The new approach to quantize the gravity based on the notion of differential algebra is suggested. It is shown that the differential geometry of this object can not be described in terms of points. The spatialization procedure giving rise to points by loosing a part of the entire structure is discussed. The counterpart of the traditional objects of differential geometry are studied. FOREWORD In...

We present the construction of an associative, commutative algebra Ĝ(X) of generalized functions on a manifold X satisfying the following optimal set of permanence properties: (i) D(X) is linearly embedded into Ĝ(X), f(p) ≡ 1 is the unity in Ĝ(X). (ii) For every smooth vector field ξ on X there exists a derivation operator L̂ξ : Ĝ(X) → Ĝ(X) which is linear and satisfies the Leibniz rule. (iii) L...

Quantum field planes furnish a noncommutative differential algebra Ω which substitutes for the commutative algebra of functions and forms on a contractible manifold. The data requirered in their construction come from a quantum field theory. The basic idea is to replace the ground field C of quantum planes by the noncommutative algebra A of observables of the quantum field theory. HUTMP 94-B335...

We study general relativity in the framework of non-commutative differential geometry. In particular, we introduce a gravity action for a space-time which is the product of a four dimensional manifold by a two-point space. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gr...

A one parameter set of noncommutative complex algebras is given. These may be considered deformation quantisation algebras. The commutative limit of these algebras correspond to the algebra of polynomial functions over a manifold or variety. The topology of the manifold or variety depends on the parameter, varying from nothing, to a point, a sphere, a certain variety and finally a torus. The ir...

We consider the Yang-Mills problem for a quantum Heisenberg manifold, which is a C∗-algebra defined by the (strict) deformation quantization of the ordinary Heisenberg manifold, in the setting of non-commutative differential geometry following Connes and Rieffel [Co] [Co1]. 1. Preliminaries Classical Yang-Mills theory is concerned with the set of connections (i.e. gauge potentials) on a vector ...

We argue that for a certain class of symplectic manifolds the category of A-branes (which includes the Fukaya category as a full subcategory) is equivalent to a noncommutative deformation of the category of B-branes (which is equivalent to the derived category of coherent sheaves) on the same manifold. This equivalence is different from Mirror Symmetry and arises from the Seiberg-Witten transfo...

We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of c...

The non-commutative algebraic analog of the moduli of vector and covector fields is built. The structure of moduli of derivations of non-commutative algebras are studied. The canonical coupling is introduced and the conditions for appropriate moduli to be reflexive are obtained. FOREWORD The duality problem we are going to tackle stems from the non-commutative generalization of differential geo...

The paper gives a definition of exponential arcs in the manifold non-degenerate density matrices and uses it as starting point to develop parameter-free version non-commutative Information Geometry finite-dimensional case. Given Bogoliubov metric m- e-connections are each other dual. Convex potentials introduced. They allow introduce dual charts. Affine coordinates introduced at end make connec...