Quantum group gauge theory on quantum spaces

TL;DR

This paper develops a framework for quantum gauge theories on non-commutative spaces, constructing quantum group connections and a q-deformed Dirac monopole within a general quantum principal bundle setting.

Contribution

It introduces a general theory of quantum principal bundles with quantum group fibers, enabling the construction of connections and monopoles on quantum spaces.

Findings

Constructed quantum group-valued connections on quantum homogeneous spaces.

Presented a q-deformed Dirac monopole on the quantum sphere.

Established a comprehensive framework for quantum gauge theories on non-commutative geometries.

Abstract

We construct quantum group-valued canonical connections on quantum homogeneous spaces, including a q-deformed Dirac monopole on the quantum sphere of Podles quantum differential coming from the 3-D calculus of Woronowicz on $SU_q(2)$ . The construction is presented within the setting of a general theory of quantum principal bundles with quantum group (Hopf algebra) fiber, associated quantum vector bundles and connection one-forms. Both the base space (spacetime) and the total space are non-commutative algebras (quantum spaces).