Coupling Supersymmetric Nonlinear Sigma Models to Supergravity

It is known that supersymmetric nonlinear sigma models for the compact Kahler manifolds G/H cannot be consistently coupled to supergravity, since the Kahler potentials are not invariant under the G transformation. We show that the supersymmetric nonlinear sigma models can be deformed such that the Kahler potential be exactly G-invariant if and only if one enlarges the manifolds by dropping all the U(1)s in the unbroken subgroup H. Then, those nonlinear sigma models can be coupled to supergravity without losing the G invariance.