Local Grand Unification

TL;DR

The paper tackles how to realize grand unification within a string-theoretic framework without introducing light exotics, addressing the doublet–triplet splitting problem. It advocates a local grand unification approach in heterotic orbifolds, where matter lives at fixed points with local $\mathrm{SO}(10)$ symmetry while Higgs doublets live in the bulk, yielding the SM as an intersection of local groups. A concrete $\mathbb{Z}_{3}\times\mathbb{Z}_{2}$ heterotic model is presented that produces three chiral generations, a pair of Higgs doublets, and gauge–Yukawa unification $y_t \simeq g$ at $M_{\mathrm{GUT}}$, with all other states vector-like and heavy. Orbifold GUT limits show that gauge coupling unification is maintained up to threshold corrections across different anisotropic compactifications, supporting a realistic string-derived MSSM and offering a mechanism to discuss flavour structure via the local fixed-point geometry.

Abstract

In the standard model matter fields form complete representations of a grand unified group whereas Higgs fields belong to incomplete `split' multiplets. This remarkable fact is naturally explained by `local grand unification' in higher-dimensional extensions of the standard model. Here, the generations of matter fields are localized in regions of compact space which are endowed with a GUT gauge symmetry whereas the Higgs doublets are bulk fields. We realize local grand unification in the framework of orbifold compactifications of the heterotic string, and we present an example with SO(10) as a local GUT group, which leads to the supersymmetric standard model as an effective four-dimensional theory. We also discuss different orbifold GUT limits and the unification of gauge and Yukawa couplings.

Figures (6)

• Figure 1: The picture of local grand unification. The gauge group $G$ is broken locally to different subgroups. Each of the local subgroups contains the standard model gauge group $G_\mathrm{SM}$ which emerges as an intersection of the local groups. 'Brane' fields which are confined to a region with certain symmetry have to come in complete matter multiplets of that symmetry. Hence, localized $\boldsymbol{16}$-plets of $\mathrm{SO}(10)$ are an attractive explanation of complete matter generations. Higgs doublets, on the other hand, are states which are not confined to an $\mathrm{SO}(10)$ region, and can therefore appear as 'split multiplets' in the low--energy spectrum.
• Figure 2: Root lattice $\Lambda_{G_2\times\mathrm{SU}(3)\times\mathrm{SO}(4)}$ and fixed points of the $\mathbbm{Z}_{6}$ action.
• Figure 3: (a) orbifolds can be constructed by combining 'corners' carrying a local gauge group emerging from the action of a local shift. (b) In $2+1$ family models, two families appear as $\boldsymbol{16}$--plets residing on fixed points with local $\mathrm{SO}(10)$ symmetry. The third family comes from elsewhere.
• Figure 4: Local gauge groups up to $\mathrm{U}(1)$ factors and subgroups of the second $\mathrm{E}_{8}$. As indicated, the fixed point come in six pairs.
• Figure 5: Illustration of gauge--Yukawa unification. The plot shows the MSSM evolution of $\alpha_i=g_i^2/(4\pi)$ and $\alpha_t=y_t^2/(4\pi)$ where $g_i$ denotes the gauge couplings and $y_t$ is the top Yukawa coupling.