The four-fermi coupling of the supersymmetric non-linear sigma-model on G/S circle times {U(1)}(k)

The reducible Kahler coset space G/S circle times {U(1)}(k) is discussed in a geometrical approach. We derive a formula which expresses the Riemann cu rvature of the reducible Kahler coset space in terms of its Killing vectors . The formula manifests the group structure of G. On the basis of this form ula we establish an algebraic method to evaluate the four-fermi coupling of the supersymmetric non-linear sigma-model on G/S circle times {U(1)}(k) at the low-energy limit. As an application we consider the supersymmetric non -linear a-model on E7/SU(5) circle times {U(1)}(3) which contains the three families of 10 + 5* + 1 of SU(5) as pseudo-NG fermions. The four-fermi cou pling constants among different families of the fermions are explicitly giv en at the low-energy limit. (C) 2000 Elsevier Science B.V. All rights reser ved.