{
  "schema_version": 1,
  "artifact_id": "qft.artifact.supersymmetry-duality.moduli.flatness-quotient-stratification",
  "title": "Flatness, stability, and quotient strata in one exact fixture",
  "generated_by": "figures-src/supersymmetry-duality/flatness-quotient-stratification.mjs",
  "source_revision": 1,
  "generated_on": "2026-08-24",
  "registry_lifecycle": {
    "status": "prototype",
    "public_route": null,
    "note": "This exact artifact alone is promoted from planned to prototype after materialization and review; prototype is not release acceptance."
  },
  "figure_kind": "original monochrome stability-chamber, invariant-ring, and orbit-stratification flow with an explicit white canvas",
  "quantitative_status": "exact for the displayed classical U(1) fixture, equations, chamber quotients, invariant relation, dimensions, anomalies, stabilizers, and mass diagnostics; schematic and not to scale",
  "reader_question": "How can one supersymmetric vacuum problem be read as F- and D-flat equations, a compact quotient, a stability-selected complex quotient, and an invariant variety without losing the stabilizer and low-energy field data?",
  "takeaway": "For the anomaly-free U(1) fixture with charges (+1,+1,-1,-1), the site convention P=g mu+xi makes the geometric level r=-xi/g; positive and negative r give the two smooth small resolutions, while r=0 gives the rank-one determinantal cone whose origin restores the U(1) vector multiplet and all four chiral multiplets.",
  "alt_text": "A vertical black-and-gray flow begins with an anomaly-free U(1) theory containing two charge-plus-one fields x and two charge-minus-one fields y. It writes W equals zero, P equals g mu plus xi, and r equals minus xi over g, then equates the compact quotient with the stability-selected complex quotient. The flow splits into the r-positive chamber with x nonzero and the r-negative chamber with y nonzero; each is a smooth total space of two copies of O(-1) over CP1. Dashed contraction arrows rejoin at r equals zero, where the invariant matrix M equals x times y transpose obeys det M equals zero. The final comparison contrasts the smooth rank-one stratum, with trivial stabilizer and massive vector, against the singular origin, with U(1) stabilizer, vanishing Jacobian, massless vector, and four massless chirals. The diagram is classical, schematic, and not to scale.",
  "caption_semantics": "Exact classical quotient fixture in the site convention. For G=U(1), charges (+1,+1,-1,-1), canonical K, W=0, and P=g mu+xi, D-flatness is mu=r with r=-xi/g. Stability requires x nonzero for r>0 and y nonzero for r<0, giving the two smooth small resolutions Tot(O(-1) plus O(-1) over CP1). At r=0 the invariant matrix M_ij=x_i y_j obeys det M=0. Its nonzero rank-one stratum has complex dimension three and trivial stabilizer. At M=0, the determinant Jacobian vanishes, the closed representative x=y=0 has U(1) stabilizer, and both the vector multiplet and all four charged chiral multiplets are massless. The drawing is schematic and not to scale; the adjacent JSON supplies the exact equations, chamber records, adjacency list, and independent checks.",
  "conventions": {
    "spacetime": "four-dimensional rigid N=1 gauge theory used as a classical two-derivative fixture",
    "metric": "eta=diag(+1,-1,-1,-1)",
    "gauge_group": "compact U(1) with minimal electric charges normalized to plus or minus one",
    "complexified_gauge_group": "C* acting by (x,y)->(lambda x,lambda^{-1} y)",
    "gauge_covariant_derivative": "D_mu=partial_mu-i g q A_mu",
    "kahler_gauging": "Phi_dagger exp(-2 g q V) Phi",
    "fayet_iliopoulos_term": "-2 xi V",
    "moment_map_sign": "iota_k omega=-d mu with mu=sum_i q_i |phi_i|^2 for canonical K",
    "shifted_moment_map": "P=g mu+xi",
    "d_flatness": "P=0",
    "geometric_level": "r=-xi/g, so D-flatness is mu=r",
    "gauge_auxiliary_metric": "h=1 in the displayed fixture",
    "quotient_rule": "for nonzero r, mu^{-1}(r)/U(1) equals the C* GIT quotient only after the r-dependent graded semi-invariants and semistable locus are specified; at r=0 the degree-zero invariant ring gives the affine categorical quotient"
  },
  "fixture": {
    "fields": [
      {
        "name": "x_1",
        "charge": 1
      },
      {
        "name": "x_2",
        "charge": 1
      },
      {
        "name": "y_1",
        "charge": -1
      },
      {
        "name": "y_2",
        "charge": -1
      }
    ],
    "superpotential": "W=0",
    "f_term_ideal": "I_F=(0)",
    "f_flat_locus": "C^4, one irreducible classical branch",
    "moment_map": "mu=|x_1|^2+|x_2|^2-|y_1|^2-|y_2|^2",
    "shifted_moment_map": "P=g mu+xi",
    "geometric_level": "r=-xi/g",
    "anomaly_checks": {
      "cubic_u1": "sum_i q_i^3=0",
      "mixed_gravitational_u1": "sum_i q_i=0"
    },
    "invariants": [
      "M_11=x_1 y_1",
      "M_12=x_1 y_2",
      "M_21=x_2 y_1",
      "M_22=x_2 y_2"
    ],
    "relation": "det M=M_11 M_22-M_12 M_21=0",
    "coordinate_ring": "C[M_11,M_12,M_21,M_22]/(det M)"
  },
  "chambers": [
    {
      "sign": "positive",
      "condition": "r>0",
      "semistable_locus": "x is nonzero",
      "compact_level": "|x|^2-|y|^2=r",
      "quotient": "Tot(O(-1) direct-sum O(-1) -> CP^1_[x])",
      "exceptional_locus": "y=0 gives CP^1_[x]",
      "stabilizer": "trivial everywhere",
      "geometry": "smooth small resolution",
      "contraction": "the exceptional CP1 maps to M=0"
    },
    {
      "sign": "zero",
      "condition": "r=0",
      "semistable_locus": "affine categorical quotient of all C^4",
      "compact_level": "|x|^2=|y|^2",
      "quotient": "Spec C[M_11,M_12,M_21,M_22]/(det M)",
      "exceptional_locus": "none; the exceptional curve is contracted",
      "stabilizer": "trivial on M nonzero and U(1) at the closed representative of M=0",
      "geometry": "singular conifold",
      "contraction": "common affine target of both small resolutions"
    },
    {
      "sign": "negative",
      "condition": "r<0",
      "semistable_locus": "y is nonzero",
      "compact_level": "|x|^2-|y|^2=r",
      "quotient": "Tot(O(-1) direct-sum O(-1) -> CP^1_[y])",
      "exceptional_locus": "x=0 gives CP^1_[y]",
      "stabilizer": "trivial everywhere",
      "geometry": "smooth flopped small resolution",
      "contraction": "the exceptional CP1 maps to M=0"
    }
  ],
  "invariant_map": {
    "map": "(x,y)->M=x y^T",
    "image": "rank M at most one",
    "relation": "det M=M_11 M_22-M_12 M_21=0",
    "loss_of_information": "the affine invariant point M=0 does not retain the U(1) stabilizer or the complete microscopic massless field list"
  },
  "strata": [
    {
      "id": "rank_one_nonzero",
      "condition": "M is nonzero and rank M=1",
      "complex_dimension": 3,
      "jacobian_rank": 1,
      "representative": "x nonzero and y nonzero",
      "stabilizer": "identity",
      "vector_mass": "M_V^2=2 g^2 (|x|^2+|y|^2)>0",
      "massless_content": "three quotient chiral coordinates",
      "smoothness": "smooth"
    },
    {
      "id": "origin",
      "condition": "M=0 at its closed orbit representative x=y=0",
      "complex_dimension": 3,
      "zariski_tangent_dimension": 4,
      "jacobian_rank": 0,
      "stabilizer": "U(1)",
      "vector_mass": "M_V^2=0",
      "chiral_hessian": "W_ij=0",
      "massless_content": "one U(1) vector multiplet and all four charged chiral multiplets",
      "smoothness": "singular"
    }
  ],
  "branch_and_stratum_adjacency": [
    {
      "from": "positive chamber exceptional CP1 y=0",
      "to": "r=0 origin M=0",
      "map": "small-resolution contraction"
    },
    {
      "from": "negative chamber exceptional CP1 x=0",
      "to": "r=0 origin M=0",
      "map": "flopped small-resolution contraction"
    },
    {
      "from": "r=0 rank-one stratum",
      "to": "r=0 origin",
      "map": "closure inclusion"
    }
  ],
  "dimension_checks": {
    "field_space_complex_dimension": 4,
    "generic_complexified_orbit_dimension": 1,
    "quotient_complex_dimension": 3,
    "field_space_real_dimension": 8,
    "moment_map_real_constraints": 1,
    "compact_orbit_real_dimension": 1,
    "quotient_real_dimension": 6
  },
  "inference_boundaries": [
    "The fixture has W=0 and one irreducible F-flat branch; a nonzero superpotential requires a new ideal decomposition before quotienting.",
    "The equality between compact and complex quotients requires the displayed r-dependent stability condition.",
    "The affine invariant cone does not by itself determine stabilizers, masses, or the correct low-energy field content.",
    "The figure establishes a classical two-derivative quotient and does not establish a quantum-exact metric or infrared fixed point.",
    "Changing the global gauge group, charge normalization, FI admissibility, anomaly status, or added matter changes the quotient record."
  ],
  "scientific_references": [
    {
      "authors": "George Kempf and Linda Ness",
      "title": "The Length of Vectors in Representation Spaces",
      "container": "Algebraic Geometry, Lecture Notes in Mathematics 732",
      "pages": "233-243",
      "year": 1979,
      "doi": "10.1007/BFb0066647",
      "url": "https://doi.org/10.1007/BFb0066647",
      "use": "compact quotient versus stability-selected complex quotient"
    },
    {
      "authors": "Markus A. Luty and Washington Taylor IV",
      "title": "Varieties of Vacua in Classical Supersymmetric Gauge Theories",
      "journal": "Physical Review D",
      "volume": "53",
      "pages": "3399-3405",
      "year": 1996,
      "arxiv": "hep-th/9506098",
      "url": "https://arxiv.org/abs/hep-th/9506098",
      "use": "classical supersymmetric vacuum varieties and complexified gauge orbits"
    },
    {
      "authors": "David R. Morrison and M. Ronen Plesser",
      "title": "Non-Spherical Horizons, I",
      "journal": "Advances in Theoretical and Mathematical Physics",
      "volume": "3",
      "pages": "1-81",
      "year": 1999,
      "doi": "10.4310/ATMP.1999.v3.n1.a1",
      "url": "https://doi.org/10.4310/ATMP.1999.v3.n1.a1",
      "locator": "section 4.2, table 3, page 24; figure 3 and discussion, page 26",
      "use": "the (1,1,-1,-1) conifold charge row, its two D-term small resolutions, and the flop between them"
    },
    {
      "authors": "Edward Witten",
      "title": "Phases of N=2 Theories in Two Dimensions",
      "journal": "Nuclear Physics B",
      "volume": "403",
      "pages": "159-222",
      "year": 1993,
      "doi": "10.1016/0550-3213(93)90033-L",
      "url": "https://doi.org/10.1016/0550-3213(93)90033-L",
      "use": "FI chambers and toric quotient phases"
    },
    {
      "authors": "Steven Weinberg",
      "title": "The Quantum Theory of Fields, Volume III: Supersymmetry",
      "publisher": "Cambridge University Press",
      "year": 2000,
      "doi": "10.1017/CBO9781139644198",
      "url": "https://doi.org/10.1017/CBO9781139644198",
      "locator": "chapter 27",
      "use": "four-dimensional N=1 auxiliary fields, moment maps, and scalar potential"
    }
  ],
  "accessibility_encoding": {
    "color_independence": "Black outlines, gray fill, solid flow arrows, dashed contraction arrows, explicit r labels, and written smooth/singular and stabilizer labels encode every distinction without color.",
    "structured_equivalent": "This JSON supplies the exact field and charge table, equations, chamber rows, invariant generators and relation, dimensions, stabilizers, mass data, adjacency list, limitations, and source chain."
  }
}