Quantum Principal Bundles

Gauge Transformations

Interestingly, the above two approaches to gauge transformations are essentially different at the quantum level. Let us briefly discuss both of them.

Gauge transformations as vertical automorphisms

the -linear

action .

Such a definition of gauge transformations gives surprising results. This is because quantum bundles possess much more 'apriorically given' geometrical structure than the classical ones, and by definition gauge transformations have to preserve all this structure. This gives additional restrictions to possible candidates for gauge transformations. To illustrate this phenomena, let us consider locally-trivial quantum bundles (with a quantum structure group G) over a classical smooth manifold M. The classification of such bundles is reduced [D2] to the classification of the classical G_cl-bundles over M, where G_cl is the classical part of G, consisting of the points of G. In fact, the classical bundle that corresponds to the quantum bundle P is simply the classical part P_cl of P.

The correspondence P::P_cl has a simple geometrical explanation. Each truly quantum group G is inherently inhomogeneous, because it always possesses a nontrivial classical part G_cl consisting of points of G and a nontrivial quantum part, imaginable as 'the complement' to G_cl in G. It is clear that 'quantum transition functions' for P, being diffeomorphisms at the level of spaces, preserve this intrinsic decomposition. As a result, because of the right covariance, transition functions are completely determined by their 'restrictions' on G_cl. And these restrictions give us the classical bundle P_cl.

A similar argumentation leads to the conclusion that gauge transformations of the quantum bundle P coincide with gauge transformations of its classical part.

Another surprising phenomena appear at the level of differential calculus. It is natural to consider differential calculi over P that are invariant under all gauge transformations. We can also consider the calculi that are locally trivial (for a classical base M). Both assumptions are automatically fulfilled in classical geometry. In the quantum case, they lead to the class of calculi that are locally trivialized whenever the bundle is locally trivialized. It turns out [D2] that this property gives very strong restrictions on the possible calculus on the quantum structure group G. Moreover, there always exists the minimal admissible calculus on G and P.

In classical geometry, this minimal admissible calculus is precisely the classical calculus on the structure group and the bundle. In the quantum case, however, some very surprising things appear. For example, if we assume G is the quantum SU(2) group [W1] and if the deformation parameter (with its natural range) is different from -1,1, then the minimal admissible calculus over this group is infinite-dimensional with the space naturally identified with the algebra of functions over a quantum sphere [P]. The classical part of the quantum SU(2) group is U(1). In the classical case the minimal admissible calculus is just the classical 3-dimensional one, and the classical part of the group is the whole group itself.

The quantum adjoint bundle

map .

of into ,

,

action .

The quantum gauge bundle is a very interesting geometrical object, in particular because its properties are connected with an intrinsic braid structure that exists on every quantum principal bundle [D12]. Moreover, the algebra possesses a canonical braided quantum group [D13] structure over , represented by a natural coproduct map