As a mathematical theory per se, noncommutative geometry (NCG) is by now well established. From the beginning, its progress has been crucially influenced by quantum physics: we briefly review this development in recent years. The Standard Model of fundamental interactions, with its central role for the Dirac operator, led to several formulations culminating in the concept of a real spectral tri...

Various approaches by the author and collaborators to define gravitational fluctuations associated with a noncommutative space are reviewed. Geometry of a noncommutative space is defined by the data (A, H,D) where A is a noncommutative involutive algebra, H is a separable Hilbert space and D a self-adjoint operator on H referred to as Dirac operator [1]. Geometry on Riemannian manifolds could b...

We study the 3D-calculus on SUq(2) (c.f. [W1]) from the point of view of noncommutative Riemannian geometry as formulated in [F1]. In particular, we show how we can obtain the Haar state from the ”Laplacian” on SUq(2) using a formula very similar to what is used in [F1] but with an appropriate modification. Furthermore, we recast the 3D-calculus along the line of [F1], showing that the complex ...

We extend standard path–integral techniques of bosonization and duality to the setting of noncommutative geometry. We start by constructing the bosonization prescription for a free Dirac fermion living in the noncommutative plane Rθ. We show that in this abelian situation the fermion theory is dual to a noncommutative Wess–Zumino–Witten model. The non–abelian situation is also constructed along...

We present an introduction to the geometry of higher order vector and co–vector bundles (including higher order generalizations of the Finsler geometry and Kaluza– Klein gravity) and review the basic results on Clifford and spinor structures on spaces with generic local anisotropy modeled by anholonomic frames with associated nonlinear connection structures. We emphasize strong arguments for ap...

In recent publications Alain Connes [1] and John Barrett [2] proposed to change the KO-dimension of the internal space of the standard model in its noncommutative representation [3] from zero to six. This apparently minor modification allowed to resolve the fermion doubling problem [4], and the introduction of Majorana mass terms for the right-handed neutrino. The price which had to be paid was...

Exponential regularization of orthogonal and Anti-de Sitter (AdS) space is presented based on noncommutative geometry. We show that an adequately adopted noncommutative deformation of geometry makes the holography of higher dimensional quantum theory of gravity and lower dimensional theory possible. Detail counting for observable degrees of freedom of quantum system of gravity in the bulk of no...

We make some brief remarks on the relation of the mixmaster universe model of chaotic cosmology to the geometry of modular curves and to noncommutative geometry. We show that the full dynamics of the mixmaster universe is equivalent to the geodesic ow on the modular curve X 0(2). We then consider a special class of solutions, with bounded number of cycles in each Kasner era, and describe their ...