The four-fermi coupling of the supersymmetric non-linear σ -model on G/S⊗{U(1)}k Origi

The reducible Kähler coset space G/S⊗{U(1)}k is discussed in a geometrical approach. We derive a formula which expresses the Riemann curvature of the reducible Kähler coset space in terms of its Killing vectors. The formula manifests the group structure of G . On the basis of this formula we establish an algebraic method to evaluate the four-fermi coupling of the supersymmetric non-linear σ -model on G/S⊗{U(1)}k at the low-energy limit. As an application we consider the supersymmetric non-linear σ -model on E7/SU(5)⊗{U(1)}3 which contains the three families of 10+5∗+1 of SU(5) as pseudo-NG fermions. The four-fermi coupling constants among different families of the fermions are explicitly given at the low-energy limit.