Quantum Group Gauge Theory on Classical Spaces

We study the quantum group gauge theory developed elsewhere in the limit when the base space (spacetime) is a classical space rather than a general quantum space. We show that this limit of the theory for gauge quantum group Uq(g) is isomorphic to usual gauge theory with Lie algebra g. Thus a new kind of gauge theory is not obtained in this way, although we do find some differences in the coupling to matter. Our analysis also illuminates certain inconsistencies in previous work on this topic where a different conclusion had been reached. In particular, we show that the use of the quantum trace in defining a Yang-Mills action in this setting as claimed in [14][9] is not appropriate. 1. Recently a number of physicists have tried to develop a q-analogue or quantumgroup gauge theory in which the gauge fields have values in Uq(g) but spacetime is an ordinary manifold [2][14][9], [8]. Here we want to clarify some of the difficulties that arise in this context in the light of our own general quantum group gauge theory developed in [3]. In our work both the gauge group and spacetime were allowed to be quantum spaces and non-trivial examples constructed. We now study the limit of this theory in the case when the base space becomes a usual classical one and show that the resulting gauge theory in this case is necessarily isomorphic to usual gauge theory at least at the level of the gauge fields. There can nevertheless be some subtle differences when the coupling to matter fields is considered. We then contrast our result with some of the above-mentioned previous literature, concentrating on [14][9]. We show that the Supported by St. John’s College, Cambridge & KBN grant 2 0218 91 01 SERC Fellow and Drapers Fellow of Pembroke College, Cambridge