{
  "schema_version": 1,
  "artifact_id": "qft.artifact.supersymmetry-duality.n2.su3-genus-two-discriminant-period-system",
  "split_from_artifact_id": "qft.artifact.supersymmetry-duality.n2.sw-singularities-monodromies-bps-chambers",
  "title": "Pure SU(3): a genus-two fiber over intersecting discriminant divisors",
  "generated_by": "figures-src/supersymmetry-duality/su3-genus-two-discriminant-period-system.mjs",
  "source_revision": 1,
  "generated_on": "2026-08-24",
  "figure_kind": "original monochrome real discriminant-slice and genus-two branch-cut/cycle diagram with an explicit white canvas",
  "quantitative_status": "exact for the curve, polynomial discriminant, component parameterizations, cusp and intersection loci, frozen branch roots, genus, and symplectic pairing; planar contour geometry is schematic and numerical periods are not implemented",
  "reader_question": "How does one concrete pure SU(3) Seiberg-Witten curve exhibit genus two, two discriminant components, local vanishing cycles, and exceptional cusp loci that have no rank-one analogue?",
  "takeaway": "At rank two, the singular set is a pair of intersecting complex divisors rather than isolated Coulomb-branch points, and a regular fiber requires a full polarized basis of four one-cycles patched by Sp(4,Z).",
  "alt_text": "Two-panel monochrome diagram for the pure SU(3) Seiberg-Witten family. The left panel plots a real slice of the two semicubical discriminant components. Their cusps occur at u2 equals zero and u3 equals minus or plus Lambda cubed. One real transverse intersection is marked, and notes distinguish one primitive vanishing cycle on a smooth divisor, two disjoint local nodes at the intersection, and two nonlocally intersecting vanishing cycles at a cusp. The right panel freezes Lambda equals one, u2 equals three, and u3 equals zero. Six directly labelled real branch points form three gray cuts. Solid A1 and A2 loops and dashed B1 and B2 contours display one declared symplectic cut convention. Equations state genus two, the canonical intersection pairing, the period vector ordered as aD1, aD2, a1, a2, Sp4Z patching, the exact polynomial discriminant including its factor 64, and a convention-dependent Seiberg-Witten differential. The diagram explicitly says that the contours are schematic, not to scale, and that numerical genus-two periods have not been evaluated.",
  "conventions": {
    "theory": "pure four-dimensional N=2 supersymmetric Yang-Mills theory with gauge group SU(3)",
    "scale": "Lambda is nonzero and has mass dimension one",
    "coulomb_coordinates": "P_3(x)=x^3-u_2 x-u_3, with [x]=1, [u_2]=2 and [u_3]=3",
    "curve": "y^2=(P_3-Lambda^3)(P_3+Lambda^3)=P_3^2-Lambda^6",
    "discriminant": "Disc_x means the monic degree-six polynomial discriminant, including its nonzero overall factor 64 Lambda^18",
    "discriminant_labels": {
      "D_plus": "4u_2^3-27(u_3+Lambda^3)^2=0; its cusp is (0,-Lambda^3)",
      "D_minus": "4u_2^3-27(u_3-Lambda^3)^2=0; its cusp is (0,+Lambda^3)"
    },
    "differential": "lambda_SW=C x P_3'(x) dx/y; C is intentionally left convention-dependent",
    "regular_base_point": "Lambda=1, u_2=3, u_3=0",
    "branch_point_order": "increasing real x at the regular base point",
    "cuts": "C_1=[e_1,e_2], C_2=[e_3,e_4], C_3=[e_5,e_6] on the drawn real x-plane",
    "cycle_orientation": "A_i are counterclockwise around C_i for i=1,2; B_i run toward C_3 on the first sheet and return on the second sheet",
    "intersection": "<A_i,B_j>=delta_ij, <B_i,A_j>=-delta_ij, and all A-A and B-B pairings vanish",
    "period_vector": "Pi=(a_D,1,a_D,2,a_1,a_2)^T with a_i=integral over A_i and a_D,i=integral over B_i",
    "patching": "Pi maps to M Pi for M in Sp(4,Z), M^T J M=J",
    "global_form_boundary": "the curve and local homology data do not by themselves classify the complete genuine-line lattice for other global forms"
  },
  "equations": {
    "polynomial": "P_3=x^3-u_2 x-u_3",
    "curve": "y^2=(P_3-Lambda^3)(P_3+Lambda^3)",
    "discriminant": "Disc_x(P_3^2-Lambda^6)=64 Lambda^18 [4u_2^3-27(u_3+Lambda^3)^2][4u_2^3-27(u_3-Lambda^3)^2]",
    "differential": "lambda_SW=C x P_3'(x) dx/y",
    "genus": "g=(6-2)/2=2 for six distinct finite branch points",
    "periods": "a_i=integral_(A_i) lambda_SW, a_D,i=integral_(B_i) lambda_SW",
    "patching": "Pi -> M Pi with integral M satisfying M^T J M=J"
  },
  "discriminant": {
    "polynomial": "Disc_x(P_3^2-Lambda^6)=64 Lambda^18 [4u_2^3-27(u_3+Lambda^3)^2][4u_2^3-27(u_3-Lambda^3)^2]",
    "derivation": "For q_plus=P_3-Lambda^3 and q_minus=P_3+Lambda^3, Disc(q_plus q_minus)=Disc(q_plus) Disc(q_minus) Res(q_plus,q_minus)^2. The two cubic discriminants are the displayed bracketed factors and Res(q_plus,q_minus)^2=64 Lambda^18.",
    "literature_normalization_note": "Klemm et al. eq. (19) suppresses the irrelevant nonzero overall factor 64; this record uses the exact monic polynomial discriminant.",
    "components": [
      {
        "label": "D_plus",
        "equation": "4u_2^3-27(u_3+Lambda^3)^2=0",
        "real_parameterization": "u_2=3t^2, u_3=-Lambda^3+2t^3",
        "parameter_domain": "t is real for the displayed slice; complex t normalizes the complex component",
        "cusp": {
          "u_2": "0",
          "u_3": "-Lambda^3"
        },
        "generic_local_cycle": "one primitive cycle delta_plus vanishes"
      },
      {
        "label": "D_minus",
        "equation": "4u_2^3-27(u_3-Lambda^3)^2=0",
        "real_parameterization": "u_2=3t^2, u_3=+Lambda^3+2t^3",
        "parameter_domain": "t is real for the displayed slice; complex t normalizes the complex component",
        "cusp": {
          "u_2": "0",
          "u_3": "+Lambda^3"
        },
        "generic_local_cycle": "one primitive cycle delta_minus vanishes"
      }
    ],
    "component_intersections": {
      "complex_equations": "u_3=0 and u_2^3=27 Lambda^6/4",
      "count_for_nonzero_lambda": 3,
      "real_intersection_for_positive_lambda": "(u_2,u_3)=(3 Lambda^2/2^(2/3),0)",
      "local_geometry": "two separated double roots produce two disjoint local vanishing cycles with zero intersection pairing"
    },
    "cusp_geometry": {
      "exact_factorization": "on either normalized component, the degenerating cubic is x^3-3t^2 x-2t^3=(x-2t)(x+t)^2; t=0 produces a triple root",
      "local_model": "y^2 is proportional to x^3 plus lower deformation terms",
      "vanishing_cycles": "two local vanishing cycles have intersection magnitude one; their overall signs and global charge labels require based transport paths",
      "evidence_status": "the local A_2 topology is exact; no global charge label is inferred without a declared base path"
    }
  },
  "regular_fiber": {
    "parameters": {
      "Lambda": 1,
      "u_2": 3,
      "u_3": 0
    },
    "equation": "y^2=(x^3-3x)^2-1",
    "polynomial_discriminant": "419904",
    "roots": [
      {
        "index": 1,
        "exact_x": "2 cos(8 pi/9)",
        "numeric_x": -1.879385241571817,
        "factor_equation": "P_3(x)=-1",
        "target": -1,
        "residual_abs": 1.776357e-15,
        "derivative_abs": 7.596266658714
      },
      {
        "index": 2,
        "exact_x": "2 cos(7 pi/9)",
        "numeric_x": -1.532088886237956,
        "factor_equation": "P_3(x)=+1",
        "target": 1,
        "residual_abs": 4.440892e-16,
        "derivative_abs": 4.041889066002
      },
      {
        "index": 3,
        "exact_x": "2 cos(5 pi/9)",
        "numeric_x": -0.347296355333861,
        "factor_equation": "P_3(x)=+1",
        "target": 1,
        "residual_abs": 6.661338e-16,
        "derivative_abs": 2.638155724715
      },
      {
        "index": 4,
        "exact_x": "2 cos(4 pi/9)",
        "numeric_x": 0.347296355333861,
        "factor_equation": "P_3(x)=-1",
        "target": -1,
        "residual_abs": 6.661338e-16,
        "derivative_abs": 2.638155724715
      },
      {
        "index": 5,
        "exact_x": "2 cos(2 pi/9)",
        "numeric_x": 1.532088886237956,
        "factor_equation": "P_3(x)=-1",
        "target": -1,
        "residual_abs": 4.440892e-16,
        "derivative_abs": 4.041889066002
      },
      {
        "index": 6,
        "exact_x": "2 cos(pi/9)",
        "numeric_x": 1.879385241571817,
        "factor_equation": "P_3(x)=+1",
        "target": 1,
        "residual_abs": 1.776357e-15,
        "derivative_abs": 7.596266658714
      }
    ],
    "target_sequence_in_increasing_x": [
      -1,
      1,
      1,
      -1,
      -1,
      1
    ],
    "cuts": [
      {
        "id": "C_1",
        "endpoints": [
          1,
          2
        ],
        "role": "first paired cut"
      },
      {
        "id": "C_2",
        "endpoints": [
          3,
          4
        ],
        "role": "second paired cut"
      },
      {
        "id": "C_3",
        "endpoints": [
          5,
          6
        ],
        "role": "reference cut"
      }
    ],
    "genus": {
      "degree": 6,
      "finite_simple_branch_points": 6,
      "branch_at_infinity": false,
      "riemann_hurwitz": "2g-2=2(-2)+6=2",
      "result": 2
    }
  },
  "cycles_and_pairings": {
    "ordered_basis": [
      "A_1",
      "A_2",
      "B_1",
      "B_2"
    ],
    "cycles": [
      {
        "id": "A_1",
        "contour": "counterclockwise loop around C_1 on the double cover"
      },
      {
        "id": "A_2",
        "contour": "counterclockwise loop around C_2 on the double cover"
      },
      {
        "id": "B_1",
        "contour": "lift of a path from C_1 toward C_3 on sheet one and back on sheet two"
      },
      {
        "id": "B_2",
        "contour": "lift of a path from C_2 toward C_3 on sheet one and back on sheet two"
      }
    ],
    "intersection_matrix": [
      [
        0,
        0,
        1,
        0
      ],
      [
        0,
        0,
        0,
        1
      ],
      [
        -1,
        0,
        0,
        0
      ],
      [
        0,
        -1,
        0,
        0
      ]
    ],
    "cut_pairing_is_conventional": true,
    "planar_contours_are_schematic": true
  },
  "period_system": {
    "differential": "lambda_SW=C x P_3'(x) dx/y",
    "normalization": "C is convention-dependent and must be fixed before comparing numerical period values",
    "mass_dimension_check": "[x]+[P_3']+[dx]-[y]=1+2+1-3=1",
    "periods": {
      "a_1": "integral over A_1 of lambda_SW",
      "a_2": "integral over A_2 of lambda_SW",
      "a_D_1": "integral over B_1 of lambda_SW",
      "a_D_2": "integral over B_2 of lambda_SW"
    },
    "vector_order": [
      "a_D,1",
      "a_D,2",
      "a_1",
      "a_2"
    ],
    "polarization_matrix": [
      [
        0,
        0,
        1,
        0
      ],
      [
        0,
        0,
        0,
        1
      ],
      [
        -1,
        0,
        0,
        0
      ],
      [
        0,
        -1,
        0,
        0
      ]
    ],
    "patching": "Pi -> M Pi with integral M satisfying M^T J M=J",
    "numerical_periods_implemented": false,
    "publication_limitation": "No robust numerical genus-two period integration, analytic continuation, or monodromy transport is implemented in this artifact; no numerical period or coupling matrix is claimed."
  },
  "limitations": [
    "The Coulomb-branch panel is a real two-dimensional slice of a complex two-dimensional base; it shows only one of the three Z_3-related transverse intersections.",
    "Labels delta_plus and delta_minus are local labels on smooth discriminant patches, not globally transported electromagnetic charge vectors.",
    "At each cusp the magnitude of the local intersection pairing is fixed, but its sign and any global charge name depend on cycle orientations and based transport paths.",
    "The branch-cut drawing is a convention at one regular base point and is schematic, not to scale; a different cut system is related by an integral symplectic transformation.",
    "The differential normalization C is deliberately unfixed, so the artifact cannot be used for cross-source numerical period comparisons.",
    "Numerical genus-two periods, the coupling matrix, explicit monodromy matrices, BPS indices, and an integrable-system Lax realization are not computed here.",
    "The artifact does not determine the spectrum of genuine line operators or resolve global-form choices beyond the stated SU(3) theory label."
  ],
  "sources": [
    {
      "authors": "Albrecht Klemm, Wolfgang Lerche, Stefan Theisen, and Shimon Yankielowicz",
      "title": "Simple Singularities and N=2 Supersymmetric Yang-Mills Theory",
      "journal": "Physics Letters B",
      "volume": "344",
      "pages": "169-175",
      "year": 1995,
      "arxiv": "hep-th/9411048",
      "doi": "10.1016/0370-2693(94)01558-F",
      "url": "https://arxiv.org/abs/hep-th/9411048",
      "locator": "eqs. (15)-(19), pp. 5-7, and eqs. (21)-(25), pp. 9-10 of the arXiv PDF",
      "use": "genus-(n-1) SU(n) curve, SU(3) discriminant factors, branch cuts, periods, Sp(4,Z) homology, and Picard-Lefschetz vanishing cycles"
    },
    {
      "authors": "A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, and A. Morozov",
      "title": "Integrability and Seiberg-Witten Exact Solution",
      "journal": "Physics Letters B",
      "volume": "355",
      "pages": "466-474",
      "year": 1995,
      "arxiv": "hep-th/9505035",
      "doi": "10.1016/0370-2693(95)00723-X",
      "url": "https://arxiv.org/abs/hep-th/9505035",
      "locator": "eqs. (16), (28)-(31), pp. 5-7 of the arXiv PDF",
      "use": "periods of the generating differential and the integrable-system interpretation"
    },
    {
      "authors": "Ron Donagi and Edward Witten",
      "title": "Supersymmetric Yang-Mills Systems and Integrable Systems",
      "journal": "Nuclear Physics B",
      "volume": "460",
      "pages": "299-334",
      "year": 1996,
      "arxiv": "hep-th/9510101",
      "doi": "10.1016/0550-3213(95)00609-5",
      "url": "https://arxiv.org/abs/hep-th/9510101",
      "locator": "sections 2-3",
      "use": "holomorphic algebraic-integrable-system interpretation and spectral-curve context"
    },
    {
      "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 2-3",
      "use": "exceptional SU(3) cusp loci with mutually nonlocal massless states"
    }
  ],
  "accessibility": {
    "color_independence": "The two discriminant components use solid and dashed strokes with direct labels; branch-factor membership uses filled and open markers plus a written legend; A and B cycles use solid and dashed strokes with direct labels.",
    "structured_equivalent": "This JSON provides every curve equation, discriminant factor and cusp, intersection locus, branch root and residual, cut and cycle convention, pairing matrix, period definition, source locator, and limitation.",
    "explicit_white_canvas": true,
    "animation": false,
    "schematic_elements": "the real-slice and planar cycle drawings are not to scale; numeric root positions are listed separately"
  }
}