Operator-valued pseudo-differential operators and the twisted index pairing

TL;DR

This paper develops a theory of operator-valued pseudo-differential operators with continuous trace algebra coefficients, establishing an index formula that generalizes Atiyah-Singer index theory to twisted settings.

Contribution

It introduces a new class of pseudo-differential operators with coefficients in continuous trace algebras and derives an Atiyah-Singer type index formula for twisted operators.

Findings

Established an algebra of principal symbols as an abstract Poincaré dual

Derived index formulas for twisted pseudo-differential operators

Connected twisted index pairing with elliptic operator index theory

Abstract

The notion of pseudo-differential operators with coefficients in a continuous trace algebra over a manifold are introduced and their index theory is studied. The algebra of principal symbols in this calculus provides an abstract Poincar\'e dual to the continuous trace algebra. Index formulas for pseudo-differential operators twisted by a bundle on the opposite continuous trace algebra are obtained in terms of an Atiyah-Singer type index formula describing the twisted index pairing as index theory for elliptic operators.