{ "artifact_id": "qft.artifact.supersymmetry-duality.supercurrents.improvements-background-coupling-map", "title": "Supercurrent improvements and source packages", "artifact_class": "original monochrome qualitative decision map", "quantitative": false, "reader_question": "Which local, gauge-invariant, globally defined improvements reduce the four-dimensional N=1 S-multiplet to FZ, R, or superconformal form, what physical brane-current obstruction does each failed branch retain, and which linearized source package can consume the result?", "takeaway": "The S-multiplet is the general current package; FZ, R, and superconformal representatives require increasingly restrictive global improvements, and matching a current to an old-minimal, new-minimal, larger S, or conformal source package does not prove that any chosen curved background preserves supersymmetry.", "alt_text": "A monochrome vertical decision map. The general S-multiplet flows through one improvement gate requiring a local, gauge-invariant, globally defined real U and harmless surface terms. Three dashed conditional branches lead to the Ferrara-Zumino, R, and superconformal multiplets. The FZ branch removes the string-current class and matches an old-minimal linearized source; the R branch requires an exact anomaly-compatible continuous R symmetry, removes the domain-wall-current class, and matches a new-minimal source; the superconformal branch removes both sources. Four stop boxes identify a genuine FI term, a non-exact Kahler form, missing or anomalous R data, and boundary, brane, or bundle effects. A final warning says source compatibility is not a supersymmetric-background existence claim.", "canonical_anchor": "improvements-background-coupling-map", "owner_page": { "id": "qft.topic.susy-actions.supercurrent-multiplets-improvements", "file": "src/content/docs/supersymmetry-duality/actions-supercurrents-effective-theory/supercurrent-multiplets-improvements.md", "route": "/supersymmetry-duality/actions-supercurrents-effective-theory/supercurrent-multiplets-improvements/" }, "conventions": { "domain": "four-dimensional Lorentzian N=1 flat-space operator equations in Wess-Bagger superspace conventions", "metric": "eta=diag(+1,-1,-1,-1)", "s_equation": "bar D^dot-alpha S_alpha-dot-alpha = D_alpha X + chi_alpha", "chirality": "bar D_dot-alpha X=0 and bar D_dot-alpha chi_alpha=0", "chi_reality": "D^alpha chi_alpha = bar D_dot-alpha bar chi^dot-alpha", "improvement": "S -> S+[D,bar D]U; X -> X+(1/2)bar D^2 U; chi_alpha -> chi_alpha+(3/2)bar D^2 D_alpha U, with U real", "admissibility": "U is local, gauge invariant, globally defined up to a real constant, compatible with operator renormalization, and has improvement surface terms that do not change the charges", "fz_condition": "chi_alpha=-(3/2)bar D^2 D_alpha U", "r_condition": "X=-(1/2)bar D^2 U together with an exact continuous anomaly-compatible U(1)_R current", "superconformal_condition": "one admissible improvement removes both source superfields; no unremovable virial or quantum obstruction is asserted away", "inference_boundary": "a compatible linearized source multiplet is necessary input only; geometry, spin structure, bundles, auxiliary backgrounds, Killing-spinor equations, anomalies, contact terms, and boundaries are separate existence checks" }, "nodes": [ { "id": "s_multiplet", "status": "yes", "label": "S-multiplet", "equation": "bar D^dot-alpha S_alpha-dot-alpha = D_alpha X + chi_alpha", "contents": [ "symmetric conserved stress tensor", "conserved supersymmetry current", "string-current two-form in chi", "domain-wall-current one-form in Y=D X when X is global" ], "source_compatibility": "larger linearized S source package, with an additional chiral or equivalent linear field in the Komargodski-Seiberg construction" }, { "id": "improvement_gate", "status": "conditional", "label": "admissible real U", "tests": [ "local", "gauge invariant", "globally defined", "operator-scheme compatible", "boundary surface terms harmless" ], "transformation": "S -> S+[D,bar D]U; X -> X+(1/2)bar D^2 U; chi_alpha -> chi_alpha+(3/2)bar D^2 D_alpha U, with U real" }, { "id": "fz", "status": "conditional", "condition": "chi_alpha=-(3/2)bar D^2 D_alpha U", "equation": "bar D^dot-alpha J_alpha-dot-alpha = D_alpha X_FZ", "brane_current_consequence": "charged string-current class is removable when improvement surface terms vanish", "source_compatibility": "old-minimal linearized source package" }, { "id": "r", "status": "conditional", "condition": "X=-(1/2)bar D^2 U together with an exact continuous anomaly-compatible U(1)_R current", "equation": "bar D^dot-alpha R_alpha-dot-alpha = chi_alpha", "brane_current_consequence": "charged domain-wall-current class is removable when improvement surface terms vanish", "source_compatibility": "new-minimal linearized source package" }, { "id": "superconformal", "status": "conditional", "condition": "one admissible improvement removes both source superfields; no unremovable virial or quantum obstruction is asserted away", "equation": "bar D^dot-alpha J_alpha-dot-alpha = 0", "brane_current_consequence": "both charged brane-current classes are removable when improvement surface terms vanish", "source_compatibility": "conformal current source package; old- and new-minimal embeddings may coexist but are not unique background choices" } ], "edges": [ { "from": "s_multiplet", "to": "improvement_gate", "status": "required", "meaning": "classify only well-defined global improvements" }, { "from": "improvement_gate", "to": "fz", "status": "conditional", "meaning": "remove chi" }, { "from": "improvement_gate", "to": "r", "status": "conditional", "meaning": "remove X and identify the exact R current" }, { "from": "improvement_gate", "to": "superconformal", "status": "conditional", "meaning": "one admissible improvement removes both sources" } ], "obstructions": [ { "id": "fi", "status": "obstructed", "blocks": "fz", "reason": "for a genuine Abelian FI term the required U contains the vector prepotential V and is not gauge invariant" }, { "id": "nonexact_kahler", "status": "obstructed", "blocks": "fz", "reason": "local Kahler potentials do not assemble into one globally defined U when the Kahler form is non-exact" }, { "id": "r_missing_or_anomalous", "status": "no_or_obstructed", "blocks": "r", "reason": "no exact continuous R current, a quantum R anomaly, or an incompatible boundary blocks the R-multiplet export" }, { "id": "boundary_brane_bundle", "status": "conditional", "blocks": "any purported charge-preserving improvement", "reason": "surface terms, defect currents, nontrivial bundles, or brane junctions can change integrated charges even when a local formula exists" } ], "source_compatibility_matrix": { "statuses": [ "yes", "no", "conditional", "obstructed", "unknown" ], "columns": [ "current package", "availability", "charged string-current class", "charged domain-wall-current class", "linearized source package", "curved-background verdict" ], "rows": [ [ "S", "yes", "conditional", "conditional", "larger S / 16-16 candidate", "unknown" ], [ "FZ", "conditional", "no under stated boundary assumptions", "conditional", "old-minimal", "unknown" ], [ "FZ with FI or non-exact Kahler data", "obstructed", "conditional", "conditional", "no FZ / old-minimal source from the obstructed representative", "unknown" ], [ "R", "conditional", "conditional", "no under stated boundary assumptions", "new-minimal", "unknown" ], [ "superconformal", "conditional", "no under stated boundary assumptions", "no under stated boundary assumptions", "conformal; minimal embeddings nonunique", "unknown" ] ] }, "scope_limits": [ "The map classifies flat-space operator multiplets and linearized source compatibility only.", "It does not assert a nonlinear supergravity completion, quantum-gravity consistency, or a supersymmetric solution on a named manifold.", "No absence of an actual brane follows solely from a formal local improvement; the global and boundary hypotheses printed in the map are essential.", "An anomaly-free exact R current is stronger than a classical R-charge assignment." ], "accessibility_encoding": { "color_independence": "All logic is carried by explicit status words, solid versus dashed or barred lines, equations, and labels; color is not used.", "structured_equivalent": "Every node, edge, obstruction, status, source match, boundary, and inference limit is present in this JSON record." }, "scientific_references": [ { "citation": "Zohar Komargodski and Nathan Seiberg, Comments on Supercurrent Multiplets, Supersymmetric Field Theories and Supergravity, JHEP 07 (2010) 017", "arxiv": "https://arxiv.org/abs/1002.2228", "doi": "https://doi.org/10.1007/JHEP07(2010)017", "locator": "arXiv v4, eqs. (1.11), (1.12), (2.1), (2.6), and sections 4-5", "use": "S, FZ, and R equations; improvement coefficients; FI and Kahler obstructions; linearized old-minimal, new-minimal, and larger S couplings" }, { "citation": "Thomas T. Dumitrescu and Nathan Seiberg, Supercurrents and Brane Currents in Diverse Dimensions, JHEP 07 (2011) 095", "arxiv": "https://arxiv.org/abs/1106.0031", "doi": "https://doi.org/10.1007/JHEP07(2011)095", "locator": "arXiv v4, eqs. (1.10), (2.10), (2.16)-(2.19), (3.2), (3.5), and sections 2-3", "use": "well-defined improvement requirement, smaller multiplets, string and domain-wall currents, and boundary qualifications" }, { "citation": "Guido Festuccia and Nathan Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP 06 (2011) 114", "arxiv": "https://arxiv.org/abs/1105.0689", "doi": "https://doi.org/10.1007/JHEP06(2011)114", "locator": "arXiv v2, sections 1-2 and 5-7", "use": "rigid-limit construction and the separate requirement that background auxiliary fields and geometry preserve a supercharge" } ] }