Anomaly of discrete symmetries can be defined as the Jacobian of the
path-integral measure. We assume that an anomalous discrete symmetry at low
energy is remnant of an anomaly free discrete symmetry, and that its anomaly is
cancelled by the Green-Schwarz(GS) mechanism at a more fundamental scale. If
the Kac-Moody levels k_i assume non-trivial values, the GS cancellation
conditions of anomaly modify the ordinary unification of gauge couplings. This
is most welcome, because for a renormalizable model to be realistic any
non-abelian family symmetry, which should not be hardly broken low-energy,
requires multi SU(2)_L doublet Higgs fields. As an example we consider a
recently proposed supersymmetric model with Q_6 family symmetry. In this
example, k_2=1, k_3=3 satisfies the GS conditions and the gauge coupling
unification appears close to the Planck scale.Comment: Talk given at the Summer Institute 2006 (SI2006), APCTP, Pohang,
Korea. 23-30 Aug 200
