{
  "schema_version": "1.0.0",
  "record_version": "1.0.0",
  "theory_card_version": "1.0.0",
  "artifact_id": "qft.artifact.supersymmetry-duality.n2.argyres-douglas-cusp-class-s-evidence-map",
  "split_from_artifact_id": "qft.artifact.supersymmetry-duality.n2.sw-singularities-monodromies-bps-chambers",
  "title": "From an Argyres-Douglas cusp to justified claims",
  "reader_question": "Which facts follow exactly from the local (A1,A2) Seiberg-Witten cusp, and which existence, protected-sector, class-S, global-form, and interface claims require additional construction data or evidence?",
  "takeaway": "The local normal form and Seiberg-Witten differential fix the scaling weights and discriminant cusp exactly, while anomaly values, the proposed Yang-Lee chiral-algebra identification, a class-S realization, and duality-wall claims enter at successively stronger and explicitly bounded evidence levels.",
  "scope": "Original status-bounded local-cusp, protected-theory-card, class-S-input, and interface-qualification figure for the Chapter 11 Argyres-Douglas page.",
  "theory_card": {
    "theory": "(A1,A2) Argyres-Douglas theory, also called H0",
    "coulomb_branch_rank": {
      "value": 1,
      "status": "protected datum"
    },
    "local_seiberg_witten_data": {
      "curve": "x^2=z^3+v z+u",
      "differential": "lambda_SW=x dz",
      "interpretation": "local normal form near the fixed point; not a complete ultraviolet curve",
      "scaling_dimensions": {
        "z": {
          "numerator": 2,
          "denominator": 5,
          "exact": "2/5"
        },
        "x": {
          "numerator": 3,
          "denominator": 5,
          "exact": "3/5"
        },
        "v": {
          "numerator": 4,
          "denominator": 5,
          "exact": "4/5"
        },
        "u": {
          "numerator": 6,
          "denominator": 5,
          "exact": "6/5"
        }
      },
      "discriminant": {
        "resultant_of_cubic_and_derivative": [
          {
            "coefficient": 4,
            "v_power": 3,
            "u_power": 0
          },
          {
            "coefficient": 27,
            "v_power": 0,
            "u_power": 2
          }
        ],
        "cubic_discriminant": [
          {
            "coefficient": -4,
            "v_power": 3,
            "u_power": 0
          },
          {
            "coefficient": -27,
            "v_power": 0,
            "u_power": 2
          }
        ],
        "exact_expression": "-4 v^3-27 u^2",
        "homogeneous_weight": {
          "numerator": 12,
          "denominator": 5,
          "exact": "12/5"
        },
        "parameterization": "(v,u)=(-3t^2,2t^3)",
        "substituted_polynomial_terms": [],
        "vanishes_identically": true,
        "unique_singular_point_of_plane_curve": "v=u=0 from dDelta/dv=-12v^2 and dDelta/du=-54u",
        "irreducibility_check": "Over C[v], reducibility as a quadratic in u would require v^3 to be a square polynomial; its odd v-adic order forbids this. Equivalently gcd(2,3)=1 for the cusp binomial.",
        "one_irreducible_complex_cusp": true,
        "displayed_curve_scope": "real t slice only"
      }
    },
    "colliding_vanishing_cycles": {
      "gamma_1": [
        1,
        0
      ],
      "gamma_2": [
        0,
        1
      ],
      "symplectic_matrix": [
        [
          0,
          1
        ],
        [
          -1,
          0
        ]
      ],
      "pairing_gamma_1_gamma_2": 1,
      "pairing_gamma_2_gamma_1": -1,
      "physical_status": "The nonlocal assignment to the colliding vanishing cycles is source-dependent physical input; the generator checks its exact symplectic consistency but does not derive the charge identification from the plane curve alone."
    },
    "continuous_flavor_symmetry": {
      "value": "none",
      "status": "source-backed protected theory-card datum, corroborated by the absence of a flavor-current contribution in the Schur-index data; not derived from the plane curve alone",
      "source_keys": [
        "argyres-douglas-1995",
        "cordova-shao-2016"
      ]
    },
    "conformal_anomalies": {
      "a": {
        "numerator": 43,
        "denominator": 120,
        "exact": "43/120"
      },
      "c_4d": {
        "numerator": 11,
        "denominator": 30,
        "exact": "11/30"
      },
      "two_a_minus_c": {
        "numerator": 7,
        "denominator": 20,
        "exact": "7/20"
      },
      "rank_one_relation": "2a-c=(2 Delta(u)-1)/4=7/20",
      "status": "source-backed intrinsic 4d data; the generator independently verifies the rational identities",
      "source_keys": [
        "shapere-tachikawa-2008"
      ]
    },
    "higgs_branch": {
      "value": "point",
      "status": "qualified literature classification; not inferred from the local curve, the anomaly relation, absence of a continuous flavor current, or the Schur index alone",
      "release_gate": "retain an adjacent dedicated Higgs-branch classification source before promoting this entry beyond prototype"
    },
    "protected_chiral_algebra": {
      "proposed_correspondence": "Yang-Lee Virasoro minimal model M(2,5)",
      "c_2d": {
        "numerator": -22,
        "denominator": 5,
        "exact": "-22/5"
      },
      "general_4d_2d_relation": "c_2d=-12 c_4d",
      "evidence_status": "proposed for (A1,A2) and strongly supported by the explicit Schur-index equality with the M(2,5) vacuum character and Rogers-Ramanujan q-series",
      "non_implication": "The general Beem-et-al. protected 4d/2d construction alone does not identify this specific chiral algebra, and the correspondence does not reconstruct the full unprotected 4d operator algebra.",
      "source_keys": [
        "beem-et-al-2015",
        "cordova-shao-2016"
      ]
    }
  },
  "class_s_input_card": {
    "realization_status": "one irregular class-S construction of the local (A1,A2) system, not a statement that the local curve fixes every compactification datum",
    "six_dimensional_type": "A1 (2,0) theory",
    "surface": "C is the Riemann sphere CP1",
    "regular_punctures": [],
    "irregular_punctures": [
      {
        "location": "z=infinity",
        "count": 1,
        "local_spectral_curve": "x^2=z^3+v z+u",
        "required_extra_data": "the irregular boundary condition, including its allowed leading behavior and Stokes/sector data when applicable; pole order alone is not a complete fixture"
      }
    ],
    "seiberg_witten_differential": "lambda_SW=x dz",
    "twist": "partial topological twist along C preserving four-dimensional N=2 supersymmetry",
    "polarization_and_discrete_data": "choice controlling the global form and the spectrum of genuine line operators; not fixed by the local spectral equation",
    "defects": "inserted defects and their labels are additional construction inputs",
    "source_keys": [
      "gaiotto-2012"
    ]
  },
  "evidence_and_construction_ladder": [
    {
      "order": 1,
      "label": "local Seiberg-Witten scaling",
      "status": "exactly derived in this artifact",
      "establishes": [
        "quasi-homogeneous scaling weights",
        "discriminant homogeneity",
        "real slice of the irreducible cusp"
      ],
      "does_not_establish": [
        "existence of a complete four-dimensional SCFT",
        "anomaly coefficients",
        "global form or genuine line spectrum"
      ]
    },
    {
      "order": 2,
      "label": "UV embedding or class-S irregular-puncture construction",
      "status": "additional construction input",
      "establishes": [
        "a physical origin for the local singularity",
        "identification of deformation parameters within a complete construction"
      ],
      "does_not_establish": [
        "equality of every protected and unprotected observable"
      ]
    },
    {
      "order": 3,
      "label": "anomaly, index, and chiral-algebra tests",
      "status": "protected evidence",
      "establishes": [
        "nontrivial cross-checks among protected quantities",
        "the specific Schur-index q-series match"
      ],
      "does_not_establish": [
        "long-multiplet spectrum",
        "generic correlators",
        "complete BPS spectrum in every chamber"
      ]
    },
    {
      "order": 4,
      "label": "global form, genuine line lattice, and interface data",
      "status": "separate global input and tests",
      "establishes": [
        "which line operators are genuine",
        "how lines end on or cross a wall",
        "the scoped action of a duality interface"
      ],
      "does_not_establish": [
        "complete unprotected equivalence without further evidence"
      ]
    }
  ],
  "interface_qualification": {
    "wall_theory": "T[SU(2)]",
    "supersymmetry": "three-dimensional N=4",
    "faithful_global_symmetry": "SO(3)_H x SO(3)_C",
    "nonfaithful_cover_often_written": "SU(2)_H x SU(2)_C",
    "faithful_group_source_key": "gaiotto-witten-2009",
    "bulk_relation": "an S-wall couples to bulk descriptions associated with G and its Langlands dual G^vee",
    "required_global_inputs": [
      "global form on each side",
      "theta/discrete-theta data when relevant",
      "lattice of genuine Wilson-'t Hooft lines",
      "rule for line ending, transport, and junction operators at the interface"
    ],
    "limitation": "the Lie-algebra action or protected kernel does not by itself define an invertible full quantum interface for every global form"
  },
  "exact_science_checks": {
    "scaling_system": {
      "equations": [
        "2[x]=3[z]",
        "[x]+[z]=1"
      ],
      "solution": {
        "z": {
          "numerator": 2,
          "denominator": 5,
          "exact": "2/5"
        },
        "x": {
          "numerator": 3,
          "denominator": 5,
          "exact": "3/5"
        },
        "v": {
          "numerator": 4,
          "denominator": 5,
          "exact": "4/5"
        },
        "u": {
          "numerator": 6,
          "denominator": 5,
          "exact": "6/5"
        }
      },
      "all_curve_terms_weight": {
        "numerator": 6,
        "denominator": 5,
        "exact": "6/5"
      },
      "lambda_weight": {
        "numerator": 1,
        "denominator": 1,
        "exact": "1"
      }
    },
    "discriminant": {
      "resultant_of_cubic_and_derivative": [
        {
          "coefficient": 4,
          "v_power": 3,
          "u_power": 0
        },
        {
          "coefficient": 27,
          "v_power": 0,
          "u_power": 2
        }
      ],
      "cubic_discriminant": [
        {
          "coefficient": -4,
          "v_power": 3,
          "u_power": 0
        },
        {
          "coefficient": -27,
          "v_power": 0,
          "u_power": 2
        }
      ],
      "exact_expression": "-4 v^3-27 u^2",
      "homogeneous_weight": {
        "numerator": 12,
        "denominator": 5,
        "exact": "12/5"
      },
      "parameterization": "(v,u)=(-3t^2,2t^3)",
      "substituted_polynomial_terms": [],
      "vanishes_identically": true,
      "unique_singular_point_of_plane_curve": "v=u=0 from dDelta/dv=-12v^2 and dDelta/du=-54u",
      "irreducibility_check": "Over C[v], reducibility as a quadratic in u would require v^3 to be a square polynomial; its odd v-adic order forbids this. Equivalently gcd(2,3)=1 for the cusp binomial.",
      "one_irreducible_complex_cusp": true,
      "displayed_curve_scope": "real t slice only"
    },
    "vanishing_cycle_fixture": {
      "gamma_1": [
        1,
        0
      ],
      "gamma_2": [
        0,
        1
      ],
      "symplectic_matrix": [
        [
          0,
          1
        ],
        [
          -1,
          0
        ]
      ],
      "pairing_gamma_1_gamma_2": 1,
      "pairing_gamma_2_gamma_1": -1,
      "physical_status": "The nonlocal assignment to the colliding vanishing cycles is source-dependent physical input; the generator checks its exact symplectic consistency but does not derive the charge identification from the plane curve alone."
    },
    "anomaly_identities": {
      "a": {
        "numerator": 43,
        "denominator": 120,
        "exact": "43/120"
      },
      "c_4d": {
        "numerator": 11,
        "denominator": 30,
        "exact": "11/30"
      },
      "two_a_minus_c": {
        "numerator": 7,
        "denominator": 20,
        "exact": "7/20"
      },
      "rank_one_expression": {
        "numerator": 7,
        "denominator": 20,
        "exact": "7/20"
      },
      "c_2d": {
        "numerator": -22,
        "denominator": 5,
        "exact": "-22/5"
      },
      "two_a_minus_c_identity": true,
      "c_2d_identity": true,
      "evidence_status": "The arithmetic is exact once a and c are supplied; their values are literature input, not consequences of the local normal form."
    }
  },
  "evidence_status": {
    "exactly_derived_here": [
      "scaling dimensions from the two linear homogeneity equations",
      "cubic discriminant from the exact Sylvester resultant",
      "homogeneous discriminant weight",
      "cusp parameterization lies on the discriminant",
      "symplectic consistency of the displayed charge-pairing fixture",
      "rational anomaly and c_2d identities after literature values are supplied"
    ],
    "source_dependent_physics": [
      "the identification of the two colliding cycles as physical mutually nonlocal BPS states",
      "existence and identity of the H0 SCFT",
      "values of a and c and absence of continuous flavor symmetry",
      "the irregular class-S construction",
      "the T[SU(2)] wall interpretation"
    ],
    "proposed_and_strongly_supported": [
      "the H0 chiral algebra as the Yang-Lee M(2,5) vacuum sector, supported by the Schur-index and Rogers-Ramanujan q-series equality"
    ],
    "explicitly_qualified": [
      "Higgs branch equals a point is retained as a literature-classification entry and is not derived by this artifact"
    ]
  },
  "limitations": [
    "The displayed cusp is a real slice of one irreducible complex discriminant curve, not two independent complex components.",
    "The plane curve does not determine the integral charge local system or prove which physical cycles vanish; that identification is source input.",
    "Local quasi-homogeneity does not establish a positive special-Kahler metric or a UV-complete four-dimensional theory.",
    "Protected anomalies, indices, and chiral algebras do not determine long multiplets, generic correlators, or every BPS chamber.",
    "The class-S curve does not fix puncture boundary conditions, polarization, global form, genuine lines, or inserted defects by itself.",
    "An interface action on protected data or the charge lattice does not by itself prove invertibility on the full Hilbert space."
  ],
  "accessibility": {
    "color_independence": "All distinctions use direct status words, double or dashed borders, solid or dashed cycle paths, and black/gray fills; no relation is encoded by color alone.",
    "structured_equivalent": "This JSON supplies the exact local equations, scaling solution, discriminant derivation, cusp parameterization, theory-card status, class-S inputs, evidence ladder, interface qualifications, source chain, and limitations.",
    "svg_title_and_long_description_required": true,
    "explicit_white_canvas_required": true
  },
  "source_chain": [
    {
      "key": "argyres-douglas-1995",
      "authors": "Philip C. Argyres and Michael R. Douglas",
      "title": "New Phenomena in SU(3) Supersymmetric Gauge Theory",
      "journal": "Nuclear Physics B",
      "volume": "448",
      "pages": "93-126",
      "year": 1995,
      "arxiv": "hep-th/9505062",
      "doi": "10.1016/0550-3213(95)00281-V",
      "url": "https://arxiv.org/abs/hep-th/9505062",
      "locator": "sections 2-4",
      "use": "mutually nonlocal massless states and the interacting fixed-point mechanism"
    },
    {
      "key": "shapere-tachikawa-2008",
      "authors": "Alfred D. Shapere and Yuji Tachikawa",
      "title": "Central Charges of N=2 Superconformal Field Theories in Four Dimensions",
      "journal": "Journal of High Energy Physics",
      "issue": "09",
      "article": "109",
      "year": 2008,
      "arxiv": "0804.1957",
      "doi": "10.1088/1126-6708/2008/09/109",
      "url": "https://arxiv.org/abs/0804.1957",
      "locator": "sections 3-5",
      "use": "rank-one Coulomb-dimension relation and four-dimensional anomaly data"
    },
    {
      "key": "beem-et-al-2015",
      "authors": "Christopher Beem, Madalena Lemos, Pedro Liendo, Wolfger Peelaers, Leonardo Rastelli, and Balt C. van Rees",
      "title": "Infinite Chiral Symmetry in Four Dimensions",
      "journal": "Communications in Mathematical Physics",
      "volume": "336",
      "pages": "1359-1433",
      "year": 2015,
      "arxiv": "1312.5344",
      "doi": "10.1007/s00220-014-2272-x",
      "url": "https://arxiv.org/abs/1312.5344",
      "locator": "sections 3.2 and 4.4",
      "use": "general protected 4d/2d chiral-algebra construction and c_2d=-12 c_4d; not by itself an H0-to-M(2,5) identification"
    },
    {
      "key": "cordova-shao-2016",
      "authors": "Clay Cordova and Shu-Heng Shao",
      "title": "Schur Indices, BPS Particles, and Argyres-Douglas Theories",
      "journal": "Journal of High Energy Physics",
      "issue": "01",
      "article": "040",
      "year": 2016,
      "arxiv": "1506.00265",
      "doi": "10.1007/JHEP01(2016)040",
      "url": "https://arxiv.org/abs/1506.00265",
      "locator": "section 4.1, especially equations (4.13)-(4.14)",
      "use": "proposed (A1,A2) chiral algebra and explicit Schur-index match to the M(2,5) vacuum character and Rogers-Ramanujan q-series"
    },
    {
      "key": "gaiotto-2012",
      "authors": "Davide Gaiotto",
      "title": "N=2 Dualities",
      "journal": "Journal of High Energy Physics",
      "issue": "08",
      "article": "034",
      "year": 2012,
      "arxiv": "0904.2715",
      "doi": "10.1007/JHEP08(2012)034",
      "url": "https://arxiv.org/abs/0904.2715",
      "locator": "sections 2-4",
      "use": "class-S construction from the six-dimensional (2,0) theory, punctures, spectral curves, and duality frames"
    },
    {
      "key": "gaiotto-witten-2009",
      "authors": "Davide Gaiotto and Edward Witten",
      "title": "S-Duality of Boundary Conditions in N=4 Super Yang-Mills Theory",
      "journal": "Advances in Theoretical and Mathematical Physics",
      "volume": "13",
      "pages": "721-896",
      "year": 2009,
      "arxiv": "0807.3720",
      "doi": "10.4310/ATMP.2009.v13.n3.a5",
      "url": "https://arxiv.org/abs/0807.3720",
      "locator": "section 2.4.1 and section 4.1",
      "use": "T[SU(2)], its faithful SO(3)xSO(3) global symmetry, and its role in S-duality walls"
    }
  ]
}