On Diff(M)-pseudo-differential operators and the geometry of non linear grassmannians

TL;DR

This paper studies the structure groups of tangent bundles on spaces of embeddings modulo diffeomorphisms, revealing connections to Fourier integral operators, pseudo-differential operators, and potential applications to knot invariants.

Contribution

It introduces a new framework for the structure group of tangent bundles on embedding spaces using Fourier integral operators, extending previous pseudo-differential operator approaches.

Findings

The structure group is a central extension of diffeomorphisms by pseudo-differential operators.

The groups are shown to be regular, enabling geometric analysis.

Connections to knot theory and potential for knot invariants are identified.

Abstract

We consider two principal bundles of embeddings with total space $Emb(M,N),$ with structure groups $Diff(M)$ and $Diff_+(M),$ where $Diff_+(M)$ is the groups of orientation preserving diffeomorphisms. The aim of this paper is to describe the structure group of the tangent bundle of the two base manifolds: $$ B(M,N) = Emb(M,N)/Diff(M) \hbox{ and } B_+(M,N)= Emb(M,N)/Diff_+(M).$$ From the various properties described, an adequate group seems to be a group of Fourier integral operators, which is carefully studied. This is the main goal of this paper to analyze this group, which is a central extension of a group of diffeomorphisms by a group of pseudo-differential operators which is slightly different from the one developped in \cite{OMYK4}. We show that these groups are regular, and develop the necessary properties for applications to the geometry of $ B(M,N) .$ A case of particular interest is $M=S^1,$ where connected components of $B_+(S^1,N)$ are deeply linked with homotopy classes of oriented knots. In this example, the structure group of the tangent space $TB_+(S^1,N)$ is a subgroup of some group $GL_{res},$ following the classical notations of \cite{PS}. These constructions suggest some approaches in the spirit of \cite{Ma2006} that could lead to knot invariants through a theory of Chern-Weil forms.