String theory in four dimensions with three families of chiral fermions and standard gauge symmetry

A heterotic string theory in four dimensions is constructed by compactifying the ${\mathrm{E}}_8$×${\mathrm{E}}_8^{\mathcal{’}}$ heterotic strings from 26 to 4 dimensions via a manifold $T^{22}$/G. The theory is modular invariant. In the low-energy limit, there are three families of chiral fermions with exactly the same quantum numbers as in the standard SU(3${)}_C$×SU(2${)}_L$×U(1) model. They couple to gravity and gauge fields with the gauge symmetry SU(3${)}_F$×SU(3${)}_C$×SU(2${)}_L$×U(1 ${)}^3$. The ${\mathrm{E}}_8^{\mathcal{’}}$ is broken down to SU(3)’ ×SU(4)’×U(1)${\mathcal{’}}^3$. The SU(4)’ group can play the role of the technicolor group to break the SU(2${)}_L$ ×U(1${)}^3$ symmetry.