Dual Models of Gauge Unification in Various Dimensions

TL;DR

<p>We present a $\mathbb{Z}_6$ heterotic orbifold model that yields the Standard Model gauge group with three families embedded as $\mathrm{SO}(10)$ $16$-plets and accompanied by vector-like matter. The 4D spectrum arises from untwisted and twisted sectors, with three fixed-point $16$-plets and various localized states arranged to realize $SU(3)_c\times SU(2)_L\times U(1)^5$ before embedding into a larger GUT structure. By allowing certain compactification radii to be large, the model naturally accommodates intermediate higher-dimensional GUTs—$E_6\times SU(3)$, $SU(4)\times SU(4)\times U(1)^2$, or $SO(8)\times SO(8)$—which are dual descriptions at different points in moduli space but share the same massless spectrum and UV completion. Gauge coupling unification remains viable across these setups, though the unification scale and Yukawa couplings depend on the bulk gauge group and the number of extra dimensions. This framework highlights how distinct higher-dimensional GUTs can emerge coherently from a single string construction while preserving the underlying UV physics.</p>

Abstract

We construct a compactification of the heterotic string on an orbifold T^6/Z_6 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 E_8. 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 E_6 \times SU(3), SU(4) \times SU(4) \times U(1)^2 or SO(8) \times 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.

Figures (10)

• Figure 1: $\mathrm{SO}(10)$ breaking patterns by a $\mathbbm{Z}_{2}$ twist. The action of the Pati-Salam twist is indicated by crosses, while that of the Georgi-Glashow twist is indicated by a slash.
• Figure 2: $\mathrm{E}_{6}$ breaking patterns under $\mathbbm{Z}_{2}$ twisting. Three different $\mathbbm{Z}_{2}$ twists are indicated by crosses, a slash and a backslash, respectively.
• Figure 3: $\mathrm{E}_{8}$ breaking to $\mathrm{E}_{6}\times \mathrm{SU}(3)$.
• Figure 4: Fixed points and invariant planes (hatched) under the $\mathbbm{Z}_{6}$ twist and $\mathbbm{Z}_{3},\mathbbm{Z}_{2}$ subtwists, describing localization of different twisted sectors.
• Figure 5: Local gauge symmetries in the $\mathrm{SO}(4)$-plane.