The anomalous

In string compactifications, frequently the anomalous U(1) gauge symmetry appears which belongs to $$E_8 \times E_8'$$ of the heterotic string. This anomalous U(1) gauge boson obtains mass at the compactification scale ($$\approx 10^{18\,}\,\text {GeV}$$) by absorbing one pseudoscalar (corresponding to the model-independent axion) from the second rank antisymmetric tensor field $$B_{MN}$$. Below the compactification scale a global symmetry $$U(1)_{\mathrm{anom}}$$ results whose charge $$Q_\mathrm{anom}$$ is the original gauge U(1) charge. This is the most natural global symmetry, realizing the “invisible” axion. This global symmetry $$U(1)_{\mathrm{anom}}$$ is suitable for a flavor symmetry. In the simplest compactification model with the flipped SU(5) grand unification, all the low energy parameters are calculated in terms of the vacuum expectation values of the standard model singlets