Grading of spinor bundles and gravitating matter in noncommutative geometry

The gravitating matter is studied within the framework of noncommutative geometry. The noncommutative Einstein-Hilbert action on the product of a four-dimensional manifold with discrete space gives models of matter fields coupled to the standard Einstein gravity. The matter multiplet is encoded in the Dirac operator which yields a representation of the algebra of universal forms. The general form of the Dirac operator depends on a choice of the grading of the corresponding spinor bundle. A choice is given, which leads to the nonlinear vectorσ-model coupled to the Einstein gravity.