Three-Family $SU(5)$ Grand Unification in String Theory

We present two $3$-family $SU(5)$ grand unified models in the heterotic string theory. One model has $3$ chiral families and $9$ pairs of ${\bf 5} + {\overline {\bf 5}}$ Higgs fields, and an asymptotically-free $SU(2) \otimes SU(2)$ hidden sector, where the two $SU(2)$s have different matter contents. The other model has $6$ left-handed and $3$ right-handed ${\bf 10}$s, $12$ left-handed and $9$ right-handed ${\overline {\bf 5}}$s, and an asymptotically-free $SU(3)$ hidden sector. At the string scale, the gauge couplings $g^2$ of the hidden sectors are three times as big as that of $SU(5)$. In addition, both models have an anomalous $U(1)$.