{
  "schema_version": 1,
  "artifact_id": "qft.artifact.supersymmetry-duality.bps.charges-defects-wall-crossing",
  "title": "One rank-two BPS wall: charges, defects, and protected transport",
  "generated_by": "figures-src/supersymmetry-duality/charges-defects-wall-crossing.mjs",
  "source_revision": 2,
  "generated_on": "2026-08-24",
  "registry_lifecycle": {
    "current_status": "read from the governed registry when provenance is generated",
    "recommended_status": "prototype",
    "public_route": null,
    "note": "Materialization does not itself promote the governed record or assert release acceptance."
  },
  "figure_kind": "original monochrome charge-role boundary, central-charge-ray chamber flow, mass threshold, and KS identity diagram with an explicit white canvas",
  "quantitative_status": "exact for the frozen lattice, central charges, alignment diagnostic, radius sign, primitive indices, masses, and KS identity; ray angles and spacing are schematic and not to scale",
  "reader_question": "How does one exact rank-two path connect particle central charges, a line-defect core-halo torsor, and wall crossing without confusing particle electromagnetic charges with tensorial wall or string charges?",
  "takeaway": "For gamma_1 paired once with gamma_2, Z_1=1+i t and Z_2=1-i t give a stable gamma_1+gamma_2 particle only for t>0; the protected index drops from one to zero when t decreases through zero, while the rightmost-first KS pentagon preserves the ordered transport. Line-defect sectors are an affine torsor over the particle lattice, whereas wall and string charges are tensorial extensions outside that particle lattice.",
  "alt_text": "A monochrome vertical diagram first separates three charge roles. Particle electromagnetic charges lie in a rank-two lattice, enter a Lorentz-scalar central charge, and carry bulk protected indices. A line defect instead has an affine charge torsor with a reference core plus particle halos. Walls and strings carry tensorial topological charges and tension bounds outside the particle electromagnetic lattice. The exact particle fixture then sets the pairing of gamma one with gamma two to one, Z one to one plus i t, and Z two to one minus i t for absolute t below one. Solid, dashed, and dotted rays show the stable-composite chamber t positive, the aligned wall t zero, and the composite-absent chamber t negative. The seed states gamma one and gamma two remain on both sides, while the composite index Omega one-two changes from one to zero. The conditional composite mass is two, while the constituent sum is two times square root of one plus t squared and agrees only at the wall. The final panel states K two K one equals K one K one-two K two for the displayed rightmost-first coordinate-function action. The ray angles are schematic; every equation and index is exact only for the frozen primitive composite sector.",
  "caption_semantics": "Exact primitive rank-two BPS fixture. The bulk particle lattice is Gamma=Z gamma_1 plus Z gamma_2 with pairing one, Z_1=1+i t, Z_2=1-i t, and -1<t<1 on the principal continuous phase branch. The two-center separation is positive only in the stable-composite chamber C_+ with t>0, so Omega(gamma_1+gamma_2) changes from one to zero along the oriented crossing C_+ to the composite-absent chamber C_-. The seed indices Omega_1 and Omega_2 remain one. The conditional composite BPS mass is two, below the constituent threshold away from the wall and equal to it only at t=0. The KS equality uses K_gamma(X_beta)=X_beta(1-X_gamma) to the pairing beta,gamma, with rightmost-first coordinate-function pullback action. The upper bands mark a type boundary: particle scalar central charges use the electromagnetic lattice; line sectors form a torsor over it; wall and string charges are tensorial extensions. Ray angles are schematic and not to scale; every spectrum statement is restricted to the displayed primitive composite sector. The adjacent JSON supplies the exact chamber table, charge-role table, equations, source locators, and independent checks.",
  "conventions": {
    "spacetime": "four-dimensional rigid N=2 for the particle and framed sectors; generic supersymmetric wall and string charges are shown only as a type boundary",
    "particle_charge_lattice": "Gamma=Z gamma_1 direct-sum Z gamma_2 with antisymmetric pairing <gamma_1,gamma_2>=+1",
    "particle_central_charge": "Z_gamma is a Lorentz scalar, additive in the electromagnetic charge gamma",
    "index": "second-helicity-supertrace convention of the owner page; primitive composite-present-minus-composite-absent jump is (-1)^(kappa-1)|kappa| Omega_1 Omega_2",
    "path_orientation": "t decreases from C_+ (0<t<1) through t=0 to C_- (-1<t<0)",
    "ks_variables": "X_gamma X_gamma_prime=(-1)^(<gamma,gamma_prime>) X_(gamma+gamma_prime)",
    "ks_automorphism": "K_gamma(X_beta)=X_beta(1-X_gamma)^(<beta,gamma>)",
    "ks_composition": "rightmost-first coordinate-function pullback action, (A B)(X)=A(B(X)); tuple point maps compose in the reverse order",
    "phase_branch": "principal continuous branch arg Z_i in (-pi/2,pi/2) for -1<t<1, with arg Z_12=0",
    "supercharge_normalization": "the particle inequality is displayed in the owner-page normalization M_gamma>=|Z_gamma|; no R charge enters this fixture",
    "global_form_scope": "an abstract primitive rank-two local electromagnetic sublattice is frozen; the fixture does not select a gauge-group global form or a complete spectrum of genuine lines",
    "line_defect": "Gamma_L is an affine torsor gamma_c+Gamma; gamma_c is a reference core and gamma_h is a bulk-particle halo charge",
    "extended_objects": "wall and string charges are tensorial topological extensions tied to oriented tension bounds and are not elements of the particle electromagnetic lattice"
  },
  "fixture": {
    "parameter_domain": "-1<t<1",
    "basis": [
      "gamma_1",
      "gamma_2"
    ],
    "pairing_matrix": [
      [
        0,
        1
      ],
      [
        -1,
        0
      ]
    ],
    "central_charges": {
      "gamma_1": "Z_1=1+i t",
      "gamma_2": "Z_2=1-i t",
      "gamma_1_plus_gamma_2": "Z_12=2"
    },
    "alignment_diagnostic": "Im(Z_1 conjugate(Z_2))=2t",
    "positive_alignment_control": "Re(Z_1 conjugate(Z_2))=1-t^2>0",
    "constituent_indices": {
      "gamma_1": 1,
      "gamma_2": 1
    },
    "primitive_pairing": 1,
    "oriented_crossing": "C_+ -> wall -> C_- as t decreases",
    "composite_index_jump": "Omega_12: 1 -> 0"
  },
  "charge_roles": [
    {
      "role": "bulk_particle",
      "charge_space": "Gamma_em",
      "algebraic_charge": "Lorentz-scalar Z_gamma",
      "observable": "particle mass bound M_gamma>=|Z_gamma| and bulk Omega(gamma)",
      "participates_in_fixture_ks_product": true
    },
    {
      "role": "line_defect",
      "charge_space": "Gamma_L=gamma_c+Gamma_em, an affine torsor",
      "algebraic_charge": "reference core plus a bulk-particle halo charge",
      "observable": "framed index underlined Omega(L,gamma)",
      "participates_in_fixture_ks_product": "indirectly, through the action of bulk-particle halo factors"
    },
    {
      "role": "wall_or_string",
      "charge_space": "tensorial topological extension",
      "algebraic_charge": "Z_(mu nu ...) adapted to the unbroken worldvolume symmetry",
      "observable": "oriented tension bound and worldvolume projector",
      "participates_in_fixture_ks_product": false
    }
  ],
  "chambers": [
    {
      "chamber": "C_+",
      "sector_status": "stable composite present",
      "parameter": "0<t<1",
      "phase_order": "arg Z_1>arg Z_12=0>arg Z_2",
      "im_z1_conj_z2": "2t>0",
      "two_center_radius": "r_12=1/(2t)>0",
      "stable_charges": [
        "gamma_1",
        "gamma_1+gamma_2",
        "gamma_2"
      ],
      "protected_indices": {
        "gamma_1": 1,
        "gamma_2": 1,
        "gamma_1_plus_gamma_2": 1
      },
      "ks_factors": "K_1 K_12 K_2"
    },
    {
      "chamber": "wall",
      "sector_status": "aligned marginal-stability wall",
      "parameter": "t=0",
      "phase_order": "arg Z_1=arg Z_12=arg Z_2=0",
      "im_z1_conj_z2": "0",
      "re_z1_conj_z2": "1>0",
      "two_center_radius": "diverges",
      "threshold": "M_12=M_1+M_2=2"
    },
    {
      "chamber": "C_-",
      "sector_status": "composite absent; the two seed states remain",
      "parameter": "-1<t<0",
      "phase_order": "arg Z_2>arg Z_12=0>arg Z_1",
      "im_z1_conj_z2": "2t<0",
      "two_center_radius": "negative in the displayed orientation; no two-center state",
      "stable_charges": [
        "gamma_1",
        "gamma_2"
      ],
      "protected_indices": {
        "gamma_1": 1,
        "gamma_2": 1,
        "gamma_1_plus_gamma_2": 0
      },
      "ks_factors": "K_2 K_1"
    }
  ],
  "mass_threshold": {
    "conditional_composite_bps_mass": "M_12=|Z_1+Z_2|=2 in C_+",
    "constituent_threshold": "M_1+M_2=2 sqrt(1+t^2)",
    "strict_binding": "2<2 sqrt(1+t^2) for t nonzero",
    "marginal_equality": "M_12=M_1+M_2=2 only at t=0",
    "inference_boundary": "|Z_12|=2 is defined throughout the path, but it is a particle mass only where a gamma_1+gamma_2 BPS state exists"
  },
  "wall_crossing": {
    "oriented_jump": "C_+ -> C_- sends Omega_12 from 1 to 0",
    "primitive_composite_present_minus_absent_jump": 1,
    "constituent_indices_unchanged": true,
    "ks": {
      "identity": "K_2 K_1=K_1 K_12 K_2",
      "automorphism": "K_gamma(X_beta)=X_beta(1-X_gamma)^(<beta,gamma>)",
      "composition": "rightmost-first coordinate-function pullback action, (A B)(X)=A(B(X)); tuple point maps compose in the reverse order",
      "left_chamber_factors": "C_-: K_2 K_1",
      "right_chamber_factors": "C_+: K_1 K_12 K_2",
      "gmn_translation": "GMN eq. (2.16) uses the same <beta,gamma> exponent. Their eq. (2.21) is written in point-map order; reversing every factor gives the displayed rightmost-first coordinate-pullback automorphism identity."
    }
  },
  "semantic_table": [
    {
      "chamber": "C_+",
      "sector_status": "stable composite present",
      "parameter": "0<t<1",
      "phase_order": "arg Z_1>arg Z_12=0>arg Z_2",
      "radius": "r_12=1/(2t)>0",
      "stable_charges": [
        "gamma_1",
        "gamma_1+gamma_2",
        "gamma_2"
      ],
      "protected_indices": {
        "gamma_1": 1,
        "gamma_2": 1,
        "gamma_1_plus_gamma_2": 1
      }
    },
    {
      "chamber": "wall",
      "sector_status": "aligned marginal-stability wall",
      "parameter": "t=0",
      "phase_order": "arg Z_1=arg Z_12=arg Z_2=0",
      "radius": "diverges",
      "stable_charges": "threshold continuum",
      "protected_indices": "wall value not assigned"
    },
    {
      "chamber": "C_-",
      "sector_status": "composite absent; the two seed states remain",
      "parameter": "-1<t<0",
      "phase_order": "arg Z_2>arg Z_12=0>arg Z_1",
      "radius": "negative in the displayed orientation; no two-center state",
      "stable_charges": [
        "gamma_1",
        "gamma_2"
      ],
      "protected_indices": {
        "gamma_1": 1,
        "gamma_2": 1,
        "gamma_1_plus_gamma_2": 0
      }
    }
  ],
  "inference_boundaries": [
    "The parameter band -1<t<1 keeps Re(Z_1 conjugate Z_2)>0 and isolates the aligned marginal-stability wall at t=0 from the anti-aligned regime.",
    "The primitive index jump assumes no additional aligned rays, nonprimitive states, or scaling solutions in the chosen sector.",
    "The mass formula is conditional on state existence; the charge lattice and central charge do not populate a BPS state automatically.",
    "The line-defect panel classifies charge spaces and halo action but does not identify a line core with a dynamical bulk particle.",
    "The tensorial wall or string panel is a type boundary only; it does not place extended charges in the particle KS lattice.",
    "Protected wall crossing does not determine unprotected excitations or prove completeness outside the displayed charge sector."
  ],
  "scientific_references": [
    {
      "authors": "Edward Witten and David Olive",
      "title": "Supersymmetry Algebras That Include Topological Charges",
      "journal": "Physics Letters B",
      "volume": "78",
      "pages": "97-101",
      "year": 1978,
      "doi": "10.1016/0370-2693(78)90357-X",
      "url": "https://doi.org/10.1016/0370-2693(78)90357-X",
      "use": "BPS positivity and topological extensions of supersymmetry algebras"
    },
    {
      "authors": "Frederik Denef",
      "title": "Supergravity Flows and D-Brane Stability",
      "journal": "Journal of High Energy Physics",
      "issue": "08",
      "article": "050",
      "year": 2000,
      "arxiv": "hep-th/0005049",
      "url": "https://arxiv.org/abs/hep-th/0005049",
      "locator": "eq. (6.5), p. 24, and eqs. (6.6)-(6.7), p. 28",
      "use": "sign of the two-center radius across marginal stability"
    },
    {
      "authors": "Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke",
      "title": "Four-Dimensional Wall-Crossing via Three-Dimensional Field Theory",
      "journal": "Communications in Mathematical Physics",
      "volume": "299",
      "pages": "163-224",
      "year": 2010,
      "arxiv": "0807.4723",
      "url": "https://arxiv.org/abs/0807.4723",
      "locator": "eqs. (2.13)-(2.16) and (2.20)-(2.21)",
      "use": "twisted coordinates, elementary K transformation, and pentagon identity"
    },
    {
      "authors": "Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke",
      "title": "Framed BPS States",
      "journal": "Advances in Theoretical and Mathematical Physics",
      "volume": "17",
      "pages": "241-397",
      "year": 2013,
      "arxiv": "1006.0146",
      "url": "https://arxiv.org/abs/1006.0146",
      "locator": "section 3.1, eqs. (3.1)-(3.3); sections 3.3-3.4, especially eqs. (3.23), (3.32), and (3.36)",
      "use": "line-defect charge torsors, core-halo sectors, and framed halo transformations"
    }
  ],
  "accessibility_encoding": {
    "color_independence": "Black outlines, white and gray fills, solid gamma_1 rays, dashed gamma_2 rays, dotted composite rays, direct labels, named chambers, and written stable-composite, wall, and composite-absent states encode every distinction without color.",
    "structured_equivalent": "This JSON supplies the exact charge-role table, lattice and pairing, central charges, chamber table, stable charges and protected indices, mass threshold, KS convention and identity, limitations, source locators, and independent checks."
  }
}