{ "schema_version": "1.0.0", "artifact_id": "qft.artifact.supersymmetry-duality.n2.coulomb-branch-special-kahler-period-map", "title": "The period vector is the bridge from Coulomb-branch geometry to observables", "evidence_date": "2026-08-24", "record_kind": "exact-data-backed schematic with structured semantic equivalent", "reader_question": "How do a regular Coulomb-branch patch, a polarized fiber and a normalized differential produce local low-energy data without mistaking one duality frame for a global description?", "dominant_point": "A declared cycle basis turns the Seiberg-Witten differential into a period vector; local prepotential, coupling, metric and central charges follow, while contragredient charge transport makes the central charge frame invariant and the singular locus remains an explicit stop.", "governance": { "registry_status_observed": "prototype", "registry_public_route_observed": null, "target_owner_page_ids": [ "qft.topic.seiberg-witten.coulomb-branches-abelian-low-energy-theory", "qft.topic.seiberg-witten.prepotentials-special-kahler-geometry", "qft.topic.seiberg-witten.charge-lattices-duality-local-systems", "qft.topic.seiberg-witten.curve-differential-periods" ], "registry_owner_page_ids_observed": [ "qft.topic.seiberg-witten.coulomb-branches-abelian-low-energy-theory", "qft.topic.seiberg-witten.prepotentials-special-kahler-geometry", "qft.topic.seiberg-witten.charge-lattices-duality-local-systems", "qft.topic.seiberg-witten.curve-differential-periods" ], "registry_scope_is_final": true, "retired_owners_still_observed": [], "owner_routes": [ { "id": "qft.topic.seiberg-witten.coulomb-branches-abelian-low-energy-theory", "route": "/supersymmetry-duality/n2-seiberg-witten/coulomb-branches-abelian-low-energy-theory/" }, { "id": "qft.topic.seiberg-witten.prepotentials-special-kahler-geometry", "route": "/supersymmetry-duality/n2-seiberg-witten/prepotentials-special-kahler-geometry/" }, { "id": "qft.topic.seiberg-witten.charge-lattices-duality-local-systems", "route": "/supersymmetry-duality/n2-seiberg-witten/charge-lattices-duality-local-systems/" }, { "id": "qft.topic.seiberg-witten.curve-differential-periods", "route": "/supersymmetry-duality/n2-seiberg-witten/curve-differential-periods/" } ] }, "regular_patch_pipeline": [ { "stage": 1, "label": "regular Coulomb-branch patch", "data": [ "u in B^o=B minus Delta", "generic gauge group U(1)^r", "scale, base point and path system", "integral polarization and basis" ], "stop": "the Abelian derivative expansion is not asserted on Delta" }, { "stage": 2, "label": "polarized fiber and differential", "data": [ "exact curve Sigma_u or polarized Abelian fiber", "A_I intersect B^J=delta_I^J", "normalized Seiberg-Witten differential lambda", "cycles transported from the fixed base point" ] }, { "stage": 3, "label": "period vector", "equations": [ "a^I=integral_(A_I) lambda", "a_(D,I)=integral_(B^I) lambda", "Pi=(a_D,a)^T", "Z_gamma=gamma^T Pi for gamma=(n_m,n_e)^T" ] }, { "stage": 4, "label": "local low-energy data", "equations": [ "a_D=partial F/partial a", "tau=partial^2 F/partial a^2", "ds^2=(Im tau)_(IJ) da^I d conjugate(a^J)", "Im tau positive definite" ] } ], "frame_change_table": { "principal_polarization": "M=[[A,B],[C,D]] in Sp(2r,Z)", "period_action": "Pi'=M Pi", "charge_action": "gamma'=M^-T gamma", "coupling_action": "tau'=(A tau+B)(C tau+D)^-1", "invariant": "gamma'^T Pi'=gamma^T Pi", "polarized_generalization": "for a nonprincipal polarization, replace Sp(2r,Z) by the integral group preserving the declared polarization form" }, "pure_su2_anchor": { "theory": "pure four-dimensional N=2 SU(2) Yang-Mills", "scale_and_base_point": "Lambda>0, u_b=2 Lambda^2", "curve": "y^2=(x-u)(x^2-Lambda^4)", "differential": "lambda=sqrt(2)/(2 pi) (x-u) dx/y", "differential_derivative": "partial_u lambda=-sqrt(2)/(4 pi) dx/y", "cuts": [ "[-Lambda^2,+Lambda^2]", "[u_b,infinity]" ], "cycles": "A intersect B=+1; a=integral_A lambda>0 and a_D=integral_B lambda lies on positive imaginary axis at u_b", "period_order": "Pi=(a_D,a)^T", "exact_periods": { "definitions": "s=u+Lambda^2, m=2 Lambda^2/s", "a": "2 sqrt(2s) E(m)/pi", "a_D": "2 i sqrt(2s) [K(1-m)-E(1-m)]/pi", "tau": "i K(1-m)/K(m)" }, "normalized_base_point_values": { "u_over_Lambda_squared": 2, "m": 0.6666666666666666, "a_over_Lambda": 1.96668530155033, "aD_over_Lambda": "0.4734344365355367 i", "tau": "0.8545844432787435 i" }, "singular_limit": { "a_at_u_equals_Lambda_squared": "4 Lambda/pi", "a_D": "i (u-Lambda^2)/(2 Lambda)+higher orders", "du_daD": "-2 i Lambda" } }, "scientific_verification": { "status": "passed", "elliptic_integral_algorithm": "arithmetic-geometric mean with the exact complementary-parameter formulas", "base_point": { "u_over_Lambda_squared": 2, "m": 0.6666666666666666, "a_over_Lambda": 1.96668530155033, "aD_over_Lambda": "0.4734344365355367 i", "tau": "0.8545844432787435 i" }, "local_monopole_limit": { "formula": "a_D=i(u-Lambda^2)/(2 Lambda)+higher orders", "epsilon_over_Lambda_squared": 1e-7, "relative_error": 2.1447543741004438e-9 }, "exact_checks": [ "for the displayed representative, d lambda/du=-sqrt(2) dx/(4 pi y) exactly", "det M=1 and M^T J M=J for the frozen rank-one transitions", "Pi -> M Pi, gamma -> M^-T gamma preserves gamma^T Pi", "a(Lambda^2)=4 Lambda/pi" ] }, "sources": [ { "id": "seiberg-witten-1994", "locator": "Sections 2-5 and Appendix B", "url": "https://arxiv.org/abs/hep-th/9407087", "use": "pure-SU(2) curve, periods, singularities, monodromies and the low-energy solution" }, { "id": "freed-1999", "locator": "Sections 1-3", "url": "https://arxiv.org/abs/hep-th/9712042", "use": "intrinsic special-Kahler and symplectic-frame structure" }, { "id": "donagi-witten-1995", "locator": "Sections 2-4", "url": "https://arxiv.org/abs/hep-th/9510101", "use": "integrable-system and polarized-fibration interpretation" } ], "visual_encodings": { "solid_arrow": "licensed local construction or exact frame transformation", "dashed_box": "regular-patch inference stop", "gray_box": "frame-change relation", "color_dependence": "none" }, "accessibility": { "alt_text": "A four-stage pipeline starts with a regular Coulomb-branch patch, then a polarized fiber with oriented cycles and a normalized differential, then the period vector Pi equals a_D over a, and finally the local prepotential, coupling, positive metric and BPS central charge. A symplectic frame-change panel sends Pi to M Pi, charges to M inverse transpose gamma and tau by its fractional-linear rule, leaving Z invariant. A dashed stop says the construction is only valid away from the discriminant. An exact pure SU(2) anchor gives the curve, differential, elliptic-integral periods and numerical values at u equals two Lambda squared.", "semantic_equivalent": "/figures/supersymmetry-duality/coulomb-branch-special-kahler-period-map.json", "narrow_width": "the full-size path-based SVG can pan without page-wide overflow; every equation and relationship is reproduced in this JSON" }, "limitations": [ "A local prepotential is not asserted to exist in one frame over the full regular Coulomb branch.", "At the discriminant, extra light states must be restored and the Abelian vector-multiplet-only action is incomplete.", "The rank-r pipeline is structural; the numerical fixture is only the declared pure-SU(2) rank-one convention.", "The artifact registry lifecycle observed during generation was prototype." ] }