Bruno Iochum

Moyal Planes are Spectral Triples

Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces R2N endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes–Lott functional action, are given for these noncommutative hyperplanes.

The spectral action for Moyal planes

Extending a result of Vassilevich, we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ(f)e−tΔΘ, where ΔΘ is a generalized Laplacian defined with Moyal products and LΘ(f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymmetric matrix Θ being spectral triples, the spectral action introduced in noncommutative geometry by Chamseddine and Connes is computed. This result generalizes the Connes–Lott action previously computed by Gayral for symplectic Θ.

DISTANCES IN FINITE SPACES FROM NONCOMMUTATIVE GEOMETRY.

Following the general principles of noncommutative geometry, it is possible to define a metric on the space of pure states of the noncommutative algebra generated by the coordinates. This metric generalizes the usual Riemannian one. We investigate some general properties of this metric in finite commutative cases corresponding to a metric on a finite set, and also compute explicitly some distances associated to commutative or noncommutative algebras.

On the universal Chamseddine–Connes action. I. Details of the action computation

We give a detailed computation of the bosonic action of the Chamseddine–Connes model which we performed using different techniques.

Yang-Mills-Higgs versus Connes-Lott

By a suitable choice of variables we show that every Connes-Lott model is a Yang-Mills-Higgs model. The contrary is far from being true. Necessary conditions are given. Our analysis is pedestrian and illustrated by examples.

Spectral action on noncommutative torus

The spectral action on noncommutative torus is obtained, using a Chamseddine--Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series of holomorphic functions are obtained in this context.

Arens-regularity of Banach algebras and the geometry of Banach spaces

We show that a Banach algebra A such that A∗ has the property (V∗) of A. Pelczynski is Arens-regular. A strong uniqueness property of the extension of the product on a unital C∗-algebra A to A∗∗ is proved. The algebraic structure of the bidual of a C∗-algebra can be obtained through the local reflexivity principle. We give examples which show that the results are sharp.

Fuzzy Mass Relations for the Higgs

The noncommutative approach of the standard model produces a relation between the top and the Higgs masses. We show that, for a given top mass, the Higgs mass is constrained to lie in an interval. The length of this interval is of the order of m2τ/mt.

The noncommutative constraints on the standard model à la Connes

Noncommutative geometry applied to the standard model of electroweak and strong interactions was shown to produce fuzzy relations among masses and gauge couplings. We refine these relations and show then that they are exhaustive.

A left-right symmetric model à la Connes-Lott

We present a left-right symmetric model with the gauge group U(2)L × U(2)R within the Connes-Lott noncommutative framework. Its gauge symmetry is spontaneously broken, although parity remains unbroken.