4,339 research outputs found
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
-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 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 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
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