the magnetic theory Higgses and reproduces the lower-rank scale relation, with a broken-group instanton at the exceptional endpoint.",
  "scope": {
    "theory": "four-dimensional massless N=1 SQCD with simply connected SU(Nc), Nc at least 3, Nc+2 <= Nf < 3Nc, and non-Abelian magnetic rank Nf-Nc at least 2; Nc=2 has enhanced SU(2Nf) flavor symmetry and requires a separate card",
    "normalization": "B(Q)=1; the electric composite-normalized meson M_e=Q Q-tilde is related to the engineering-dimension-one magnetic singlet M_m by M_e=mu M_m; W_mag=M_m q q-tilde",
    "scale_relation": "Lambda^(3Nc-Nf) Lambda_tilde^(3(Nf-Nc)-Nf)=(-1)^(Nf-Nc) mu^Nf in the declared common holomorphic scheme",
    "claim_ceiling": "exact protected matches and closing deformation squares strongly test but do not prove the full infrared equivalence or fixed-point existence"
  },
  "theory_cards": {
    "electric": {
      "gauge_group": "SU(Nc)",
      "global_form": "simply connected",
      "matter": [
        "Q: (fundamental color, fundamental SU(Nf)_L)",
        "Q-tilde: (antifundamental color, antifundamental SU(Nf)_R)"
      ],
      "superpotential": "zero",
      "one_loop_holomorphic_coefficient": "b_e=3Nc-Nf"
    },
    "magnetic": {
      "gauge_group": "SU(Nf-Nc)",
      "global_form": "simply connected",
      "matter": [
        "q: antifundamental SU(Nf)_L",
        "q-tilde: fundamental SU(Nf)_R",
        "M_m: (fundamental, antifundamental) flavor bifundamental"
      ],
      "superpotential": "W=M_m q q-tilde",
      "one_loop_holomorphic_coefficient": "b_m=2Nf-3Nc"
    }
  },
  "field_charge_table": [
    {
      "field": "Q",
      "gauge": "fundamental SU(Nc)",
      "flavor": "(Nf,1)",
      "B": "1",
      "R_scalar": "1-Nc/Nf"
    },
    {
      "field": "Q-tilde",
      "gauge": "antifundamental SU(Nc)",
      "flavor": "(1,Nf-bar)",
      "B": "-1",
      "R_scalar": "1-Nc/Nf"
    },
    {
      "field": "q",
      "gauge": "fundamental SU(Nf-Nc)",
      "flavor": "(Nf-bar,1)",
      "B": "Nc/(Nf-Nc)",
      "R_scalar": "Nc/Nf"
    },
    {
      "field": "q-tilde",
      "gauge": "antifundamental SU(Nf-Nc)",
      "flavor": "(1,Nf)",
      "B": "-Nc/(Nf-Nc)",
      "R_scalar": "Nc/Nf"
    },
    {
      "field": "M_m",
      "gauge": "singlet",
      "flavor": "(Nf,Nf-bar)",
      "B": "0",
      "R_scalar": "2(1-Nc/Nf)"
    }
  ],
  "operator_dictionary": [
    {
      "electric": "Q^i Q-tilde_j=M_e^i_j",
      "magnetic": "mu M_m^i_j",
      "status": "protected map with declared normalization"
    },
    {
      "electric": "B=epsilon Q^Nc",
      "magnetic": "C epsilon q^(Nf-Nc) with complementary flavor epsilon",
      "coefficient": "C^2=-(-mu)^(Nc-Nf) Lambda^(3Nc-Nf)",
      "coefficient_dimension": "2Nc-Nf",
      "status": "exact in the declared M_e=mu M_m convention, up to the stated epsilon orientation and square-root phase"
    },
    {
      "electric": "B-tilde=epsilon Q-tilde^Nc",
      "magnetic": "C epsilon q-tilde^(Nf-Nc) with complementary flavor epsilon",
      "coefficient": "C^2=-(-mu)^(Nc-Nf) Lambda^(3Nc-Nf)",
      "coefficient_dimension": "2Nc-Nf",
      "status": "same convention and phase qualification"
    }
  ],
  "anomaly_fixture": {
    "parameters": {
      "Nc": 3,
      "Nf": 6,
      "magnetic_rank": 3
    },
    "generator_normalization": "T(fundamental)=1/2; cubic fundamental coefficient=+1",
    "declared_set": "left and right flavor cubic anomalies; left and right flavor^2 times B and R; Tr(B^2 R), Tr(B R^2), Tr(B^3), Tr(B), Tr(R), Tr(R^3); and the gauge^2 times B and R consistency constraints",
    "coefficients": [
      {
        "anomaly": "SU(Nf)_L^3",
        "electric": "3",
        "magnetic": "3",
        "equal": true
      },
      {
        "anomaly": "SU(Nf)_L^2 U(1)_B",
        "electric": "3/2",
        "magnetic": "3/2",
        "equal": true
      },
      {
        "anomaly": "SU(Nf)_L^2 U(1)_R",
        "electric": "-3/4",
        "magnetic": "-3/4",
        "equal": true
      },
      {
        "anomaly": "SU(Nf)_R^3",
        "electric": "-3",
        "magnetic": "-3",
        "equal": true
      },
      {
        "anomaly": "SU(Nf)_R^2 U(1)_B",
        "electric": "-3/2",
        "magnetic": "-3/2",
        "equal": true
      },
      {
        "anomaly": "SU(Nf)_R^2 U(1)_R",
        "electric": "-3/4",
        "magnetic": "-3/4",
        "equal": true
      },
      {
        "anomaly": "Tr B^2 R",
        "electric": "-18",
        "magnetic": "-18",
        "equal": true
      },
      {
        "anomaly": "Tr B R^2",
        "electric": "0",
        "magnetic": "0",
        "equal": true
      },
      {
        "anomaly": "Tr B^3",
        "electric": "0",
        "magnetic": "0",
        "equal": true
      },
      {
        "anomaly": "Tr B",
        "electric": "0",
        "magnetic": "0",
        "equal": true
      },
      {
        "anomaly": "Tr R",
        "electric": "-10",
        "magnetic": "-10",
        "equal": true
      },
      {
        "anomaly": "Tr R^3",
        "electric": "7/2",
        "magnetic": "7/2",
        "equal": true
      },
      {
        "anomaly": "SU(gauge)^2 U(1)_B",
        "electric": "0",
        "magnetic": "0",
        "equal": true
      },
      {
        "anomaly": "SU(gauge)^2 U(1)_R",
        "electric": "0",
        "magnetic": "0",
        "equal": true
      }
    ]
  },
  "global_data": {
    "faithful_zero_form_flavor_group": "[SU(Nf)_L x SU(Nf)_R x U(1)_B]/[Z_Nf x Z_Nc], with kernel elements (z,z,exp(i beta))=(z,z,omega z^(-1)), z^Nf=1 and omega^Nc=1; their action on Q and Q-tilde is the SU(Nc) gauge-center action, so correlated quotient bundles belong to the background record",
    "one_form_symmetry": "none for either simply connected gauge theory because dynamical fundamental matter screens the center Wilson line",
    "discrete_involution": "charge conjugation exchanges Q with Q-tilde, left with right, q with q-tilde, and B with -B on both sides",
    "discrete_anomaly_scope": "the canonical card includes no additional independent Abelian discrete generator beyond subgroups of its declared continuous symmetries; torsion or global anomalies of nontrivial quotient backgrounds have not been computed here and are not counted as matched evidence"
  },
  "mass_flow_fixture": {
    "parent": {
      "Nc": 3,
      "Nf": 6,
      "magnetic_rank": 3,
      "b_e": 3,
      "b_m": 3
    },
    "deformation": "delta W_el=m M_e[Nf,Nf]; delta W_mag=m mu M_m[Nf,Nf]",
    "magnetic_f_term": "q_Nf q-tilde_Nf=-m mu",
    "daughter": {
      "Nc": 3,
      "Nf": 5,
      "magnetic_rank": 2,
      "b_e": 4,
      "b_m": 1
    },
    "thresholds": [
      "Lambda_5^4=m Lambda_6^3",
      "Lambda_tilde_5^1=Lambda_tilde_6^3/(-m mu)"
    ],
    "exceptional_endpoint": "when the parent has Nf=Nc+2, the magnetic SU(2) is completely Higgsed and a broken-group instanton generates the determinant term required by the Nf=Nc+1 s-confining daughter"
  },
  "rank_and_moduli_checks": [
    "generic electric meson rank is at most Nc",
    "magnetic classical F-terms impose q q-tilde=0 and M_m q=M_m q-tilde=0",
    "rank(M)>Nc is excluded quantum mechanically by the low-energy magnetic dynamics rather than by those classical F-terms alone",
    "at rank(M)=Nc the magnetic quantum constraint reproduces the meson-baryon relation",
    "the generic fully Higgsed complex dimension is 2NcNf-(Nc^2-1), with stabilizer corrections on lower strata"
  ],
  "sources": [
    {
      "id": "seiberg-1995",
      "locator": "Sections 2-4, arXiv PDF pp. 4-14",
      "url": "https://arxiv.org/pdf/hep-th/9411149",
      "use": "dual theory cards, operator maps, anomalies, moduli and deformations"
    },
    {
      "id": "intriligator-seiberg-1996",
      "locator": "Section 5.3-5.5, arXiv PDF pp. 21-27",
      "url": "https://arxiv.org/pdf/hep-th/9509066",
      "use": "mu normalization, scale relation, dual-of-dual, mass/Higgs flows and quantum rank constraints"
    },
    {
      "id": "csaki-murayama-1998",
      "locator": "Section 5 and conclusion",
      "url": "https://arxiv.org/abs/hep-th/9710105",
      "use": "discrete anomaly matching scope in supersymmetric dualities"
    },
    {
      "id": "gaiotto-kapustin-seiberg-willett-2015",
      "locator": "Section 1",
      "url": "https://arxiv.org/abs/1412.5148",
      "use": "screening criterion for one-form symmetry"
    }
  ],
  "visual_encodings": {
    "double_arrow": "conditional infrared claim or protected dictionary map",
    "solid_arrow": "one-way deformation operation",
    "gray_box": "dictionary or closure gate",
    "color_dependence": "none"
  },
  "accessibility": {
    "alt_text": "For Nc at least three and Nc plus two no greater than Nf below three Nc, electric SU(Nc) SQCD and magnetic SU(Nf minus Nc) SQCD are joined by meson, convention-fixed baryon, charge, declared-anomaly, quantum-stratum and scale maps. Beneath them, an electric one-flavor mass lowers Nf while a magnetic quark expectation value lowers both flavor and magnetic color rank. The daughter scale relation closes the square, with a separate warning for complete magnetic SU(2) breaking at Nf equals Nc plus two.",
    "semantic_equivalent": "/figures/supersymmetry-duality/seiberg-map-operator-dictionary.json",
    "narrow_width": "the SVG remains unclipped and has a full-size link; this JSON preserves every field, qualification and equation at 320 pixels"
  },
  "limitations": [
    "The drawing treats Nc>=3 and generic non-Abelian magnetic rank; the Nc=2 enhanced-flavor card and all named endpoint qualifications remain separate.",
    "The baryon coefficient is exact only in the declared composite, epsilon and square-root phase convention; it is not a unit-normalized correlator coefficient.",
    "Protected matches do not determine the unprotected spectrum or prove the infrared equivalence.",
    "The artifact registry lifecycle observed during generation was prototype."
  ]
}