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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.
New Global Ftheory GUTs with U(1) symmetries
"... We construct global Ftheory GUTs with SU(5)×U(1) gauge group defined by specifying a fully resolved CalabiYau fourfold and consistent fourform Gflux. Its specific U(1) charged matter spectrum allows the desired Yukawa couplings, but forbids dangerous proton decay operators. The model we find: (1 ..."
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Cited by 27 (5 self)
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We construct global Ftheory GUTs with SU(5)×U(1) gauge group defined by specifying a fully resolved CalabiYau fourfold and consistent fourform Gflux. Its specific U(1) charged matter spectrum allows the desired Yukawa couplings, but forbids dangerous proton decay operators. The model we find: (1) does not follow from an underlying higgsed E8 gauge group (2) leaves the class of theories that can be analyzed with current splitspectral cover techniques. This avoids recently proposed nogo theorems for models with hypercharge flux, as required to break the GUT group. The appearance of additional fields is related geometrically to considering a more general class of sections and 4–1 splits. We show explicitly that the fourdimensional chiral matter index can still be computed using threedimensional oneloop ChernSimons terms.
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.
The Higgs mass from a StringTheoretic Perspective, Nucl.Phys
 B874
"... The Higgs quartic coupling λ has now been indirectly measured at the electroweak scale. Assuming no new lowscale physics, its running is known and, together with gauge and Yukawa couplings, it is a crucial new piece of information constraining UV completions of the Standard Model. In particular, su ..."
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Cited by 11 (2 self)
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The Higgs quartic coupling λ has now been indirectly measured at the electroweak scale. Assuming no new lowscale physics, its running is known and, together with gauge and Yukawa couplings, it is a crucial new piece of information constraining UV completions of the Standard Model. In particular, supersymmetry broken at an intermediate or high energy scale with tan β = 1 (i.e. λ = 0) is consistent with present data and has an independent theoretical appeal. We analyze the possible stringtheoretic motivations for tan β = 1 (including both the shiftsymmetry and the more economical variant of a Z2 symmetry) in a Higgs sector realized on either 6 or 7branes. We identify specific geometries where λ ' 0 may arise naturally and specify the geometrical problems which need to be solved to determine its precise value in the generic case. We then analyze the radiative corrections to λ. Finally we show that, in contrast to naive expectations, λ < 0 at the SUSY breaking scale is also possible. Specifically, string theory may produce an MSSM plus chiral singlet at a very high scale, which immediately breaks to a nonSUSY Standard Model with λ < 0. This classically unstable theory then becomes metastable through running towards the IR. ar
Explicit de Sitter Flux Vacua for Global String Models with Chiral Matter
"... We address the open question of performing an explicit stabilisation of all closed string moduli (including dilaton, complex structure and Kähler moduli) in fluxed type IIB CalabiYau compactifications with chiral matter. Using toric geometry we construct CalabiYau manifolds with del Pezzo singula ..."
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Cited by 5 (3 self)
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We address the open question of performing an explicit stabilisation of all closed string moduli (including dilaton, complex structure and Kähler moduli) in fluxed type IIB CalabiYau compactifications with chiral matter. Using toric geometry we construct CalabiYau manifolds with del Pezzo singularities. Dbranes located at such singularities can support the Standard Model gauge group and matter content. In order to control complex structure moduli stabilisation we consider CalabiYau manifolds which exhibit a discrete symmetry that reduces the effective number of complex structure moduli. We calculate the corresponding periods in the symplectic basis of invariant threecycles and find explicit flux vacua for concrete examples. We compute the values of the flux superpotential and the string coupling at these vacua. Starting from these explicit complex structure solutions, we obtain AdS and dS minima where the Kähler moduli are stabilised by a mixture of Dterms, nonperturbative and perturbative α ′ corrections as in the LARGE Volume Scenario. In the considered example the visible sector lives at a dP6 singularity which can be higgsed to the phenomenologically interesting class of models at the dP3 singularity. 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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Cited by 4 (0 self)
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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
Mtheory Compactifications to Three Dimensions with
 M2brane Potentials ”, JHEP 1404 (2014) 026, [arXiv:1311.5901 [hepth
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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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Cited by 2 (0 self)
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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