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FTheory Compactifications with Multiple U(1)Factors: Constructing Elliptic Fibrations with Rational Sections
, 2013
"... We study Ftheory compactifications with U(1)×U(1) gauge symmetry on elliptically fibered CalabiYau manifolds with a rank two MordellWeil group. We find that the natural presentation of an elliptic curve E with two rational points and a zero point is the generic CalabiYau onefold in dP2. We dete ..."
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We study Ftheory compactifications with U(1)×U(1) gauge symmetry on elliptically fibered CalabiYau manifolds with a rank two MordellWeil group. We find that the natural presentation of an elliptic curve E with two rational points and a zero point is the generic CalabiYau onefold in dP2. We determine the birational map to its Tate and Weierstrass form and the coordinates of the two rational points in Weierstrass form. We discuss its resolved elliptic fibrations over a general base B and classify them in the case of B = P2. A thorough analysis of the generic codimension two singularities of these elliptic CalabiYau manifolds is presented. This determines the general U(1)×U(1)charges of matter in corresponding Ftheory compactifications. The matter multiplicities for the fibration over P2 are determined explicitly and shown to be consistent with anomaly cancellation. Explicit toric examples are constructed, both with U(1)×U(1) and SU(5)×U(1)×U(1) gauge symmetry. As a byproduct, we prove the birational equivalence of the two elliptic fibrations with elliptic fibers in the two blowups Bl(1,0,0)P²(1, 2, 3) and Bl(0,1,0)P2(1, 1, 2) employing birational maps and extremal transitions.
Elliptic Fibrations with Rank Three MordellWeil Group: Ftheory with U(1)×U(1)×U(1) Gauge Symmetry
, 2013
"... We analyze general Ftheory compactifications with U(1)xU(1)xU(1) Abelian gauge symmetry by constructing the general elliptically fibered CalabiYau manifolds with a rank three MordellWeil group of rational sections. The general elliptic fiber is shown to be a complete intersection of two nongen ..."
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Cited by 22 (2 self)
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We analyze general Ftheory compactifications with U(1)xU(1)xU(1) Abelian gauge symmetry by constructing the general elliptically fibered CalabiYau manifolds with a rank three MordellWeil group of rational sections. The general elliptic fiber is shown to be a complete intersection of two nongeneric quadrics in P3 and resolved elliptic fibrations are obtained by embedding the fiber as the generic CalabiYau complete intersection into Bl3P3, the blowup of P3 at three points. For a fixed base B, there are finitely many CalabiYau elliptic fibrations. Thus, Ftheory compactifications on these CalabiYau manifolds are shown to be labeled by integral points in reflexive polytopes constructed from the nefpartition of Bl3P3. We determine all 14 massless matter representations to six and four dimensions by an explicit study of the codimension two singularities of the elliptic fibration. We obtain three matter representations charged under all three U(1)factors, most notably a trifundamental representation. The existence of these representations, which are not present in generic perturbative Type II compactifications, signifies an intriguing universal structure of codimension two singularities of the elliptic fibrations with higher rank MordellWeil groups. We also compute explicitly the corresponding 14 multiplicities of massless hypermultiplets of a sixdimensional Ftheory compactification for a general base B.
Box Graphs and Singular Fibers
"... drm physics.ucsb.edu We determine the higher codimension fibers of elliptically fibered CalabiYau fourfolds with section by studying the threedimensional N = 2 supersymmetric gauge theory with matter which describes the low energy effective theory of Mtheory compactified on the associated Weierst ..."
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drm physics.ucsb.edu We determine the higher codimension fibers of elliptically fibered CalabiYau fourfolds with section by studying the threedimensional N = 2 supersymmetric gauge theory with matter which describes the low energy effective theory of Mtheory compactified on the associated Weierstrass model, a singular model of the fourfold. Each phase of the Coulomb branch of this theory corresponds to a particular resolution of the Weierstrass model, and we show that these have a concise description in terms of decorated box graphs based on the representation graph of the matter multiplets, or alternatively by a class of convex paths on said graph. Transitions between phases have a simple interpretation as “flopping ” of the path, and in the geometry correspond to actual flop transitions. This description of the phases enables us to enumerate and determine the entire network between them, with various matter representations for all reductive Lie groups. Furthermore, we observe that each network of phases carries the structure of a (quasi)minuscule representation of a specific Lie algebra. Interpreted from a geometric point of view, this analysis determines the generators of the cone of effective curves as well as the network of flop transitions between crepant resolutions of singular elliptic CalabiYau fourfolds. From the box graphs we determine all fiber types in codimensions two and three, and we find new, nonKodaira, fiber types for E6, E7 and E8. ar X iv
Chiral FourDimensional FTheory Compactifications With SU(5) and Multiple U(1)Factors
, 2014
"... We develop geometric techniques to determine the spectrum and the chiral indices of matter multiplets for fourdimensional Ftheory compactifications on elliptic CalabiYau fourfolds with rank two MordellWeil group. The general elliptic fiber is the CalabiYau onefold in dP2. We classify its resol ..."
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We develop geometric techniques to determine the spectrum and the chiral indices of matter multiplets for fourdimensional Ftheory compactifications on elliptic CalabiYau fourfolds with rank two MordellWeil group. The general elliptic fiber is the CalabiYau onefold in dP2. We classify its resolved elliptic fibrations over a general base B. The study of singularities of these fibrations leads to explicit matter representations, that we determine both for U(1)×U(1) and SU(5)×U(1)×U(1) constructions. We determine for the first time certain matter curves and surfaces using techniques involving prime ideals. The vertical cohomology ring of these fourfolds is calculated for both cases and general formulas for the Euler numbers are derived. Explicit calculations are presented for a specific base B = P3. We determine the general G4flux that belongs to H(2,2)V of the resolved CalabiYau fourfolds. As a byproduct, we derive for the first time all conditions on G4flux in general Ftheory compactifications with a nonholomorphic zero section. These conditions have to be formulated after a circle reduction in terms of ChernSimons terms on the 3D Coulomb branch and invoke Mtheory/Ftheory duality. New ChernSimons terms are generated by KaluzaKlein states of the circle compactification. We explicitly perform the relevant field theory computations, that yield nonvanishing results precisely for fourfolds with a nonholomorphic zero section. Taking into account the new ChernSimons terms, all 4D matter chiralities are determined via 3D Mtheory/Ftheory duality. We independently check these chiralities using the subset of matter surfaces we determined. The presented techniques are general and do not rely on toric data.
KCLmth1301 IPMU120235 On Singular Fibres in FTheory
"... In this paper, we propose a connection between the field theory local model (Katz–Vafa field theory) and the type of singular fibre in flat crepant resolutions of elliptic CalabiYau fourfolds, a class of fourfolds considered by Esole and Yau. We review the analysis of degenerate fibres for models w ..."
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In this paper, we propose a connection between the field theory local model (Katz–Vafa field theory) and the type of singular fibre in flat crepant resolutions of elliptic CalabiYau fourfolds, a class of fourfolds considered by Esole and Yau. We review the analysis of degenerate fibres for models with gauge groups SU(5) and SO(10) in detail, and observe that the naively expected fibre type is realized if and only if the Higgs vev in the field theory local model is unramified. To test this idea, we implement a linear (unramified) Higgs vev for the “E6 ” Yukawa point in a model with gauge group SU(5) and verify that this indeed leads to a fibre of Kodaira type IV∗. Based on this observation, we argue i) that the singular fibre types appearing in the fourfolds studied by Esole–Yau are not puzzling at all, (so that this class of fourfolds does not have to be excluded from the candidate of input data of some yetunknown formulation of Ftheory) and ii) that such fourfold geometries also contain more information than just the eigenvalues of the Higgs field vev configuration in the field theory local models. ar X iv
and its Higgs Branches
, 2014
"... We consider Ftheory compactifications on genusone fibered CalabiYau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a baseindependent analysis of the codimension one, two and three singularities of these fibrati ..."
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We consider Ftheory compactifications on genusone fibered CalabiYau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a baseindependent analysis of the codimension one, two and three singularities of these fibrations. We use these geometric results to determine the gauge groups, matter representations, 6D matter multiplicities and 4D Yukawa couplings of the corresponding effective theories. All these theories have a nontrivial gauge group and matter content. We explore the network of Higgsings relating these theories. Such Higgsings geometrically correspond to extremal transitions induced by blowups in the 2D toric varieties. We recover the 6D effective theories of all 16 toric hypersurface fibrations by repeatedly Higgsing the theories that exhibit MordellWeil torsion. We find that the three CalabiYau manifolds without section, whose fibers are given by the toric hypersurfaces in P2, P1 × P1 and the recently studied P2(1, 1, 2), yield Ftheory realizations of SUGRA theories with discrete gauge groups Z3, Z2 and Z4. This opens up a whole new arena for model building with discrete global symmetries in Ftheory. In these three manifolds, we also find codimension two I2fibers supporting matter charged only under these discrete gauge groups. Their 6D matter multiplicities are computed employing ideal techniques and the associated Jacobian fibrations. We also show that the Jacobian of the biquadric fibration has one rational section, yielding one U(1)gauge field in Ftheory. Furthermore, the elliptically fibered CalabiYau manifold based on dP1 has a U(1)gauge field induced by a nontoric rational section. In this model, we find the first Ftheory realization of matter with U(1)charge q = 3.
NonHiggsable clusters for 4D Ftheory models
, 2015
"... Abstract: We analyze nonHiggsable clusters of gauge groups and matter that can arise at the level of geometry in 4D Ftheory models. NonHiggsable clusters seem to be generic features of Ftheory compactifications, and give rise naturally to structures that include the nonabelian part of the standa ..."
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Abstract: We analyze nonHiggsable clusters of gauge groups and matter that can arise at the level of geometry in 4D Ftheory models. NonHiggsable clusters seem to be generic features of Ftheory compactifications, and give rise naturally to structures that include the nonabelian part of the standard model gauge group and certain specific types of potential dark matter candidates. In particular, there are nine distinct single nonabelian gauge group factors, and only five distinct products of two nonabelian gauge group factors with matter, including SU(3) × SU(2), that can be realized through 4D nonHiggsable clusters. There are also more complicated configurations involving more than two gauge factors; in particular, the collection of gauge group factors with jointly charged matter can exhibit branchings, loops, and long linear chains.
the University of Liverpool for the degree of Doctor in Philosophy
"... Thesis submitted in accordance with the requirements of ..."
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