Local SU(5) Unification from the Heterotic String

TL;DR

This work embeds the Standard Model in a string-derived local SU(5) GUT by constructing a 6D N=2 supergravity from heterotic string theory, then compactifying on a Z_2 orbifold to four dimensions. The two inequivalent fixed points support distinct local symmetries (SU(5) and SU(2)×SU(4)), enabling a split-multiplet structure where two quark–lepton families live on SU(5) branes and a third family arises from split bulk multiplets, with a top Yukawa generated by the 6D gauge coupling due to partial gauge–Higgs unification. Anomalies are consistently cancelled by the Green–Schwarz mechanism, with bulk and brane contributions factorizing appropriately, and exotic states are largely decoupled via vacuum expectation values of SM singlets, yielding a phenomenologically motivated local SU(5) GUT with controlled decoupling. The analysis also discusses supersymmetric vacua, Fayet–Iliopoulos terms, and the stabilization challenges for extra dimensions, outlining a path toward more realistic compactifications and Yukawa structures. Overall, the paper demonstrates a coherent string-theoretic realization of local SU(5) unification with explicit anomaly cancellation and a mechanism for decoupling exotics, while highlighting the need for further work on moduli stabilization and fully realistic Yukawa textures.

Abstract

We construct a 6D supergravity theory which emerges as intermediate step in the compactification of the heterotic string to the supersymmetric standard model in four dimensions. The theory has N=2 supersymmetry and a gravitational sector with one tensor and two hypermultiplets in addition to the supergravity multiplet. Compactification to four dimensions occurs on a T^2/Z_2 orbifold which has two inequivalent pairs of fixed points with unbroken SU(5) and SU(2)xSU(4) symmetry, respectively. All gauge, gravitational and mixed anomalies are cancelled by the Green-Schwarz mechanism. The model has partial 6D gauge-Higgs unification. Two quark-lepton generations are localized at the SU(5) branes, the third family is composed of split bulk hypermultiplets. The top Yukawa coupling is given by the 6D gauge coupling, all other Yukawa couplings are generated by higher-dimensional operators at the SU(5) branes. The presence of the SU(2)xSU(4) brane breaks SU(5) and generates split gauge and Higgs multiplets with N=1 supersymmetry in four dimensions. The third generation is obtained from two split \bar{5}-plets and two split 10-plets, which together have the quantum numbers of one \bar{5}-plet and one 10-plet. This avoids unsuccessful SU(5) predictions for Yukawa couplings of ordinary 4D SU(5) grand unified theories.

Figures (2)

• Figure 1: The tori of the orbifold $T^6/\mathbbm{Z}_{6}$. Red crosses mark fixed points of the $\mathbbm{Z}_{3}$ twist used for the first step of compactification. The $\text{SO}\!\left(4\right)$ torus is invariant, while the other tori contain three fixed points each. The fixed points in the $\text{G}_{2}$ torus are equivalent, while the $\text{SU}\!\left(3\right)$ torus contains a Wilson line, and the fixed points are inequivalent and labelled by $n_3$. The blue circles mark the $\mathbbm{Z}_{2}$ fixed points in the $\text{SO}\!\left(4\right)$ plane which are labelled by $(n_2,n_2')$. There are further $\mathbbm{Z}_{2}$ fixed points in the $\text{G}_{2}$ torus which are not shown.
• Figure 2: The orbifold $T^2/\mathbbm{Z}_{2}$. The blue dots (on the left) label the fixed points with $n_2=0$, the red ones (right) have $n_2=1$. Two quark-lepton generations live at the $n_2=0$ fixed points, the third one originates from two $\text{SU}\!\left(5\right)$$\boldsymbol{\bar{5}}$ and $\boldsymbol{10}$ multiplets in the bulk, half of which is projected out due to the boundary conditions at $n_2=1$.