The Connes–Lott Program on the Sphere

We describe a classical Schwinger-type model as a study of the projective modules over the algebra of complex-valued functions on the sphere. On these modules, classified by π₂(S²), we construct hermitian connections 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 the Hilbert space of complexified inhomogeneous forms with its Atiyah–Kähler structure. This Hilbert space splits in two minimal left ideals of the Clifford algebra preserved by the Dirac–Kähler operator D=i(d-δ). The induced representation of the universal differential envelope, in order to recover its differential structure, is divided by the unwanted differential ideal and the obtained quotient is the usual complexified de Rham exterior algebra with Clifford action on the "spinors" of the Hilbert space. The subsequent steps of the Connes–Lott program allow to define a matter action, and the field action is obtained using the Dixmier trace which reduces to the integral of the curvature squared.