The Heisenberg Calculus, Index Theory and Cyclic (Co)homology

TL;DR

This paper develops an index formula for hypoelliptic operators in the Heisenberg calculus on contact manifolds, linking noncommutative geometry, cyclic cohomology, and classical index theory.

Contribution

It constructs a local cyclic cocycle that computes the index of Heisenberg calculus operators, extending classical index theory to a noncommutative setting.

Findings

Constructed a cyclic cocycle for Heisenberg symbols.

Derived a local index formula involving principal symbols and curvature.

Reduced the index problem to Boutet de Monvel's theorem for Toeplitz operators.

Abstract

A hypoelliptic operator in the Heisenberg calculus on a compact contact manifold is a Fredholm operator. Its symbol determines an element in the K-theory of the noncommutative algebra of Heisenberg symbols. We construct a periodic cyclic cocycle which, when paired with the Connes-Chern character of the principal Heisenberg symbol, calculates the index. Our index formula is local, i.e. given as a local expression in terms of the principal symbol of the operator and a connection on TM and its curvature. We prove our index formula by reduction to Boutet de Monvel's index theorem for Toeplitz operators.