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FTheory and the MordellWeil Group of EllipticallyFibered CalabiYau Threefolds
 JHEP 1210 (2012) 128 [arXiv:1208.2695 [hepth
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Anomaly Cancellation And Abelian Gauge Symmetries
 In Ftheory, JHEP 1302
, 2013
"... cvetic at cvetic.hep.upenn.edu, grimm at mpp.mpg.de, klevers at sas.upenn.edu We study 4D Ftheory compactifications on singular CalabiYau fourfolds with fluxes. The resulting N = 1 effective theories can admit nonAbelian and U(1) gauge groups as well as charged chiral matter. In these setups we a ..."
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Cited by 33 (9 self)
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cvetic at cvetic.hep.upenn.edu, grimm at mpp.mpg.de, klevers at sas.upenn.edu We study 4D Ftheory compactifications on singular CalabiYau fourfolds with fluxes. The resulting N = 1 effective theories can admit nonAbelian and U(1) gauge groups as well as charged chiral matter. In these setups we analyze anomaly cancellation and the generalized GreenSchwarz mechanism. This requires the study of 3D N = 2 theories obtained by a circle compactification and their Mtheory duals. Reducing Mtheory on resolved CalabiYau fourfolds corresponds to considering the effective theory on the 3D Coulomb branch in which certain massive states are integrated out. Both 4D gaugings and 3D oneloop corrections of these massive states induce ChernSimons terms. All 4D anomalies are captured by the oneloop terms. The ones corresponding to the mixed gaugegravitational anomalies depend on the KaluzaKlein vector and are induced by integrating out KaluzaKlein modes of the U(1) charged matter. In Mtheory all ChernSimons terms classically arise from G4flux. We find that Ftheory fluxes implement automatically the 4D GreenSchwarz mechanism if nontrivial geometric relations for the resolved CalabiYau fourfold are satisfied. We confirm these relations in various explicit examples and elucidate the general construction of U(1) symmetries in Ftheory. We also compare anomaly cancellation in Ftheory with its analog in Type IIB orientifold setups.
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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Constraints on 6D Supergravity Theories with Abelian Gauge Symmetry,” [arXiv:1110.5916 [hepth
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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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Cited by 1 (0 self)
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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.
Tate Form and Weak Coupling Limits in Ftheory
"... We consider the weak coupling limit of Ftheory in the presence of nonAbelian gauge groups implemented using the traditional ansatz coming from Tate’s algorithm. We classify the types of singularities that could appear in the weak coupling limit and explain their resolution. In particular, the weak ..."
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We consider the weak coupling limit of Ftheory in the presence of nonAbelian gauge groups implemented using the traditional ansatz coming from Tate’s algorithm. We classify the types of singularities that could appear in the weak coupling limit and explain their resolution. In particular, the weak coupling limit of SU(n) gauge groups leads to an orientifold theory which suffers from conifold singulaties that do not admit a crepant resolution compatible with the orientifold involution. We present a simple resolution to this problem by introducing a new weak coupling regime that admits singularities compatible with both a crepant resolution and an orientifold symmetry. We also comment on possible applications of the new limit to model building. We finally discuss other unexpected phenomena as for example the existence of several nonequivalent directions to flow from strong to weak coupling leading to different gauge groups. ♠Email: esole at math.harvard.edu ♦Email: savelli at mpp.mpg.de ar X iv
Gauge Fluxes in Ftheory and Type IIB Orientifolds
"... We provide a detailed correspondence between G4 gauge fluxes in Ftheory compactifications with SU(n) and SU(n) × U(1) gauge symmetry and their Type IIB orientifold limit. Based on the resolution of the relevant Ftheory Tate models we classify the factorisable G4fluxes and match them with the se ..."
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We provide a detailed correspondence between G4 gauge fluxes in Ftheory compactifications with SU(n) and SU(n) × U(1) gauge symmetry and their Type IIB orientifold limit. Based on the resolution of the relevant Ftheory Tate models we classify the factorisable G4fluxes and match them with the set of universal D5tadpole free U(1)fluxes in Type IIB. Where available, the global version of the universal spectral cover flux corresponds to Type IIB gauge flux associated with a massive diagonal U(1). In U(1)restricted Tate models extra massless abelian fluxes exist which are associated with specific linear combinations of Type IIB fluxes. Key to a quantitative match between Ftheory and Type IIB is a proper treatment of the conifold singularity encountered in the Sen limit of generic Ftheory models. We also collect evidence that the Type IIB orientifold limit of the considered Tate models involves a nontrivial Bfield and shed further light on the brane recombination process relating generic and U(1)restricted Tate models. ar X iv