Dual Models of Gauge Unification in Various Dimensions

We construct a compactification of the heterotic string on an orbifold T 6/Z6 leading to the standard model spectrum plus vector–like matter. The standard model gauge group is obtained as an intersection of three SO(10) subgroups of E8. Three families of SO(10) 16-plets are localized at three equivalent fixed points. Gauge coupling unification favours existence of an intermediate GUT which can have any dimension between five and ten. Various GUT gauge groups occur. For example, in six dimensions one can have E6×SU(3), SU(4)×SU(4)×U(1) 2 or SO(8)×SO(8), depending on which of the compact dimensions are large. The different higher– dimensional GUTs are ‘dual’ to each other. They represent different points in moduli space, with the same massless spectrum and ultraviolet completion. 1 Embedding the standard model in E8 The symmetries and the particle content of the standard model point towards grand unified theories (GUTs) as the next step in the unification of all forces. Leftand right-handed quarks and leptons can be grouped in three SU(5) multiplets [1], 10 = (qL, u c R, e c R), 5 = (d c R, lL) and 1 = ν c R. Here we have added right-handed neutrinos which are suggested by the evidence for neutrino masses. All quarks and leptons of one generation can be unified in a single multiplet of the GUT group SO(10) [2], 16 = 10 + 5+ 1 . (1) The group SO(10) contains as subgroups the Pati-Salam group [3], GPS = SU(4)× SU(2) × SU(2), the Georgi-Glashow group SU(5), GGG = SU(5) × U(1), and the ‘flipped’ SU(5) group, Gfl = SU(5) ′ × U(1) [4], where the right-handed upand down-quarks are interchanged, yielding another viable GUT group. It is a remarkable property of the standard model that the matter fields form complete SO(10) multiplets whereas the gauge and Higgs fields are ‘split multiplets’. They have to be combined with other split multiplets, not contained in the standard model, in order to obtain a complete unified theory. It is also well known [5] that exceptional groups play an exceptional role in grand unification, and the embedding SO(10) ⊂ E6 ⊂ E8 (2) appears, in particular, in compactifications of the heterotic string [6] on Calabi-Yau manifolds [7]. As we shall see, complete SO(10) matter multiplets together with split gauge and Higgs multiplets arise naturally in orbifold compactifications of higherdimensional unified theories. Orbifold compactifications have first been considered in string theory [8, 9] and subsequently in effective higher-dimensional field theories [10, 11]. They provide a simple and elegant way to break GUT symmetries, while avoiding the notorious doublet-triplet splitting problem. More recently, it has been shown how orbifold GUTs can occur in orbifold string compactifications [12–14]. In the following we shall first search for a scheme of ZN twists which allows to break E8, a common ingredient of string models, to the standard model group. A ZN twist is an element of the gauge group G, with P = exp (−2πiVN · H) , P N = 1 . (3) Here the generators H i form the (Abelian) Cartan subalgebra of G, and VN is a real vector. The twist P acts on the Cartan and step generators Eα as follows: P Hi P −1 = H i P Eα P −1 = exp (−2πiVN · α)Eα , (4) 2