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Group representations and the Euler characteristic of elliptically fibered CalabiYau threefolds
 J. Alg. Geom
"... To every elliptic Calabi–Yau threefold with a section X there can be associated a Lie group G and a representation ρ of that group, determined from the Weierstrass model and the types of singular fibers. We explain this construction, which first arose in physics. The requirement of anomaly cancellat ..."
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To every elliptic Calabi–Yau threefold with a section X there can be associated a Lie group G and a representation ρ of that group, determined from the Weierstrass model and the types of singular fibers. We explain this construction, which first arose in physics. The requirement of anomaly cancellation in the associated physical theory makes some surprising predictions about the connection between X and ρ, including an explicit formula (in terms of ρ) for the Euler characteristic of X. We give a purely mathematical proof of that formula in this paper, introducing along the way a new invariant of elliptic Calabi–Yau threefolds. We also verify the other geometric predictions which are consequences of anomaly cancellation, under some mild hypotheses. As a byproduct we discover a novel relation between the Coxeter number and the rank in the case of the simply laced groups in the “exceptional series ” studied by Deligne. It was noted by Du Va [11] that certain surface singularities, now known as rational double points, are classified by the Dynkin diagrams of the simply laced Lie groups 1 of type An,Dn,E6,E7,E8. Du Val pointed out that the Dynkin diagram is the dual diagram to the intersection configuration of the exceptional divisors in the minimal resolution of the singularities. Further connections between these singularities and Lie groups were subsequently discovered by Brieskorn and Grothendieck [5].
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.
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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Ftheory on GenusOne Fibrations
"... We argue that Mtheory compactified on an arbitrary genusone fibration, that is, an elliptic fibration which need not have a section, always has an Ftheory limit when the area of the genusone fiber approaches zero. Such genusone fibrations can be easily constructed as toric hypersurfaces, and va ..."
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We argue that Mtheory compactified on an arbitrary genusone fibration, that is, an elliptic fibration which need not have a section, always has an Ftheory limit when the area of the genusone fiber approaches zero. Such genusone fibrations can be easily constructed as toric hypersurfaces, and various SU(5) × U(1)n and E6 models are presented as examples. To each genusone fibration one can associate a τfunction on the base as well as an SL(2,Z) representation which together define the IIB axiodilaton and 7brane content of the theory. The set of genusone fibrations with the same τfunction and SL(2,Z) representation, known as the TateShafarevich group, supplies an important degree of freedom in the corresponding Ftheory model which has not been studied carefully until now. Sixdimensional anomaly cancellation as well as Witten’s zeromode count on wrapped branes both imply corrections to the usual Ftheory dictionary for some of these models. In particular, neutral hypermultiplets which are localized at codimensiontwo fibers can arise. (All previous known examples of localized hypermultiplets were charged under the gauge group of the theory.) Finally, in the absence of a section some novel monodromies of Kodaira fibers are allowed which lead to new breaking patterns of nonAbelian gauge groups.
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