{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.supersymmetry-duality.n4.charge-lattice-global-form-s-duality-orbits",
  "split_from_artifact_id": "qft.artifact.supersymmetry-duality.duality.global-form-lines-anomalies-map",
  "artifact_class": "finite charge-lattice orbit and marked-torus transport figure",
  "title": "The line-spectrum orbit of su(2) N=4 super-Yang–Mills",
  "evidence_date": "2026-08-24",
  "lifecycle_status": "materialized governed prototype; publication promotion remains a registry decision",
  "reader_question": "Why does SL(2,Z) act on a family of globally distinct su(2) N=4 theories rather than automatically as a self-duality of one theory?",
  "dominant_point": "The three maximal-isotropic subgroups of the screened Z2 electric-magnetic charge group define SU(2), SO(3)+, and SO(3)- line-spectrum nodes; S and T permute them, so a node's line-lattice stabilizer is only a necessary candidate internal subgroup, and full quantum equivalence additionally requires the background, counterterm, and observable dictionary.",
  "conventions": {
    "spacetime": "four-dimensional Lorentzian QFT.org (+---) unless compactification geometry is stated",
    "charge_order": "gamma_bar=(e,m)^T, electric first and magnetic second",
    "finite_charge_group": "Z_2 times Z_2 after root and coroot screening",
    "dirac_pairing": "<(e,m),(e_prime,m_prime)>=e m_prime-m e_prime modulo 2",
    "complex_coupling": "tau=theta/(2 pi)+4 pi i/g_YM^2",
    "modular_action": "tau maps to (a tau+b)/(c tau+d)",
    "passive_charge_action": "(e,m)^T maps by [[a,-b],[-c,d]] so e_prime+tau_prime m_prime=(e+tau m)/(c tau+d)",
    "torus_orientation": "A dot B=+1",
    "word_order": "rightmost modular generator acts first",
    "manifold_scope": "spin four-manifolds for the displayed SO(3) discrete-theta labels"
  },
  "charge_classes": {
    "zero": [
      0,
      0
    ],
    "W": [
      1,
      0
    ],
    "H": [
      0,
      1
    ],
    "D": [
      1,
      1
    ]
  },
  "theory_objects": [
    {
      "id": "su2",
      "display": "SU(2)",
      "genuine_subgroup": [
        [
          0,
          0
        ],
        [
          1,
          0
        ]
      ],
      "nonzero_generator": "W",
      "condition": "m=0 modulo 2",
      "global_form": "simply connected SU(2)",
      "discrete_theta": "not an independent SO(3) label",
      "stabilizer": "line-lattice stabilizer Gamma_0(2): c=0 modulo 2"
    },
    {
      "id": "so3_plus",
      "display": "SO(3)_+",
      "genuine_subgroup": [
        [
          0,
          0
        ],
        [
          0,
          1
        ]
      ],
      "nonzero_generator": "H",
      "condition": "e=0 modulo 2",
      "global_form": "SO(3)=SU(2)/Z_2",
      "discrete_theta": "n=0 on spin four-manifolds",
      "stabilizer": "line-lattice stabilizer Gamma^0(2): b=0 modulo 2"
    },
    {
      "id": "so3_minus",
      "display": "SO(3)_-",
      "genuine_subgroup": [
        [
          0,
          0
        ],
        [
          1,
          1
        ]
      ],
      "nonzero_generator": "D",
      "condition": "e=m modulo 2",
      "global_form": "SO(3)=SU(2)/Z_2",
      "discrete_theta": "n=1 on spin four-manifolds",
      "stabilizer": "line-lattice stabilizer Gamma_theta=<S,T^2>"
    }
  ],
  "transformations": {
    "S": {
      "coupling_matrix": [
        [
          0,
          -1
        ],
        [
          1,
          0
        ]
      ],
      "passive_charge_matrix": [
        [
          0,
          1
        ],
        [
          -1,
          0
        ]
      ],
      "coupling_rule": "tau maps to -1/tau",
      "charge_rule": "(e,m) maps to (m,-e)"
    },
    "T": {
      "coupling_matrix": [
        [
          1,
          1
        ],
        [
          0,
          1
        ]
      ],
      "passive_charge_matrix": [
        [
          1,
          -1
        ],
        [
          0,
          1
        ]
      ],
      "coupling_rule": "tau maps to tau+1",
      "charge_rule": "(e,m) maps to (e-m,m)"
    },
    "action_table": [
      {
        "source": "su2",
        "S_target": "so3_plus",
        "T_target": "su2"
      },
      {
        "source": "so3_plus",
        "S_target": "su2",
        "T_target": "so3_minus"
      },
      {
        "source": "so3_minus",
        "S_target": "so3_minus",
        "T_target": "so3_plus"
      }
    ]
  },
  "invariants": {
    "dirac_pairing": "e m_prime-m e_prime",
    "modular_bps_norm": "|e+tau m|^2/Im(tau)",
    "group_relations": "S^2=(ST)^3=-I before reduction modulo 2"
  },
  "six_dimensional_transport": {
    "input": "type A_1 (2,0) relative theory on an oriented marked torus",
    "defect_group": "Z_2",
    "homology_basis": "A dot B=+1",
    "wrapped_charge": "gamma=e A+m B",
    "polarization": "a maximal-isotropic choice selects which reduced line classes are genuine",
    "outputs": [
      "SU(2)",
      "SO(3)_+",
      "SO(3)_-"
    ],
    "mapping_class_status": "SL(2,Z) naturally acts on the family of polarizations; a node's line-lattice stabilizer is a necessary candidate internal subgroup, while a full quantum duality also requires compatible background, counterterm, and observable data"
  },
  "owner_integration": {
    "canonical_owner_id": "qft.topic.n4-duality.line-operators-global-forms-discrete-theta",
    "canonical_route": "/supersymmetry-duality/n4-s-duality-higher-dimensional/line-operators-global-forms-discrete-theta/",
    "canonical_anchor": "n4-charge-lattice-global-form-s-duality-orbits",
    "contextual_owner_ids": [
      "qft.topic.n4-duality.field-content-action-superconformal-data",
      "qft.topic.duality.montonen-olive-s-duality",
      "qft.topic.n4-duality.duality-groupoids-walls",
      "qft.topic.n4-duality.higher-dimensional-origin-duality-frames"
    ]
  },
  "accessibility": {
    "alt_text": "Three-panel monochrome diagram. Panel A shows the four screened center-charge classes zero, W=(1,0), H=(0,1), and D=(1,1), and the three maximal-isotropic genuine-line subgroups selecting SU(2), SO(3) plus, and SO(3) minus. Panel B shows solid S arrows exchanging SU(2) with SO(3) plus and fixing SO(3) minus, while dashed T arrows fix SU(2) and exchange the two SO(3) theories; each node displays its line-lattice stabilizer. Panel C shows a marked torus with A and B cycles, the passive coupling and charge transformations, preservation of the Dirac pairing and modular BPS norm, and the polarization needed to select a four-dimensional line-spectrum node; background and counterterm data are additional.",
    "semantic_equivalent": "/figures/supersymmetry-duality/n4-charge-lattice-global-form-s-duality-orbits.json",
    "monochrome": true,
    "direct_labels": true,
    "color_only_encoding": false,
    "narrow_width_behavior": "natural-width SVG inside a page-local horizontal scroller",
    "full_size_route": "/figures/supersymmetry-duality/n4-charge-lattice-global-form-s-duality-orbits.svg"
  },
  "limitations": [
    "The Z_2^2 quotient suppresses full weight and coweight labels, spin refinements, monopole bubbling, and supersymmetric line dressing.",
    "The displayed orbit is the simply-laced su(2) example; non-simply-laced algebras use Langlands-dual data and a lacing-dependent modular transformation.",
    "The displayed nodes omit background one-form fields, their local counterterms, possible gravitational phases, line spin, and the rest of the observable dictionary; returning the finite lattice is necessary but not sufficient for an internal quantum duality.",
    "A mapping-class construction transports a specified six-dimensional input; it does not prove an intrinsic construction of that input or equality of every unprotected observable.",
    "The figure classifies line-spectrum nodes and exact lattice transport, not the evidential strength of the full nonperturbative S-duality conjecture."
  ],
  "scientific_sources": [
    {
      "authors": "Ofer Aharony, Nathan Seiberg, and Yuji Tachikawa",
      "year": 2013,
      "title": "Reading between the lines of four-dimensional gauge theories",
      "publication": "Journal of High Energy Physics 08 (2013) 115",
      "doi": "10.1007/JHEP08(2013)115",
      "url": "https://arxiv.org/abs/1305.0318",
      "locator": "arXiv v5, section 1.2, printed pp. 4-5; section 2.4, printed pp. 17-18",
      "use": "The three su(2) line spectra, theta-shift relation, complete line-set criterion, and S-duality transport of global theory data."
    },
    {
      "authors": "Yuji Tachikawa",
      "year": 2014,
      "title": "On the 6d origin of discrete additional data of 4d gauge theories",
      "publication": "Journal of High Energy Physics 05 (2014) 020",
      "doi": "10.1007/JHEP05(2014)020",
      "url": "https://arxiv.org/abs/1309.0697",
      "locator": "sections 2-4",
      "use": "Polarization and maximal-isotropic data in six dimensions, their reduction to four-dimensional global form and discrete theta choices, and mapping-class transport."
    }
  ],
  "verification": {
    "determinant_and_pairing_checks": [
      {
        "matrix": [
          [
            0,
            -1
          ],
          [
            1,
            0
          ]
        ],
        "determinant": 1,
        "passive_charge_matrix": [
          [
            0,
            1
          ],
          [
            -1,
            0
          ]
        ],
        "pairing_preserved": true
      },
      {
        "matrix": [
          [
            1,
            1
          ],
          [
            0,
            1
          ]
        ],
        "determinant": 1,
        "passive_charge_matrix": [
          [
            1,
            -1
          ],
          [
            0,
            1
          ]
        ],
        "pairing_preserved": true
      },
      {
        "matrix": [
          [
            0,
            -1
          ],
          [
            1,
            1
          ]
        ],
        "determinant": 1,
        "passive_charge_matrix": [
          [
            0,
            1
          ],
          [
            -1,
            1
          ]
        ],
        "pairing_preserved": true
      }
    ],
    "group_relations": {
      "S_squared": "-I",
      "ST_cubed": "-I"
    },
    "word_order_check": {
      "word": "ST",
      "convention": "T acts first, then S",
      "source": "so3_plus",
      "after_rightmost_T": "so3_minus",
      "target": "so3_minus",
      "left_first_counterexample_target": "su2"
    },
    "action_table": [
      {
        "source": "su2",
        "S_target": "so3_plus",
        "T_target": "su2"
      },
      {
        "source": "so3_plus",
        "S_target": "su2",
        "T_target": "so3_minus"
      },
      {
        "source": "so3_minus",
        "S_target": "so3_minus",
        "T_target": "so3_plus"
      }
    ],
    "stabilizer_checks_over_SL2_Z2": [
      {
        "matrix": [
          [
            0,
            1
          ],
          [
            1,
            0
          ]
        ],
        "targets": {
          "su2": "so3_plus",
          "so3_plus": "su2",
          "so3_minus": "so3_minus"
        },
        "c_zero": false,
        "b_zero": false,
        "in_Gamma_theta": true
      },
      {
        "matrix": [
          [
            0,
            1
          ],
          [
            1,
            1
          ]
        ],
        "targets": {
          "su2": "so3_plus",
          "so3_plus": "so3_minus",
          "so3_minus": "su2"
        },
        "c_zero": false,
        "b_zero": false,
        "in_Gamma_theta": false
      },
      {
        "matrix": [
          [
            1,
            0
          ],
          [
            0,
            1
          ]
        ],
        "targets": {
          "su2": "su2",
          "so3_plus": "so3_plus",
          "so3_minus": "so3_minus"
        },
        "c_zero": true,
        "b_zero": true,
        "in_Gamma_theta": true
      },
      {
        "matrix": [
          [
            1,
            0
          ],
          [
            1,
            1
          ]
        ],
        "targets": {
          "su2": "so3_minus",
          "so3_plus": "so3_plus",
          "so3_minus": "su2"
        },
        "c_zero": false,
        "b_zero": true,
        "in_Gamma_theta": false
      },
      {
        "matrix": [
          [
            1,
            1
          ],
          [
            0,
            1
          ]
        ],
        "targets": {
          "su2": "su2",
          "so3_plus": "so3_minus",
          "so3_minus": "so3_plus"
        },
        "c_zero": true,
        "b_zero": false,
        "in_Gamma_theta": false
      },
      {
        "matrix": [
          [
            1,
            1
          ],
          [
            1,
            0
          ]
        ],
        "targets": {
          "su2": "so3_minus",
          "so3_plus": "su2",
          "so3_minus": "so3_plus"
        },
        "c_zero": false,
        "b_zero": false,
        "in_Gamma_theta": false
      }
    ],
    "maximal_isotropic_checks": [
      {
        "theory": "su2",
        "internal_pairings_mod_2": [
          0,
          0,
          0,
          0
        ],
        "excluded": [
          {
            "charge": [
              0,
              1
            ],
            "pairing_with_generator_mod_2": 1
          },
          {
            "charge": [
              1,
              1
            ],
            "pairing_with_generator_mod_2": 1
          }
        ]
      },
      {
        "theory": "so3_plus",
        "internal_pairings_mod_2": [
          0,
          0,
          0,
          0
        ],
        "excluded": [
          {
            "charge": [
              1,
              0
            ],
            "pairing_with_generator_mod_2": 1
          },
          {
            "charge": [
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              1
            ],
            "pairing_with_generator_mod_2": 1
          }
        ]
      },
      {
        "theory": "so3_minus",
        "internal_pairings_mod_2": [
          0,
          0,
          0,
          0
        ],
        "excluded": [
          {
            "charge": [
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            ],
            "pairing_with_generator_mod_2": 1
          },
          {
            "charge": [
              0,
              1
            ],
            "pairing_with_generator_mod_2": 1
          }
        ]
      }
    ],
    "modular_bps_norm_fixtures": [
      {
        "matrix": [
          [
            0,
            -1
          ],
          [
            1,
            0
          ]
        ],
        "tau": [
          0.37,
          1.41
        ],
        "charge": [
          3,
          -2
        ],
        "transformed_tau": [
          -0.17411764705882357,
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        ],
        "transformed_charge": [
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        ],
        "before": 9.26241134751773,
        "after": 9.262411347517732,
        "absolute_error": 1.7763568394002505e-15
      },
      {
        "matrix": [
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            1,
            1
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          [
            0,
            1
          ]
        ],
        "tau": [
          -0.21,
          0.93
        ],
        "charge": [
          -4,
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        ],
        "transformed_tau": [
          0.79,
          0.93
        ],
        "transformed_charge": [
          -9,
          5
        ],
        "before": 50.67204301075269,
        "after": 50.67204301075269,
        "absolute_error": 0
      },
      {
        "matrix": [
          [
            0,
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          ],
          [
            1,
            1
          ]
        ],
        "tau": [
          0.49,
          2.17
        ],
        "charge": [
          7,
          3
        ],
        "transformed_tau": [
          -0.21503824505700678,
          0.3131765045461105
        ],
        "transformed_charge": [
          3,
          -4
        ],
        "before": 52.590322580645164,
        "after": 52.590322580645164,
        "absolute_error": 0
      }
    ],
    "maximum_bps_norm_error": 1.7763568394002505e-15
  }
}