The Connes-Lott program on the sphere

TL;DR

This paper applies the Connes-Lott noncommutative geometry framework to the sphere, constructing a model of the Schwinger gauge theory using projective modules, Clifford algebras, and the Dirac-Kaehler operator, leading to a geometric formulation of matter and field actions.

Contribution

It extends the Connes-Lott program to the sphere, explicitly constructing the geometric and algebraic structures needed for the Schwinger model in this setting.

Findings

Constructed hermitian connections on projective modules over the sphere.

Derived the usual de Rham algebra and Clifford action for spinors on the sphere.

Defined matter and field actions using the Dixmier trace, reproducing curvature squared integral.

Abstract

We describe the classical Schwinger model as a study of the projective modules over the algebra of complex-valued functions on the sphere. On these modules, classified by $\pi_2(S^2)$, we construct hermitian connections with values in the universal differential envelope which leads us to the Schwinger model on the sphere. The Connes-Lott program is then applied using the Hilbert space of complexified inhomogeneous forms with its Atiyah-Kaehler structure. It splits in two minimal left ideals of the Clifford algebra preserved by the Dirac-Kaehler operator D=i(d-delta). 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 over the sphere 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.