Yang-Mills-Connes Theory and Quantum Principal Bundles

TL;DR

This paper extends Connes' Dirac theory to quantum principal bundles, defining a Yang-Mills functional for quantum connections and illustrating it with a noncommutative torus example.

Contribution

It generalizes Connes' Dirac framework to quantum principal bundles and introduces a new Yang-Mills functional for quantum connections.

Findings

The Dirac theory applies to all quantum principal bundles with various quantum groups.

A new Yang-Mills functional for quantum connections is defined and analyzed.

An example with the noncommutative n-torus illustrates the theory's application.

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.