We study the non-linear sigma model realization of a heterotic vacuum with N=2 space-time supersymmetry. We examine the requirements of (0,2) + (0,4) worldsheet supersymmetry and show that a geometric vacuum must be described by a principal two-torus bundle over a K3 manifold.

We show that a principal G bundle on a smooth projective curve over a finite field is strongly semistable if and only if it is defined by a representation of the fundamental group scheme of the curve into G. MSC 2000 classification:14-XX

The multidimensional gravity on principal bundle with structural SU (2) gauge group is considered. The Lorentzian wormhole solution disposed between two event horizons is founded in this case. In some sense this wormhole is dual to the black hole in 4D Einstein-Yang-Mills gravity (sphaleron): they are located at different sides of the event horizon.

We consider the L 2-gradient ow associated with the Yang-Mills functional, the so-called Yang-Mills heat ow. In the setting of a trivial principal SO(n)-bundle over R n in dimension n greater than 4, we show blow-up in nite time for a class of SO(n)-equivariant initial connections.

We clarify some aspects of quantum group gauge theory and its recent generalisations (by T. Brzezinski and the author) to braided group gauge theory and coalgebra gauge theory. We outline the diagrammatic version of the braided case. We study the bosonisation of any braided group provides as a trivial principal bundle in three ways.

In the present paper we describe topological obstructions to embedding a (complex) matrix algebra bundle into a trivial one under some additional arithmetic condition on their dimensions. We explain a relation between this problem and some principal bundles with groupoid structure. Finally, we briefly discuss a relation to the Brauer group and its generalization.

The problem treated here is to find the Hamiltonian structure for an ideal gauge-charged fluid. Using a Kaluza-Klein point of view, we obtain the non-canonical Poisson bracket and the motion equations by a Poisson reduction involving the automorphism group of a principal bundle.

Using the description of a fiber product preserving bundle functor F in terms of Weil algebras, we deduce several geometric properties of the Fprolongations of principal and associated bundles. Then we clarify that the flow prolongation with respect to F of a projectable vector field can be constructed by using a natural morphism.

We consider a first order linear partial differential operator of principal type on a closed connected orientable two-dimensional manifold sending sections of one complex line bundle to sections of another. We prove that the assumption of global hypoellipticity of the operator implies a relation between the degrees of the line bundles and the Euler characteristic of the manifold.

We show that diffeomorphisms with a dominated splitting of the form Es⊕Ec⊕Eu, where Ec is a nonhyperbolic central bundle that splits in a dominated way into 1-dimensional subbundles, are entropy-expansive. In particular, they have a principal symbolic extension and equilibrium states.