{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.supersymmetry-duality.lower-d.three-dimensional-monopole-contact-duality-web",
  "artifact_type": "figure",
  "title": "Representative records behind three-dimensional duality arrows",
  "reader_question": "Which parity, monopole, global-form, contact-term, deformation, and evidence data must accompany a three-dimensional duality arrow before it can be used?",
  "takeaway": "A representative three-dimensional duality map must carry a regulator-complete theory card and a gauge-invariant monopole sector; the displayed supersymmetric dictionaries have protected checks, their threshold flow is conditional on the parent claim, and the nonsupersymmetric descendant needs an additional dynamical conjecture.",
  "alt_text": "Black-and-gray two-panel diagram. The first panel sends a theory and regulator card through effective-level and monopole-gauge-invariance checks into an infrared duality record. The second panel keeps the N=4 U(1) SQED/free-twisted-hyper mirror fixture separate. Below it, a generic Aharony pair undergoes a selected equal-sign threshold flow to a Giveon-Kutasov pair; the listed shifts are only the electric heavy-pair contribution, so matched magnetic-vacuum contacts remain required. A dashed arrow from that flow reaches conjectural finite-rank non-Abelian bosonization descendants. The separate Abelian particle-vortex route is not drawn.",
  "long_description": "Panel A is a certification pipeline, not an RG flow. The theory card fixes the compact global gauge group, allowed bundles and lines, tangential structure, dynamical and background Chern-Simons levels, gravitational contact, masses, FI terms, regulator, and vacuum chamber. Its displayed parity check is explicitly the compact U(1) charge-one fixture with k bare minus one half: the positive- and negative-mass effective levels are zero and minus one. The monopole gate chooses a cocharacter modulo the Weyl group, applies the BPS completion, counts matter and root zero modes, evaluates chamber-dependent Chern-Simons electric and global charges, and requires dressing under the unbroken stabilizer. Panel B keeps the N=4 mirror fixture separate from the Aharony flow. In the basic mirror, U(1) SQED with one hypermultiplet maps to a free twisted hyper: V plus maps to X, V minus to Y, Phi maps to X Y, and the free-hyper ordinary mass equals the SQED topological mass minus two pi times its FI parameter in the frozen current convention. In the generic Aharony fixture, Nf is at least Nc plus one. A parent with Nf plus p at least Nc plus one undergoes selected positive equal-sign thresholds in a matched vacuum, yielding U(Nc) at level p and U(Nf plus p minus Nc) at level minus p. Per heavy electric pair, the frozen single-trace U(Nc) level shifts by one, so the central U(1) coefficient shifts by Nc; the displayed electric axial, axial-R, R-R, and gravitational contacts are Nc, Nc times r minus one, Nc times r minus one squared, and two Nc. These are not the full relative Giveon-Kutasov contact matrix: magnetic determinants, Higgsed blocks, topological contacts, and invertible sectors remain to be computed in the matched vacuum. The dashed arrow from this flow soft-breaks supersymmetry and retunes independent relevant couplings toward a selected non-Abelian bosonization card with positive N and k and integer Nf between zero and N. Quantization and massive-response checks survive, but equality of the finite-rank nonsupersymmetric critical points is an additional dynamical conjecture, and Nf greater than N requires a separate phase-structure analysis. The distinct Abelian particle-vortex route is described on the canonical page but is not drawn in this representative artifact.",
  "schematic": true,
  "scale_status": "not to scale",
  "conventions": {
    "spacetime": "oriented Lorentzian three-manifold with metric (+--), epsilon^012=+1; Euclidean claims require their stated continuation",
    "gauge_fields": "Hermitian compact connections with integral charge-one normalization",
    "abelian_actions": "CS[a]=(1/4pi) integral a wedge da and BF[a;A]=(1/2pi) integral a wedge dA",
    "parity_regulator": "a complex two-component fermion contributes Delta kappa_IJ=(1/2) Q_I Q_J sign(M) and Delta kappa_g=sign(M)",
    "gravitational_action": "S_grav=(kappa_g/192pi) integral Tr(omega wedge d omega+(2/3) omega cubed)",
    "monopole_flux": "for connected compact G, m is a cocharacter modulo the Weyl group and (1/2pi) integral over S2 of F equals m; disconnected groups require additional bundle and twisted-sector data",
    "monopole_gauss_law": "with D=partial-iq a, positive flux, and positive CS/BF action coefficients, Q_I(V_m)=+(k_eff)_Ia m_a before matter dressing",
    "fi_topological_mass": "the action contains -zeta D and the normalized topological real mass is m_J=-2pi zeta",
    "unitary_level": "U(N)_k means [SU(N)_k x U(1)_(Nk)]/Z_N with the single fundamental-trace action",
    "status_encoding": "single-headed solid arrows in panel A are required record-assembly gates, not physical implications; the displayed one-fermion level calculation is exact, but its adjacent arrow is only a gate; double-headed solid arrows are explicitly labelled supersymmetric infrared claims with protected checks; single-headed solid arrows into the mass-flow card are exact threshold bookkeeping conditional on the parent claim; dashed arrows denote an additional unprotected descendant assumption"
  },
  "ordered_semantic_equivalent": [
    {
      "order": 1,
      "record": "theory and regulator card",
      "inputs": [
        "compact global gauge group, bundles, genuine lines, and magnetic cocharacter lattice",
        "spin or spin-c structure",
        "dynamical and background Chern-Simons matrix, BF terms, FI terms, and gravitational contact",
        "matter representations, real masses, trial R charges, regulator convention, and selected vacuum chamber"
      ],
      "exact_check": "for one compact U(1), one charge-one fermion, integral charges, and the stated spin regulator, k_bare=-1/2 gives k_eff=0 at M>0 and k_eff=-1 at M<0",
      "boundary": "k_eff=0 alone does not establish an empty or gapped infrared phase for a remaining dynamical gauge field"
    },
    {
      "order": 2,
      "record": "gauge-invariant BPS monopole sector",
      "inputs": [
        "for connected compact G, m in Hom(U(1),T)/W_G and the correlated supersymmetric scalar boundary condition; disconnected groups require additional bundle and twisted-sector data",
        "matter zero modes, root zero modes, classical and induced Chern-Simons charges"
      ],
      "equations": [
        "R_trial(V_m)=(1/2) sum_i (1-r_i) sum_rho |rho(m)|-(1/2) sum_alpha |alpha(m)|, with r_i trial scalar R charges",
        "Q_I(V_m)=+(k_eff)_Ia m_a for every gauge or background current I in the frozen positive-CS/BF Abelian Gauss-law convention"
      ],
      "output": "a gauge-neutral operator dressed under every factor of the stabilizer G_m together with all induced flavor, R, and topological charges, or an explicit statement that the flux sector supplies no proposed local chiral generator",
      "boundary": "zero-mode vacuum charges and effective Chern-Simons contributions must be partitioned in one convention rather than double counted"
    },
    {
      "order": 3,
      "record": "basic N=4 mirror fixture",
      "left": "N=4 U(1) SQED with one hypermultiplet (Q,Q-tilde), adjoint chiral Phi, and W=Phi Q Q-tilde",
      "right": "a free twisted hypermultiplet with N=2 chirals X,Y and W=0",
      "maps": [
        "V_+ <-> X",
        "V_- <-> Y",
        "Phi <-> X Y, consistent with V_+ V_- proportional to Phi",
        "SU(2)_H <-> SU(2)_C and m_free=m_J,SQED=-2pi zeta_SQED in the frozen current basis"
      ],
      "exact_checks": [
        "Delta(V_+)=Delta(V_-)=1/2",
        "the SQED Higgs branch and the free theory Coulomb branch are points",
        "the SQED quantum Coulomb ring and free-hyper Higgs ring are both generated by two unconstrained complex coordinates after eliminating Phi through V_+V_- proportional to Phi"
      ],
      "status": "supersymmetric infrared mirror claim with protected operator, branch, deformation, partition-function, and zero-mode evidence; not equality of ultraviolet Lagrangians"
    },
    {
      "order": 4,
      "record": "Aharony duality fixture",
      "left": "U(N_c)_0 with N_f fundamental-antifundamental pairs, W=0, N_c an integer at least one, N_f a nonnegative integer, N_f>=N_c+1 for the displayed generic card, and electric-quark trial scalar R charge r",
      "right": "U(N_f-N_c)_0 with N_f dual pairs, meson M, electric-monopole singlets v_+,v_-, and magnetic monopoles Vhat_+,Vhat_-",
      "equations": [
        "W_mag=M q q-tilde+v_+ Vhat_-+v_- Vhat_+",
        "Q_J(v_+,-)=(+1,-1), Q_A(v_+,-)=-N_f, R_trial(v_+,-)=N_f(1-r)-N_c+1",
        "R_trial(Vhat_+,-)=N_c+1-N_f(1-r), so each monopole cross-term has trial R=2"
      ],
      "boundary": "all displayed R formulas are trial charges and become exact dimensions only after current mixing, accidental symmetries, and decoupled free sectors are resolved; N_f=N_c and lower-flavor confining or runaway cases require separate endpoint cards",
      "status": "supersymmetric infrared duality supported by chiral rings, monopole charges, real-mass flows, indices, sphere partition functions, and massive phases"
    },
    {
      "order": 5,
      "record": "equal-sign mass flow to Giveon-Kutasov",
      "parent": "N_c is an integer at least one, N_f is a nonnegative integer, p is a positive integer, and the generic Aharony parent has N_f+p>=N_c+1; the Giveon-Kutasov bound N_c<=N_f+p includes equality only through the separate confining parent",
      "deformation": "give p electric pairs a large positive axial mass and select the correlated supersymmetric magnetic vacuum before the heavy-mass limit",
      "endpoint": "U(N_c)_p with N_f pairs <-> U(N_f+p-N_c)_(-p) with N_f pairs, meson M, and W=M q q-tilde",
      "per_heavy_electric_pair": {
        "delta_single_trace_U_Nc_level": 1,
        "delta_central_U1_level_coefficient": "N_c",
        "delta_kappa_AA": "N_c",
        "delta_kappa_AR": "N_c(r-1)",
        "delta_kappa_RR": "N_c(r-1)^2",
        "delta_kappa_gravitational": "2N_c"
      },
      "boundary": "the displayed contacts are only the heavy electric-pair contribution; the magnetic determinants, Higgsed blocks, topological and non-Abelian flavor contacts, relative gravitational term, and residual invertible sectors must be computed in the matched vacuum; a rank-zero magnetic gauge factor can still leave meson singlets when N_f>0",
      "status": "exact threshold bookkeeping in the selected chamber, conditional on the supersymmetric parent duality claim and its separately completed relative contact matrix"
    },
    {
      "order": 6,
      "record": "non-Abelian bosonization descendant edge from the Giveon-Kutasov flow",
      "domain": "for the displayed SU(N)_k scalar versus U(k) fermion local core, N and k are positive integers and N_f is an integer satisfying 0<=N_f<=N",
      "local_core": "SU(N)_k with N_f critical scalars <-> U(k)_(-N+N_f/2,-N+N_f/2) with N_f critical Dirac fermions in the stated Yang-Mills regulator convention",
      "deformation": "softly break the Giveon-Kutasov parent, tune scalar and fermion masses independently, and retain the inherited spin or spin-c prescription and all background and gravitational contacts",
      "inherited_checks": [
        "level quantization and level-rank endpoint checks",
        "massive topological phases, framing response, baryon-dressed-monopole quantum numbers, and planar correlator tests"
      ],
      "failure_modes": [
        "first-order transition",
        "intervening symmetry-breaking phase",
        "missed relevant singlet or dangerously irrelevant coupling",
        "accidental symmetry or omitted transparent sector"
      ],
      "status": "the equality of finite-rank non-Abelian bosonization critical points is a dynamical conjecture; the planar double-trace relation is conditional on the supersymmetric parent and does not establish finite-rank global data",
      "boundary": "N_f>N lies outside the displayed conjectural card and requires a separate phase-structure analysis rather than extrapolation of its simple two-phase formula"
    }
  ],
  "verification_test_vectors": {
    "compact_u1_parity": {
      "inputs": {
        "integral_charge": 1,
        "k_bare": -0.5,
        "mass_signs": [
          1,
          -1
        ]
      },
      "expected": {
        "k_eff_positive_mass": 0,
        "k_eff_negative_mass": -1,
        "change_positive_to_negative": -1
      }
    },
    "mixed_abelian_monopole_gauss_law": {
      "inputs": {
        "magnetic_flux": 1,
        "k_eff_gauge_gauge": 2,
        "k_eff_flavor_gauge": -1
      },
      "expected": {
        "gauge_charge": 2,
        "flavor_charge": -1
      }
    },
    "n4_sqed_one_gradings": {
      "inputs": {
        "flavors": 1,
        "magnetic_flux": 1,
        "topological_charges": [
          1,
          -1
        ],
        "phi_dimension": 1,
        "phi_topological_charge": 0
      },
      "expected": {
        "monopole_dimension": 0.5,
        "product_dimension": 1,
        "product_topological_charge": 0
      }
    },
    "aharony_trial_r": {
      "inputs": {
        "colors": 2,
        "flavors": 3,
        "electric_quark_trial_r": 0.5
      },
      "expected": {
        "electric_monopole_trial_r": 0.5,
        "magnetic_monopole_trial_r": 1.5,
        "cross_term_trial_r": 2
      }
    },
    "equal_sign_mass_flow": {
      "inputs": {
        "colors": 2,
        "light_flavors": 3,
        "heavy_pairs": 2,
        "trial_r": 0.5
      },
      "expected": {
        "parent_flavors": 5,
        "magnetic_rank": 3,
        "single_trace_U_N_level": 2,
        "central_U1_level_coefficient": 4,
        "kappa_AA": 4,
        "kappa_AR": -2,
        "kappa_RR": 1,
        "kappa_gravitational": 8
      }
    },
    "abelian_k_matrix_response": {
      "inputs": {
        "K": 1,
        "background_charge": 1
      },
      "expected": {
        "induced_background_level": -1
      }
    }
  },
  "evidence_ceiling": "The deterministic fixtures verify the displayed compact-U(1) parity jump, one mixed Abelian monopole Gauss-law charge vector, N=4 dimension and grading consistency, Aharony trial-R superpotential sum, selected equal-sign electric thresholds, and Abelian massive response. They do not verify a general monopole lattice or dressing problem, supply the full relative Giveon-Kutasov contact matrix, turn an infrared duality into a theorem, or promote a supersymmetry-breaking descendant to a protected result.",
  "accessibility_encoding": {
    "color_independence": "Black, white, and gray only; solid versus dashed arrows, direct labels, numbered records, and explicit status words redundantly encode every distinction.",
    "mobile": "The canonical owner uses horizontal panning at narrow widths and supplies a reflowing ordered semantic edge list.",
    "print": "The SVG has an explicit white canvas and monochrome line-style encoding; the adjacent ordered semantic edge list is the print-reflow equivalent."
  },
  "primary_sources": [
    {
      "source_id": "redlich-1984-parity",
      "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",
      "version": "published article",
      "locators": [
        "full article"
      ],
      "use": "parity anomaly and half-integer fermion-induced Chern-Simons term"
    },
    {
      "source_id": "closset-et-al-2012-contacts",
      "citation": "Cyril Closset, Thomas 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/1206.5218v2",
      "version": "arXiv v2",
      "locators": [
        "§§2-4"
      ],
      "use": "background and gravitational contact terms, integer ambiguity, and supersymmetric completion"
    },
    {
      "source_id": "borokhov-kapustin-wu-2002-monopoles",
      "citation": "Vadim Borokhov, Anton Kapustin, and Xinkai Wu, Monopole Operators and Mirror Symmetry in Three Dimensions, JHEP 12 (2002) 044",
      "doi": "10.1088/1126-6708/2002/12/044",
      "url": "https://arxiv.org/abs/hep-th/0207074v2",
      "version": "arXiv v2",
      "locators": [
        "§§3-4"
      ],
      "use": "radial-quantization monopoles, exact N=4 charges, and the basic N_f=1 mirror fixture"
    },
    {
      "source_id": "cremonesi-hanany-zaffaroni-2014",
      "citation": "Stefano Cremonesi, Amihay Hanany, and Alberto Zaffaroni, Monopole Operators and Hilbert Series of Coulomb Branches of 3d N=4 Gauge Theories, JHEP 01 (2014) 005",
      "doi": "10.1007/JHEP01(2014)005",
      "url": "https://arxiv.org/abs/1309.2657v3",
      "version": "arXiv v3, published version",
      "locators": [
        "§2, Eqs. (2.1)-(2.5)"
      ],
      "use": "GNO cocharacters, Weyl quotient, magnetic sectors, and monopole Hilbert-series charge bookkeeping"
    },
    {
      "source_id": "intriligator-seiberg-1996-mirror",
      "citation": "Kenneth Intriligator and Nathan Seiberg, Mirror Symmetry in Three Dimensional Gauge Theories, Physics Letters B 387 (1996) 513-519",
      "doi": "10.1016/0370-2693(96)01088-X",
      "url": "https://arxiv.org/abs/hep-th/9607207v1",
      "version": "arXiv v1",
      "locators": [
        "§§2-3"
      ],
      "use": "N=4 mirror theory cards, Higgs-Coulomb exchange, and mass-FI dictionary"
    },
    {
      "source_id": "aharony-hanany-intriligator-seiberg-strassler-1997",
      "citation": "Ofer Aharony, Amihay Hanany, Kenneth Intriligator, Nathan Seiberg, and Matthew J. Strassler, Aspects of N=2 Supersymmetric Gauge Theories in Three Dimensions, Nuclear Physics B 499 (1997) 67-99",
      "doi": "10.1016/S0550-3213(97)00323-4",
      "url": "https://arxiv.org/abs/hep-th/9703110v1",
      "version": "arXiv v1",
      "locators": [
        "§§2-5"
      ],
      "use": "three-dimensional N=2 trial R charges, FI and topological central-charge conventions, monopoles, and vacuum structure"
    },
    {
      "source_id": "aharony-1997-duality",
      "citation": "Ofer Aharony, IR Duality in d=3 N=2 Supersymmetric USp(2N_c) and U(N_c) Gauge Theories, Physics Letters B 404 (1997) 71-76",
      "doi": "10.1016/S0370-2693(97)00530-3",
      "url": "https://arxiv.org/abs/hep-th/9703215v1",
      "version": "arXiv v1",
      "locators": [
        "pp. 71-76"
      ],
      "use": "Aharony electric and magnetic theory cards, monopole singlets, and superpotential"
    },
    {
      "source_id": "benini-closset-cremonesi-2011",
      "citation": "Francesco Benini, Cyril Closset, and Stefano Cremonesi, Comments on 3d Seiberg-like Dualities, JHEP 10 (2011) 075",
      "doi": "10.1007/JHEP10(2011)075",
      "url": "https://arxiv.org/abs/1108.5373v4",
      "version": "arXiv v4",
      "locators": [
        "§§2-3",
        "§5.2"
      ],
      "use": "global Chern-Simons terms, Aharony dictionary, and level-changing real-mass flows"
    },
    {
      "source_id": "giveon-kutasov-2009",
      "citation": "Amit Giveon and David Kutasov, Seiberg Duality in Chern-Simons Theory, Nuclear Physics B 812 (2009) 1-11",
      "doi": "10.1016/j.nuclphysb.2008.09.045",
      "url": "https://arxiv.org/abs/0808.0360v1",
      "version": "arXiv v1",
      "locators": [
        "§§2-3"
      ],
      "use": "Giveon-Kutasov ranks, levels, matter, superpotential, and supersymmetry bound"
    },
    {
      "source_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/1606.01989v2",
      "version": "arXiv v2",
      "locators": [
        "§§1-2, especially §2.1 and Eq. (2.2)"
      ],
      "use": "positive-CS/BF monopole-charge sign, regulated Abelian bosonization seed, spin-c qualification, massive response, and descendant evidence boundary"
    },
    {
      "source_id": "gur-ari-yacoby-2015-bosonization",
      "citation": "Guy Gur-Ari and Ran Yacoby, Three Dimensional Bosonization From Supersymmetry, JHEP 11 (2015) 013",
      "doi": "10.1007/JHEP11(2015)013",
      "url": "https://arxiv.org/abs/1507.04378v2",
      "version": "arXiv v2, published version",
      "locators": [
        "§§2-5, especially §5.2"
      ],
      "use": "planar double-trace flow from supersymmetric Chern-Simons matter and the conditional evidence ceiling for nonsupersymmetric bosonization"
    },
    {
      "source_id": "aharony-2016-baryons-monopoles",
      "citation": "Ofer Aharony, Baryons, Monopoles and Dualities in Chern-Simons-Matter Theories, JHEP 02 (2016) 093",
      "doi": "10.1007/JHEP02(2016)093",
      "url": "https://arxiv.org/abs/1512.00161v1",
      "version": "arXiv v1, published version",
      "locators": [
        "§§1-4"
      ],
      "use": "finite-rank global forms, baryon-monopole maps, mass-deformed level-rank endpoints, and the distinction between evidence and a proof of critical duality"
    },
    {
      "source_id": "benini-hsin-seiberg-2017-global-symmetries",
      "citation": "Francesco Benini, Po-Shen Hsin, and Nathan Seiberg, Comments on Global Symmetries, Anomalies, and Duality in (2+1)d, JHEP 04 (2017) 135",
      "doi": "10.1007/JHEP04(2017)135",
      "url": "https://arxiv.org/abs/1702.07035v2",
      "version": "arXiv v2, published version",
      "locators": [
        "Eq. (1.1) and pp. 2-3"
      ],
      "use": "finite-rank SU(N)-U(k) bosonization card, its N_f<=N domain, global forms, and the conditional critical-point interpretation"
    }
  ]
}