Gravity in non-commutative geometry

Abstract

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 gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.* Supported by the Swiss National Foundation (SNF)

Quantum Hall effect in higher dimensions, matrix models and fuzzy geometry

“…In fact, the SU (4) potential can be written as (6) and P α are a basis for the complement of O(5) in O (6). E α are the frame fields and ω αβ is the spin connection.…”

An Introduction to Noncommutative Spaces and their Geometries

“…with λ, µ ∈ C, k 34,2 ∈ K H 32 ⊕H 42 and k 34,21 ∈ K (H 32 ⊕H 42 )⊗H 21 . The representations are the following ones,…”

“…A different approach to gravity theory, developed in [21,22], is based on a theory of linear connections on an analogue of the cotangent bundle in the noncommutative setting. It turns out the the analogue of the cotangent bundle is more appropriate that the one of tangent bundle.…”

“…A theory of linear connections and Riemannian geometry, culminating in the analogue of the Hilbert-Einstein action in the context of noncommutative geometry has been proposed in [21]. Again, for the canonical triple one recovers the usual Einstein gravity.…”

“…Again, for the canonical triple one recovers the usual Einstein gravity. When computed for a Connes-Lott space M × Y as in [21], the action produces a KaluzaKlein model which contains the usual integral of the scalar curvature of the metric on M, a minimal coupling for the scalar field to such a metric, and a kinetic term for the scalar field. A somewhat different model of geometry on the space M × Y produced an action which is is just the Kaluza-Klein action of unified gravity-electromagnetism consisting of the usual gravity term, a kinetic term for a minimally coupled scalar field and an electromagnetic term [71].…”