On elliptic differential operators with shifts: II. The cohomological index formula

TL;DR

This paper extends the index theory for elliptic differential operators with shifts by establishing a cohomological Atiyah-Singer type formula, involving the construction of closed graded traces on relevant differential algebras.

Contribution

It introduces a cohomological index formula for elliptic operators with shifts acting on sections of vector bundles, generalizing previous local index results.

Findings

Established a cohomological index formula of Atiyah-Singer type.

Constructed closed graded traces on differential algebras over the symbol algebra.

Extended index theory to operators acting between sections of arbitrary vector bundles.

Abstract

This paper is a continuation of arXiv:0706.3511, where we obtained a local index formula for matrix elliptic operators with shifts. Here we establish a cohomological index formula of Atiyah-Singer type for elliptic differential operators with shifts acting between section spaces of arbitrary vector bundles. The key step is the construction of closed graded traces on certain differential algebras over the symbol algebra for this class of operators.