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33
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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Cited by 30 (4 self)
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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.
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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Cited by 27 (3 self)
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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.
Effective action of 6D FTheory with U(1) factors: Rational sections make ChernSimons terms jump,” JHEP 1307
, 2013
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The Tate Form on Steroids: Resolution and Higher Codimension Fibers,” arXiv:1212.2949 [hepth
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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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Cited by 2 (0 self)
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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