Yang--Mills--Connes Theory and Quantum Principal $SU(N)$--Bundles over Toric Non--Commutative Manifolds

Abstract

This paper has two main objectives. The first one is to show that the Connes formulation of Dirac theory can be applied in the framework of quantum principal bundles for any n dimensional spectral triple, any quantum group, any quantum principal connection and any finite dimensional corepresentation of the quantum group. The second objective is to demonstrate that, under certain conditions, one can define a Yang Mills functional that measures the squared norm of the curvature of a quantum principal connection, in contrast to the Yang Mills functional proposed by Connes, which measures the squared norm of the curvature of a compatible quantum linear connection. An illustrative example based on the noncommutative n torus is presented, highlighting the differences and similarities between the two functionals.