An index formula for the extended Heisenberg algebra of Epstein, Melrose and Mendoza

TL;DR

This paper derives a unified index formula for the extended Heisenberg algebra on contact manifolds, combining classical and Heisenberg pseudodifferential operator theories.

Contribution

It provides a new index formula that integrates Atiyah-Singer and Boutet de Monvel's formulas within the extended Heisenberg algebra framework.

Findings

Unified index formula for extended Heisenberg algebra

Incorporates Atiyah-Singer and Boutet de Monvel's formulas

Advances symbolic calculus on contact manifolds

Abstract

The extended Heisenberg algebra for a contact manifold has a symbolic calculus that accommodates both Heisenberg pseudodifferential operators as well as classical pseudodifferential operators. We derive here a formula for the index of Fredholm operators in this extended calculus. This formula incorporates in a single expression the Atiyah-Singer formula for elliptic operators, as well as Boutet de Monvel's Toeplitz index formula.