{ "schema_version": "1.0.0", "artifact_id": "qft.artifact.supersymmetry-duality.lower-d.two-dimensional-glsm-phase-mirror-ttstar", "artifact_type": "figure", "title": "Two controlled fixtures for phases, mirrors, and ground states", "reader_question": "Which relations in a two-dimensional GLSM phase diagram, mirror Landau-Ginzburg model, soliton spectrum, and tt-star bundle are actually derived in one fixed convention, and where does each description stop?", "takeaway": "The quintic certifies two semiclassical FI chambers only after charges, exclusions, anomalies, and scale hierarchies are supplied, while the CP1 fixture gives an exact protected chain from the dimensionless Coulomb relation to mirror vacua, critical values, monodromy, and a rank-two gapped tt-star bundle away from q=0.", "alt_text": "Two-panel black-and-gray map. The upper quintic panel starts from a U(1) charge vector with five plus-one fields and one minus-five field and branches to the r much greater than zero quintic sigma model and the r much less than zero Z5 Landau-Ginzburg orbifold; dashed arrows enter the complex q-plane only after the Coulomb discriminant is removed. The lower CP1 panel maps the exact relation (sigma over mu) squared equals q to the dimensionless mirror equation W tilde over mu equals x plus q over x, whose two vacua and critical values define soliton data and a rank-two tt-star bundle; a loop around q equals zero exchanges the vacua, while q equals zero is a gap boundary.", "long_description": "Panel A freezes the two-dimensional N=(2,2) quintic fixture: global gauge group U(1), charges (1,1,1,1,1,-5), superpotential P G5 with transverse G5, t=2 pi r-i theta, q=exp(-t), theta periodic by 2 pi, and zero charge sum. Solving D- and F-flatness gives a geometric chamber at r much greater than zero, with the all-X-zero locus excluded and low-energy target G5=0 in CP4 at energies much smaller than e sqrt(r). At r much less than zero, P cannot vanish, U(1) is broken to Z5, and the low-energy theory is the G5 Landau-Ginzburg orbifold at energies much smaller than e sqrt(abs(r)). Dashed arrows emphasize that continuation between these real chambers uses complex q and must remove the scheme-fixed Coulomb discriminant q star=(-5)^(-5). Panel B uses a separate exact CP1 fixture: a U(1) GLSM with two charge-one chirals and no superpotential. On the valid Coulomb patch sigma nonzero, its protected relation (sigma over mu) squared=q maps to the twisted-chiral mirror on C star with W tilde over mu=x+q/x. The vacua x plus or minus equal plus or minus square-root q have critical values plus or minus 2 mu square-root q and critical-value difference of magnitude 4 abs(mu square-root q). For nonzero q with a gap these two states form a rank-two bundle obeying D bar-q C q=0 and [D q,D bar-q]=-[C q,C bar-q]. A full loop around q=0 exchanges the two square-root branches. At q=0 the critical values and mass scale coalesce, so the displayed finite-rank gapped chart reaches its boundary.", "schematic": true, "scale_status": "not to scale", "conventions": { "spacetime": "two-dimensional Lorentzian metric (+,-), analytically continued locally where needed", "supersymmetry": "N=(2,2); Sigma and the Hori-Vafa mirror variables are twisted chiral", "gauge_global_form": "compact U(1) with integral charges and minimal allowed charge one", "fi_theta": "L_FI,theta=-r D+(theta/2pi) F_01; t=2pi r-i theta; q=exp(-t); theta is periodic by 2pi", "cp1_scale": "q is the renormalized dimensionless parameter at scale mu; sigma/mu and the mirror coordinate x are dimensionless, while W_tilde has mass dimension one", "logarithm_branch": "vacuum equations are exponentiated; square-root labels are local and exchange under one counterclockwise loop around q=0", "quintic_discriminant": "in the displayed one-loop convention product_i (Q_i sigma/mu)^(Q_i)=q, hence q_star=(-5)^(-5)", "tt_star": "D is the Berry/Chern connection and C_q is chiral-ring multiplication in a finite normalizable gapped ground-state bundle", "status_encoding": "solid arrows are derived protected relations; dashed arrows are conditional regime changes or analytic continuations" }, "ordered_semantic_equivalent": [ { "panel": "A", "fixture": "quintic U(1) GLSM", "inputs": [ "charges Q=(1,1,1,1,1,-5)", "W=P G5(X), with G5 transverse", "t=2pi r-i theta, q=exp(-t), theta~theta+2pi", "sum_i Q_i=0, so the perturbative FI running and axial gauge anomaly coefficient vanish" ], "relationships": [ { "condition": "r>>0", "excluded_locus": "X_1=...=X_5=0", "residual_group": "none on a smooth generic vacuum", "low_energy_description": "G5=0 subset CP4", "control": "E<x for the displayed protected ring and vacuum comparison", "status": "object-level mirror dictionary" }, { "step": 3, "object": "vacua and soliton central data", "equation": "x_+=+sqrt(q), x_-=-sqrt(q); W_+=+2mu sqrt(q), W_-=-2mu sqrt(q); abs(Delta W)=4 abs(mu sqrt(q))", "status": "exact critical-point and critical-value calculation; no convention-independent soliton-mass factor is asserted" }, { "step": 4, "object": "tt-star ground-state bundle", "equation": "D_barq C_q=0; [D_q,D_barq]=-[C_q,Cbar_barq]", "status": "valid where two normalizable supersymmetric ground states remain separated from the rest of the spectrum by a gap" }, { "step": 5, "object": "monodromy and boundary", "equation": "q->exp(2pi i)q sends sqrt(q)->-sqrt(q), exchanging the two local vacuum labels", "failure_boundary": "at q=0 the critical values and mass scale coalesce; the displayed finite-rank gapped chart is not continued through that point" } ] } ], "evidence_ceiling": "The diagram establishes the displayed semiclassical phase certificates and protected CP1 calculations. It does not promote a partial ring/vacuum match or a real-axis phase sketch to an unqualified equivalence of complete infrared QFTs.", "accessibility_encoding": { "color_independence": "Black and gray only; solid versus dashed arrows, direct labels, panel letters, inequalities, and explicit status words redundantly encode every distinction.", "mobile": "The owner page uses horizontal panning at narrow widths and supplies this ordered semantic equivalent as a reflowing table.", "print": "The SVG has an explicit white canvas and monochrome line-style encoding; the semantic table is the print-reflow equivalent." }, "primary_sources": [ { "source_id": "witten-1993-phases", "citation": "Edward Witten, Phases of N=2 Theories in Two Dimensions, Nuclear Physics B 403 (1993) 159-222", "doi": "10.1016/0550-3213(93)90033-L", "url": "https://arxiv.org/abs/hep-th/9301042v3", "version": "arXiv v3", "revision_date": "1993-02-18", "locators": [ "§§3-4" ], "use": "GLSM vacuum equations, quintic phases, and Coulomb singularity logic" }, { "source_id": "hori-vafa-2000", "citation": "Kentaro Hori and Cumrun Vafa, Mirror Symmetry, arXiv:hep-th/0002222", "doi": "10.48550/arXiv.hep-th/0002222", "url": "https://arxiv.org/abs/hep-th/0002222v3", "version": "arXiv v3", "revision_date": "2000-03-31", "locators": [ "§3" ], "use": "Abelian dualization, affine Y constraints, and the CP1 Toda mirror" }, { "source_id": "cecotti-vafa-1991-ttstar", "citation": "Sergio Cecotti and Cumrun Vafa, Topological-Antitopological Fusion, Nuclear Physics B 367 (1991) 359-461", "doi": "10.1016/0550-3213(91)90021-O", "url": "https://doi.org/10.1016/0550-3213(91)90021-O", "version": "published article", "revision_date": "1991-12-23", "locators": [ "§§3-4" ], "use": "ground-state metric, Berry connection, ring action, and tt-star compatibility" }, { "source_id": "cecotti-vafa-1991-cpn", "citation": "Sergio Cecotti and Cumrun Vafa, Exact Results for Supersymmetric Sigma Models, Physical Review Letters 68 (1992) 903-906", "doi": "10.1103/PhysRevLett.68.903", "url": "https://arxiv.org/abs/hep-th/9111016v1", "version": "arXiv v1", "revision_date": "1991-11-07", "locators": [ "full article; CP1 example" ], "use": "CPn ground-state metric and affine-Toda/tt-star specialization" } ] }