Related papers: Algebraic backgrounds: a framework for noncommutat…
The present paper is a short survey on the mathematical basics of Classical Field Theory including the Serre-Swan' theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spin^C-structures, Dirac operators,…
This paper establishes a link between Noncommutative Geometry and canonical quantum gravity. A semi-finite spectral triple over a space of connections is presented. The triple involves an algebra of holonomy loops and a Dirac type operator…
We give a geometrical construction of Connes spectral triples or noncommutative Dirac operators $D$ starting with a bimodule connection on the proposed spinor bundle. The theory is applied to the example of $M_2(\Bbb C)$, and also applies…
A Dirac operator is presented that will yield a 1+ summable regular even spectral triple for all noncommutative compact surfaces defined as subalgebras of the Toeplitz algebra. Connes' conditions for noncommutative spin geometries are…
The twined almost commutative structure of the standard spectral triple on the noncommutative torus with rational parameter is exhibited, by showing isomorphisms with a spectral triple on the algebra of sections of certain bundle of…
The purpose of this article is to apply the concept of the spectral triple, the starting point for the analysis of noncommutative spaces in the sense of A.~Connes, to the case where the algebra $\cA$ contains both bosonic and fermionic…
We compute the metric associated to noncommutative spaces described by a tensor product of spectral triples. Well known results of the two-sheets model (distance on a sheet, distance between the sheets) are extended to any product of two…
We present a general formalism based on the framework of non-commutative geometry, suitable to the study the standard model of electroweak interactions, as well as that of more general gauge theories. Left- and right-handed chiral fields…
We construct a spectral-triple framework for a noncommutative planar system associated with a fixed nondegenerate irreducible unitary sector of the kinematical symmetry group $G_{\mathrm{NC}}$, labelled by central parameters…
A new approach to a unified theory of quantum gravity based on noncommutative geometry and canonical quantum gravity is presented. The approach is built around a *-algebra generated by local holonomy-diffeomorphisms on a 3-manifold and a…
In this paper we construct a candidate for a spectral triple on a quotient space of gauge connections modulo gauge transformations and show that it is related to a Kasparov type bi-module over two canonical algebras: the HD-algebra, which…
We consider a four-dimensional space-time supplemented by two discrete points assigned to a $Z_2$ algebraic structure and develop the formalism of noncommutative geometry. By setting up a generalised vielbein, we study the metric structure.…
Exceptional field theories yield duality-covariant formulations of higher-dimensional supergravity. They have proven to be an efficient tool for the construction of consistent truncations around various background geometries. In this paper,…
Spectral triples describe and generalize Riemannian spin geometries by converting the geometrical information into algebraic data, which consist of an algebra $A$, a Hilbert space $H$ carrying a representation of $A$ and the Dirac operator…
It is well-known that a compact Riemannian spin manifold can be reconstructed from its canonical spectral triple which consists of the algebra of smooth functions, the Hilbert space of square integrable spinors and the Dirac operator. It…
The notion of a K\"ahler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any…
The Higgs field is a connection one-form as the other bosonic fields, provided one describes space no more as a manifold M but as a slightly non-commutative generalization of it. This is well encoded within the theory of spectral triples:…
We present a spectral triple for $\kappa$-Minkowski space in two dimensions. Starting from an algebra naturally associated to this space, a Hilbert space is built using a weight which is invariant under the $\kappa$-Poincar\'e algebra. The…
Using some elementary methods from noncommutative geometry a structure is given to a point of space-time which is different from and simpler than that which would come from extra dimensions. The structure is described by a supplementary…
We give a derivation of the Dirac operator on the noncommutative $2$-sphere within the framework of the bosonic fuzzy sphere and define Connes' triple. It turns out that there are two different types of spectra of the Dirac operator and…