Gauge anomaly with vector and axial-vector fields in six dimensional curved space

Imposing the conservation equation of the vector current for a fermion of spin $\frac{1}{2}$ at the quantum level, a gauge anomaly for the fermion coupling with non-Abelian vector and axial–vector fields in six–dimensional curved space is expressed in tensorial form. The anomaly consists of terms that resemble the chiral U(1) anomaly and the commutator terms that disappear if the axial–vector field is Abelian.