Strong Connections on Quantum Principal Bundles

Abstract

A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and non-strong connections are provided. In particular, such connections are constructed on a quantum deformation of the fibration $S^2->R{P}^2$. A certain class of strong $U_q(2)$-connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with the q-dependent hermitian metric. A particular form of the Yang-Mills action on a trivial $U\sb q(2)$-bundle is investigated. It is proved to coincide with the Yang-Mills action constructed by A.Connes and M.Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent of q.

AMS-LaTeX, 40 pages, major revision including examples of connections over a quantum real projective space

References in corpus (4)

• Quantum group gauge theory on quantum spaces
• Geometry of Quantum Principal Bundles I
• Quantum Group Gauge Theory on Classical Spaces
• Quantum Groups on Fibre Bundles

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