F-theory family unification

Abstract

We propose a new geometric mechanism for naturally realizing unparallel three families of flavors in string theory, using the framework of F-theory. We consider a set of coalesced local 7-branes of a particular Kodaira singularity type and allow some of the branes to bend and separate from the rest, so that they meet only at an intersection point. Such a local configuration can preserve supersymmetry. Its matter spectrum is investigated by studying string junctions near the intersection, and shown to coincide, after an orbifold projection, with that of a supersymmetric coset sigma model whose target space is a homogeneous Kähler manifold associated with a corresponding painted Dynkin diagram. In particular, if one starts from the E 7 singularity, one obtains the E 7 /(SU(5) × U(1) 3 ) model yielding precisely three generations with an unparallel family structure. Possible applications to string phenomenology are also discussed.

F-theory models with U(1) × ℤ2, ℤ4 and transitions in discrete gauge groups

“…3 See, e.g. [48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76] for discussions of F-theory on elliptic fibrations with a global section. 4 When F-theory is compactified on a Calabi-Yau genus-one fibration Y , the discrete gauge group arising in this compactification is identified with the discrete part of the "Tate-Shafarevich group," X(J(Y )), of the Jacobian, J(Y ) [47,19].…”

$$ \frac{1}{2} $$ Calabi-Yau 3-folds, Calabi-Yau 3-folds as double covers, and F-theory with U(1)s

“…12 The correspondence between the types of the singular fibers of the elliptic fibration and non-Abelian gauge symmetries forming on the 7-branes in F-theory is discussed in [3,65]. 13 Methods to determine the singular fibers of the elliptic surfaces are described in [68,69]. 14 The cubic curves in P 2 have 10 dimensions, and among these the cubic curves that pass through the seven points that are blown up to yield the degree two del Pezzo surface have 10 − 7 = 3 dimensions.…”

“…The rank of the group formed by the set of the global sections of the compactification space -or the "Mordell-Weil group"-yields the number of U (1) factors forming in F-theory on the space [3]. The rank 1 Recent studies of F-theory models on elliptic fibrations that admit a global section can be found, for example, in [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34].…”

Family unification in special grand unification

“…In addition, their hierarchical mass structures generated by the Higgs mechanism via their corresponding Yukawa couplings strongly suggest the existence of a hidden structure of nature. There are many attempts to understand the origin of chiral generations and/or their hierarchical masses structures by considering e.g., so-called horizontal symmetry (or family symmetry) in four-dimensional (4D) theories [1][2][3][4][5][6][7], geometrical structures in higherdimensional theories [8][9][10] and string theories [11][12][13].…”