A note on traces for the Heisenberg calculus

TL;DR

This paper clarifies the construction of a crucial trace used in the index formula for Heisenberg elliptic operators on contact manifolds, simplifying previous complex constructions.

Contribution

It provides a clearer understanding of the trace on Heisenberg pseudodifferential operators, essential for the index formula in the Heisenberg calculus.

Findings

Explicit description of the trace on Heisenberg pseudodifferential operators

Simplification of the trace construction process

Enhanced understanding of the index formula for Heisenberg elliptic operators

Abstract

In previous work, we gave a local formula for the index of Heisenberg elliptic operators on contact manifolds. We constructed a cocycle in periodic cyclic cohomology which, when paired with the Connes-Chern character of the principal Heisenberg symbol, calculates the index. A crucial ingredient of our index formula was a new trace on the algebra of Heisenberg pseudodifferential operators. The construction of this trace was rather involved. In the present paper, we clarify the nature of this trace.