Francois Gieres

Translation-invariant models for non-commutative gauge fields

Motivated by the recent construction of a translation-invariant renormalizable non-commutative model for a scalar field [1], we introduce models for non-commutative U(1) gauge fields along the same lines. More precisely, we include some extra terms into the action with the aim of getting rid of the UV/IR mixing.

CONFORMALLY COVARIANT OPERATORS ON RIEMANN SURFACES (WITH APPLICATIONS TO CONFORMAL AND INTEGRABLE MODELS)

Following the standard procedure for gauging in Yang-Mills and gravitational theories, we introduce projective connections to covariantize differential operators on Riemann surfaces. We present applications in integrable models (Lax pairs, Poisson operators) and conformal models (conformal Ward identity, diffeomorphism anomaly, Krichever-Novikov algebra, Virasoro algebra and its representations, Kac-Moody algebras, W-algebras, WZW model, twistor theory). The generalization to higher dimensions is indicated and the whole discussion is generalized to the supersymmetric case (both in superspace and in component field formalism).

Magnetic fields in noncommutative quantum mechanics

We discuss various descriptions of a quantum particle on noncommutative space in a (possibly non-constant) magnetic field. We have tried to present the basic facts in a unified and synthetic manner, and to clarify the relationship between various approaches and results that are scattered in the literature.

Quantum Corrections for Translation-Invariant Renormalizable Non-Commutative Phi**4 Theory

In this paper we elaborate on the translation-invariant renormalizable 4 theory in 4-dimensional non-commu\-ta\-tive space which was recently introduced by the Orsay group. By explicitly performing Feynman graph calculations at one loop and higher orders we illustrate the mechanism which overcomes the UV/IR mixing problem and ultimately leads to a renormalizable model. The obtained results show that the IR divergences are also suppressed in the massless case, which is of importance for the gauge field theoretic generalization of the scalar field model.

On the renormalization of non-commutative field theories

This paper addresses three topics concerning the quantization of non-commutative field theories (as defined in terms of the Moyal star product involving a constant tensor describing the non-commutativity of coordinates in Euclidean space). To start with, we discuss the Quantum Action Principle and provide evidence for its validity for non-commutative quantum field theories by showing that the equation of motion considered as insertion in the generating functional Zc[j] of connected Green functions makes sense (at least at one-loop level). Second, we consider the generalization of the BPHZ renormalization scheme to non-commutative field theories and apply it to the case of a self-interacting real scalar field: Explicit computations are performed at one-loop order and the generalization to higher loops is commented upon. Finally, we discuss the renormalizability of various models for a self-interacting complex scalar field by using the approach of algebraic renormalization.

Classical N=1 and N=2 super W algebras from a zero-curvature condition

Starting from superdifferential operators in an N=1 superfield formulation, we present a systematic prescription for the derivation of classical N=1 and N=2 super W algebras by imposing a zero-curvature condition on the connection of the corresponding first-order system. We illustrate the procedure on the first nontrivial example (beyond the N=1 superconformal algebra) and also comment on the relation with the Gelfand-Dickey construction of W algebras.

BPHZ renormalization and its application to non-commutative field theory

In a recent work a modified BPHZ scheme has been introduced and applied to one-loop Feynman graphs in non-commutative ϕ4-theory. In the present paper, we first review the BPHZ method and then we apply the modified BPHZ scheme as well as Zimmermann’s forest formula to the sunrise graph, i.e. a typical higher-loop graph involving overlapping divergences. Furthermore, we show that the application of the modified BPHZ scheme to the IR-singularities appearing in non-planar graphs (UV/IR mixing problem) leads to the introduction of a 1/p2 term and thereby to a renormalizable model. Finally, we address the application of this approach to gauge field theories.

OBSERVABLES IN TOPOLOGICAL YANG–MILLS THEORIES

Using topological Yang–Mills theory as example, we discuss the definition and determination of observables in topological field theories (of Witten-type) within the superspace formulation proposed by Horne. This approach to the equivariant cohomology leads to a set of bi-descent equations involving the BRST and supersymmetry operators as well as the exterior derivative. This allows us to find the most general solution to the problem of equivariant cohomology, and to recover the Donaldson–Witten polynomials when choosing a Wess–Zumino-type gauge.

ANOMALIES AND EFFECTIVE ACTIONS IN TWO-DIMENSIONAL SUPERSPACE

We derive a chirally split expression for the superdiffeomorphism anomaly in the two-dimensional superplane. The effective action from which this anomaly was obtained is related to the WZ-action associated with the superdiffeomorphism group. Furthermore, we show that the anomalous Ward identities generated by supercoordinate transformations encompass the wellknown OPE’s for the stress-energy supertensor. In conclusion, the extension to generic super Riemann surfaces is discussed.

Topological Yang-Mills Theories and their Observables: A Superspace Approach

Wittens observables of topological Yang-Mills theory, defined as classes of an equivariant cohomology, are reobtained as the BRST cohomology classes of a superspace version of the theory.