{ "schema_version": "1.0.0", "artifact_id": "qft.artifact.supersymmetry-duality.higher-dimensional.status-branch-anomaly-ledger", "evidence_date": "2026-08-24", "title": "Intrinsic status records for rank-one 5D and 6D supersymmetric fixed-point inputs", "reader_question": "Which intrinsic branch data, quantized checks, independent evidence, cutoffs, and unresolved claims belong to each proposed ultraviolet endpoint?", "dominant_point": "The two pure 5D SU(2) theories share one local Coulomb-branch prepotential but differ by a discrete theta choice and ultraviolet flavor symmetry; the 6D A1 (2,0) input is instead controlled by a tensor lattice, self-dual strings, and anomaly matching. None of these infrared records alone is an existence proof.", "conventions": { "metric": "QFT.org mostly-minus Lorentzian convention in each dimension", "five_dimensional_scalar_chamber": "phi >= 0 with SU(2) roots alpha=plus or minus 2", "five_dimensional_prepotential": "F(phi)=phi^2/(2 g0^2)+(4/3) phi^3", "five_dimensional_metric": "g_eff^{-2}=d^2F/dphi^2=g0^{-2}+8 phi", "five_dimensional_instanton": { "trace": "Hermitian generators obey Tr_2(T_a T_b)=delta_ab/2", "action": "L_YM=-Tr_2(F_mu_nu F^mu_nu)/(2 g5^2)", "charge": "Q=(1/(8 pi^2)) integral_R4 Tr_2(F wedge F) is an integer", "mass": "M_inst=8 pi^2 |Q|/g5^2" }, "six_dimensional_tensor_pairing": "Omega=(2) for the A1 root lattice", "six_dimensional_anomaly": "I8[A1]=I8_tensor+(1/4)p2(N)", "green_schwarz_branch": "on Spin(4)_R subset Spin(5)_R, p2(N)=chi4(N)^2 and Delta I8=(1/2) Omega I4^2 with I4=chi4(N)/2", "evidence_language": "necessary consistency, protected evidence, external construction, and intrinsic existence are distinct fields" }, "records": [ { "id": "5d-pure-su2-theta-zero", "system": "pure 5D SU(2), theta5=0", "dimension": 5, "poincare_supercharges": 8, "candidate_superconformal_algebra": "F(4) with bosonic SO(5,2) times SU(2)_R", "candidate_endpoint": "E1", "branch": { "type": "rank-one Coulomb branch", "coordinate": "phi >= 0", "infrared_fields": "one Abelian vector multiplet away from the origin", "prepotential": "F=phi^2/(2 g0^2)+(4/3)phi^3", "first_derivative": "F'=g0^{-2}phi+4phi^2", "metric": "F''=g0^{-2}+8phi", "positivity": "nonnegative for phi >= 0 and g0^{-2} >= 0; strictly positive except at the joint endpoint phi=g0^{-2}=0", "continuous_cubic_casimir": "absent for SU(2)" }, "quantized_or_discrete_data": { "global_form": "SU(2)=Sp(1)", "discrete_theta": "theta5=0 in pi4(SU(2))=Z2", "chern_simons": "no independent continuous SU(2) cubic Chern-Simons coefficient", "instanton_number": "integer in the stated SU(2) normalization" }, "bps_probe": { "object": "instanton particle", "branch_quantity": { "kind": "particle mass", "expression": "M_inst=8 pi^2 |Q|/g5^2 in the declared trace, action, and charge normalization", "mass_dimension": 1 }, "ultraviolet_signal": "additional protected states can join a conserved-current multiplet as g0^{-2} approaches zero" }, "ultraviolet_global_symmetry": "SU(2), conventionally called E1", "deformation_to_branch": "the inverse gauge coupling g0^{-2} is a relevant deformation of the proposed fixed point", "cutoff": { "nonabelian_ym": "weakly coupled 5D Yang-Mills requires E much less than g0^{-2}", "abelian_branch": "the Abelian truncation also requires E much less than the W-boson mass M_W=2phi and energies below instanton and every other charged threshold", "at_origin": "neither weak-coupling inequality supplies a description of the interacting endpoint" }, "intrinsic_evidence": [ "one-loop-exact cubic prepotential in each Weyl chamber", "nonnegative Coulomb-branch metric with its unique zero identified at the strongly coupled endpoint", "instanton-particle charge and protected-current enhancement", "protected partition-function and index checks" ], "external_construction": "type I′, five-brane-web, or Calabi-Yau engineering after an explicit decoupling limit", "decoupled_sector": "must be checked in each external construction; none is inferred away by the prepotential", "existence_status": "strong interacting-SCFT evidence; no ordinary ultraviolet Lagrangian and no theorem-first construction claimed here", "open_fields": [ "regulator-independent intrinsic construction", "general unprotected correlation functions at finite rank" ], "safe_exports": [ "theta5=0 label", "prepotential and Hessian in the stated chamber", "E1=SU(2) protected flavor enhancement", "instanton-particle probe", "evidence and cutoff status" ], "forbidden_inference": "nonnegative F double prime alone proves the ultraviolet fixed point exists", "reader_table": { "intrinsic_system": "Pure 5D $SU(2)$, $\\theta_5=0$", "branch_calculation": "$\\mathcal F=\\phi^2/(2g_0^2)+4\\phi^3/3$ and $\\mathcal F''=g_0^{-2}+8\\phi\\geq0$ for $g_0^{-2},\\phi\\geq0$, with equality only at their joint zero", "quantized_data": "$\\theta_5=0$ in $\\pi_4(SU(2))=\\mathbb Z_2$; no continuous cubic $SU(2)$ Chern–Simons parameter", "intrinsic_checks": "The instanton-current multiplet and protected indices support the $E_1$ endpoint and $U(1)_I\\to SU(2)$ enhancement.", "external_and_decoupled": "Calabi–Yau degeneration, type I′ engineering, or a five-brane web supplies independent support only after its decoupling limit is controlled; the prepotential alone removes no extra sector.", "branch_regime": "The Abelian branch requires $E\\ll m_W=2\\phi$ and energies below instanton and other charged thresholds; 5D Yang–Mills is effective below a scale of order $g_0^{-2}$.", "open_field_and_ceiling": "A regulator-independent intrinsic construction and general unprotected correlators are not supplied here; the mutually reinforcing evidence is strong, not a theorem of existence." } }, { "id": "5d-pure-su2-theta-pi", "system": "pure 5D SU(2), theta5=pi", "dimension": 5, "poincare_supercharges": 8, "candidate_superconformal_algebra": "F(4) with bosonic SO(5,2) times SU(2)_R", "candidate_endpoint": "tilde E1", "branch": { "type": "rank-one Coulomb branch", "coordinate": "phi >= 0", "infrared_fields": "one Abelian vector multiplet away from the origin", "prepotential": "F=phi^2/(2 g0^2)+(4/3)phi^3", "first_derivative": "F'=g0^{-2}phi+4phi^2", "metric": "F''=g0^{-2}+8phi", "positivity": "nonnegative for phi >= 0 and g0^{-2} >= 0; strictly positive except at the joint endpoint phi=g0^{-2}=0", "continuous_cubic_casimir": "absent for SU(2)" }, "quantized_or_discrete_data": { "global_form": "SU(2)=Sp(1)", "discrete_theta": "theta5=pi in pi4(SU(2))=Z2", "chern_simons": "no independent continuous SU(2) cubic Chern-Simons coefficient", "instanton_number": "integer in the stated SU(2) normalization" }, "bps_probe": { "object": "instanton particle with theta-dependent projection", "branch_quantity": { "kind": "particle mass", "expression": "M_inst=8 pi^2 |Q|/g5^2 in the declared trace, action, and charge normalization", "mass_dimension": 1 }, "ultraviolet_signal": "protected spectra distinguish this endpoint from E1 and do not supply the same SU(2) enhancement" }, "ultraviolet_global_symmetry": "U(1), conventionally called tilde E1", "deformation_to_branch": "the inverse gauge coupling g0^{-2} is a relevant deformation of the proposed fixed point", "cutoff": { "nonabelian_ym": "weakly coupled 5D Yang-Mills requires E much less than g0^{-2}", "abelian_branch": "the Abelian truncation also requires E much less than the W-boson mass M_W=2phi and energies below instanton and every other charged threshold", "at_origin": "neither weak-coupling inequality supplies a description of the interacting endpoint" }, "intrinsic_evidence": [ "one-loop-exact cubic prepotential in each Weyl chamber", "nonnegative Coulomb-branch metric with its unique zero identified at the strongly coupled endpoint", "discrete-theta dependence of the instanton sector", "protected superconformal-index evidence for the absence of E1-type enhancement" ], "external_construction": "type I′, five-brane-web, or Calabi-Yau engineering after an explicit decoupling limit", "decoupled_sector": "must be checked in each external construction; none is inferred away by the prepotential", "existence_status": "strong interacting-SCFT evidence; no ordinary ultraviolet Lagrangian and no theorem-first construction claimed here", "open_fields": [ "whether any higher-instanton sector could supply a current absent at one instanton", "regulator-independent intrinsic construction and general unprotected data" ], "safe_exports": [ "theta5=pi label", "prepotential and Hessian in the stated chamber", "tilde E1=U(1) protected flavor status", "theta-dependent instanton probe", "evidence and cutoff status" ], "forbidden_inference": "the shared local prepotential makes the theta5=0 and theta5=pi ultraviolet theories identical", "reader_table": { "intrinsic_system": "Pure 5D $SU(2)$, $\\theta_5=\\pi$", "branch_calculation": "The same local $\\mathcal F$ and nonnegative Hessian as the $\\theta_5=0$ theory, with the same zero at the joint endpoint", "quantized_data": "$\\theta_5=\\pi$ changes the instanton-sector projection while leaving the local prepotential unchanged.", "intrinsic_checks": "Protected spectra distinguish the $\\widetilde E_1$ endpoint, whose flavor symmetry remains $U(1)_I$.", "external_and_decoupled": "The same controlled geometric or brane constructions provide independent support; their decoupling limits must again be checked explicitly.", "branch_regime": "The same Abelian and five-dimensional Yang–Mills cutoffs apply as for $E_1$.", "open_field_and_ceiling": "The missing one-instanton multiplet alone cannot exclude a current first appearing at higher instanton number; a complete intrinsic construction and unprotected data are not supplied here." } }, { "id": "6d-a1-20-relative", "system": "6D A1 (2,0) relative theory", "dimension": 6, "poincare_supercharges": 16, "candidate_superconformal_algebra": "OSp(8*|4) with bosonic Spin(6,2) times USp(4)_R, with USp(4)_R isomorphic to Spin(5)_R", "candidate_endpoint": "A1 (2,0) interacting relative sector", "branch": { "type": "rank-one tensor branch", "coordinate": "one relative tensor-multiplet scalar radius together with its Spin(5)_R orbit", "infrared_fields": "one free relative (2,0) tensor multiplet at a generic branch point", "prepotential": "not applicable: the protected branch data are a self-dual tensor lattice and Green-Schwarz coupling", "first_derivative": "not applicable", "metric": "positive tensor-scalar kinetic term in the chosen relative normalization", "positivity": "necessary but not sufficient for an interacting conformal origin", "continuous_cubic_casimir": "not applicable" }, "quantized_or_discrete_data": { "string_lattice": "A1 root lattice", "pairing_matrix": [ [ 2 ] ], "discriminant_group": "Lambda_string^*/Lambda_string=Z2", "polarization_status": "compactification requires a maximal-isotropic global completion rather than silently selecting one partition-function component" }, "bps_probe": { "object": "self-dual string of primitive A1 charge", "branch_quantity": { "kind": "string tension", "expression": "T_str is proportional to the relative tensor-branch scalar in the declared normalization", "mass_dimension": 2 }, "ultraviolet_signal": "T_str approaches zero at the conformal origin" }, "ultraviolet_global_symmetry": "Spin(5)_R R symmetry; the Z2 discriminant datum is global defect data, not an ordinary flavor group", "anomaly": { "free_tensor": "I8_tensor=[p2(N)-p2(T)+(p1(T)-p1(N))^2/4]/48", "uv_relative": "I8[A1]=I8_tensor+(1/4)p2(N)", "ir_relative": "I8_tensor", "mismatch": "Delta I8=(1/4)p2(N)=(1/4)chi4(N)^2 after restricting the R bundle to Spin(4)_R", "green_schwarz": "Delta I8=(1/2)Omega I4^2 with Omega=2 and I4=chi4(N)/2", "two_m5_subtraction": { "total": "I8[2 M5]=2 I8_tensor+(1/4)p2(N)", "free_center_of_mass": "I8_tensor", "relative": "I8[2 M5]-I8_tensor=I8[A1]=I8_tensor+(1/4)p2(N)" } }, "deformation_to_branch": "a tensor-branch expectation value gives the self-dual strings nonzero tension", "cutoff": { "tensor_branch": "the free-tensor effective description requires E much less than sqrt(T_str) and below every other massive threshold", "dimension_check": "T_str has mass dimension 2, so sqrt(T_str) rather than T_str is the energy scale", "at_origin": "the branch effective fields do not define the interacting tensionless-string point" }, "intrinsic_evidence": [ "integral A1 string pairing and Z2 discriminant group", "self-dual-string BPS tension", "UV anomaly polynomial", "factorized tensor-branch anomaly mismatch" ], "external_construction": "two coincident M5-branes in a decoupling limit", "decoupled_sector": "subtract the free center-of-mass tensor before identifying the interacting A1 relative anomaly", "existence_status": "strong mutually consistent field-theory and M-theory evidence; no ordinary local Lorentz-covariant non-Abelian Lagrangian and no theorem-first construction claimed here", "open_fields": [ "complete intrinsic construction of the interacting origin", "general unprotected observables and operator products" ], "safe_exports": [ "A1 string lattice and Z2 discriminant group", "relative anomaly polynomial and explicit free-tensor subtraction", "self-dual-string charge and tension scaling", "polarization requirement", "evidence and cutoff status" ], "forbidden_inference": "anomaly factorization or the M5-brane realization alone proves every intrinsic observable or compactification limit", "reader_table": { "intrinsic_system": "Relative 6D $A_1$ $(2,0)$ theory", "branch_calculation": "One free relative tensor multiplet away from the origin; a primitive self-dual string has $T_{\\rm str}\\to0$ at the origin.", "quantized_data": "$A_1$ string lattice with $\\Omega=2$ and discriminant group $\\mathbb Z_2$; compactification requires a polarization.", "intrinsic_checks": "$I_8[A_1]=I_8^{\\rm tens}+p_2(N)/4$ and $\\Delta I_8=\\Omega I_4^2/2$ with $I_4=\\chi_4(N)/2$ test anomaly matching.", "external_and_decoupled": "Two coincident M5-branes supply independent support only after subtracting the decoupled free center-of-mass tensor.", "branch_regime": "The tensor-branch EFT requires $E\\ll\\sqrt{T_{\\rm str}}$ and energies below every other massive threshold.", "open_field_and_ceiling": "Lattice integrality, anomaly matching, and M-theory give strong evidence, but no complete construction of every intrinsic or unprotected observable is supplied here." } } ], "sources": [ { "id": "seiberg-1996-5d-fixed-points", "authors": "Nathan Seiberg", "year": 1996, "title": "Five dimensional SUSY field theories, non-trivial fixed points and string dynamics", "publication": "Physics Letters B 388 (1996) 753-760", "doi": "10.1016/S0370-2693(96)01215-4", "url": "https://arxiv.org/abs/hep-th/9608111", "locator": "sections 2-5", "use": "exact Coulomb metric, E1 fixed point, inverse-coupling deformation, protected enhancement, and external construction status" }, { "id": "ims-1997-5d-prepotential", "authors": "Kenneth Intriligator, David R. Morrison, and Nathan Seiberg", "year": 1997, "title": "Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces", "publication": "Nuclear Physics B 497 (1997) 56-100", "doi": "10.1016/S0550-3213(97)00279-4", "url": "https://arxiv.org/abs/hep-th/9702198", "locator": "sections 2-4, especially equations 2.22 and 4.1-4.2", "use": "one-loop-exact piecewise-cubic prepotential, Hessian metric, convexity criterion, and the two pure Sp(1) endpoints" }, { "id": "bergman-rodriguez-gomez-zafrir-2014", "authors": "Oren Bergman, Diego Rodriguez-Gomez, and Gabi Zafrir", "year": 2014, "title": "Discrete theta and the 5d superconformal index", "publication": "Journal of High Energy Physics 01 (2014) 079", "doi": "10.1007/JHEP01(2014)079", "url": "https://arxiv.org/abs/1310.2150", "locator": "sections 2-4", "use": "discrete theta dependence and protected-index evidence distinguishing E1 from tilde E1" }, { "id": "ohmori-shimizu-tachikawa-yonekura-2014", "authors": "Kantaro Ohmori, Hiroyuki Shimizu, Yuji Tachikawa, and Kazuya Yonekura", "year": 2014, "title": "Anomaly polynomial of general 6D SCFTs", "publication": "Progress of Theoretical and Experimental Physics 2014, 103B07", "doi": "10.1093/ptep/ptu140", "url": "https://arxiv.org/abs/1408.5572", "locator": "sections 2.2-2.3, especially equations 2.3, 2.14, 2.15, and 2.16", "use": "ADE string lattice, A1 anomaly arithmetic, tensor-branch Green-Schwarz matching, and the two-M5/free-center-of-mass subtraction" }, { "id": "tachikawa-2014-discrete-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": "partition-vector interpretation, maximal-isotropic polarization, and global completion of the A1 discriminant data" }, { "id": "seiberg-witten-1996-6d", "authors": "Nathan Seiberg and Edward Witten", "year": 1996, "title": "Comments on string dynamics in six dimensions", "publication": "Nuclear Physics B 471 (1996) 121-134", "doi": "10.1016/0550-3213(96)00189-7", "url": "https://arxiv.org/abs/hep-th/9603003", "locator": "sections 2-3", "use": "tensor branches, tensionless strings, and the boundary between infrared descriptions and interacting six-dimensional inputs" } ], "limitations": [ "The 5D coefficients use the explicitly stated SU(2) scalar and root normalization; another normalization must be translated before comparison.", "The two 5D rows deliberately have identical local prepotentials because the discrete theta choice lives in global and instanton-sector data.", "The A1 row is the interacting relative sector; a two-M5-brane construction also contains a free center-of-mass tensor.", "Necessary positivity, integrality, and anomaly tests can exclude candidates but are not sufficient existence theorems.", "External geometric, brane, or M-theory engineering is recorded as an independent evidence class and never silently promoted to an intrinsic Lagrangian construction." ], "lifecycle_status": "materialized governed prototype; registry promotion and release acceptance remain separate decisions", "semantic_role": "structured equivalent of the reader-facing comparison table and complete versioned evidence record", "verification": { "record_count": 3, "required_record_ids": [ "5d-pure-su2-theta-zero", "5d-pure-su2-theta-pi", "6d-a1-20-relative" ], "blank_field_count": 0, "five_dimensional": { "shared_local_branch_record": true, "distinct_discrete_theta": true, "endpoint_symmetries": [ "SU(2), conventionally called E1", "U(1), conventionally called tilde E1" ], "prepotential_fixtures": [ { "phi": 0, "inverse_coupling": 0, "F": 0, "first_derivative": 0, "exact_second_derivative": 0, "numerical_second_derivative": 0, "difference_scheme": "four-point forward at phi=0", "absolute_hessian_error": 0, "nonnegative_metric": true, "strictly_positive_metric": false }, { "phi": 0, "inverse_coupling": 2.25, "F": 0, "first_derivative": 0, "exact_second_derivative": 2.25, "numerical_second_derivative": 2.250000000000001, "difference_scheme": "four-point forward at phi=0", "absolute_hessian_error": 8.881784197001252e-16, "nonnegative_metric": true, "strictly_positive_metric": true }, { "phi": 0.125, "inverse_coupling": 2.25, "F": 0.020182291666666668, "first_derivative": 0.34375, "exact_second_derivative": 3.25, "numerical_second_derivative": 3.2499999996771223, "difference_scheme": "three-point central in phi>0", "absolute_hessian_error": 3.228777245567471e-10, "nonnegative_metric": true, "strictly_positive_metric": true }, { "phi": 1, "inverse_coupling": 2.25, "F": 2.458333333333333, "first_derivative": 6.25, "exact_second_derivative": 10.25, "numerical_second_derivative": 10.250000048728225, "difference_scheme": "three-point central in phi>0", "absolute_hessian_error": 4.872822501056362e-8, "nonnegative_metric": true, "strictly_positive_metric": true }, { "phi": 3.5, "inverse_coupling": 2.25, "F": 70.94791666666666, "first_derivative": 56.875, "exact_second_derivative": 30.25, "numerical_second_derivative": 30.250001259446435, "difference_scheme": "three-point central in phi>0", "absolute_hessian_error": 0.0000012594464351423085, "nonnegative_metric": true, "strictly_positive_metric": true } ], "maximum_hessian_error": 0.0000012594464351423085 }, "six_dimensional": { "rank": 1, "string_pairing_determinant": 2, "discriminant_group_order": 2, "dh_vee_over_24": 0.25, "green_schwarz_coefficient": 0.25, "tension_mass_dimension": 2, "cutoff_energy_scale": "sqrt(T_str)", "two_m5_total_free_tensor_count": 2, "center_of_mass_tensor_count": 1, "relative_free_tensor_count": 1, "two_m5_interacting_anomaly_coefficient": 0.25, "center_of_mass_subtraction_explicit": true }, "source_count": 6, "reader_table_projection_sha256": "f89741f768ffbc3731798aa6eb2fd4dd489a745080a35d48cb69017615691faa", "necessary_not_sufficient_boundary": true } }