ar X iv : h ep - t h / 96 11 05 8 v 1 8 N ov 1 99 6 UFRJ - IF - FPC - 012 / 96 The Connes - Lott Program on the Sphere
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چکیده
We describe the classical Schwinger model as a study of the pro-jective modules over the algebra of complex-valued functions on the sphere. On these modules, classified by π 2 (S 2), we construct her-mitian 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 inho-mogeneous 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 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.
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تاریخ انتشار 1996