The Heisenberg Calculus, Index Theory and Cyclic (Co)homology
Operator Algebras
2020-10-07 v1 Functional Analysis
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.
Keywords
Cite
@article{arxiv.2010.02884,
title = {The Heisenberg Calculus, Index Theory and Cyclic (Co)homology},
author = {Alexander Gorokhovsky and Erik van Erp},
journal= {arXiv preprint arXiv:2010.02884},
year = {2020}
}