THE CONNES-LOTT PROGRAM ON THE SPHERE

We describe a classical Schwinger-type model as a study of the project ive modules over the algebra of complex-valued functions on the sphere . On these modules, classified by pi(2)(S-2), we construct hermitian c onnections with values in the universal differential envelope. Instead of describing matter by the usual Dirac spinors yielding the standard Schwinger model on the sphere, we apply the Connes-Lott program to th e Hilbert space of complexified inhomogeneous forms with its Atiyah-Ka hler structure. This Hilbert space splits in two minimal left ideals o f the Clifford algebra preserved by the Dirac-Kahler operator D = i(d- delta). The induced representation of the universal differential envel ope, in order to recover its differential structure, is divided by the unwanted differential ideal and the obtained quotient is the usual co mplexified de Rham exterior algebra with Clifford action on the ''spin ors'' of the Hilbert space. The subsequent steps of the Connes-Lott pr ogram allow to define a matter action, and the field action is obtaine d using the Dixmier trace which reduces to the integral of the curvatu re squared.