We construct a non-formal deformation machinery for the actions of the Heisenberg supergroup analogue to the one developed by M. Rieffel for the actions of R d . However, the method used here differs from Rieffel's one: we obtain a Universal Deformation Formula for the actions of R m|n as a byproduct of Weyl ordered Kirillov's orbit method adapted to the graded setting. To do so, we have to introduce the notion of C*-superalgebra, which is compatible with the deformation, and which can be seen as corresponding to noncommutative superspaces. We also use this construction to interpret the renormalizability of a noncommutative Quantum Field Theory.

Abstract. The harmonic term in the scalar field theory on the Moyal space removes the UV-IR mixing, so that the theory is renormalizable to all orders. In this paper, we review the three principal interpretations of this harmonic term: the Langmann-Szabo duality, the superalgebraic approach and the noncommutative scalar curvature interpretation. Then, we show some deep relationship between these interpretations.

We investigate symmetries of the scalar field theory with harmonic term on the Moyal space with euclidean scalar product and general symplectic form. The classical action is invariant under the orthogonal group if this group acts also on the symplectic structure. We find that the invariance under the orthogonal group can be restored also at the quantum level by restricting the symplectic structures to a particular orbit. * Work supported by the Belgian Interuniversity Attraction Pole (IAP) within the framework "Nonlinear systems, stochastic processes, and statistical mechanics" (NOSY). arXiv:0911.2645v2 [math-ph] 6 Jan 2011In the past few years, there has been a growing interest in the noncommutative quantum fields theories (for a review, see [1]). These theories on "spaces" coming from noncommutative geometry [2] are indeed strong candidates for new physics behind the Standard Model of particle physics. Moreover, fields theories defined on the Moyal space [3], one of the simplest example of noncommutative space, can be seen as an effective regime of string theory [4] and matrix theory [5].The simplest generalization of the ϕ 4 commutative scalar theory gives rise to a new type of divergence, the ultraviolet-infrared (UV/IR) mixing [6], which is responsible for the nonrenormalizability of this model. The first solution to this problem of UV/IR mixing, due to Grosse and Wulkenhaar, was to add some harmonic term in the action [7], and the resulting theory is then renormalizable up to all orders in perturbation [7,8,9]. The renormalizability of this model seems to be related to a new symmetry thanks to this harmonic term, the Langmann-Szabo duality [10], which exchanges positions and impulsions at the level of the quadratic terms of the action. This Langmann-Szabo duality has also been interpreted in the framework of superalgebras as a grading symmetry [11,12] and an adapted differential calculus has been exhibited (see also [13,14]). Note that another interpretation of the harmonic term has been given in [15]. Moreover, the vacuum solutions of this theory have been studied in [16], and it has been proved that the beta function vanishes up to irrelevant terms [17,18]. Associated to this scalar field theory with harmonic term, a gauge invariant action, candidate to renormalizability, has been exhibited in [19] (see also [20,21]), but its vacua are always non-trivial [22].However, since a symplectic structure Σ is necessary to define the Moyal product, the group of rotations is no longer a symmetry group for this theory on the Moyal space as soon as the dimension D > 2. The question of rotational invariance of the theory is important since it corresponds to the Lorentz invariance, necessary for a physical theory, in the Minkowskian framework. This problem is related to the fact that a symplectic structure is not natural on a configuration (or position) space, contrary to a phase space. A simple standard way to restore the rotational invariance of the classical action is to consider a family of actions labeled by the s...

We define and study a noncommutative Fourier transform on every homogeneous complex bounded domain. We then give an application in noncommutative differential geometry by defining noncommutative Baumslag-Solitar tori.

In this paper, we introduce Fréchet quantum supergroups and their representations. By using the universal deformation formula of the abelian supergroups R m|n we construct various classes of Fréchet quantum supergroups that are deformation of classical ones. For such quantum supergroups, we find an analog of Kac-Takesaki operators that are superunitary and satisfy the pentagonal relation.