arXiv1999
A consistent N=1 supersymmetric $σ$-model can be constructed, given a Kähler manifold by adding chiral matter multiplets. Their scalar components are covariant tensors on the underlying Kähler manifold. The Kähler U(1)-charges can be adjusted such that the anomalies cancel, using the holomorphic functions in which the Kähler potential transforms. The arbitrariness of the U(1)-charges of matter multiplets is related to their Weyl-weights in superconformal gravity, before it is reduced to supergravity. The covariance of the Kähler potential forces the superpotential to be covariant as well. This relates the cut-off, the Planck scale and the matter charges to each other. A non-vanishing VEV of the covariant superpotential breaks the Kähler U(1) spontaneously. If this VEV vanishes, the gravitino is massless and depending on the above mentioned parameters there may be additional internal symmetry breaking. The separation of the different representations of chiral multiplets can be achieved by covariantizations of derivatives and fermions. Using non-holomorphic transformations, the full Kähler metric can be block-diagonalized and the necessary covariantizations come out naturally. Various aspects are illustrated by applying them to Grassmannian coset models. In particular a consistent model of coset $SU(5)/SU(2)\times U(1) \times SU(3)$ with the field content of the standard model is constructed. The phenomenology of this model is analyzed.