Hopf-Galois extensions are known to be the right generalizations of both Galois field extensions and principal G-bundles in the framework of non-commutative associative algebras. An abundant literature has been devoted to them by Hopf algebra specialists (see [Mg], [Sn1], [Sn2] and references therein). Recently there has been a surge of interest in Hopf-Galois extensions among mathematicians an...

We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H(BG, Z) for a compact semi-simple Lie group G. The ChernSimons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply these notions to refine the Dijkgraaf-Witten correspondence between three dimensional Ch...

Let G be a complex simply connected semisimple Lie group and let Γ torsionless uniform irreducible lattice in G. Then Γ﹨G is compact non-Kähler manifold whose tangent bundle holomorphically trivial. In this note we compute the Picard of when rank(G)≥3. When rank(G)<3, determine Pic0(Γ﹨G)⊂Pic(Γ﹨G) topologically trivial holomorphic line bundles. rank(G)≥2, also show that Pic0(PΓ) isomorphic to Pi...

The loop derivative as a curvature. Abstract Recently, a set of tools has been developed with the purpose of the study of Quantum Gravity. Until now, there have been very few attempts to put these tools into a rigorous mathematical framework. This is the case, for example, of the so called path bundle of a manifold. It is well known that this topological principal bundle plays the role of a uni...

We develop the theory of smooth principal bundles for a group G, using framework diffeological spaces. After giving new examples showing why arbitrary cannot be classified, we define D-numerable bundles, analogs numerable from topology, and prove that pulling back bundle along smoothly homotopic maps gives isomorphic pullbacks. then structures on Milnor’s spaces EG BG, show → BG is bundle, it c...

We study the Ehresmann--Schauenburg bialgebroid of a noncommutative principal bundle as quantization gauge groupoid classical bundle. show that group is isomorphic to bisections bialgebroid, and we give crossed module structure for automorphisms bialgebroid. Examples illustrating these constructions include: Galois objects Taft algebras, monopole over quantum spheres not faithfully flat Hopf--G...

We reexamine the results on the global properties of T-duality for principal circle bundles in the context of a dimensionally reduced Gysin sequence. We will then construct a Gysin sequence for principal torus bundles and examine the consequences. In particular, we will argue that the T-dual of a principal torus bundle with nontrivial H-flux is, in general, a continuous field of noncommutative,...

A reductive structure is associated here with Lagrangian canonically defined conserved quantities on gauge-natural bundles. Infinitesimal parametrized transformations defined by the gauge-natural lift of infinitesimal principal automorphisms induce a variational sequence such that the generalized Jacobi morphism is naturally self-adjoint. As a consequence, its kernel defines a reductive split s...

We consider generalized (possibly depending on fields as well as on space-time variables) gauge transformations and gauge symmetries in the context of general – that is, possibly non variational nor covariant – differential equations. In this case the relevant principal bundle admits the first jet bundle (of the phase manifold) as an associated bundle, at difference with standard Yang-Mills the...

A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category F of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theorem. AMS SC : 55R55 (Fiberings with singularities); 55R65 (Generalizations o...