We define a quantum generalization of the algebra of functions over an associated vector bundle of a principal bundle. Here the role of a quantum principal bundle is played by a Hopf-Galois extension. Smash products of an algebra times a Hopf algebra H are particular instances of these extensions, and in these cases we are able to define a differential calculus over their associated vector bund...

We make the category BGrb M of bundle gerbes on a manifold M into a 2-category by providing 2-cells in the form of transformations of bundle gerbe morphisms. This description of BGrb M as a 2-category is used to define the notion of a bundle 2-gerbe. To every bundle 2-gerbe on M is associated a class in H 4 (M ; Z). We define the notion of a bundle 2-gerbe connection and show how this leads to ...

A Q-manifold is a graded manifold endowed with a vector field of degree one squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of “gauge fields” (sections in the category of graded manifolds) and each cohomology class of a certain subcomplex of forms on the fiber we associate a cohomology class on the base. Any ...

Let X be a smooth projective variety over C. Let G be the group O(r,C), or Sp(r,C). We find the natural notion of semistable principal G-bundle and construct the moduli space, which we compactify by considering also principal G-sheaves, i.e., pairs (E,φ), where E is a torsion free sheaf on X and φ is a symmetric (if G is orthogonal) or antisymmetric (if G is symplectic) bilinear form on E. If G...

Let X be a smooth projective variety over C. Let G be the group O(r,C), or Sp(r,C). We find the natural notion of semistable principal G-bundle and construct the moduli space, which we compactify by considering also principal G-sheaves, i.e., pairs (E,φ), where E is a torsion free sheaf on X and φ is a symmetric (if G is orthogonal) or antisymmetric (if G is symplectic) bilinear form on E. If G...

LetM be an n-dimensional Riemannian manifold. ThenM is a differentiable manifold with a smoothly varying inner product 〈, 〉x on each tangent space TxM . An orthonormal frame at the point x ∈ M is an orthonormal basis {e1, . . . , en} of the tangent space TxM . The set of orthonormal frames at each point is isomorphic to the Lie group O(n), and the set of orthonormal frames on M forms a principa...

In this paper we review the concepts of the superalgebra, superderivation and some properties of them. We will define algebraic and differential superderivations on a superalgebra and will prove some theorems about them, Then we consider a superalgebra bundle, that is an algebra bundle which its fibers are superalgebras and then characterize the superderivations of the algebra of sections of th...

We introduce a symplectic structure on the space of connections in a G-principal bundle over a four-manifold and the Hamiltonian action on it of the group of gauge transformations which are trivial on the boundary. The symplectic reduction becomes the moduli space of flat connections over the manifold. On the moduli space of flat connections we shall construct a hermitian line bundle with conne...

We investigate relative holomorphic connections on a principal bundle over family of compact complex manifolds. A sufficient condition is given for the existence connection analytic family. also introduce notion equivariant bundles and establish its relation with bundles.