Quantum Principal BundlesTable of Contents
Introduction Quantum Classifying SpacesLet us assume that G is a classical Lie group. The classification problem of classical principal G-bundles is solved by constructing the classifying space BG together with a principal G-bundle EG over BG. The pair (BG, EG) has the following universal property: Every classical principal G-bundle P over a given classical space M can be obtained as a pull back of the bundle EG via an appropriately chosen map f:M-to-BG. Characteristic classes of P can be obtained as pull-backs of the cohomology classes of BG, via these classification maps.Moreover, isomorphism classes of principal G-bundles over M are in a natural one-to-one correspondence with homotopy classes of continuous maps from M to BG. The construction of classifying spaces can be incorporated in the quantum context [D7]. Starting from an arbitrary compact quantum group G, we can construct a quantum space QBG and a quantum principal G-bundle QEG over QBG such that every quantum principal bundle P over a quantum space M can be obtained as a pull back of the bundle QEG via a classifying map from M to QBG A quantum version of the classification theorem is [D7] that there exists a natural bijection between homotopic classes of quantum principal G-bundles P over M and homotopic classes of classifying maps from M to QBG Furthermore, all quantum characteristic classes of P can be obtained from the cohomology classes of QBG via the corresponding pull-backs, as in the classical case (however it is necessary to take care of differential calculi over various bundles figuring in the game). Let us observe the difference in the formulations of classical and quantum classification theorems: In the quantum case we have homotopy classes of quantum bundles, and in the classical case we have isomorphism classes of classical bundles. The explanation is that in classical geometry homotopic bundles over the same space and with the same structure groups are always isomorphic. This is not true in the quantum case. For instance, deformation quantization theory gives us examples of homotopical but not isomorphic quantum spaces and bundles. In particular, a given classical bundle could be homotopically equivalent to a truly quantum bundle (the base and the structure group remain the same-classical). Furthermore, there exist quantum bundles over a classical manifold M and with the classical structure group G that are not homotopically equivalent to any classical bundle. This means that the classification problem, for classical Lie group G and a classical base manifold M, is essentially different depending on the context--whether we are considering the classification within the framework of classical or quantum geometry. Indeed, the quantum classifying space QBG associated to a classical Lie group G is still an intrinsically quantum object--described by a highly non-commutative algebra! The classical classifying space BG can be viewed as the classical part of QBG consisting of all the points of QBG (*-characters of its function algebra). [Previous Segment]: Quantum Characteristic Classes [Next Segment]: Quantum Frame Bundles |