J. Madore

The Fuzzy sphere

A model of Euclidean spacetime is presented in which at scales less than a certain length kappa the notion of a point does not exist. At scales larger then kappa the model resembles the 2-sphere S2. The algebra which determines the structure of the model, and which replaces the algebra of functions, is an algebra of matrices. The order of n of the matrices is connected with the length kappa and the radius r of the sphere by the relation kappa approximately r/n. The elements of differential calculus are sketched as well as the possible definitions of a metric and linear connection. A definition of the path integral is given and a few examples of field theory on a fuzzy sphere are finally referred to.

Gauge theory on noncommutative spaces

Abstract. We introduce a formulation of gauge theory on noncommutative spaces based on the notion of covariant coordinates. Some important examples are discussed in detail. A Seiberg-Witten map is established in all cases.

Noncommutative differential geometry of matrix algebras

The noncommutative differential geometry of the algebra Mn (C) of complex n×n matrices is investigated. The role of the algebra of differential forms is played by the graded differential algebra C(sl(n,C),Mn (C))=Mn (C)⊗Λsl(n,C)*,sl(n,C) acting by inner derivations on Mn (C). A canonical symplectic structure is exhibited for Mn (C) for which the Poisson bracket is, to within a factor i, the commutator. Also, a canonical Riemannian structure is described for Mn (C). Finally, the analog of the Maxwell potential is constructed and it is pointed out that there is a potential with a vanishing curvature that is not a pure gauge.

Noncommutative Differential Geometry and New Models of Gauge Theory

The noncommutative differential geometry of the algebra C∞(V)⊗Mn(C) of smooth Mn(C)‐valued functions on a manifold V is investigated. For n≥2, the analog of Maxwell’s theory is constructed and interpreted as a field theory on V. It describes a U(n)–Yang–Mills field minimally coupled to a set of fields with values in the adjoint representation that interact among themselves through a quartic polynomial potential. The Euclidean action, which is positive, vanishes on exactly two distinct gauge orbits, which are interpreted as two vacua of the theory. In one of the corresponding vacuum sectors, the SU(n) part of the Yang–Mills field is massive. For the case n=2, analogies with the standard model of electroweak theory are pointed out. Finally, a brief description is provided of what happens if one starts from the analog of a general Yang–Mills theory instead of Maxwell’s theory, which is a particular case.

Gauge bosons in a noncommutative geometry

A noncommutative extension of abelian gauge theory is proposed and is compared with standard nonabelian gauge theory.

A noncommutative version of the Schwinger model

Abstract A model of euclidean space-time has been given in which at scales less than a certain length ϰ the notion of a point does not exist. At scales larger than ϰ the model resembles the two-sphere S2. The model is defined by a noncommutative matrix geometry. We use this version of two-dimensional space to study a noncommutative generalization of the Schwinger model.

Field theory on the q-deformed fuzzy sphere I

We study the q–deformed fuzzy sphere, which is related to D-branes on SU(2) WZW models, for both real q and q a root of unity. We construct for both cases a differential calculus which is compatible with the star structure, study the integral, and find a canonical frame of one–forms. We then consider actions for scalar field theory, as well as for Yang–Mills and Chern–Simons–type gauge theories. The zero curvature condition is solved.

Quantum Space-time and Classical Gravity

A method has been recently proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example a version of quantized space–time is considered here. It is found that there is a natural differential calculus over this version of quantized space–time using which the only possible torsion-free, metric-compatible, linear connection has zero curvature. It is then the noncummutative version of Minkowski space–time. Perturbations of this calculus are shown to give rise to nontrivial gravitational fields.

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.

Differential calculi and linear connections

A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a one‐to‐one correspondence, between the module structure of the 1‐forms and the metric torsion‐free connections on it. In the commutative limit the connection remains as a shadow of the algebraic structure of the 1‐forms.