On the geometry of $Diff(S^1)$-pseudodifferential operators based on renormalized traces.

Abstract


 In this article, we examine the geometry of a group of Fourier-integral operators, which is the central extension of $Diff(S^1)$ with a group of classical pseudo-differential operators of any order.
 Several subgroups are considered, and the corresponding groups with formal pseudodifferential operators are defined.
 We investigate the relationship of this group with the restricted general linear group $GL_{res}$, we define a right-invariant pseudo-Riemannian metric on it that extends the Hilbert-Schmidt Riemannian metric by the use of renormalized traces of pseudo-differential operators, and we describe classes of remarkable connections.

On diffeological principal bundles of non-formal pseudo-differential operators over formal ones

“…These exact sequences have actually no local slice FCl ˚(M, E) Ñ Cl ˚(M, E) or FCl 0,˚( M, E) Ñ Cl 0,˚( M, E), in other words the diffeological G-principal bundle Cl 0,˚o ver FCl 0,˚h as actually no local trivialization (according to the Adams, Ratiu and Schmid topology which is the only one studied explicitely). This is at this point that we carry now a new element, that generalize the constructions given in [20] which treats the case M = S 1 .…”

“…If there exists diffeologies on Cl ˚(M, E) and FCl ˚(M, E) such that Theorem 2.17 applies to at least one P DO ´8(M, E)-valued connection, then there exists local trivializations for the diffeological principal bundle Cl ˚(M, E) over FCl ˚(M, E) and hence local slices to the exact sequence (3.1). [20]. If the diffeology on Cl ˚has enough 2-dimensional plots, one can define the curvature of the connection [19].…”

On a class of closed cocycles for algebras of non-formal, possibly unbounded, pseudodifferential operators

“…Then, in order to deal rigorously with them, one has to consider Gelfand's formal geometry or, from another viewpoint, diffeological Lie groups. • As mentioned in [24] there is no exponential map from paths on Cl(S 1 , V ) to paths on Cl * (S 1 , V ). The same obstruction holds obviously with P DO(S 1 , V ) and P DO * (S 1 , V ).…”

“…Since tr Q is not tracial, let us give one more property on the renormalized trace of the bracket, from e.g. [24]. Proposition 1.11.…”

“…• The groups P DO * (S 1 , V ) and Cl * (S 1 , V ) carry no atlas, as is was first mentioned in [1], see e.g. [24]. Then, in order to deal rigorously with them, one has to consider Gelfand's formal geometry or, from another viewpoint, diffeological Lie groups.…”