Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition f...

We consider a vector bundle E →M and the principal bundle PE of frames of E. We determine all natural transformations of the connection bundle of the first order principal prolongation of principal bundle PE into itself. Unless otherwise specified, we use the terminology and notation from the book [7]. All manifolds and maps are assumed to be infinitely differentiable. In the monograph [7], the...

We define the notion of a principal S-bundle where S is a groupoid group bundle and show that there is a one-to-one correspondence between principal S-bundles and elements of a sheaf cohomology group associated to S. We also define the notion of a locally unitary action and show that the spectrum of the crossed product is a principal b S-bundle. Furthermore, we prove that the isomorphism class ...

Let G/P be a rational homogeneous variety, where P is a parabolic subgroup of a simple and simply connected linear algebraic group G defined over an algebraically closed field of characteristic zero. A homogeneous principal bundle over G/P is semistable (respectively, polystable) if and only if it is equivariantly semistable (respectively, equivariantly polystable). A stable homogeneous princip...

We define locally trivial quantum vector bundles (QVB) and construct such QVB associated to locally trivial quantum principal fibre bundles. The construction is quite analogous to the classical construction of associated bundles. A covering of such bundles is induced from the covering of the subalgebra of coinvariant elements of the principal bundle. There exists a differential structure on the...

The notion of a translation map in a quantum principal bundle is introduced. A translation map is then used to prove that the cross sections of a quantum fibre bundle E(B, V,A) associated to a quantum principal bundle P (B,A) are in bijective correspondence with equivariant maps V → P , and that a quantum principal bundle is trivial if it admits a cross section which is an algebra map. The vert...

It was argued in [25], [5] that in the presence of a nontrivial Bfield, D-brane charges in type IIB string theories are classified by twisted Ktheory. In [4], it was proved that twisted K-theory is canonically isomorphic to bundle gerbe K-theory, whose elements are ordinary Hilbert bundles on a principal projective unitary bundle, with an action of the bundle gerbe determined by the principal p...