{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.supersymmetry-duality.lower-d.anomaly-contact-duality-ledger",
  "artifact_type": "table",
  "evidence_date": "2026-08-24",
  "title": "Dimension-labelled anomaly, contact, and duality fixtures",
  "reader_question": "Which anomaly-like coefficients, R-current procedures, operator maps, deformation tests, and evidence claims may be compared between two- and three-dimensional supersymmetric field theories without conflating their meanings?",
  "dominant_point": "Two-dimensional chiral anomalies control Jacobi phases and c-extremization, whereas three-dimensional parity-odd contact terms control quantized Chern–Simons response; both constrain infrared claims, but neither datum may be substituted for the other.",
  "conventions": {
    "two_dimensional": "For Weyl fermions, k^{AB}=Tr(gamma^3 Q^A Q^B), with gamma^3=+1 on right-movers and -1 on left-movers; k_grav=Tr(gamma^3)=c_R-c_L. Integral flavor charges give z~z+1, while z~z+lambda tau+mu carries the anomaly multiplier.",
    "three_dimensional": "On a closed oriented spin three-manifold, compact U(1) fields have integral flux F/(2 pi), CS[A]=(1/(4 pi)) integral A dA, and BF[b;A]=(1/(2 pi)) integral b dA. Levels are dimensionless. A background integer shift is a scheme choice, while a dynamical level must obey the exact global quantization law.",
    "regulator": "A charge-one complex two-component fermion of real mass M shifts the declared three-dimensional Abelian level by Delta k=(1/2) sgn(M). Reversing orientation reverses every Chern–Simons sign; changing regulator changes the integer counterterm representative.",
    "evidence": "Exact anomaly arithmetic and massive-response checks are distinguished from a conjectured equality of interacting nonsupersymmetric fixed points."
  },
  "records": [
    {
      "id": "2d-ak-zero-two-lg",
      "dimension": 2,
      "algebra": "N=(0,2) presentation of the chapter's N=(2,2) A_k Landau-Ginzburg model; the same potential is often indexed A_(k+1) in singularity terminology",
      "fields": {
        "multiplets": "one (0,2) chiral Phi and one Fermi Gamma",
        "interactions": "E=0 and J=Phi^(k+1), equivalent to W=Phi^(k+2)/(k+2)",
        "fermions": [
          {
            "name": "psi_+",
            "chirality_sign": 1,
            "trial_R": "r-1",
            "flavor_F": 1
          },
          {
            "name": "gamma_-",
            "chirality_sign": -1,
            "trial_R": "1-(k+1)r",
            "flavor_F": "-(k+1)"
          }
        ]
      },
      "anomaly_data": {
        "k_FF": "-k(k+2)",
        "k_RF": "r-1+(k+1)[1-(k+1)r]",
        "k_RR": "(r-1)^2-[1-(k+1)r]^2",
        "k_grav": 0,
        "gauge_anomaly": "not applicable: this fixture is ungauged"
      },
      "r_current": {
        "procedure": "c-extremize c_R^tr=3 k_RR over the eligible flavor mixing direction",
        "stationary_point": "r_*=1/(k+2), where k_RF=0",
        "central_charges": "c_R=c_L=3k/(k+2)"
      },
      "operators_and_deformations": {
        "protected_operator_ring": "C[Phi]/(Phi^(k+1))",
        "relevant_deformations": "lower monomials in J split or collide critical points while preserving the declared supersymmetry",
        "index": "the unrefined elliptic genus and Witten index equal k+1"
      },
      "evidence_status": "Exact for the compact isolated unitary LG fixed point after the complete infrared mixing family is included. The flavor Jacobi index is m_FF=k(k+2)/2, so half-integral cases require the declared multiplier system. No duality is asserted. Accidental currents, nonnormalizable currents, or continuum sectors invalidate naive use of the ultraviolet trial family.",
      "reader_table": {
        "dimension_and_fixture": "**2d — the chapter’s $A_k$ LG convention in a $(0,2)$ presentation** (often called $A_{k+1}$ in singularity indexing)",
        "algebra_fields_normalization": "$\\Phi$ chiral and $\\Gamma$ Fermi, $E=0$, $J=\\Phi^{k+1}$. Right-moving $\\psi_+$ has $(R,F)=(r-1,1)$; left-moving $\\gamma_-$ has $(R,F)=(1-(k+1)r,-(k+1))$.",
        "anomaly_or_contact_data": "$k^{AB}=\\operatorname{Tr}\\gamma^3Q^AQ^B$ is dimensionless, with $\\gamma^3=+1$ right and $-1$ left. Thus $k^{FF}=-k(k+2)$, $k^{RF}=r-1+(k+1)[1-(k+1)r]$, and $k_{\\mathrm{grav}}=0$. There is no gauge field. Integral $F$ charges give $z\\sim z+1$ with anomaly phases under elliptic shifts.",
        "r_current_operator_deformation": "c-extremization sets $k^{RF}=0$: $r_*=1/(k+2)$ and $c_R=c_L=3k/(k+2)$. Local chiral operators are classes in $\\mathbb C[\\Phi]/(\\Phi^{k+1})$; lower monomials in $J$ deform the critical points.",
        "protected_check_and_evidence": "Exact compact-LG result if the fixed point is unitary and isolated and the mixing family is complete: $m^{FF}=k(k+2)/2$ and the unrefined genus and Witten index are $k+1$. Half-integral $m^{FF}$ uses a multiplier system. No duality is asserted; accidental or nonnormalizable currents and continua are stopping conditions."
      }
    },
    {
      "id": "3d-n2-charge-one-parity",
      "dimension": 3,
      "algebra": "N=2 charge-one chiral coupled to a compact dynamical U(1)",
      "fields": {
        "multiplet": "one N=2 chiral with scalar R charge r and fermion R charge r-1",
        "global_setting": "closed oriented spin three-manifold; integral compact-U(1) flux",
        "bare_level": "k_bare=-1/2 in the declared gauge-invariant regulator presentation"
      },
      "contact_data": {
        "fermion_shift": "Delta k=(1/2) sgn(M)",
        "gauge_quantization": "k_bare+(1/2) sum_i q_i^2 is an integer",
        "positive_mass": "k_eff=0",
        "negative_mass": "k_eff=-1",
        "background_rule": "fractional contact classes are invariant; integer representatives depend on a shared counterterm scheme"
      },
      "r_current": {
        "procedure": "no two-dimensional c-extremization applies",
        "three_dimensional_rule": "at an interacting N=2 fixed point the exact R current is determined by three-dimensional F-maximization, subject to accidental-symmetry and convergence checks",
        "role_of_contacts": "R and flavor contact terms constrain response and duality matching but do not by themselves determine the exact R charge"
      },
      "operators_and_deformations": {
        "deformation": "the sign of the real mass M selects two matter-mass chambers with different induced levels",
        "endpoint_difference": "the induced levels differ by one; the negative-mass chamber contains a level -1 spin Chern-Simons sector, while the zero-level dynamical gauge theory is not automatically a gapped empty phase"
      },
      "evidence_status": "The regulated one-loop induced-level shift and large-gauge quantization test are exact. The infrared fate of the zero-level dynamical gauge field depends on Maxwell, FI, monopole, and global data. This row asserts no duality and no unorientable or boundary extension.",
      "reader_table": {
        "dimension_and_fixture": "**3d — $\\mathcal N=2$ charge-one chiral induced-level fixture**",
        "algebra_fields_normalization": "Closed oriented spin three-manifold; compact $U(1)$ with $\\int_\\Sigma F/(2\\pi)\\in\\mathbb Z$; $\\operatorname{CS}[A]=(4\\pi)^{-1}\\int A\\wedge dA$. One charge-one chiral uses $k_{\\mathrm{bare}}=-1/2$.",
        "anomaly_or_contact_data": "A massive fermion shifts $\\Delta k=\\tfrac12\\operatorname{sgn}M$, so $k_{\\mathrm{eff}}=0$ for $M>0$ and $-1$ for $M<0$. Levels are dimensionless. Background integer shifts are scheme choices; dynamical quantization $k_{\\mathrm{bare}}+\\tfrac12\\sum_iq_i^2\\in\\mathbb Z$ is compulsory.",
        "r_current_operator_deformation": "The scalar has R charge $r$ and its fermion $r-1$. The sign of the real mass $M$ selects the chamber. Two-dimensional c-extremization does not apply; 3d $\\mathcal N=2$ fixed points use F-maximization, while contact terms constrain response rather than determine $r$.",
        "protected_check_and_evidence": "The regulated one-loop induced-level shift and large-gauge test are exact. A remaining dynamical $U(1)_0$ field is not automatically a gapped empty phase; its fate needs Maxwell, FI, monopole, and global data. No duality is asserted. Boundary inflow and Pin refinements require extra data."
      }
    },
    {
      "id": "3d-abelian-bosonization-seed",
      "dimension": 3,
      "algebra": "nonsupersymmetric critical descendant used in three-dimensional duality webs",
      "fields": {
        "fermionic_side": "one charge-one Dirac fermion with the counterterm -(1/2) CS[A]",
        "bosonic_side": "Wilson-Fisher scalar coupled to a compact U(1) field b with CS[b]+BF[b;A]",
        "normalization": "CS[A]=(1/(4 pi)) integral A dA and BF[b;A]=(1/(2 pi)) integral b dA"
      },
      "contact_data": {
        "fermion_responses": "k_A=0 for M>0 and k_A=-1 for M<0",
        "boson_responses": "the Higgs chamber has zero response; integrating out the unHiggsed U(1)_1 field gives -CS[A] plus the convention-matched invertible spin/framing response",
        "orientation_and_regulator": "orientation reversal flips all Chern-Simons signs; a regulator change shifts the shared integer background counterterm"
      },
      "r_current": {
        "procedure": "none: the displayed seed is nonsupersymmetric",
        "parent_boundary": "a supersymmetric parent can motivate a deformation map but does not prove survival of the descendant critical point"
      },
      "operators_and_deformations": {
        "current_map": "j_f^mu maps to (1/(2 pi)) epsilon^{mu nu rho} partial_nu b_rho",
        "operator_map": "the fermion is represented by a gauge-invariant dressed monopole, schematically phi-dagger times M_b in this convention, whose spin and statistics include the level-one Chern-Simons sector",
        "deformation": "for the displayed bosonic term -r |phi|^2, M>0 maps to r>0 in the Higgsed zero-response chamber and M<0 maps to r<0 in the unHiggsed chamber"
      },
      "evidence_status": "The current variation and the two massive response calculations are exact algebraic checks once the regulated seed is stated. Equality of the finite-rank interacting nonsupersymmetric fixed points remains the conjectural core and can fail through a split or first-order transition or an intervening phase.",
      "reader_table": {
        "dimension_and_fixture": "**3d — regulated Abelian bosonization seed**",
        "algebra_fields_normalization": "No superalgebra in the displayed descendant. A charge-one Dirac fermion with $-\\tfrac12\\operatorname{CS}[A]$ is compared with a Wilson–Fisher scalar coupled to compact $b$ through $\\operatorname{CS}[b]+\\operatorname{BF}[b;A]$, where $\\operatorname{BF}[b;A]=(2\\pi)^{-1}\\int b\\wedge dA$.",
        "anomaly_or_contact_data": "The fermion chambers have background levels $0$ and $-1$. The bosonic Higgs chamber has zero response; the unHiggsed $U(1)_1$ chamber induces $-\\operatorname{CS}[A]$ plus its convention-matched invertible spin/framing response. Orientation and regulator choices are part of the statement.",
        "r_current_operator_deformation": "There is no R-extremization for this nonsupersymmetric seed. $j_f^\\mu\\longleftrightarrow(2\\pi)^{-1}\\epsilon^{\\mu\\nu\\rho}\\partial_\\nu b_\\rho$; the fermion maps to a gauge-invariant dressed monopole. For the displayed $-r\\lvert\\phi\\rvert^2$, $M>0\\longleftrightarrow r>0$ is the Higgsed chamber.",
        "protected_check_and_evidence": "Current variation and the two gapped response calculations are exact checks once the seed is stated. Equality of the interacting finite-rank critical points is conjectural; phase matching is necessary, not sufficient."
      }
    }
  ],
  "sources": [
    {
      "id": "benini-bobev-2013",
      "citation": "Francesco Benini and Nikolay Bobev, Exact Two-Dimensional Superconformal R-Symmetry and c-Extremization, Physical Review Letters 110 (2013) 061601",
      "doi": "10.1103/PhysRevLett.110.061601",
      "url": "https://arxiv.org/abs/1211.4030v2",
      "locators": [
        "pp. 1-3",
        "eqs. (2)-(7)"
      ],
      "use": "two-dimensional anomaly convention, trial central charge, stationarity, and applicability conditions"
    },
    {
      "id": "kawai-yamada-yang-1994",
      "citation": "Toshiya Kawai, Yasuhiko Yamada, and Sung-Kil Yang, Elliptic Genera and N=2 Superconformal Field Theory, Nuclear Physics B 414 (1994) 191-212",
      "doi": "10.1016/0550-3213(94)90428-6",
      "url": "https://arxiv.org/abs/hep-th/9306096",
      "locators": [
        "sections 2-3"
      ],
      "use": "Landau-Ginzburg elliptic genus and A_k fixed-point cross-check"
    },
    {
      "id": "jafferis-2011",
      "citation": "Daniel L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, Journal of High Energy Physics 05 (2012) 159",
      "doi": "10.1007/JHEP05(2012)159",
      "url": "https://arxiv.org/abs/1012.3210",
      "locators": [
        "sections 1 and 4"
      ],
      "use": "three-dimensional N=2 F-maximization and its relation to the superconformal R current"
    },
    {
      "id": "redlich-1984",
      "citation": "A. N. Redlich, Gauge Noninvariance and Parity Nonconservation of Three-Dimensional Fermions, Physical Review D 29 (1984) 2366-2374",
      "doi": "10.1103/PhysRevD.29.2366",
      "url": "https://doi.org/10.1103/PhysRevD.29.2366",
      "locators": [
        "pp. 2366-2374"
      ],
      "use": "regulated three-dimensional fermion determinant and half-level parity-odd response"
    },
    {
      "id": "closset-et-al-2012",
      "citation": "Cyril Closset, Thomas T. Dumitrescu, Guido Festuccia, Zohar Komargodski, and Nathan Seiberg, Comments on Chern-Simons Contact Terms in Three Dimensions, JHEP 09 (2012) 091",
      "doi": "10.1007/JHEP09(2012)091",
      "url": "https://arxiv.org/abs/1206.5218",
      "locators": [
        "sections 2-4"
      ],
      "use": "background contact classes, counterterm schemes, and supersymmetric flavor, R, and gravitational terms"
    },
    {
      "id": "seiberg-senthil-wang-witten-2016",
      "citation": "Nathan Seiberg, T. Senthil, Chong Wang, and Edward Witten, A Duality Web in 2+1 Dimensions and Condensed Matter Physics, Annals of Physics 374 (2016) 395-433",
      "doi": "10.1016/j.aop.2016.08.007",
      "url": "https://arxiv.org/abs/1606.01989",
      "locators": [
        "sections 2.1-2.2"
      ],
      "use": "regulated Abelian bosonization seed, current map, and massive response checks"
    }
  ],
  "lifecycle_status": "materialized governed table; checksum-bound registry promotion and release acceptance remain separate decisions",
  "semantic_role": "structured equivalent of the reader-facing dimension-labelled comparison",
  "verification": {
    "record_count": 3,
    "dimension_labels": [
      2,
      3,
      3
    ],
    "blank_field_count": 0,
    "ak_general_fixtures_checked": 12,
    "ak_k3_checkpoint": {
      "k": 3,
      "r_star": 0.2,
      "k_RR": 0.6,
      "k_RF": 0,
      "k_FF": -15,
      "k_grav": 0,
      "c_R": 1.8,
      "c_L": 1.8,
      "witten_index": 4
    },
    "three_dimensional_parity_fixture": {
      "k_bare": -0.5,
      "gauge_quantization_integer": 0,
      "k_eff_positive_mass": 0,
      "k_eff_negative_mass": -1
    },
    "bosonization_response_fixture": {
      "K": 1,
      "background_charge": 1,
      "induced_background_level": -1
    },
    "source_count": 6,
    "reader_table_sha256": "dd35e5ca8d816469fe74ba554ca701c211b3002316f897ea6fc1ccc86e482f96",
    "dimension_specific_meanings_kept_separate": true,
    "phase_matching_marked_necessary_not_sufficient": true
  }
}