The Tate Form on Steroids: Resolution and Higher Codimension Fibers

Abstract

F-theory on singular elliptically fibered Calabi-Yau four-folds provides a setting to geometrically study four-dimensional N=1 supersymmetric gauge theories, including matter and Yukawa couplings. The gauge degrees of freedom arise from the codimension 1 singular loci, the matter and Yukawa couplings are generated at enhanced singularities in higher codimension. We construct the resolution of the singular Tate form for an elliptic Calabi-Yau four-fold with an ADE type singularity in codimension 1 and study the structure of the fibers in codimension 2 and 3. We determine the fibers in higher codimension which in general are of Kodaira type along minimal singular loci, and are thus consistent with the low energy gauge-theoretic intuition. Furthermore, we provide a complementary description of the fibers in higher codimension, which will also be applicable to non-minimal singularities. The irreducible components in the fiber in codimension 2 correspond to weights of representations of the ADE gauge group. These can split further in codimension 3 in a way that is consistent with the generation of Yukawa couplings. Applying this reasoning, we then venture out to study non-minimal singularities, which occur for A type along codimension 3, and for D and E also in codimension 2. The fibers in this case are non-Kodaira, however some insight into these singularities can be gained by considering the splitting of fiber components along higher codimension, which are shown to be consistent with matter and Yukawa couplings for the corresponding gauge groups.

75 pages, v2: clarifications and references added, JHEP version

References in corpus (8)

• GUTs and Exceptional Branes in F-theory - II: Experimental Predictions
• T-Branes and Monodromy
• New Aspects of Heterotic--F Theory Duality
• Monodromies, Fluxes, and Compact Three-Generation F-theory GUTs
• Anomaly Cancellation And Abelian Gauge Symmetries In F-theory
• U(1) symmetries in F-theory GUTs with multiple sections
• A Global SU(5) F-theory model with Wilson line breaking
• G-flux and Spectral Divisors

Cited by in corpus (55)

• F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches
• F-Theory Compactifications with Multiple U(1)-Factors: Constructing Elliptic Fibrations with Rational Sections
• Chiral Four-Dimensional F-Theory Compactifications With SU(5) and Multiple U(1)-Factors
• Coulomb and Higgs Branches from Canonical Singularities: Part 0
• Fibers add Flavor, Part I: Classification of 5d SCFTs, Flavor Symmetries and BPS States
• Machine Learning in the String Landscape
• Physics of F-theory compactifications without section
• 5d and 4d SCFTs: Canonical Singularities, Trinions and S-Dualities
• SU(5) Tops with Multiple U(1)s in F-theory
• Fibers add Flavor, Part II: 5d SCFTs, Gauge Theories, and Dualities
• Elliptic Fibrations with Rank Three Mordell-Weil Group: F-theory with U(1) x U(1) x U(1) Gauge Symmetry
• Box Graphs and Singular Fibers
• Yukawas and discrete symmetries in F-theory compactifications without section
• Phases of 5d SCFTs from M-/F-theory on Non-Flat Fibrations
• Coulomb and Higgs Branches from Canonical Singularities, Part 1: Hypersurfaces with Smooth Calabi-Yau Resolutions
• The Global Gauge Group Structure of F-theory Compactification with U(1)s
• Matter From Geometry Without Resolution
• Scanning the skeleton of the 4D F-theory landscape
• Non-Higgsable clusters for 4D F-theory models
• Calabi-Yau threefolds with large h^{2, 1}
• ${\mathbb{P}}^1$-bundle bases and the prevalence of non-Higgsable structure in 4D F-theory models
• F-theory and All Things Rational: Surveying U(1) Symmetries with Rational Sections
• A Monte Carlo exploration of threefold base geometries for 4d F-theory vacua
• F-theory and 2d (0,2) Theories
• Up-type quark masses in SU(5) F-theory models
• Phases, Flops and F-theory: SU(5) Gauge Theories
• 5D and 6D SCFTs from ${\mathbb{C}}^3$ orbifolds
• Non-Higgsable QCD and the Standard Model Spectrum in F-theory
• Generalized Symmetries in F-theory and the Topology of Elliptic Fibrations
• Gauge Backgrounds and Zero-Mode Counting in F-Theory
• Rational F-Theory GUTs without exotics
• Enhanced gauge symmetry in 6D F-theory models and tuned elliptic Calabi-Yau threefolds
• Tate's Algorithm for F-theory GUTs with two U(1)s
• Theories of Class F and Anomalies
• 5d SCFTs from $(E_n,E_m)$ Conformal Matter
• Singular Geometry and Higgs Bundles in String Theory
• The Toric SO(10) F-Theory Landscape
• Matter in transition
• On Singular Fibres in F-Theory
• Global tensor-matter transitions in F-theory
• When Rational Sections Become Cyclic: Gauge Enhancement in F-theory via Mordell--Weil Torsion
• Non-simply-laced Symmetry Algebras in F-theory on Singular Spaces
• The duality between F-theory and the Heterotic String in $D=8$ with two Wilson lines
• The Fate of Discrete 1-Form Symmetries in 6d
• Fibration structure in toric hypersurface Calabi-Yau threefolds
• $6$D Anomaly-Free Matter Spectrum in F-theory on Singular Spaces
• Generalized Toric Polygons, T-branes, and 5d SCFTs
• Natural F-theory constructions of Standard Model structure from $E_7$ flux breaking
• Gauge symmetry breaking with fluxes and natural Standard Model structure from exceptional GUTs in F-theory
• Towards natural and realistic $E_7$ GUTs in F-theory
• Electric-Magnetic Duality in a Class of $G_2$-Compactifications of M-theory
• Higher Dimensional Elliptic Fibrations and Zariski Decompositions
• Non-flat elliptic four-folds, three-form cohomology and strongly coupled theories in four dimensions
• Non-split singularities and conifold transitions in F-theory
• On the Sen limit squared