Nguyen Ai Viet

Gravity And Electromagnetism In Noncommutative Geometry

Abstract We present a unified description of gravity and electromagnitsm in the framework of a Z 2 noncommutative differential calculus. It can be considered as a “discrete version” of Kaluza-Klein theory, where the fifth continuous dimension is replaced by two discrete points. We derive an action which coincides with the dimensionaly reduced one of the ordinary Kaluza-Klein theory.

Noncommutative geometry and a discretized version of Kaluza-Klein theory with a finite field content

We consider a four-dimensional space-time supplemented by two discrete points assigned to a Z2-algebraic structure and develop the formalism of noncommutative geometry. By setting up a generalized vielbein, we study the metric structure. Metric-compatible torsion-free connection defines a unique finite field content in the model and leads to a discretized version of Kaluza-Klein theory. We study some special cases of this model that illustrate the rich and complex structure with massive modes and the possible presence of a cosmological constant.

Chiral spinors and gauge fields in noncommutative curved space-time

The fundamental concepts of Riemannian geometry, such as differential forms, vielbein, metric, connection, torsion and curvature, are generalized in the context of non-commutative geometry. This allows us to construct the Einstein-Hilbert-Cartan terms, in addition to the bosonic and fermionic ones in the Lagrangian of an action functional on non-commutative spaces. As an example, and also as a prelude to the Standard Model that includes gravitational interactions, we present a model of chiral spinor fields on a curved two-sheeted spacetime with two distinct abelian gauge fields. In this model, the full spectrum of the generalized metric consists of pairs of tensor, vector and scalar fields. They are coupled to the chiral fermions and the gauge fields leading to possible parity violation effects triggered by gravity.

A DISCRETIZED VERSION OF KALUZA–KLEIN THEORY WITH TORSION AND MASSIVE FIELDS

We consider an internal space of two discrete points in the fifth dimension of the Kaluza–Klein theory by using the formalism of noncommutative geometry — developed in a previous paper1 — of a spacetime supplemented by two discrete points. With the non-vanishing internal torsion two-form there are no constraints implied on the vielbeins. The theory contains a pair of tensor fields, a pair of vector fields and a pair of scalar fields. Using the generalized Cartan structure equation we are able to uniquely determine not only the Hermitian and metric-compatible connection one-forms, but also the nonvanishing internal torsion two-form in terms of vielbeins. The resulting action has a rich and complex structure, a particular feature being the existence of massive modes. Thus the nonvanishing internal torsion generates a Kaluza–Klein type model with zero and massive modes.

Chern-Simons terms in noncommutative geometry and its application to bilayer quantum hall systems

Abstract Considering bilayer systems as extensions of the planar ones by an internal space of two discrete points, we use the ideas of Noncommutative Geometry to construct the gauge theories for these systems. After integrating over the discrete space we find an effective 2 + 1 action involving an extra complex scalar field, which can be interpreted as arising from the tunneling between the layers. The gauge fields are found in different phases corresponding to the different correlations due to the Coulomb interaction between the layers. In a particular phase, when the radial part of the complex scalar field is a constant, we recover the Wen-Zee model [X.G. Wen and A. Zee, Phys. Rev. Lett. 69 (1992) 1811; Phys. Rev. B 47 (1993) 2265] of Bilayer Quantum Hall systems. There are some circumstances, where this radial part may become dynamical and cause dissipation in the oscillating supercurrent between the layers.

Matter fields in curved space-time

We study the geometry of a two-sheeted space-time within the framework of non-commutative geometry. As a prelude to the Standard Model in curved space-time, we present a model of a left- and a right- chiral field living on the two sheeted-space time and construct the action functionals that describe their interactions.

Discrete Internal Space and its Visibility

In this paper we consider a version of extended space-time with the inter- nal space of two discrete points. A comptex metric structure for the internal space is suggested in order to use a common generalized differential calculus and a common wedge product for both gravity and the Standard Model. In this extended space-time there are some testable predictions near the electroweak energy scale: the Higgs mass of 350 GeV and the existence of a scalar particle from the gravity sector with mass of 440 GeV.

Structure deformation by gauge fields and higher order cocycles

Abstract Cocycles are considered as the non-integrable phase factors appearing in composition laws of the spacetime translation groups gauge deformations. In this context, we can classify the non-associative structures and calculate any cocycle of arbitrary order. The highest order cocycles are interpreted in relations with the chiral anomalies.