Connections on central bimodules

We define and study the theory of derivation-based connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also discuss the different noncommutative versions of differential forms based on derivations. Then we investigate reality conditions and a noncommutative generalization of pseudo-riemannian structures.Comment: 27 pages, AMS-LaTe

Higgs Mass and Noncommutative Geometry

We show that the description of the electroweak interactions based on noncommutative geometry of a continuous and a discrete space gives no special relations between the Higgs mass and other parameters of the model. We prove that there exists a gauge invariant term, linear in the curvature, which is trivial in the standard differential geometry but nontrivial in the case of the discrete geometry. The relations could appear only if one neglects this term, otherwise one gets the Lagrangian of the Standard model with the exact number of free parameters.Comment: 23 pages LaTeX, TPJU 4/93, (minor text misprints corrected

A bigraded version of the Weil algebra and of the Weil homomorphism for Donaldson invariants

We describe a bigraded generalization of the Weil algebra, of its basis and of the characteristic homomorphism which besides ordinary characteristic classes also maps on Donaldson invariants.Comment: 19 page

Non Commutative Differential Geometry, and the Matrix Representations of Generalised Algebras

The underlying algebra for a noncommutative geometry is taken to be a matrix algebra, and the set of derivatives the adjoint of a subset of traceless matrices. This is sufficient to calculate the dual 1-forms, and show that the space of 1-forms is a free module over the algebra of matrices. The concept of a generalised algebra is defined and it is shown that this is required in order for the space of 2-forms to exist. The exterior derivative is generalised for higher order forms and these are also shown to be free modules over the matrix algebra. Examples of mappings that preserve the differential structure are given. Also given are four examples of matrix generalised algebras, and the corresponding noncommutative geometries, including the cases where the generalised algebra corresponds to a representation of a Lie algebra or a $q$-deformed algebra.Comment: 16 pages Latex, No figures. Accepted for publication: Journal of Physics and Geometry, March 199

Exceptional quantum geometry and particle physics

Based on an interpretation of the quark-lepton symmetry in terms of the unimodularity of the color group $SU(3)$ and on the existence of 3 generations, we develop an argumentation suggesting that the "finite quantum space" corresponding to the exceptional real Jordan algebra of dimension 27 (the Euclidean Albert algebra) is relevant for the description of internal spaces in the theory of particles. In particular, the triality which corresponds to the 3 off-diagonal octonionic elements of the exceptional algebra is associated to the 3 generations of the Standard Model while the representation of the octonions as a complex 4-dimensional space $\mathbb C\oplus\mathbb C^3$ is associated to the quark-lepton symmetry, (one complex for the lepton and 3 for the corresponding quark). More generally it is is suggested that the replacement of the algebra of real functions on spacetime by the algebra of functions on spacetime with values in a finite-dimensional Euclidean Jordan algebra which plays the role of "the algebra of real functions" on the corresponding almost classical quantum spacetime is relevant in particle physics. This leads us to study the theory of Jordan modules and to develop the differential calculus over Jordan algebras, (i.e. to introduce the appropriate notion of differential forms). We formulate the corresponding definition of connections on Jordan modules.Comment: 37 pages ; some minor typo corrections. To appear in Nucl. Pays. B (2016), http://dx.doi.org/10.1016/j.nuclphysb.2016.04.01

On Curvature in Noncommutative Geometry

A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.Comment: 16 pages, PlainTe

Classical Gravity on Fuzzy Space-Time

A review is made of recent efforts to find relations between the commutation relations which define a noncommutative geometry and the gravitational field which remains as a shadow in the commutative limit.Comment: Lecture given at the 30th International Symposium Ahrenshoop on the Theory of Elementary Particles, Buckow, Germany, August 27-31, 1996; 11 Pages LaTe

Connes' Model Building Kit

Alain Connes' applications of non-commutative geometry to interaction physics are described for the purpose of model building.Comment: 35 pages, LATeX, CPT-93/P.296