{
  "schema_version": 1,
  "artifact_id": "qft.artifact.supersymmetry-duality.actions.superspace-component-reduction-flow",
  "title": "Superspace to components: a reproducible reduction",
  "generated_by": "figures-src/supersymmetry-duality/superspace-component-reduction-flow.mjs",
  "source_revision": 2,
  "revision_note": "Align the gauge-matter branch with the Chapter-3 prepotential, charged-matter exponential, FI sign, covariant derivative, positive D solution, and phase-bearing gauge Yukawa convention.",
  "generated_on": "2026-08-24",
  "registry_lifecycle": {
    "status": "planned",
    "public_route": null,
    "note": "Materialization does not advance the checksum-bound registry lifecycle or assign a public route."
  },
  "figure_kind": "original vertically stacked monochrome superspace-to-component reduction flow with an explicit white canvas",
  "quantitative_status": "exact for the displayed convention package, coefficient selections, algebraic auxiliary equations, Wess-Zumino anchor, gauge-matter anchor, dimensions, and normalized cancellation checks; schematic and not to scale",
  "reader_question": "How can a four-dimensional N=1 superspace action be reduced without losing a Berezin sign, a boundary term, an off-shell cancellation, or the algebraic F and D equations that determine masses, Yukawa couplings, and the scalar potential?",
  "takeaway": "Freeze the measure and derivative order, select the complete Grassmann coefficient, retain the boundary current, verify supersymmetry before eliminating auxiliaries, and only then complete the F and D squares; this single workflow reproduces the Wess-Zumino mass and Yukawa data and the positive gauge-matter F-plus-D potential.",
  "alt_text": "A vertical black-and-gray eight-stage reduction flow. It fixes the chiral, antichiral, and full-superspace projectors; writes the canonical chiral and gauge-matter inputs; selects the raw D-term and holomorphic F-term components; records the one discarded boundary current and the off-shell supersymmetry cancellations; splits into a cubic Wess-Zumino branch and a general gauge-matter branch; rejoins at the exact F and D completing-the-square equations; and ends with the positive scalar potential, the Wess-Zumino Yukawa coupling, and boson-fermion mass agreement. A final box limits the result to regular positive-metric two-derivative branches with the declared boundary assumptions.",
  "caption_semantics": "Exact convention-controlled reduction transcript for the displayed rigid two-derivative systems. The raw canonical D-term differs from the kinetic representative by the printed divergence, while the chiral coefficient is F^i W_i minus one half W_ij psi^i psi^j. The off-shell action is checked before solving F or D. With V=-theta sigma bar_theta A+i theta^2 bar_theta bar_lambda-i bar_theta^2 theta lambda+(1/2)theta^2 bar_theta^2 D, charged matter Phi_dagger exp(-2gV)Phi, and FI term -2 xi V, completing both positive quadratic forms yields F^i=-g^{i bar j} bar W_bar j, D^a=+h^{-1,ab} P_b, and V=g^{-1} W bar W plus one half P h^{-1} P. For W=m Phi squared over two plus y Phi cubed over three, the same transcript gives the displayed Yukawa coefficient and scalar and fermion masses. The drawing is schematic and not to scale; the adjacent JSON is the complete semantic transcript.",
  "intended_canonical_placement": {
    "page_route": "/supersymmetry-duality/actions-supercurrents-effective-theory/action-principles-component-reduction/",
    "anchor": "superspace-component-reduction-flow",
    "placement_is_not_a_registry_public_route": true
  },
  "conventions": {
    "spacetime": "four-dimensional Lorentzian spacetime",
    "metric": "eta=diag(+1,-1,-1,-1)",
    "supersymmetry": "rigid N=1",
    "sigma_matrices": "sigma^mu=(identity,sigma-vector), bar_sigma^mu=(identity,-sigma-vector)",
    "spinor_epsilon": "epsilon^{12}=epsilon_{21}=+1",
    "odd_derivatives": "all odd derivatives act from the left",
    "derivative_order": "D^2=D^alpha D_alpha, bar_D^2=bar_D_dot_alpha bar_D^dot_alpha, and full projection is D^2 bar_D^2 in that order",
    "chiral_coordinate": "y^mu=x^mu+i theta sigma^mu bar_theta",
    "chiral_expansion": "Phi(y,theta)=phi(y)+sqrt(2) theta psi(y)+theta^2 F(y)",
    "berezin_orientation": {
      "chiral": "integral d^2 theta theta^2=1",
      "antichiral": "integral d^2 bar_theta bar_theta^2=1",
      "full_measure_order": "d^4 theta=d^2 theta d^2 bar_theta",
      "full": "integral d^4 theta theta^2 bar_theta^2=1"
    },
    "component_projection": "vertical bar means theta=bar_theta=0",
    "vector_prepotential": "Wess-Zumino representative V=-theta sigma^mu bar_theta A_mu+i theta^2 bar_theta bar_lambda-i bar_theta^2 theta lambda+(1/2)theta^2 bar_theta^2 D",
    "gauge_covariant_derivative": "D_mu=partial_mu-i g A_mu",
    "charged_matter_d_term": "Phi_dagger exp(-2gV) Phi",
    "fayet_iliopoulos_d_term": "-2 xi_a V^a",
    "field_normalization": "canonical chiral K=Phi_dagger Phi; gauge auxiliary sector uses h_ab=Re f_ab, P_a=g mu_a+xi_a, and L_Daux=(1/2)h_ab D^a D^b-D^a P_a",
    "boundary_rule": "a spacetime divergence is removed only after decay, a boundary condition, or a compensating boundary action makes its integral vanish",
    "complex_variation": "F^i and bar_F^bar_i are varied independently before imposing Lorentzian conjugation"
  },
  "projection_operators": [
    {
      "measure": "d^2 theta",
      "integrand_class": "chiral X",
      "projector": "-D^2 X|/4",
      "selected_monomial": "theta^2 coefficient",
      "measure_dimension": 1
    },
    {
      "measure": "d^2 bar_theta",
      "integrand_class": "antichiral bar_X",
      "projector": "-bar_D^2 bar_X|/4",
      "selected_monomial": "bar_theta^2 coefficient",
      "measure_dimension": 1
    },
    {
      "measure": "d^4 theta",
      "integrand_class": "real U",
      "projector": "D^2 bar_D^2 U|/16",
      "selected_monomial": "theta^2 bar_theta^2 coefficient",
      "measure_dimension": 2
    }
  ],
  "numbered_reduction_transcript": [
    {
      "stage": 1,
      "id": "freeze_projectors",
      "action": "freeze metric, spinor, odd-derivative, measure, and projection conventions",
      "inputs": [
        "conventions",
        "projectors"
      ],
      "output": "unambiguous coefficient-selection operators",
      "required_check": "D^2 theta^2|=-4 and bar_D^2 bar_theta^2|=-4"
    },
    {
      "stage": 2,
      "id": "record_superspace_input",
      "action": "write K, W, h, P, reality, gauge, and boundary data before simplifying",
      "inputs": [
        "canonical K=Phi_dagger Phi",
        "holomorphic W",
        "gauge auxiliary metric h=Re f",
        "charged matter Phi_dagger exp(-2gV)Phi and FI term -2 xi V",
        "P=g mu+xi"
      ],
      "output": "complete superspace and auxiliary input",
      "required_check": "chirality, reality, gauge invariance, and mass dimension"
    },
    {
      "stage": 3,
      "id": "select_grassmann_coefficients",
      "action": "apply the frozen projectors and retain every selected sign and coefficient",
      "inputs": [
        "canonical_d_term",
        "holomorphic_f_term",
        "gauge_auxiliary_term"
      ],
      "output": "selectedComponents",
      "required_check": "D-term coefficients are -1, -i, +1 in the raw representative; F-term coefficients are +1 and -1/2; D quadratic and source coefficients are +1/2 and -1"
    },
    {
      "stage": 4,
      "id": "boundary_and_off_shell_check",
      "action": "record the unique aggregate divergence, state exactly why its integral vanishes, and vary the complete action before auxiliary elimination",
      "inputs": [
        "B_D^mu",
        "off-shell chiral transformations"
      ],
      "output": "delta L_off=partial_mu K^mu with no Euler-Lagrange equation",
      "required_check": "both normalized variation-route sums vanish and the divergence is not set to zero locally"
    },
    {
      "stage": 5,
      "id": "wess_zumino_anchor",
      "action": "differentiate the cubic W and expose the F and Yukawa coefficients",
      "inputs": [
        "m",
        "y",
        "phi",
        "psi",
        "F"
      ],
      "output": "wessZuminoAnchor",
      "required_check": "W_prime=m phi+y phi^2 and W_double_prime=m+2y phi"
    },
    {
      "stage": 6,
      "id": "gauge_matter_anchor",
      "action": "assemble the positive F and D quadratic forms with the moment-map source",
      "inputs": [
        "g_i_bar_j",
        "h_ab",
        "W_i",
        "P_a",
        "V=-theta sigma bar_theta A+i theta^2 bar_theta bar_lambda-i bar_theta^2 theta lambda+(1/2)theta^2 bar_theta^2 D",
        "D_mu=partial_mu-i g A_mu"
      ],
      "output": "gaugeMatterAnchor",
      "required_check": "g and h are positive and nonsingular on the selected algebraic branch"
    },
    {
      "stage": 7,
      "id": "solve_auxiliaries",
      "action": "complete both squares and solve the independent algebraic equations",
      "inputs": [
        "off-shell F sector",
        "off-shell D sector"
      ],
      "output": [
        "F^i=-g^{i bar_j}bar_W_bar_j",
        "D^a=+h^{-1,ab}P_b"
      ],
      "required_check": "substitution solves g F+bar_W=0 and h D-P=0"
    },
    {
      "stage": 8,
      "id": "read_interactions_and_checks",
      "action": "substitute once, then read potential, Yukawa matrix, pole masses, and closure status",
      "inputs": [
        "auxiliary solutions",
        "W_double_prime",
        "positive g and h"
      ],
      "output": "V=g^{-1}W bar_W+(1/2)P h^{-1}P, Wess-Zumino Yukawa, and mass degeneracy",
      "required_check": "positive potential, m_fermion=m, m_scalar_squared=|m|^2 at phi=0, and reduced action invariance is not confused with off-shell closure"
    }
  ],
  "selected_component_transcript": {
    "canonical_d_term": {
      "integrand": "Phi_dagger Phi",
      "raw_coefficient_representative": "C_D_raw=-phi* box phi-i(partial_mu bar_psi) bar_sigma^mu psi+F*F",
      "kinetic_representative": "L_D=partial_mu phi* partial^mu phi+i bar_psi bar_sigma^mu partial_mu psi+F*F",
      "exact_relation": "C_D_raw=L_D+partial_mu B_D^mu",
      "boundary_current": "B_D^mu=-phi* partial^mu phi-i bar_psi bar_sigma^mu psi",
      "selected_terms": [
        {
          "object": "scalar raw kinetic term",
          "coefficient": -1,
          "expression": "-phi* box phi"
        },
        {
          "object": "fermion raw kinetic term",
          "coefficient": "-i",
          "expression": "-i(partial_mu bar_psi) bar_sigma^mu psi"
        },
        {
          "object": "chiral auxiliary norm",
          "coefficient": 1,
          "expression": "F*F"
        }
      ],
      "discarded_total_derivatives": [
        {
          "expression": "partial_mu B_D^mu",
          "coefficient": 1,
          "integrated_statement": "integral d^4x partial_mu B_D^mu=0",
          "condition": "sufficient decay on R^{1,3}, a boundary condition, or a boundary action that cancels the flux",
          "forbidden_local_inference": "partial_mu B_D^mu is not set to zero as a local operator"
        }
      ]
    },
    "holomorphic_f_term": {
      "integrand": "W(Phi^i)",
      "selected_coefficient": "[W(Phi)]_{theta^2}=F^i W_i-(1/2)W_ij psi^i psi^j",
      "selected_terms": [
        {
          "object": "holomorphic auxiliary coupling",
          "coefficient": 1,
          "expression": "F^i W_i"
        },
        {
          "object": "holomorphic Yukawa coupling",
          "coefficient": -0.5,
          "expression": "-(1/2)W_ij psi^i psi^j"
        },
        {
          "object": "Lorentzian conjugate auxiliary coupling",
          "coefficient": 1,
          "expression": "bar_F^bar_i bar_W_bar_i"
        },
        {
          "object": "Lorentzian conjugate Yukawa coupling",
          "coefficient": -0.5,
          "expression": "-(1/2)bar_W_bar_i_bar_j bar_psi^bar_i bar_psi^bar_j"
        }
      ],
      "discarded_total_derivatives": []
    },
    "gauge_auxiliary_term": {
      "expression": "(1/2)h_ab D^a D^b-D^a P_a",
      "selected_terms": [
        {
          "object": "positive gauge auxiliary quadratic form",
          "coefficient": 0.5,
          "expression": "(1/2)h_ab D^a D^b"
        },
        {
          "object": "moment-map and FI source",
          "coefficient": -1,
          "expression": "-D^a P_a"
        }
      ],
      "discarded_total_derivatives": []
    }
  },
  "off_shell_variation_cancellation_checks": [
    {
      "id": "canonical_F_partial_psi",
      "common_monomial": "the F* epsilon sigma^mu partial_mu bar_psi term after one integration by parts, together with its conjugate",
      "contributions": [
        {
          "source": "variation of the fermion kinetic term through delta psi=sqrt(2) epsilon F+...",
          "normalized_coefficient": 1
        },
        {
          "source": "variation of F*F through delta F=i sqrt(2) bar_epsilon bar_sigma^mu partial_mu psi",
          "normalized_coefficient": -1
        }
      ],
      "sum": 0,
      "eom_used": false
    },
    {
      "id": "superpotential_W_partial_psi",
      "common_monomial": "the W_i partial_mu psi^i term after the chain rule and one integration by parts, together with its conjugate",
      "contributions": [
        {
          "source": "variation of F^i W_i",
          "normalized_coefficient": 1
        },
        {
          "source": "variation of -(1/2)W_ij psi^i psi^j using symmetric W_ij",
          "normalized_coefficient": -1
        }
      ],
      "sum": 0,
      "eom_used": false
    }
  ],
  "wess_zumino_anchor": {
    "superfield_input": "K=Phi_dagger Phi, W(Phi)=(1/2)m Phi^2+(1/3)y Phi^3",
    "coupling_dimensions": {
      "m": 1,
      "y": 0
    },
    "derivatives": {
      "W_prime": "m phi+y phi^2",
      "W_double_prime": "m+2y phi"
    },
    "off_shell_auxiliary_sector": "F*F+F W_prime+F* bar_W_prime",
    "completing_the_square": "F*F+F W_prime+F* bar_W_prime=|F+bar_W_prime|^2-|W_prime|^2",
    "auxiliary_solution": "F=-bar_W_prime",
    "on_shell_potential": "V=|m phi+y phi^2|^2",
    "on_shell_yukawa": "L_Yukawa=-(1/2)(m+2y phi)psi psi+h.c.",
    "vacua_for_nonzero_m_y": [
      "phi_0=0",
      "phi_1=-m/y"
    ],
    "mass_check": {
      "at_phi_0": "W_double_prime=m, fermion pole mass |m|, scalar mass-squared |m|^2",
      "at_phi_1": "W_double_prime=-m, fermion pole mass |m|, scalar mass-squared |m|^2",
      "inference": "the sign change is a phase convention; the singular value and pole masses agree"
    },
    "supersymmetry_status": {
      "before_elimination": "the component action is invariant and the chiral transformations close off shell",
      "after_elimination": "the reduced action remains invariant under the induced transformations, while their algebra generally closes only after the surviving fermion equation"
    }
  },
  "gauge_matter_anchor": {
    "data": {
      "kahler_metric": "g_i_bar_j positive and nonsingular on the selected branch",
      "gauge_kinetic_metric": "h_ab=Re f_ab positive and nonsingular",
      "moment_map_source": "P_a=g mu_a+xi_a"
    },
    "auxiliary_lagrangian": "g_i_bar_j F^i bar_F^bar_j+F^i W_i+bar_F^bar_j bar_W_bar_j+(1/2)h_ab D^a D^b-D^a P_a",
    "equations": {
      "F": "F^i=-g^{i bar_j}bar_W_bar_j",
      "D": "D^a=+h^{-1,ab}P_b"
    },
    "completed_squares": {
      "F": "g_i_bar_j(F^i+g^{i bar_k}bar_W_bar_k)(bar_F^bar_j+g^{l bar_j}W_l)-g^{i bar_j}W_i bar_W_bar_j",
      "D": "(1/2)(D-h^{-1}P)^T h(D-h^{-1}P)-(1/2)P^T h^{-1}P"
    },
    "potential": "V=g^{i bar_j}W_i bar_W_bar_j+(1/2)P_a h^{-1,ab}P_b",
    "canonical_fermion_interactions": [
      "-(1/2)W_ij psi^i psi^j+h.c.",
      "-i sqrt(2)g phi_dagger T^a psi lambda^a+h.c."
    ],
    "u1_anchor": {
      "fields": "Phi_+ and Phi_- of charges +1 and -1",
      "superpotential": "W=m Phi_+ Phi_-",
      "moment_map_source": "P=g(|phi_+|^2-|phi_-|^2)+xi",
      "auxiliary_solution": "D=+P when h=1",
      "potential": "V=|m|^2(|phi_+|^2+|phi_-|^2)+(1/2)[g(|phi_+|^2-|phi_-|^2)+xi]^2",
      "boundary": "the FI term is included only for a globally admissible Abelian direction with the stated charge normalization and anomaly conditions"
    }
  },
  "engineering_dimension_records": [
    {
      "term": "partial phi* partial phi",
      "factors": [
        1,
        1,
        1,
        1
      ],
      "total": 4
    },
    {
      "term": "i bar_psi bar_sigma partial psi",
      "factors": [
        1.5,
        1,
        1.5
      ],
      "total": 4
    },
    {
      "term": "F*F",
      "factors": [
        2,
        2
      ],
      "total": 4
    },
    {
      "term": "F W_i",
      "factors": [
        2,
        2
      ],
      "total": 4
    },
    {
      "term": "W_ij psi^i psi^j",
      "factors": [
        1,
        1.5,
        1.5
      ],
      "total": 4
    },
    {
      "term": "(1/2)h_ab D^a D^b",
      "factors": [
        0,
        2,
        2
      ],
      "total": 4
    },
    {
      "term": "-D^a P_a",
      "factors": [
        2,
        2
      ],
      "total": 4
    },
    {
      "term": "-i sqrt(2)g phi_dagger T psi lambda",
      "factors": [
        0,
        1,
        0,
        1.5,
        1.5
      ],
      "total": 4
    }
  ],
  "assumptions_and_inference_boundaries": [
    "The result uses flat four-dimensional Lorentzian rigid N=1 supersymmetry, the (+---) metric, left odd derivatives, and the exact Berezin and derivative order recorded above.",
    "The raw canonical D-term representative and the positive kinetic representative differ by exactly the recorded divergence; that divergence is discarded only after its integrated flux vanishes under declared boundary data.",
    "Off-shell action invariance and off-shell algebra closure are checked before F or D elimination. After elimination, the reduced action remains invariant under induced transformations, but the reduced algebra generally closes only after surviving field equations.",
    "Positive scalar-potential conclusions require positive nonsingular Kahler and gauge kinetic matrices on the selected branch.",
    "Higher-derivative operators can make a nominal auxiliary propagate or can generate multiple algebraic branches; those theories require a new reduction.",
    "The FI contribution is local data only when xi lies along an admissible Abelian direction; global gauge form, bundle, anomaly, and supercurrent conditions are separate requirements.",
    "The gauge-matter formula suppresses fermion-dependent shifts in a nonlinear Kahler model; the displayed bosonic F solution and potential remain the named anchor.",
    "The Wess-Zumino mass equality is stated around a supersymmetric vacuum and concerns pole-mass parameters after canonical normalization.",
    "Materialization does not promote the planned registry record or give the artifact a registry public route."
  ],
  "accessibility_encoding": {
    "color_independence": "stage numbers, direct text labels, solid arrows, a double-border final result, and explicit operation words carry every relationship; gray fill is redundant",
    "canvas": "explicit white background",
    "motion": "static; no animation or reduced-motion exception is needed",
    "structured_equivalent": "/figures/supersymmetry-duality/superspace-component-reduction-flow.json",
    "semantic_scope": "the JSON preserves every stage, projection, selected coefficient, total derivative, boundary condition, off-shell cancellation, auxiliary equation, Wess-Zumino check, gauge-matter check, dimension, assumption, and inference boundary"
  },
  "scientific_references": [
    {
      "authors": "Julius Wess and Bruno Zumino",
      "title": "A Lagrangian Model Invariant under Supergauge Transformations",
      "journal": "Physics Letters B",
      "volume": "49",
      "issue": "1",
      "pages": "52-54",
      "year": 1974,
      "doi": "10.1016/0370-2693(74)90578-4",
      "url": "https://doi.org/10.1016/0370-2693(74)90578-4",
      "use": "interacting chiral multiplet and cubic Wess-Zumino anchor"
    },
    {
      "authors": "S. James Gates Jr., Marcus T. Grisaru, Martin Rocek, and Warren Siegel",
      "title": "Superspace, or One Thousand and One Lessons in Supersymmetry",
      "series": "Frontiers in Physics 58",
      "year": 1983,
      "corrected_open_edition": 2001,
      "url": "https://arxiv.org/abs/hep-th/0108200",
      "locator": "chapters 4-5",
      "use": "superspace measures, component reduction, component actions, and supersymmetry checks"
    },
    {
      "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": "sections 26.2-26.4, 26.6, 27.1, and 27.4",
      "use": "projection conventions, chiral component actions, Wess-Zumino model, and gauge-matter auxiliaries"
    }
  ]
}