B.R.S. renormalisation of some on-shell closed algebras of symmetry transformations : N=2 and 4 supersymmetric non-linear sigma models

We analyse with the algebraic, regularisation independant, cohomological B.R.S. methods, the renormalisability of torsionless N=2 and N= 4 supersymmetric non-linear \si models built on K\"ahler spaces. Surprisingly enough with respect to the common wisdom, in the case of N=2 supersymmetry, we obtain an anomaly candidate, at least in the compact K\"ahler Ricci-flat case. If its coefficient does differ from zero, such anomaly would imply the breaking of global N=2 supersymmetry and get into trouble some schemes of superstring compactification as such non-linear \si models offer candidates for the superstring vacuum state. In the compact homogeneous K\"ahler case, as expected, the anomaly candidate disappears. The same phenomena occurs when one enforces N=4 supersymmetry : in that case, we obtain the first rigorous proof of the expected all-orders renormalisability -`` in the space of metrics"- of the corresponding non-linear \si models. PAR/LPTHE/94-11Comment: 22 pages, latex, PAR/LPTHE/94-1

Lorentz and CPT violations in QED : a short comment on recent controversies

We rediscuss the recent controversy on a possible Chern-Simons like term generated through radiative corrections in QED with a CPT violating term. We emphasize the fact that any absence of an {\sl a priori} divergence should be explained by some symmetry or some non-renormalisation theorem : otherwise, no prediction can be made on the corresponding quantity.Comment: 7 pages, LaTex file, 2 references adde

Regularisation : many recipes, but a unique principle : Ward identities and Normalisation conditions. The case of CPT violation in QED

We analyse the recent controversy on a possible Chern-Simons like term generated through radiative corrections in QED with a CPT violating term : we prove that, if the theory is correctly defined through Ward identities and normalisation conditions, no Chern-Simons term appears, without any ambiguity. This is related to the fact that such a term is a kind of minor modification of the gauge fixing term, and then no renormalised. The past year literature on that subject is discussed, and we insist on the fact that any absence of an {\sl a priori} divergence should be explained by some symmetry or some non-renormalisation theorem.Comment: 14 pages, Latex fil

Einstein-Weyl structures and Bianchi metrics

We analyse in a systematic way the (non-)compact four dimensional Einstein-Weyl spaces equipped with a Bianchi metric. We show that Einstein-Weyl structures with a Class A Bianchi metric have a conformal scalar curvature of constant sign on the manifold. Moreover, we prove that most of them are conformally Einstein or conformally K\"ahler ; in the non-exact Einstein-Weyl case with a Bianchi metric of the type $VII_0, VIII$ or $IX$, we show that the distance may be taken in a diagonal form and we obtain its explicit 4-parameters expression. This extends our previous analysis, limited to the diagonal, K\"ahler Bianchi $IX$ case.Comment: Latex file, 12 pages, a minor modification, accepted for publication in Class. Quant. Gra

La royauté biblique : regards sur l’utilisation du thème dans la musique, la liturgie et le théâtre

Parmi les personnages et les thèmes de la Bible souvent exploités dans l’art, la littérature, les réflexions philosophique, théologique, politique et liturgique, le roi et la royauté se taillent la part du lion : par exemple, l’institution de la royauté en Israël, Saül le roi fou, le jeune David qui devient le roi selon le cœur de Dieu, Salomon le roi sage, les deux royaumes d’Israël et de Juda, la reine Jézabel et bien d’autres rois encore, accompagnés de prophètes qui les conseillent et les critiquent, le roi Hérode, mégalomane et massacreur d’enfants, mais aussi grand bâtisseur, l’espérance de la venue du Messie, celui qui a reçu l’onction royale, Jésus le Messie des chrétiens, nommé dans le Nouveau Testament roi des Juifs ou roi des rois. La présente étude à trois voix s’intéresse d’abord à la figure du roi Salomon en comparaison avec celle du roi George II d’Angleterre à partir de l’oratorio Salomon de Haendel. L’analyse se poursuit avec les rapports entre la critique rituelle, la légitimation, la liturgie et la royauté, c’est-à-dire le pouvoir. Enfin sont examinées les figures bibliques du roi et de la princesse dans le théâtre de Paul Claudel, plus particulièrement dans la pièce Tête d’or .

Compact Einstein-Weyl four-dimensional manifolds

We look for four dimensional Einstein-Weyl spaces equipped with a regular Bianchi metric. Using the explicit 4-parameters expression of the distance obtained in a previous work for non-conformally-Einstein Einstein-Weyl structures, we show that only four 1-parameter families of regular metrics exist on orientable manifolds : they are all of Bianchi $IX$ type and conformally K\"ahler ; moreover, in agreement with general results, they have a positive definite conformal scalar curvature. In a Gauduchon's gauge, they are compact and we obtain their topological invariants. Finally, we compare our results to the general analyses of Madsen, Pedersen, Poon and Swann : our simpler parametrisation allows us to correct some of their assertions.Comment: Latex file, 13 pages, an important reference added and a critical discussion of its claims offered, others minor modification