GRADING OF SPINOR BUNDLES AND GRAVITATING MATTER IN NONCOMMUTATIVE GEOMETRY

The gravitating matter is studied within the framework of noncommutati ve geometry. The noncommutative Einstein-Hilbert action on the product of a four-dimensional manifold with discrete space gives models of ma tter fields coupled to the standard Einstein gravity. The matter multi plet is encoded in the Dirac operator which yields a representation of the algebra of universal forms. The general form of the Dirac operato r depends on a choice of the grading of the corresponding spinor bundl e. A choice is given, which leads to the nonlinear vector sigma-model coupled to the Einstein gravity.