{ "artifact_id": "qft.artifact.supersymmetry-duality.localization.background-twist-locus-contour-map", "schema_version": "localization-validity-chain/v1", "fixture_revision": 1, "title": "From a rigid background to a qualified localized observable", "reader_question": "What must be established between choosing a supersymmetric background and reporting a localized path integral, and where can the argument fail?", "takeaway": "Localization is a conditional chain. A global odd symmetry, a valid Ward identity, a complete fixed locus, a gauge-fixed fluctuation complex, a regulated zero-mode measure, and a named contour or residue chamber are all independent obligations.", "scope": "Euclidean localization workflow whose source-compatibility branch is illustrated in four-dimensional N=1 language. Later stages apply more broadly only after translating the odd symmetry, bundles, contours, auxiliaries, signs, and normalizations to the theory under study.", "conventions": { "spacetime": "Euclidean with positive-definite metric; the Lorentzian continuation and field-space reality conditions are page-local data.", "odd_symmetry": "Q is a globally defined odd symmetry. Q squared may generate an isometry, gauge transformation, R rotation, and flavor transformation rather than vanish.", "supercharge_normalization": "No universal numerical normalization is imposed. Rescaling Q changes the displayed closure parameters and must be fixed in the theory-specific convention card.", "r_charge_normalization": "The R charge of the supersymmetry parameter and the normalization and global form of the R group are theory-specific. The symbol S_Q denotes the actual spin-R associated bundle, including any common central quotient; it does not assume a factorization as an ordinary spin bundle times an R line on a non-spin manifold.", "orientation": "Spacetime orientation, determinant-line orientation, zero-mode measure orientation, and contour orientation are inputs to stages 2, 6, 7, and 8 rather than consequences of the diagram.", "global_forms": "The global gauge group, allowed bundles and fluxes, stabilizers, and genuine boundary gauge transformations are retained in stages 5-8 even when the local Lie algebra is unchanged.", "branches_and_contours": "Every saddle branch and topological sector is enumerated at stage 5; stage 8 names the selected thimble combination or Jeffrey-Kirwan covector and its chamber.", "arrows": "Solid arrows are required construction steps, not renormalization-group flows or logical implications without the stated pass conditions.", "failure_exits": "Dashed arrows leave the validity chain when a named obstruction stops or qualifies the claim.", "status": "Schematic workflow, not a theorem asserting that an arbitrary QFT path integral exists." }, "stages": [ { "number": 1, "name": "Source compatibility", "input": "A flat-space supercurrent multiplet and the desired nondynamical supergravity formulation.", "construction": "In the four-dimensional N=1 illustration, test global improvements and choose the source branch: an FZ multiplet couples to old minimal supergravity, an R-multiplet to new minimal supergravity, while an unimproved S-multiplet requires the larger source system.", "pass_condition": "Improvements patch globally; anomalies, contact terms, defects, and boundaries are specified, with any required added-sector cancellation or inflow included.", "output": "An admissible background-source package containing the metric, R connection when available, and the branch-specific auxiliary fields." }, { "number": 2, "name": "Global odd symmetry", "input": "A complete off-shell background-source system, including every background-fermion supersymmetry variation.", "construction": "Freeze the bosonic sources, set every background fermion to zero, solve every background-fermion variation including delta_Q psi_mu = 0 for epsilon in Gamma(S_Q), the declared global spin-R associated bundle, and compute Q squared on every field.", "pass_condition": "The spin-R bundle patches globally and Q squared closes into declared even symmetries compatible with boundaries.", "output": "A global supercharge and its complete closure algebra." }, { "number": 3, "name": "Twist or equivariant Q-cohomology", "input": "Lorentz and nonanomalous R-symmetry data, or an already global supercharge.", "construction": "When a twist is used, regrade fields and identify a global Q; otherwise restrict to the Q-squared-invariant subalgebra of an already global charge. In a topological twist P_mu = {Q,G_mu}; in a holomorphic twist only P_bar_i is Q exact. Define Q-closed observables modulo Q-exact ones.", "pass_condition": "Any twisted bundles used exist; Q is nilpotent on the relevant quotient or invariant subalgebra; insertions, regulator, and boundary conditions preserve the full Q squared symmetry.", "output": "A cohomological field complex and protected insertions." }, { "number": 4, "name": "Deformation identity", "input": "A Q-invariant action, insertion O, cycle Gamma, and deformation functional V.", "construction": "Set S_t = S + t QV and rewrite the t derivative of the normalized expectation value as a Q-exact insertion.", "pass_condition": "Q squared V vanishes under the complete even symmetry, the measure and cycle are Q invariant, integration by parts has no field-space boundary term, and the finite-t integral converges.", "output": "Finite-t deformation independence, with every remainder displayed." }, { "number": 5, "name": "Complete fixed locus", "input": "The chosen cycle Gamma and the bosonic part of QV.", "construction": "Solve (QV)_bos = 0 and enumerate branches, bundles, fluxes, defects, boundary sectors, and saddle moduli.", "pass_condition": "On a chosen real cycle the bosonic deformation is real and nonnegative; on a complex cycle its real part supplies the required steepest-descent decay. The relevant zero set agrees with the declared Q-fixed locus, no saddle or contribution at infinity is omitted, and the t to infinity limit may be interchanged with integration or summation.", "output": "The full localization locus M_Q and its sector decomposition." }, { "number": 6, "name": "Gauge-fixed deformation complex", "input": "A saddle, its stabilizer, the Q algebra, and the gauge symmetry including ghosts and auxiliaries.", "construction": "Combine Q with BRST, linearize the transformations, and state the elliptic or transversely elliptic complex with boundary conditions.", "pass_condition": "Reducible saddles, residual gauge volume, ghost zero modes, grading, orientation, and boundary domains are treated.", "output": "A determinant-ready complex and an explicit cohomology of unpaired modes." }, { "number": 7, "name": "One-loop measure", "input": "Classical saddle weights, the fluctuation complex, and its zero modes.", "construction": "Pair nonzero modes, form the theory-appropriate primed determinant or Pfaffian ratio, and integrate collective coordinates separately.", "pass_condition": "The spectral cut, determinant phase, regulator scale, local counterterms, gauge volume, and benchmark normalization are fixed.", "output": "A regulated sector measure including zero modes." }, { "number": 8, "name": "Cycle, residue, and gluing prescription", "input": "The original cycle Gamma declared before deformation, the localized measure, its critical points or singular hyperplanes, and any boundary states.", "construction": "Transport and decompose the original Gamma into oriented thimbles. Only for a derived meromorphic, projective Cartan integral may the corresponding linking cycle be encoded by a Jeffrey-Kirwan covector and chamber. Include flux sums and gluing measures.", "pass_condition": "Stokes-basis jumps are transported with the full cycle; pole crossings, poles at infinity, and any resulting discontinuity are tracked; anomaly inflow and edge modes are matched; and duplicate gauge zero modes are removed.", "output": "A chamber-, boundary-, scheme-, and normalization-qualified observable." } ], "failure_exits": [ { "after_stages": [ 1, 2, 3 ], "label": "Source or global-Q obstruction", "causes": [ "obstructed current improvement", "missing spin-R patching", "uncancelled R or supersymmetry anomaly", "incompatible boundary algebra" ] }, { "after_stages": [ 4, 5 ], "label": "Ward-identity or limit obstruction", "causes": [ "measure anomaly", "field-space boundary", "moving contour", "noncompact direction", "missed sector", "nonuniform large-t limit" ] }, { "after_stages": [ 6, 7 ], "label": "Undefined fluctuation measure", "causes": [ "reducible saddle", "wrong gauge complex", "unremoved zero mode", "undetermined determinant phase", "unfixed counterterm" ] }, { "after_stages": [ 8 ], "label": "Qualified or discontinuous result", "causes": [ "untransported Stokes-basis jump", "Jeffrey-Kirwan chamber change", "pole at infinity", "boundary anomaly", "normalization ambiguity" ] } ], "output_measure": { "schematic_formula": "sum over sectors alpha of the integral over the transported Gamma_alpha in M_alpha of exp(-S_cl,alpha) times Z_1-loop,alpha times Z_rem,alpha times d mu_zero,alpha; a Jeffrey-Kirwan chamber may represent Gamma_alpha only in a derived meromorphic, projective Cartan problem", "required_qualifiers": [ "theory and background", "global bundles", "Q squared", "sector range", "integration cycle", "boundary condition", "regulator and phase", "local counterterm scheme", "normalization", "parameter chamber" ] }, "primary_sources": [ { "citation": "Komargodski and Seiberg 2010", "url": "https://arxiv.org/abs/1002.2228", "locator": "arXiv v4, Sections 1 and 5", "use": "supercurrent multiplets, improvement obstructions, and supergravity source compatibility" }, { "citation": "Festuccia and Seiberg 2011", "url": "https://arxiv.org/abs/1105.0689", "locator": "arXiv v2, Sections 1-2", "use": "rigid supersymmetry from frozen off-shell supergravity backgrounds" }, { "citation": "Dumitrescu, Festuccia, and Seiberg 2012", "url": "https://arxiv.org/abs/1205.1115", "locator": "arXiv v2, Sections 2-4", "use": "global Killing-spinor geometry and rigid supersymmetry algebra" }, { "citation": "Witten 1988", "url": "https://doi.org/10.1007/BF01223371", "locator": "Sections 2-3, especially pp. 361, 365, and 370-371", "use": "topological twisting, Q-exact stress tensor, descent, and homological observables" }, { "citation": "Closset, Dumitrescu, Festuccia, and Komargodski 2014", "url": "https://arxiv.org/abs/1407.2598", "locator": "arXiv v2, Sections 2-4", "use": "global twisted variables, holomorphic dependence, and curved-space Q-cohomology" }, { "citation": "Schwarz and Zaboronsky 1997", "url": "https://arxiv.org/abs/hep-th/9511112", "locator": "Sections 3-4, especially Theorems 1-2 and 4", "use": "finite-dimensional odd-symmetry localization hypotheses and stationary-phase exactness" }, { "citation": "Pestun 2012", "url": "https://arxiv.org/abs/0712.2824", "locator": "arXiv v3, Sections 3-4", "use": "Q-exact deformation, fixed locus, gauge fixing, and one-loop reduction in a gauge theory" }, { "citation": "Dai and Freed 1994", "url": "https://arxiv.org/abs/hep-th/9405012", "locator": "Sections 1-3", "use": "eta invariants, determinant lines, phases, global anomalies, and gluing" }, { "citation": "Assel, Cassani, and Martelli 2014", "url": "https://arxiv.org/abs/1410.6487", "locator": "Sections 3-6", "use": "supersymmetric regulator and finite local-counterterm ambiguities on rigid backgrounds" }, { "citation": "Witten 2011", "url": "https://arxiv.org/abs/1001.2933", "locator": "arXiv v4, Sections 2-3", "use": "complex integration cycles, Lefschetz thimbles, and Stokes jumps" }, { "citation": "Benini, Eager, Hori, and Tachikawa 2015", "url": "https://arxiv.org/abs/1308.4896", "locator": "arXiv v2, Sections 2.4-2.5", "use": "Jeffrey-Kirwan residues, charge arrangements, and chamber data in supersymmetric localization" }, { "citation": "Dedushenko 2018, Gluing I", "url": "https://arxiv.org/abs/1807.04274", "locator": "Sections 2-4", "use": "boundary polarization spaces, symmetries, anomaly lines, and gluing pairings" }, { "citation": "Dedushenko 2018, Gluing II", "url": "https://arxiv.org/abs/1807.04278", "locator": "Sections 2-5", "use": "boundary localization and model-dependent finite-dimensional gluing formulas" }, { "citation": "Hori, Kim, and Yi 2015", "url": "https://arxiv.org/abs/1407.2567", "locator": "Sections 4-5", "use": "wall crossing, Coulomb directions, and contributions from infinity" } ], "registry_state": { "status": "planned", "public_route": null, "lifecycle_note": "This structured record and its SVG materialize the planned artifact without promoting registry lifecycle or asserting release readiness." }, "accessibility": { "numbered_pipeline": true, "solid_arrow_meaning": "Solid arrows are required construction steps, not renormalization-group flows or logical implications without the stated pass conditions.", "dashed_exit_meaning": "Dashed arrows leave the validity chain when a named obstruction stops or qualifies the claim.", "color_dependency": false, "reader_facing_text_equivalent": "/supersymmetry-duality/rigid-backgrounds-twists-localization/q-cohomology-path-integrals/#localization-validity-chain-text", "structured_equivalent": "This JSON record preserves every input, construction, pass condition, output, failure exit, qualifier, and source from the schematic SVG." } }