Weighted Traces on Algebras of Pseudo-Differential Operators and Geometry on Loop Groups

Abstract

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.

On Diff(M)-Pseudo-Differential Operators and the Geometry of Non Linear Grassmannians

“…Such a definition is consistent for pseudo-differential operators on smooth sections of vector bundles, and applying the local formula for Wodzicki residue, one can prove [4]: This trace was first defined in the papers [50,51] by Kontesevich and Vishik. We remark that it is in particular a trace on DO(M, E) when M is odd-dimensional.…”

“…The functional tr Q is of course not a trace on Cl(M, E). Notice also that, if A and Q are pseudo-differential operators acting on sections on a real vector bundle E, they also act on E ⊗ C. The Wodzicki residue res and the renormalized traces tr Q have to be understood as functionals defined on pseudo-differential operators acting on E ⊗ C. In order to compute tr Q [A, B] and to differentiate tr Q A, in the topology of classical pseudo-differential operators, we need the following ( [4], see also [49] for the first point):…”

“…The following "covariance" property of tr Q ( [4,48]) will be useful to define renormalized traces on bundles of operators, Proposition A2. Under the previous notations, if C is a classical elliptic injective operator of order 0, tr C −1 QC C −1 AC is well-defined and equals tr Q A.…”

“…We moreover have specific properties for weighted traces of a more restricted class of pseudo-differential operators (see [4,50,51]), called odd class pseudo-differential operators following [50,51] : ∀n ∈ Z, ∀(x, ξ) ∈ T * M, σ n (A)(x, −ξ) = (−1) n σ n (A)(x, ξ).…”

“…Coming back to Emb(M, N), one could suggest that Aut(E) is sufficient as a structure group, but we refer the reader to earlier works such as [4][5][6] to see how pseudo-differential operators can arise from Levi-Civita connections of Sobolev metrics when the adequate structure group for the L 2 metric is a group of multiplication operators. Moreover, especially for M = S 1 , taking the quotient:…”