Abstract
We construct the general form of an F-theory compactification with two U(1) factors based on a general elliptically fibered Calabi-Yau manifold with Mordell-Weil group of rank two. This construction produces broad classes of models with diverse matter spectra, including many that are not realized in earlier F-theory constructions with U(1)xU(1) gauge symmetry. Generic U(1)xU(1) models can be related to a Higgsed non-Abelian model with gauge group SU(2)xSU(2)xSU(3), SU(2)^3xSU(3), or a subgroup thereof. The nonlocal horizontal divisors of the Mordell-Weil group are replaced with local vertical divisors associated with the Cartan generators of non-Abelian gauge groups from Kodaira singularities. We give a global resolution of codimension two singularities of the Abelian model; we identify the full anomaly free matter content, and match it to the unHiggsed non-Abelian model. The non-Abelian Weierstrass model exhibits a new algebraic description of the singularities in the fibration that results in the first explicit construction of matter in the symmetric representation of SU(3). This matter is realized on double point singularities of the discriminant locus. The construction suggests a generalization to U(1)^k factors with k>2, which can be studied by Higgsing theories with larger non-Abelian gauge groups.
Highlights
F-theory [1,2,3] provides a powerful nonperturbative approach to understanding large classes of string vacua in four and six space-time dimensions
While non-Abelian gauge factors in F-theory models are classified by the local Kodaira-Tate classification of singular fibers in elliptic fibrations, Abelian factors are represented by elements of the Mordell-Weil group, which are intrinsically global and more difficult to describe analytically
While we believe that the model constructed in this paper gives a fairly complete picture of the most general classes of F-theory models that can be constructed with two Abelian factors U(1)×U(1), many of the results we have found suggest generalizations that may have much broader consequences for our understanding of F-theory models and the corresponding supergravity theories
Summary
F-theory [1,2,3] provides a powerful nonperturbative approach to understanding large classes of string vacua in four and six space-time dimensions. The main result of this paper is the construction of a very general Weierstrass form for F-theory models with U(1)×U(1) gauge factors This class of models includes all those that can be realized by Higgsing the rank two groups SU(2)×SU(2) and SU(3) on adjoint matter. It is worth mentioning here the spectrum that results if an SU(3) is partially Higgsed on a single Cartan generator to give a theory with U(1)×SU(2) gauge group In this case, Higgsing for example on λ8 = diag(1, 1, −2), the fundamental representation of SU(3) breaks to the representation content (+1, 2) + (−2, 1) of U(1)×SU(2), with the anti-fundamental in the same hypermultiplet carrying the conjugate to this representation content, and the adjoint breaks to (+3, 2) + (0, 3), (−3, 2). The Higgsing on bifundamentals of a theory with SU(2)×SU(2)×SU(3) gauge group gives a different, and more general, spectrum than the rank two Higgsings on SU(2)×SU(2) or SU(3) above
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