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16
Geometric Engineering in Toric FTheory and GUTs with U(1) Gauge Factors
, 2011
"... An algorithm to systematically construct all CalabiYau elliptic fibrations realized as hypersurfaces in a toric ambient space for a given base and gauge group is described. This general method is applied to the particular question of constructing SU(5) GUTs with multiple U(1) gauge factors. The bas ..."
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An algorithm to systematically construct all CalabiYau elliptic fibrations realized as hypersurfaces in a toric ambient space for a given base and gauge group is described. This general method is applied to the particular question of constructing SU(5) GUTs with multiple U(1) gauge factors. The basic data consists of a top over each toric divisor in the base together with compactification data giving the embedding into a reflexive polytope. The allowed choices of compactification data are integral points in an auxiliary polytope. In order to ensure the existence of a lowenergy gauge theory, the elliptic fibration must be flat, which is reformulated into conditions on the top and its embedding. In particular, flatness of SU(5) fourfolds imposes additional linear constraints on the auxiliary polytope of compactifications, and is therefore nongeneric. Abelian gauge symmetries arising in toric Ftheory compactifications are studied systematically. Associated to each top, the toric MordellWeil group determining the minimal number of U(1) factors is computed. Furthermore, all SU(5)tops and their splitting types are determined and used to infer the pattern of U(1) matter charges.
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.
Physics of Ftheory compactifications without section [1406.5180
"... Abstract: We study the physics of Ftheory compactifications on genusone fibrations without section by using an Mtheory dual description. The fivedimensional action obtained by considering Mtheory on a CalabiYau threefold is compared with a sixdimensional Ftheory effective action reduced on ..."
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Abstract: We study the physics of Ftheory compactifications on genusone fibrations without section by using an Mtheory dual description. The fivedimensional action obtained by considering Mtheory on a CalabiYau threefold is compared with a sixdimensional Ftheory effective action reduced on an additional circle. We propose that the sixdimensional effective action of these setups admits geometrically massive U(1) vectors with a charged hypermultiplet spectrum. The absence of a section induces NSNS and RR threeform fluxes in Ftheory that are nontrivially supported along the circle and induce a shiftgauging of certain axions with respect to the KaluzaKlein vector. In the fivedimensional effective theory the KaluzaKlein vector and the massive U(1)s combine into a linear combination that is massless. This U(1) is identified with the massless U(1) corresponding to the multisection of the CalabiYau threefold in Mtheory. We confirm this interpretation by computing the oneloop ChernSimons terms for the massless vectors of the fivedimensional setup by integrating out all massive states. A closed formula is found that accounts for the hypermultiplets charged under the massive U(1)s. ar X iv
SU(5) Tops with Multiple U(1)s in Ftheory
"... We study Ftheory compactifications with up to two Abelian gauge group factors that are based on elliptically fibered CalabiYau 4folds describable as generic hypersurfaces. Special emphasis is put on elliptic fibrations based on generic Bl2P2[3]fibrations. These exhibit a MordellWeil group of ra ..."
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We study Ftheory compactifications with up to two Abelian gauge group factors that are based on elliptically fibered CalabiYau 4folds describable as generic hypersurfaces. Special emphasis is put on elliptic fibrations based on generic Bl2P2[3]fibrations. These exhibit a MordellWeil group of rank two corresponding to two extra rational sections which give rise to two Abelian gauge group factors. We show that an alternative description of the same geometry as a complete intersection makes the existence of a holomorphic zerosection manifest, on the basis of which we compute the U(1) generators and a class of gauge fluxes. We analyse the fibre degenerations responsible for the appearance of localised charged matter states, whose charges, interactions and chiral index we compute geometrically. We implement an additional SU(5) gauge group by constructing the four inequivalent toric tops giving rise to SU(5) × U(1) × U(1) gauge symmetry and analyse the matter content. We demonstrate that notorious nonflat points can be avoided in welldefined CalabiYau 4folds. These methods are applied to the remaining possible hypersurface fibrations with one generic Abelian gauge factor. We analyse the local limit of our SU(5) × U(1) × U(1) models and show that one of our models is not embeddable into E8 due to recombination of matter curves that cannot be described as a Higgsing of E8. We argue that such recombination forms a general mechanism that opens up new model building possibilities in Ftheory. We dedicate this work to Matthis Florian. ar X iv
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
Discrete Gauge Symmetries by Higgsing in fourdimensional FTheory Compactifications
, 2014
"... We study Ftheory compactifications to four dimensions that exhibit discrete gauge symmetries. Geometrically these arise by deforming elliptic fibrations with two sections to a genusone fibration with a bisection. From a fourdimensional field theory perspective they are remnant symmetries from a ..."
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We study Ftheory compactifications to four dimensions that exhibit discrete gauge symmetries. Geometrically these arise by deforming elliptic fibrations with two sections to a genusone fibration with a bisection. From a fourdimensional field theory perspective they are remnant symmetries from a Higgsed U(1) gauge symmetry. We implement such symmetries in the presence of an additional SU(5) symmetry and associated matter fields, giving a geometric prescription for calculating the induced discrete charge for the matter curves and showing the absence of Yukawa couplings that are forbidden by this charge. We present a detailed map between the field theory and the geometry, including an identification of the Higgs field and the massless states before and after the Higgsing. Finally we show that the Higgsing of the U(1) induces a Gflux which precisely accounts for the change in the CalabiYau Euler number so as to leave the D3 tadpole invariant. ar X iv
The NoetherLefschetz problem and gaugegroupresolved landscapes: Ftheory on K3 × K3 as a test case
 JHEP
, 2014
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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.
TABLE OF CONTENTS
, 2004
"... Paper commissioned jointly by the Urban and Rural Change Team and the Migration Team ..."
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Paper commissioned jointly by the Urban and Rural Change Team and the Migration Team