Jean-Christophe Wallet

Noncommutative induced gauge theory

We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4-dimensional Moyal space and compute in position space the one-loop Yang–Mills-type effective theory generated from the integration over the scalar field. We find that the gauge-invariant effective action involves, beyond the expected noncommutative version of the pure Yang–Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic oscillator term, which for the noncommutative ϕ4-theory on Moyal space ensures renormalisability. The expression of a possible candidate for a renormalisable action for a gauge theory defined on Moyal space is conjectured and discussed.

Vacuum configurations for renormalizable non-commutative scalar models

In this paper we find non-trivial vacuum states for the renormalizable non-commutative φ4 model. An associated linear sigma model is then considered. We further investigate the corresponding spontaneous symmetry breaking.

On the vacuum states for non-commutative gauge theory

Candidates for renormalizable gauge theory models on Moyal spaces constructed recently have non-trivial vacua. We show that these models support vacuum states that are invariant under both global rotations and symplectic isomorphisms which form a global symmetry group for the action. We compute the explicit expression in position space for these vacuum configurations in two and four dimensions.

Noncommutative Yang-Mills-Higgs actions from derivation-based differential calculus

Derivations of a (noncommutative) algebra can be used to construct various consistent differential calculi, the so-called derivation-based differential calculi. We apply this framework to the noncommutative Moyal algebras for which all the derivations are inner and analyse in detail the case where the derivation algebras generating the differential calculus are related to area preserving diffeomorphisms. The ordinary derivations corresponding to spatial dimensions are supplemented by additional derivations necessarely related to additional covariant coordinates. It is shown that these latter have a natural interpretation as Higgs fields when involved in gauge invariant actions built from the noncommutative curvature. The UV/IR mixing problem for (some of) the resulting Yang-Mills-Higgs models is discussed. A comparition to other noncommutative geometries already considered in the litterature is given.

Noncommutative ε-graded connections

We introduce the new notion of e-graded associative algebras which takes its roots from the notion of commutation factors introduced in the context of Lie algebras ([39]). We define and study the associated notion of e-derivations-based differential calculus, which generalizes the derivation-based calculus on associative algebras. A corresponding notion of noncommutative connection is also defined. We illustrate these considerations with various examples of e-graded commutative algebras, in particular some graded matrix algebras and the Moyal algebra. This last example also permits us to interpret mathematically a noncommutative gauge field theory.

One-loop Beta Functions for the Orientable Non-commutative Gross-Neveu Model

We compute at the one-loop order the β-functions for a renormalisable non-commutative analog of the Gross–Neveu model defined on the Moyal plane. The calculation is performed within the so called x-space formalism. We find that this non-commutative field theory exhibits asymptotic freedom for any number of colors. The β-function for the non-commutative counterpart of the Thirring model is found to be non vanishing.

Quantum gauge theories on noncommutative three-dimensional space

We consider a class of gauge invariant models on the noncommutative space

Symmetries of noncommutative scalar field theory

\mathbb{R}^3_\lambda

CONNES DISTANCE BY EXAMPLES: HOMOTHETIC SPECTRAL METRIC SPACES

, a deformation of

LINEAR CONNECTIONS ON THE TWO-PARAMETER QUANTUM PLANE

\mathbb{R}^3