Using weighted traces which are linear functionals of the type A → tr Q (A) := tr(AQ −z ) − z −1 tr(AQ −z ) z=0 defined on the whole algebra of (classical) pseudo-differential operators (P.D.O.s) and where Q is some positive invertible elliptic operator, we investigate the geometry of loop groups in the light of the cohomology of pseudo-differential operators. We set up a geometric framework to study a class of infinite dimensional manifolds in which we recover some results on the geometry of loop groups, using again weighted traces. Along the way, we investigate properties of extensions of the Radul and Schwinger cocycles defined with the help of weighted traces. RésuméA l'aide de traces pondérées qui sont des fonctionnelles linéaires du type:définies sur toute l'algèbre des opérateurs pseudo-différentiels classiques, Qétant un opérateur elliptique inversible, onétudie la géométrie de l'espace des lacetsà la lumière de la cohomologie des opérateurs pseudo-différentiels. On met en place un cadre géométrique afin d'étudier une classe de variétés de dimension infinie, cadre dans lequel on retrouve, toujoursà l'aide des traces pondérées, des résultats concernant la géométrie des lacets. Ces traces pondérées nous permettent aussi d'étendre la notion de cocycle de Radul et de Schwinger et d'enétudier certaines propriétés.

ζ-regularized traces, resp. super-traces, are defined on a classical pseudodifferential operator A by:where f.p. refers to the finite part and Q is an (invertible and admissible) elliptic reference operator with positive order. They are widly used in quantum field theory in spite of the fact that, unlike ordinary traces on matrices, they are neither cyclic nor do they commute with exterior differentiation, thus giving rise to tracial anomalies. The purpose of this article is to show, on two examples, how tracial anomalies can lead to anomalous phenomena in quantum field theory.

We study new regularized determinants on elliptic pseudo-differential operators. They are of the form " expTr^Log", where Tr® is a weighted trace. We express the obstruction preventing the ^-determinant from being of this type, this leading us to a general formula for the multiplicative anomaly of the ^-determinant.