SINGSTAR Conference 2017 : Index theory and Singular Structures

Local linear forms on pseudodifferential operators and index theory

30 mai 2017, 08:50

45m

Amphi Schwartz IMT building 1R3 (TOULOUSE)

Orateur

Mme Sylvie Paycha

Description

This talk discusses **local** linear forms $\mathrm{\Lambda }:A\mapsto \mathrm{\Lambda }(A)$ on classical pseudo-differential operators on a closed manifold, namely linear forms of the type $\mathrm{\Lambda }(A)={\int }_M{\lambda }_A(x)dx$ given by a density ${\lambda }_A(x)dx$ on the manifold $M$ and their relevance in index theory. Local linear forms are spanned by the well-known Wodzicki residue $A⟼\mathrm{R}\mathrm{e}\mathrm{s}(A)$ on integer order operators and the somewhat lesser known canonical trace $A⟼\mathrm{T}\mathrm{R}(A)$ on non-integer order operators (joint work with S. Azzali). For a **holomorphic perturbation** $A(z)$ of a differential operator $A(0)=A$, these two linear forms relate by (joint work with S. Scott) \[ (1)\,\,\,\,\, \,\,lim_{z\to 0}TR(A(z))= -\frac{1}{2}\, Res(A^\prime(0)), \] where the residue has been extended to logarithms. Inspired by Gilkey's approach using invariance theory, for a family $A(z)$ of geometric operators, we showed (joint work with J. Mickelsson) that the density ${\mathrm{r}\mathrm{e}\mathrm{s}}_x(A^{\prime }(0))$ arising in the r.h.s. of (1) is an invariant polynomial which can be expressed in terms of Pontryagin forms on the tangent bundle and Chern forms on the auxillary bundle. A ${\mathbb{Z}}_2$-graded generalisation of (1) applied to an appropriate holomorphic perturbation of the identity built from a Dirac operator $D=D_{+}\oplus D_{-}$ acting on a ${\mathbb{Z}}_2$-graded vector bundle, expresses the **index** of $D_{+}$ in terms of a Wodzicki residue. As a result of their locality, the canonical trace and the Wodzicki residue are preserved under lifting to the universal covering of a closed manifold; consequently formula (1) lifts to coverings. This lifted analoque of (1) yields an expression of the $L^2$-index of a lifted Dirac operator in terms of the Wodzicki residue of the logarithm of its square (joint work with S. Azzali).

Documents de présentation

Slides

paycha.pdf